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A Mathematica package for doing tensor calculations in differential

1. i i Out 7 2 Rm 2 Rm jki ljk In 8 FirstBianchiRule i Out 8 2 Rm kj1l See also SecondBianchiRule BianchiRules ContractedBianchiRules FirstStructureRule 7 5 FirstStructureRule is a rule that implements the first structure equa tion for exterior derivatives of basis covectors For example assuming WedgeConvention Alt 8 REFERENCE LIST 71 i Out 9 d Basis l In 107 FirstStructureRule k6 i 1 i Dut 10 Basis 7 Conn Tor k6 2 See also SecondStructureRule CompatibilityRule StructureRules WedgeConvention FirstUp FirstUp is a value for the RiemannConvention option of DefineBundle FlatConnection 2 1 FlatConnection is a DefineBundle option The function FlatConnection bundle returns True or False See also DefineBundle FormQ FormQ x returns True if x is a covariant alternating tensor expression and False otherwise See also DefineTensor Grad 4 4 Grad x is the gradient of the tensor expression x It is the same as Del x except that the last index is contravariant instead of covariant If x is a scalar function 0 tensor then Grad x is a vector field See also Del Laplacian Hermitian 2 1 Hermitian is a value for the Symmetries option of DefineTensor HodgeInner 3 6 HodgeInner x y represents the Hodge inner product of the alternat ing tensors x and y taken with respect to the default metric s on the bundle s with which x and y are associated It is equal to th
2. 7 4 Product tensors It often happens that a tensor is associated with the tensor product of two or more bundles In Ricci such a tensor is called a product tensor A product tensor is defined by giving a list of positive integers as the rank of the tensor in the DefineTensor command The rank of the tensor is the sum of all the numbers in the list each number in the list corresponds to a group of indices which can have its own symmetries and can be associated to a separate bundle or direct sum of bundles Indices can be inserted into a product tensor one group at a time for example if t is a product tensor whose rank is specified as 2 3 you can insert either two or five indices Once the first two indices are inserted the resulting expression is considered to be a 3 tensor with the symmetries and bundles associated with the second group of indices The Bundle and Symmetries options of DefineTensor have special interpreta tions for product tensors e Bundle gt bundle1 bundle2 the list of bundles must be the same length as the list of ranks Each bundle in the list is associated with the corresponding set of index slots Other forms are Bundle gt bundle which means that all indices are associated with the same bun dle and Bundle gt b1 b2 b3 b4 which means that each set of index slots is associated with a direct sum of bundles e Symmetries gt symi sym2 again the list of symmetries must be
3. TProd alpha beta Out 151 alpha X beta As vou can see here Ricci s output form for tensor products uses an approxi mation of the tensor product symbol Ricci automatically expands tensor products of sums and scalar multiples 3 2 Wedge Wedge products exterior products of differential forms are indicated by the Ricci operator Wedge In output form wedge products are indicated by a caret In 16 Wedgelalpha beta O0ut 16 alpha 7 beta The arguments to Wedge must be alternating tensors Ricci automatically ex pands wedge products of sums and scalar multiples and arranges the factors in lexical order inserting signs as appropriate There are two common conventions for relating wedge products to tensor prod ucts Ricci supports both conventions which one is in use at any given time is controlled by the global variable WedgeConvention You should pick one convention and set WedgeConvention at the beginning of your calculation if you change it you may get inconsistent results In Ricci the two wedge product conventions are referred to as Det and Alt Under the Alt convention the default the wedge product is defined by a b Alt a B where Alt is the projection onto the alternating part of the tensor This conven tion is used for example in the differential geometry texts by Bishop Goldberg 3 PRODUCTS CONTRACTIONS AND SYMMETRIZATIONS 21 Kobayashi Nomizu and Helgason Under this conventio
4. The default is False It can be overridden for a particular command by specifying the Quiet option as part of the command call RiemannConvention 7 2 The global variable RiemannConvention can be set by the user to spec ify a default value for the RiemannConvention option of DefineBundle The allowed values are FirstUp and SecondUp the default See also DefineBundle 8 REFERENCE LIST 90 TensorFormatting 2 2 The global variable TensorFormatting can be set to True or False by the user to turn on or off Ricci s special output formatting of tensors and indices Default is True TensorTeXFormatting The global variable TensorTeXFormatting can be set to True or False by the user to turn on or off Ricci s special formatting of tensors in TeXForm Default is True WedgeConvention 3 2 The global variable WedgeConvention can be set by the user to determine the interpretation of wedge products The allowed values are Alt and Det The default is Alt Suppose a is a p form and f a q form WedgeConvention Alt PEN san 1 aN B Alt a B and Basis A A Basis det n WedgeConvention Det aN B fol Alt a 8 and Basis A A Basis det p q See also Wedge Alt
5. into InsertIndices x p i whenever i is an index L j or ULj Ricci causes a power of a product such as a b p to be expanded into a product of powers a p b p provided a and b do not con tain any indices if they do contain indices then a b p is trans formed to Summation a b p to prevent the expression from being ex panded to a p b p Summation is not printed in output form See also SymmetricProduct Summation PowerSimplify ProductExpand PowerSimplify 5 14 PowerSimplify x attempts to simplify powers that appear in the tensor expression x PowerSimplify x n or PowerSimplify x n1 n2 applies PowerSimplify only to term n or to terms n1 n2 of x PowerSimplify transforms x by e Converting products like a p b p to Summation a b p when a and b contain indices Summation is an internal Ricci function that prevents such products from being automatically expanded by Mathematica It does not appear in output form instead Summation a b p prints as a b p e Expanding powers like a b 7c to a b c provided it knows enough about a b and c to know this is legitimate for example if c is an integer or if a is positive and b is real e Expanding and collecting constants in any expression that appears as base or exponent of a power See also TensorSimplify Power Summation ProductExpand 5 13 ProductExpand x expands symmetric products and wedge products of l tensors that occur in x and rewrites
6. it is automaticallv converted to TensorExpand See also TensorSimplifv Expand BasisExpand CovDExpand ProductExpand TensorFactor TensorFactor x returns the product of all the non constant factors in x which should be a monomial See also ConstantFactor TensorMetricQ TensorMetricQ tensorname returns True if tensorname is the metric of some bundle and False otherwise See also DefineBundle TensorProduct 3 1 TensorProduct x y z or TProd x y z represents the tensor product of x y and z Ricci automatically expands tensor products of sums and scalar multiples In output form tensor products appear as x X y X z See also SymmetricProduct Wedge TensorQ TensorQ name returns True if name is the name of a tensor and False otherwise See also DefineTensor TensorRankList TensorRankList stores the list of ranks for a product tensor Used inter nally by Ricci See also DefineTensor TensorSimplify 5 1 TensorSimplify x attempts to put the tensor expression x into a canonical form so that two expressions that are equal will usually be identical after applying TensorSimplify TensorSimplify x n or TensorSimplify x n1 n2 applies TensorSimplify to term n or to terms n1 n2 of x TensorSimplify expands products and positive integer powers sim plifies powers uses metrics to raise and lower indices tries to re name all dummy indices in a canonical order and collects all terms 8 REFERENCE LIST
7. 85 containing the same tensor factors but different constant factors When there are two or more dummy index pairs associated with the same bundle TensorSimplify tries exchanging dummy index names in pairwise choosing the lexically smallest result It does not re order indices after the use OrderCovD or CovDSimplify to do that TensorSimplify calls CorrectAllVariances TensorExpand AbsorbMetrics PowerSimplify RenameDummy OrderDummy and CollectConstants See also SuperSimplify OrderCovD CovDSimplify TensorCancel FactorConstants SimplifyConstants TensorSymmetry A tensor symmetry is an object of the form TensorSymmetry name d perm1 sgni permN sgnN where d is a positive integer permi permN are non trivial permutations of 1 2 d and sgni sgnN are constants usually 1 TensorSymmetry objects can be defined with DefineTensorSymmetries and used in the DefineTensor command See also DefineTensorSymmetries DefineTensor TeXFormat TeXFormat is an option for DefineTensor and DefineIndex Times 2 6 3 3 Tor Ricci uses ordinary multiplication to represent symmetric products of tensors Ricci transforms expressions of the form a b L il into InsertIndices a b L i Any number of upper and or lower indices can be inserted in this way provided that the number of indices is consistent with the rank of a b In output form Ricci modifies Mathematica s usual ordering of f
8. Basis In 6 ProductExpand 1 i j 1 j i Out 6 Basis X Basis Basis X Basis 2 2 You can reverse the effect of ProductExpand by applying Alt to an alternating tensor expression or Sym to a symmetric expression these operators replace tensor products by wedge or symmetric products 5 14 PowerSimplify PowerSimplify x performs various simplifications on powers that appear in x such as transforming a p b p to a b p and a b 7c to a b c when possible and expanding and collecting constants in expressions that appear as base or exponent of a power PowerSimplify is called automatically by TensorSimplify 5 SIMPLIFVING AND TRANSFORMING EXPRESSIONS 34 5 15 CorrectAllVariances CorrectAllVariances x changes the variances upper to lower or lower to upper of all indices occurring in x whose variances are not correct for their positions by inserting appropriate metric coefficients For example suppose v is a vector field i e a contravariant 1 tensor j Out 41 v i k In 42 CorrectAllVariances j K4 K3 Out 42 g g v i k3 k4 k 5 16 NewDummy NewDummy x replaces all dummy index pairs in x with computer generated dummy index names of the form kn where k is the last name in the list of indices associated with the appropriate bundle and n is a unique integer This is sometimes useful when you want to insert an index with the same name as a dummy name that appears in x For example suppose a
9. Internally Ricci represents tensors in the form Tensor name i j k 1 wherei j are the tensor indices andk 1 are indices resulting from covariant differentiation In input form you represent an unindexed tensor just by typing its name Indices are inserted by typing them in brackets after the tensor name while differentiated indices go inside a second set of brackets A tensor of rank k should be thought of as having k index slots Each index slot is associated with a bundle or list of bundles specified in the Def ineTensor command if an inserted index is not associated with one of the bundles for that slot the result is 0 Once all index slots are full indices resulting from covariant differentiation can be typed in a second set of brackets For 0 tensors all indices are assumed to result from differentiation For example suppose eta is a 2 tensor and u is a 0 tensor i e a scalar function Table 1 shows the input internal and output forms for various tensor expressions involving eta and u INPUT INTERNAL OUTPUT Tensor eta Table 1 Input internal and output forms for tensors The conjugate of a complex tensor without indices is represented in input form by Conjugate name you can type indices in brackets following this expression just as for ordinary tensors as in Conjugate name L i L j In internal form a conjugate tensor looks just as in Table 1 except that the name is
10. Ricci does not perform this transformation automatically but you can cause it to be performed by applying FirstStructureRule i Out 36 d Basis l In 37 FirstStructureRule 1 i k6 i Out 37 Tor Basis Conn 2 k6 The 1 2 appears because by Ricci s default wedge product convention and index conventions the basis expansion of Tor U i has to be i j k i j k Tor Basis X Basis Tor Basis Basis without the factor of 1 2 that appears in the standard notation used in many differential geometry texts The 1 2 does not appear in the output from FirstStructureRule when WedgeConvention Det because in that case i j k 1 i j k Tor Basis X Basis Tor Basis Basis The curvature associated with the default connection is called Curv This is a product tensor of rank 2 2 which is skew symmetric in both pairs of indices because the default connection is always assumed to be compatible with the metric The first two indices of Curv can refer to any bundle and the last two refer to its underlying tangent bundle When you insert the first two indices you get the matrix of curvature 2 forms associated with the standard basis When all four indices are inserted you get the components of the curvature tensor 7 SPECIAL FEATURES 47 defined bv ViV Basis V j Vi Basis V Basis Basis POsisn Curvz ij Basis The second structure equation relates the exterior derivatives of the connection forms to curv
11. U name Bar In output form upper and lower indices are represented by superscripts and subscripts with or without bars This special formatting can be turned off by setting TensorFormatting False Every index name must be associated with a bundle This association is estab lished either by listing the index name in the call to DefineBundle as in the examples above or by calling DefineIndex You can create new names without calling Def ineIndex just by appending digits to an alreadv defined index name For example if index j is associated with bundle x then so are ji j25 etc 2 3 Constants In the Ricci package a constant is any expression that is constant with respect to covariant differentiation in all directions You can define a symbol to be a constant by calling DefineConstant In 10 DefineConstantl c This defines c to be a real constant the default and ensures that covariant derivatives of c such as c L i are interpreted as 0 To define a complex constant use 2 RICCI BASICS 13 In 11 DefineConstantl d Type gt Complex If a constant is real you may specify that it has other attributes by giving a Type option consisting of one or more keywords in a list such as Type gt Positive Real The complete list of allowable Type keywords for constants is Complex Real Imaginary Positive Negative NonPositive NonNegative Integer Odd and Even The Type option controls how the constant behaves with
12. a and b are tensor expressions then Dot a b or a b is the tensor 3 PRODUCTS CONTRACTIONS AND SYMMETRIZATIONS 22 formed by contracting the last index of a with the first index of b It is easiest to see what this means by inserting indices For example suppose a is a 2 tensor and e is a 3 tensor In 18 a e L i L j L k k4 Out 18 a e i k4 jk Dot can be applied to three or more tensor expressions as long as all of the tensors except possibly the first and last have rank two or more Ricci expands dot products of sums and scalar multiples automatically Dot products of 1 tensors are automatically converted to inner products 3 5 Inner The Mathematica function Inner is given an additional interpretation by Ricci If a and b are tensor expressions of the same rank Inner a b is their inner product taken with respect to the default metric s on the bundle s with which a and b are associated In output form Inner a b appears as lt a b gt For example In 19 Inner a b Out 19 lt a b gt In 20 BasisExpand k5 k6 Out 20 a b k5 k6 3 6 Hodgelnner Hodgelnner is the inner product ordinarily used for differential forms This differs from the usual inner product by a constant factor determined by the global variable WedgeConvention It is defined in such a way that if e are orthonormal 1 forms then the k forms et A Me k il L Lk are orthonormal In output form HodgeInner alp
13. and CoBasisName options of DefineBundle Note that the index conventions for basis vectors are opposite those for com ponents this is so that the expansion of tensors in terms of basis vectors will be consistent with the summation convention The Ricci function BasisExpand can be used to expand any tensor expression in terms of the default basis For example In 13 BasisExpand alpha k3 Out 13 alpha Basis k3 2 5 Mathematical functions If u is a scalar function 0 tensor you may form other scalar functions by applying real or complex valued mathematical functions such as Log or Sin to u This is indicated in Ricci s input form and output form simply by typing LogLu or Sin u You may also define your own mathematical functions and use them in tensor expressions In order for Ricci to correctly interpret an expression such as f ul it must be told that f is a mathematical function and what its characteristics are You do this by calling DefineMathFunction For example to define f as a complex valued function of one variable you could type 2 RICCI BASICS 17 In 14 DefineMathFunction f Type gt Complex The Type option determines how the function behaves with respect to conju gation exponentiation and logarithms The default is Real which means that the function is always assumed to take real values Other common options are Type gt Positive Real Type gt NonNegative Real and Type gt Automati
14. and column vectors or between covariant and contravariant tensors for this purpose The identity matrix on any bundle can be represented by the tensor Kronecker whose components are the Kronecker delta symbols Ricci automatically rec ognizes some basic relations involving multiplication by Kronecker such as a Kronecker Kronecker a a The inverse of any 2 tensor expression a is represented by Inverse a When you insert indices into an inverse tensor Ricci in effect creates a new tensor whose name is Inverselal For example In 661 Inversela 1 Out 66 a In 67 U i U j 1 ij Out 67 a Ricci recognizes that multiplication by the inverse of a tensor yields the identity tensor whether as components or as pure tensors 7 SPECIAL FEATURES 43 In 68 a b Inverse b Out 68 a In 691 a L i1 L j11 Inverselal U j U x 1 k Out 69 Kronecker i You can express the trace of a 2 tensor with respect to the default metric by Tr a its transpose by Transpose a and its determinant by Det a When you insert indices into Tr a or Transpose a Ricci computes the compo nents explicitly in terms of the components of a For Det however there is no straighforward way to express the determinant of a matrix using the index conventions so Det a is maintained unevaluated Ricci s differentiation com mands compute derivatives of Det a in terms of derivatives of components of a and Inverselal
15. and x are both vector fields Lie v x is their Lie bracket and Ricci automatically puts the factors in lexical order by inserting a minus sign if necessary Applying LieRule causes Lie derivatives of differ ential forms to be expanded in terms of Int and exterior derivatives See also Del LieRule LieRule 4 9 LieRule is a rule that transforms Lie derivatives of differential forms to expressions involving exterior derivatives and Int according to the relation 8 REFERENCE LIST 75 Lie w Int dlwll dlintl w v v v See also Lie Int Extd LowerAllIndices 5 20 LowerAllIndices x lowers all of the indices in x except those appearing on metrics or Basis covectors by inserting appropriate metrics with raised indices LowerAllIndices x n or LowerAllIndices x n1 n2 applies LowerAllIndices to term n or to terms n1 n2 of x See also CorrectAllVariances Method Method is an option for the Ricci simplification command OrderDummy which specifies how hard the command should work to simplify the ex pression Metric Metric gt metricname is an option for Del Div Grad ExtdStar CovD Laplacian and LaplaceBeltrami The function Metric bundle returns the bundle s metric See also DefineBundle MetricQ MetricQ x returns True if x is a metric with or without indices and False otherwise See also DefineBundle MetricType 7 2 MetricType is a DefineBundle option which specifies whether the bun
16. are a couple of examples Suppose alpha and beta are 1 tensors Ricci can manipulate tensor products and wedge products In 4 TensorProduct alpha beta Out 4 alpha X beta In 5 Wedge alpha beta Out 5l alpha beta and exterior derivatives 1 INTRODUCTION 8 In 6 Extd Out 6 dlalphal 7 beta d beta 7 alpha To express tensor components with indices vou just tvpe the indices in brackets immediately after the tensor name Lower and upper indices are typed as L i and U il respectively in output form they appear as subscripts and super scripts Indices that result from covariant differentiation are typed in a second set of brackets For example if alpha is a 1 tensor you indicate the components of alpha and its first covariant derivative as follows In 7 alpha L i1 Out 7 alpha i In 8 alpha L i1 L j Out 8l alpha i j Ricci alwavs uses the Einstein summation convention anv index that appears as both a lower index and an upper index in the same term is considered to be implicitlv summed over and the metric is used to raise and lower indices For example if the metric is named g the components of the inner product of alpha and beta can be expressed in either of the following ways In 9 alphalL kll beta U k k Out 9 alpha beta k In 107 g U i1 U j11 alpha L i1 beta L j1 ij Out 10 alpha beta g ii J Ricci s simplification commands can recognize the equalit
17. co efficients with respect to the default basis are always assumed to be constants For one dimensional bundles the metric coefficient is explicitly set to 1 The default is False e CommutingFrame gt True or False This option is meaningful only for tangent bundles or their subbundles True means the default basis for this bundle consists of commuting vector fields as is the case for example when coordinate vector fields are used This option causes the connection coefficients for this bundle to be symmetric in their two lower indices and Lie brackets of basis vectors to be replaced by Zero e PositiveDefinite gt True or False True means the bundle s metric is positive definite False means no assumption is made about the sign of the metric This affects the simplification of complex pow ers and logarithms of inner products with respect to the metric The default is True e TorsionFree gt True or False False means that the default con nection on this bundle mav have nonvanishing torsion this makes sense only for tangent bundles or their subbundles The default is True e BasisName gt name Specifies a name to be used in input output and TEX forms for the default basis vectors for this bundle The default name is Basis Basis vectors are alwavs represented inter nallv bv the name Basis Covariant basis vectors can be named independentiv bv the CoBasisName option e CoBasisName gt name Specifies a name to be us
18. conditional relation 6 DEFINING RELATIONS BETWEEN TENSORS 39 In 44 DefineRelation u L i L j ulLljl L i I IndexOrderedQ L i L j Relation defined In 45 u L j L i Out 45 u ij 6 2 DefineRule Sometimes vou mav want to define a transformation rule that is not applied automatically but only when you request it Such transformations are done in Mathematica via rules The Ricci function DefineRule allows you to define rules whose left hand sides involve tensors For example here s how to define rules that replace the tensor a by 2 b and vice versa In 52 DefineRulel atob a 2 bl Rule defined In 53 DefineRulel btoa b a 2 Rule defined In 54 a L i L j Dut 54 a ij In 55 atob Out 55 2 b ij In 56 btoa Out 56 a ij 6 DEFINING RELATIONS BETWEEN TENSORS 40 DefineRule has another advantage over DefineRelation the left hand side of the transformation rule defined bv DefineRule can be anv tensor expression not just a single tensor with or without indices Thus suppose vou want to convert the inner product of a and b to 0 wherever it occurs You could define the following rules In 57 DefineRulel ortho Inner a b 0 Rule defined In 58 DefineRulel ortho a L i L j U i U j 0 Rule defined The symbol ortho is now a list containing these two transformation rules The first rule transforms the pure tensor expression for lt a b gt
19. disk storage including about 49K bytes of on line documentation I have tested the package on a DEC Alpha system and a Pentium 100 where it runs reasonably fast and seems to require about 6 or 7 megabytes of memory I don t have any idea how it will run on other systems but I expect that Ricci will be very slow on some platforms The source files for Ricci are available to the public free of charge either via the World Wide Web from http www math washington edu lee Ricci or bv anonvmous ftp from ftp math washington edu in directorv pub Ricci You ll need to download the Ricci source file Ricci m and put it in a directory that is accessible to Mathematica You may need to change the value of Mathe matica s Path variable in your initialization file to make sure that Mathematica can find the Ricci files see the documentation for the version of Mathematica that you re using Once you ve successfully transferred all the Ricci files to your own system start up Mathematica and load the Ricci package by typing lt lt Ricci m Once you ve loaded Ricci into Mathematica you can type name for information about any Ricci function or command If you have questions about using Ricci suggestions for improvement or a problem that you think may be caused by a bug in the program please contact the author lt lee math washington edu gt 1 3 A brief look at Ricci To give you a quick idea what typical Ricci input and output look like here
20. e Connection gt cn specifies that the divergence is to be taken with respect to the connection cn instead of the default connection Ordi narily cn should be an expression of the form Conn diff where diff is a 3 tensor expression representing the difference tensor be tween the default connection and cn e Metric gt g a symmetric 2 tensor expression indicating that the divergence is to be taken with respect to the Levi Civita connection of g instead of the default metric for x s bundle which is assumed to be Riemannian See also Del Laplacian DivGrad 4 5 DivGrad is a value for the global variable LaplacianConvention Dot 3 4 7 3 When x and y are tensor expressions x y or Dot x y represents the con traction of the last index of x with the first index of y When x and y are 2 tensors this can be interpreted as matrix multiplication If Dot is ap plied to three or more tensor expressions all arguments except possibly the first and last must have rank greater than 1 Ricci automatically expands dot products of sums and scalar multiples and replaces dot products of l tensors with inner products See also Inner Inverse Kronecker ERROR If you attempt to insert the wrong number of indices into a tensor expres sion Ricci returns ERROR expression Even 2 3 Even is a value for the Type option of DefineConstant Expand 5 3 If you apply the Mathematica function Expand to an expression that includes tensors
21. forms for the default connection on any bundle It is defined as a product tensor of rank 2 1 and variance Covariant Contravariant Covariant When the first two indices are inserted it represents an entry of the matrix of connection l forms associated with the default basis When all three indices are inserted it represents the Christoffel symbols of the default connection relative to the default basis The upper index is the second one and the index representing the direction in which the derivative is taken is the third one Thus the Christoffel symbol rk is represented by k Conn L j U k L i gt Conn j i See also Curv Tor Curvature Connection LeviCivitaConnection CovDExpand ConnToMetricRule CurvToConnRule Connection 7 6 Connection gt cnis an option for Del CovD Div Grad and Laplacian It specifies that covariant derivatives are to be taken with respect to the connection cn instead of the default connection ConnToMetricRule 7 2 ConnToMetricRule is a rule that causes indexed components of the default connection Conn on a Riemannian tangent bundle to be con verted to expressions involving directional derivatives of the metric co efficients This rule applies only to fully indexed connection compo nents whose indices all refer to a single Riemannian tangent bundle with the option CommutingFrame gt True See also CurvToConnRule Conn DefineBundle 8 REFERENCE LIST 56 ConstantFactor ConstantFac
22. is a 2 tensor After applying TensorSimplify the output from BasisExpand a becomes i j Out 31 a Basis X Basis ij If you want the i k component of this expression you cannot just type A L i L k because i would then appear twice as a lower index once in the output expression Out 31 and once in the inserted index The correct procedure is to use NewDummy In 32 NewDummy L i L k Out 33 a ik 5 SIMPLIFVING AND TRANSFORMING EXPRESSIONS 35 5 17 CommuteCovD CommuteCovD x L i L j changes all adjacent occurrences in the tensor expression x of indices L i L j in the given order after the to L j L i by adding appropriate curvature and torsion terms For example suppose a is a 2 tensor on a Riemannian tangent bundle assuming RiemannConvention SecondUp Out 35 a ij kl In 36 CommuteCovD L k L 1 1 11 12 Out 36 a a Rm a Rm ij lk 11 3 i ki i 12 j ki 5 18 OrderCovD OrderCovD x puts all derivative indices those appearing after in the ten sor expression x into lexical order by adding appropriate curvature and torsion terms OrderCovD is used by CovDSimplify 5 19 CovDsSimplifv CovDSimplify x attempts to simplify x as much as possible putting all dummy indices in lexical order including those that result from differentiation CovDSimplify calls TensorSimplify then OrderCovD then TensorSimplify again For very complicated expressions it is
23. list of index names to be associated with the bundle If only one index name is given the braces may be omitted For one dimensional bundles only one index name is allowed For higher dimensional bundles additional indices can be added to the list at any time by calling DefineIndex Options e Type gt Complex or Real Default is Real If Complex the con jugate bundle is referred to as Conjugate name and is associated with barred indices 8 REFERENCE LIST 60 e TangentBundle gt bundle Specifies the name of the underlying tangent bundle for the bundle being defined This can be a list of bundles representing the direct sum of the bundles in the list The default is the value of DefaultTangentBundle if defined otherwise this bundle itself or the direct sum of this bundle and its conjugate if the bundle is complex If DefaultTangentBundle has not been given a value the first time DefineBundle is called it is set to the first bundle defined and its conjugate if it is a complex bundle e FlatConnection gt True or False True means the default connec tion on this bundle has curvature equal to 0 The default is False e ParallelFrame gt True or False True means the default ba sis elements have vanishing covariant derivatives so the connec tion and curvature forms are set equal to 0 This option forces FlatConnection gt True The default is False e OrthonormalFrame gt True or False True means the metric
24. option by DefineTensor as the de fault value for the Bundle option and by BasisExpand and similar com mands as the default bundle for tensors such as Curv Conn and Tor that are not intrinsically associated with any particular bundle By default DefaultTangentBundle is set to the first bundle the user defines or the direct sum of this bundle and its conjugate if the bundle is complex If you want to override this default behavior you can give a value to DefaultTangentBundle before calling DefineBundle For example DefaultTangentBundle horizontal vertical specifies that the tangent bundle of all subsequently defined bundles is to be the direct sum of the bundles named horizontal and vertical See also DefineTensor DefineBundle BasisExpand LaplacianConvention 4 5 LaplacianConvention determines which sign convention is used for the covariant Laplacian on functions and tensors LaplacianConvention DivGrad means that Laplacian x Div Grad x while LaplacianConvention PositiveSpectrum means that Laplacian x Div Grad x Default is DivGrad See also Laplacian MathFunctions MathFunctions is a list of names that have been defined as scalar math ematical functions for use by Ricci See also DefineMathFunction Quiet The global variable Quiet is used by Ricci to determine whether the defining and undefining commands report on what they are doing Setting Quiet True will silence these chatty commands
25. replaced by the special form name Bar In output form the conjugate of a tensor is represented by a bar over the name Indices in any slot can be upper or lower A metric with lower indices represents the components of the metric on the bundle itself and a metric with upper in dices represents the components of the inverse matrix which can be interpreted 2 RICCI BASICS 16 invariantly as the components of the metric on the dual bundle For any other tensor if an index is inserted that is not at the natural altitude for that slot based on the Variance option when the tensor was defined it is assumed to have been raised or lowered by means of the metric For example if a is a covariant 1 tensor then the components of a ordinarily have lower indices A component with an upper index is interpreted as follows As explained above components of tensors ordinarily have lower indices in co variant slots and upper indices in contravariant slots Ricci always assumes there is a default local basis or moving frame for each bundle you define and components of tensors are understood to be computed with respect to this basis The default contravariant basis vectors i e basis vectors for the bundle itself are referred to as Basis L i and the default covariant basis vectors representing the dual basis for the dual bundle as Basis U i You may specify different names to be used in input and output forms by using the BasisName
26. to 0 while the second rule takes care of the component expression Here s how it works In 60 a L j L k bEU j U x jk Out 60 a b j k In 61 ortho Out 61 O Each time you call DefineRule the new transformation rule you give is ap pended to a list containing ones already defined under the same name To start over from scratch you can include the option NewRule gt True or just execute the assignment ortho The rule created by DefineRule recognizes any equivalent substitutions result ing from raising and lowering indices and changing index names In addition if the second argument is a single tensor name with or without indices it rec ognizes equivalent substitutions resulting from conjugation and inserting extra indices 7 SPECIAL FEATURES 41 7 Special Features 7 1 One dimensional bundles One dimensional bundles receive special treatment in Ricci Since there is only one basis element Ricci uses only one index name for all indices referring to the bundle This is the only case in which an index can legitimately appear more than once as a lower or upper index in the same term When you define a one dimensional bundle you may only give one index name When Ricci generates dummy indices for the bundle it uses only this index name instead of generating unique names If you define a one dimensional bundle with the option OrthonormalFrame gt True the metric coefficient with respect to the default
27. 4 Bundle gt name is a DefineTensor option specifying the name of the bundle or bundles the tensor is to be associated with The function Bundle i returns the name of the bundle associated with the index i BundleDummy BundleDummy bundle returns the symbol that is used for computer generated dummy indices associated with the bundle This is the last index name in the list BundleIndices bundle For most bundles new 8 REFERENCE LIST 53 dummy indices are of the form kn where k BundleDummy bundle and n is an integer For one dimensional bundles n is omitted See also BundleIndices DefineBundle Bundlelndices BundleIndices bundle returns a list of the index names currently asso ciated with bundle See also BundleDummy DefineBundle BundleQ BundleQ x returns True if x is the name of a bundle and False other wise See also DefineBundle Bundles The function Bundles x returns a list of the bundles associated with each index position in the tensor expression x each entry in the list is a sublist giving the allowed bundles for that index position If you insert an index that is not associated with one of the bundles corresponding to that index position the expression gives 0 See also DefineTensor Variance Co 2 4 Co is an abbreviation for Covariant CoBasisName 2 4 CoBasisName gt name is an option for DefineBundle It specifies the name to be used in input and output form for the basis covectors for
28. Bar In input form this is typed Conjugate name See also Tensor Conjugate L U LB UB Basis 2 4 5 10 Basis is the name used for generic basis vectors and covectors for any bundle Basis vectors are generated automatically by BasisExpand For example if index name i is associated with bundle b then Basis L i and Basis U i represent contravariant and covariant basis elements for b i e basis elements for b and its dual respectively Basis LB i and Basis UB i represent basis elements for the conjugate bundle Conjugate b and its dual Basis is defined by Ricci as a product tensor of rank 1 1 with the special option Bundle gt Same this means that the first index can be associated with any bundle but when a second index is inserted it must be associated with the same bundle as the first or else the result is 0 If you insert an index into Basis L i or Basis U i you get Kro necker delta coefficients metric coefficients which result from lowering or raising an index on the Kronecker delta or 0 as appropriate See also BasisExpand BasisName CoBasisName Kronecker BasisExpand 2 4 5 10 BasisExpand x expands all tensors appearing in x into component ex pressions multiplied by basis vectors and covectors Basis L i and Basis U j BasisExpand x n or BasisExpand x n1 n2 ap plies BasisExpand only to term n or to terms n1 n2 of x BasisExpand also inserts indices i
29. Conn Basis a k8 Because the default metric on any bundle is assumed to be compatible with its metric the coefficients of the metric satisfv the compatibility equation d gi Conn Conn ji where the second index of Conn has been lowered by the metric This relation is not applied automatically by Ricci but you can force Ricci to replace derivatives of the metric by connection forms by applying the rule CompatibilityRule A connection on the tangent bundle of a manifold may or may not have tor sion Ricci assumes that connections are torsion free unless you specify that a particular connection may have non vanishing torsion by adding the option TorsionFree gt False to the DefineBundle command when the bundle is cre ated The name of the torsion tensor on any bundle is Tor It is a product tensor 7 SPECIAL FEATURES 46 of rank 1 2 whose first index can be associated with any subbundle of a tan gent bundle and whose last two are associated with the full underlying tangent bundle or bundles if the tangent bundle is a direct sum It is skew symmetric in its last two indices The torsion is defined by V Basis Vi Basis Basis Basis Tor jp Basis When the first index is inserted Tor U i represents the torsion 2 form associ ated with the basis 1 form Basis U i The first structure equation relates the exterior derivatives of basis covectors on the tangent bundle to the connection and torsion forms
30. MetricType gt Riemannian is specified The default is Rc e ScalarCurv gt name a name to be given to the scalar curvature function for this bundle This tensor is automatically defined as a real O tensor This option takes effect only if MetricType gt Riemannian is specified The default is Sc e RiemannConvention gt SecondUp or FirstUp determines the sign convention used for the Riemannian curvature tensor for this bun dle This option takes effect only if MetricType gt Riemannian is specified The default is SecondUp or the value of the global vari able RiemannConvention if it has been set The meaning of this option is indicated by the relations below Note that the Riemann curvature tensor is always defined as a covariant 4 tensor so all four of its indices are normally down It is done this way so that all index positions are equivalent for the purpose of applying symmetries of the curvature tensor The RiemannConvention option controls only the sign of the curvature components RiemannConvention gt FirstUp VxVy VyVx Vix y Z X vIZEg Rm pri Basis Rm ix Rei RiemannConvention gt SecondUp VxVy VyVx Vix y 2 X VIZEg RMpmij Basis k Rm kj Rei 8 REFERENCE LIST 62 e Quiet gt True or False Default is False or the value of Quiet if set See also DefineTensor DefineIndex UndefineBundle Declare DefineConstant 2 3 DefineConstant c defines the symbol c to b
31. Ricci A Mathematica package for doing tensor calculations in differential geometrv User s Manual VERSION 1 32 By JOHN M LEE ASSISTED BV DALE LEAR JOHN ROTH JAY COSKEY AND LEE NAVE Ricci A Mathematica package for doing tensor calculations in differential geometrv User s Manual Version 1 32 Bv John M Lee assisted bv Dale Lear John Roth Jav Coskev and Lee Nave Copyright 1992 1998 John M Lee All rights reserved Development of this software was supported in part by NSF grants DMS 9101832 DMS 9404107 Mathematica is a registered trademark of Wolfram Research Inc This software package and its accompanying documentation are provided as is without guarantee of support or maintenance The copyright holder makes no express or implied warranty of any kind with respect to this software including implied warranties of merchantability or fitness for a particular purpose and is not liable for any damages resulting in any way from its use Everyone is granted permission to copy modify and redistribute this software package and its accompanying documentation provided that 1 All copies contain this notice in the main program file and in the supporting documentation 2 All modified copies carry a prominent notice stating who made the last modifi cation and the date of such modification 3 No charge is made for this software or works derived from it with the exception of a distribution fee to cover the cost of
32. See also DefineRule UndefineRelation DefineRule 6 2 DefineRule rulename lhs rhs condition defines a rule named rulename of the form lhs gt rhs condition The first two arguments lhs and rhs are arbitrary tensor expressions with or without indices tensors in them can be written in Ricci s input form tensor name followed optionally by one or more sets of indices in brackets Other expressions used in lhs must match Mathematica s internal form for such expressions The condition argument is optional if present it should be a true false condition that must be satisfied before the relation will be applied It mav involve the tensor name or indices that appear in lhs The rule is appended to a list containing previous rules defined with the same name unless the option NewRule gt True is specified To delete a rule entirely make the assignment rulename If index names used in 1hs are associated with bundles then the rule is applied only when indices from those bundles appear in those positions For example if i is associated with bundle a and p with bundle b DefineRule zerorule t L il L p 1 O J 8 REFERENCE LIST 65 causes rule zerorule to replace t _ by 0 whenever its first index is associated with a and its second with b In general the rule substitutes rhs suitably modified in place of any expression that matches 1hs after renaming of indices or raising or lowering of indices If 1hs is a
33. actors constants are printed first then scalars then higher rank tensor expressions and Ricci inserts explicit as terisks between tensor factors of rank greater than 0 to remind the user that multiplication is being interpreted as symmetric product See also SymmetricProduct Power ProductExpand 7 5 Tor is the name for the generic torsion tensor for the default connection in any bundle It is a product tensor of rank 1 2 inserting the first index yields the torsion 2 forms associated with the default basis Inserting all three indices yields the components of the torsion tensor TorsionFree 2 1 7 5 TorsionFree is a DefineBundle option The function TorsionFree bundle returns True or False 8 REFERENCE LIST 86 See also DefineBundle TotalRank TotalRank tensorname returns the total rank of the tensor tensorname This function is used internally by Ricci To compute the rank of an arbi trary tensor expression use Rank instead See also Rank DefineTensor TProd 3 1 TProd is an abbreviation for TensorProduct Tr 7 3 Tr x represents the trace of the 2 tensor expression x computed with respect to the default metric on x s bundle See also Dot Transpose 7 3 If x is a tensor expression Ricci interprets Transposelx as x with its index positions reversed See also Dot Type 2 1 2 3 2 4 2 6 Type is an option for DefineBundle DefineConstant DefineTensor and DefineMathFunction U 2 2 2 4 U i repre
34. ad x 4 6 Extd Extd x represents the exterior derivative of the differential form x In output form Extd x appears as d x Ricci automatically expands exterior deriva tives of sums scalar multiples powers and wedge products WARNING It is tempting to type d x in input form Because the symbol d has not been defined as a Ricci function Ricci simply returns d x unevaluated You may not be able to tell immediately from Ricci s output that it is not interpreting d x as exterior differentiation unless you happen to notice that it has not expanded derivatives of sums or products that appear in the argument to d 4 7 ExtdStar ExtdStar is the formal adjoint of Extd with respect to the Hodge inner product It is used in constructing the Laplace Beltrami operator described in Section 4 8 below If x is a differential k form ExtdStar x is a k 1 form In output form ExtdStar appears as shown here In 291 ExtdStar alpha d alpha Out 29 4 8 LaplaceBeltrami LaplaceBeltrami x is the Laplace Beltrami operator A dd d d applied to x which is assumed to be a differential form It is automatically replaced by Extd ExtdStar x ExtdStar Extd x 4 9 Lie If v is a vector field i e a contravariant l tensor and x is any tensor expression Lie v x represents the Lie derivative of x with respect to v When v and x are both vector fields Lie v x is their Lie bracket and Ricci puts v and x in lexica
35. an SkewHermitian or NoSymmetries and applies to the corresponding group of indices in the list of ranks Default is NoSymmetries The values of the Symmetries option have the following meanings Symmetric The tensor is fully symmetric in all indices Alternating The tensor changes sign if any two indices are inter changed Skew is a synonym for Alternating 8 REFERENCE LIST 66 Hermitian The tensor must be a real 2 tensor associated with a bundle and its conjugate It is symmetric in its two indices with the additional property that t L i L j and t LB i LB j are both zero whenever i j are associated with the same bundle For product tensors this can be used only for a group of two indices SkewHermitian The tensor must be a real 2 tensor associated with a bundle and its conjugate It is skew symmetric in its two indices with the additional property that t L i L j and t LB i LB j are both zero whenever i j are associated with the same bundle For product tensors this can be used only for a group of two indices RiemannSymmetries The tensor must have rank 4 The symmetries are those of the Riemann curvature tensor skew symmetric in the first two indices skew symmetric in the last two indices symmetric when the first two and last two are interchanged The first Bianchi identity is not automatically applied but can be explicitly applied to any tensor with these symmetries by means of FirstBianchiRule or BianchiR
36. aplacianConvention PositiveSpectrum Laplacian x is replaced by Div Grad x Options e Metric gt g asymmetric 2 tensor expression representing a metric to be used in place of the default metric for the underlying tangent bundle of x e Connection gt cn specifies that covariant derivatives are to be taken with respect to the connection cn instead of the default con nection Ordinarily cn should be an expression of the form Conn diff where diff is a 3 tensor expression representing the difference tensor between the default connection and cn See also Div Grad LaplaceBeltrami 2 2 2 4 LB i is the input form for a lower barred index i It is converted auto matically to the internal form L i Bar See also L U UB Bar LeviCivitaConnection 7 6 Lie LeviCivitaConnection g represents the Levi Civita connection of the arbitrary Riemannian metric g It behaves as a product tensor of rank 2 1 When indices are inserted the connection coefficients Christof fel symbols are computed in terms of the background connection of g s bundle assumed to be Riemannian and covariant derivatives of g The upper index is the second one just as for the default connection Conn See also Conn Curvature RiemannTensor RicciTensor ScalarCurv 4 9 Lie v x is the Lie derivative of the tensor expression x in the direction v The first argument v must be a vector field a contravariant 1 tensor ex pression If v
37. ative of x in the direction v this is the contraction of v with the last index of Del x In 23 Del v a Out 23 Del a v In 24 L i L j k5 Out 24 a v i j k5 4 DERIVATIVES 25 Here v can be a contravariant or covariant l tensor if it is covariant it is converted to a vector field by means of the metric If u is a scalar function i e a 0 tensor then Del v u is just the ordinary directional derivative of u in the direction v If v is a contravariant basis vector such as Basis L i Ricci uses a special shorthand output form for Del v a In 25 Dell Basis L i a Out 25 Del a i WARNING Some authors use a notation such as V a r to refer to the j k i component of the total covariant derivative Va This is not consistent with Ricci s convention to Ricci the expression Del a i jk is the directional derivative of the component ax in the i direction It is not the component of a tensor it differs from the component ajk by terms involving the connection coefficients 4 2 CovD If x is a component expression the L i component of the covariant deriva tive of x is represented in input form either by CovD x L i or by the abbreviated form x L i1 Multiple covariant derivatives are indicated by CowD x L i1 L j1 or x L i L j If x is a scalar expression with some unfilled index slots Ricci automatically inserts indices to convert it to a component expression be
38. ature forms Like the first structure equation it is applied only when you specifically request it by applying SecondStructureRule j k j Dut 15 d Conn Conn 7 Conn i i k In 16 SecondStructureRule TensorSimplify 1 j Out 16 Curv 2 i As described above for Tor the factor 1 2 appears when WedgeConvention Alt but not when WedgeConvention Det You can cause components of the curvature tensor Curv to be expanded in terms of components of Conn and their directional derivatives by applying the Ricci rule CurvToConnRule You can apply all three structure equations CompatibilityRule FirstStructureRule and SecondStructureRule at once by using the rule StructureRules If an expression involving derivatives of connection forms is converted to a com ponent expression the derivatives of the connection coefficients are maintained in a form such as j Del Conn 1 k i since covariant derivatives of connection coefficients make no invariant sense This is the onlv instance in which Ricci does not bv default express components of derivatives in terms of covariant derivatives 7 6 Non default connections and metrics You may want to compute covariant derivatives with respect to a connection or a metric other than the default one on a given bundle Manv of Ricci s differ entiation commands accept the Connection option to indicate that covariant derivatives are to be taken with respect to a non default connectio
39. b where a and b are vector bundles Suppose you define a tensor hom as follows In 35 DefineTensor hom 1 1 Bundle gt b a Variance gt Con Co Then hom is a section of b a or equivalently a homomorphism from a to b 7 SPECIAL FEATURES 45 7 5 Connections torsion and curvature Ricci assumes that every bundle comes with a default connection that is compat ible with its metric This assumption is forced by the limitations of the index convention since the notation for components of covariant derivatives gives no indication what connection is being used The name for the default connection on any bundle is Conn Conn is a product tensor of rank 2 1 whose first two indices can come from any bundle and whose third index is associated with the underlying tangent bundle of the first two When you insert the first two indices you get an entry of the matrix of connection 1 forms corresponding to the default basis defined by the relation V Basis Basis 0 Conn When you insert the third index you get the connection coefficients Christoffel symbols for the default connection with respect to the default basis These are defined by V Basis Basisi Conni p Basis For example suppose that indices a b c refer to the bundle fiber and i j K refer to the underlying tangent bundle The connection 1 forms of fiber would be expressed as Out 15 Conn a In 16 BasisExpand b k8 Out 16
40. basis is always equal to 1 TensorSimplify then puts all indices associated with this bundle at their natural altitude as specified in the corresponding call to DefineTensor 7 2 Riemannian metrics You specify that a bundle is a tangent bundle with a Riemannian metric by in cluding the option MetricType gt Riemannian in your call to DefineBundle This automatically causes the torsion of the default connection to be set to 0 and defines the Riemannian Ricci and scalar curvature tensors By de fault these are named Rm Rc and Sc you may give them any names you wish by including the DefineBundle options RiemannTensor gt name RicciTensor gt name and ScalarCurv gt name If you define more than one Riemannian tangent bundle you must give different names to the curvatures on different bundles On a Riemannian tangent bundle curvature components are automatically con verted to components of the Riemannian curvature tensor and contractions of the curvature tensor are automatically converted to the Ricci or scalar cur vature as appropriate The first and second Bianchi identities however are not automatically applied you can specifically apply them using the Ricci rules FirstBianchiRule SecondBianchiRule ContractedBianchiRules and BianchiRules described in the Reference List Chapter 8 There are two common sign conventions in use for indices of the Riemannian curvature tensor Ricci supports both you can control which on
41. built in Mathematica functions options 2 RICCI BASICS 11 for Ricci functions are typed after the required arguments and are always of the form optionname gt value This option specifies that fiber is to be a complex bundle Any time you define a complex bundle you also get its conjugate bundle In this case the conjugate bundle is referred to as Conjugate fiber the barred versions of the indices a b and c will refer to this conjugate bundle Mathematically the conjugate bundle is defined abstractly as follows If V is a complex n dimensional vector bundle let J V V denote the complex structure map of V the real linear endomorphism obtained by multiplying by i y 1 If CV denotes the complexification of V i e the tensor product over R of V with the complex numbers then J extends naturally to a complex linear endomorphism of CV CV decomposes into a direct sum CV V 9V where V V are the i and i eigenspaces of J respectively V is naturally isomorphic to V itself By convention the conjugate bundle is defined by V v The metric on a complex bundle is automatically defined as a 2 tensor with Hermitian symmetries In Ricci this is implemented as follows A Hermitian tensor h on a complex bundle is a real symmetric complex bilinear 2 tensor on the direct sum of the bundle with its conjugate with the additional property that har hag 0 Thus the only nonzero components of h are those of the f
42. c which means that Conjugate f x f Conjugate x Ricci automatically calls DefineMathFunction for the Mathematica func tions Log Sin Cos Tan Sec Csc and Cot If you wish to use any other built in mathematical functions in tensor expressions you must call DefineMathFunction yourself 2 6 Basic tensor expressions Tensor expressions are built up from tensors and constants using the arithmetic operations Plus Times and Power the Mathematica operators Conjugate Re and Im scalar mathematical functions and the Ricci product contraction symmetrization and differentiation operators described below Tensor expressions can be generally divided into three classes pure tensor ex pressions without any indices in which tensor names are combined using arith metic operations and tensor operators such as TensorProduct and Extd com ponent expressions in which all index slots are filled derivatives are indicated by components of covariant derivatives and tensor components are combined using only arithmetic operations and mixed tensor expressions in which some but not all index slots are filled and which usually arise when expressions of the preceding two types are combined Component expressions are considered to have rank 0 when they are acted upon by Ricci s tensor operators In most cases Ricci handles all three kinds of expressions in a uniform way but there are some operations such as computing components of covariant de
43. components that appear contracted with other tensors in x to be used to raise or lower indices the metric components themselves are eliminated from the expression For example jk Out 29 a g ij In 29 AbsorbMetrics k Out 30 a i AbsorbMetrics is called automatically by TensorSimplify 5 5 RenameDummy RenameDummy x changes the names of dummy indices in x to standard names choosing index names in alphabetical order from the list associated with the 5 SIMPLIFVING AND TRANSFORMING EXPRESSIONS 31 appropriate bundle and skipping those names that already appear in x as free indices When the list of index names is exhausted computer generated names of the form kn are used where k is the last index name in the list and n is an integer RenameDummy does not try to make the best choice of index names for the expression use OrderDummy to do that RenameDummy is called automatically by TensorSimplifv 5 6 OrderDummy OrderDummv x attempts to put the dummy indices occurring in the tensor expression x in a canonical form All pairs of dummy indices are ordered so that the lower member appears first whenever possible Then OrderDummy tries various permutations of the dummy index names in each term of x searching for the lexically smallest version of the expression among all equivalent versions OrderDummy has an option called Method which controls how hard it works to find the best arrangement of dummy index name
44. ct to the default metric Other differentiation operators that accept the Metric option are Grad Div ExtdStar Laplacian LaplaceBeltrami and CovD Ricci can explicitly compute the Riemannian Ricci and scalar curva tures of a non default metric h These are indicated by the expressions RiemannTensor h RicciTensor h and ScalarCurv h respectively When you insert indices into these you get expressions for the curvatures in terms 7 SPECIAL FEATURES 49 of the curvatures of the default metric and covariant derivatives of h You can also refer to the Levi Civita connection of a non default metric by LeviCivitaConnection h this produces a 3 tensor that is suitable as an input to the Connection option of the differentiation functions as described above 8 REFERENCE LIST 50 8 Reference List 8 1 Ricci commands and functions The list below describes all Ricci functions that are accessible to users All the essential information about each function is collected here in brief form including a list of all available options The numbers at the right hand sides of the function descriptions refer to section numbers in the main text where these functions are described Some less often used functions are documented only here in the Reference List For more information and examples on how to use the most common commands see the main text For descriptions of Ricci s global variables symbols starting with see the end of this cha
45. defined In 40 a L i L j Out 40 2 b ij This use of DefineRelation specifies that everv occurrence of the tensor a should be replaced bv 2 b The replacement specified in such a call to DefineRelation is made whether or not indices are inserted into a The first argument to DefineRelation must be a single tensor with or without indices it cannot be a more complicated expression The principal reason is that Mathematica requires every relation to be associated with a specific sym bol and Ricci s philosophy is to associate your definitions only with your own symbols so that they can all be saved by Ricci If you tried to define a relation involving a complicated tensor expression it would have to be associated with a Ricci or system function You can use DefineRule described below to define transformation rules whose left hand sides are arbitrary tensor expressions You can cancel any relation created by DefineRelation by calling UndefineRelation The single argument to UndefineRelation must be exactly the same as the first argument in the corresponding call to DefineRelation For example In 41 UndefineRelationla 6 DEFINING RELATIONS BETWEEN TENSORS 38 Relation undefined You can also use DefineRelation to replace a tensor only when it appears with certain indices For example suppose e is a 3 tensor In 42 DefineRelation e L i U i L j v L j 1 Relation defined This tells Ricci that the indexed
46. dle s metric has special properties Currently the only allowable values are MetricType gt Normal no special properties and MetricType gt Riemannian for a Riemannian tangent bundle The default is Normal Negative 2 3 2 4 2 6 Ricci recognizes Negative as a value for the Type option of DefineConstant DefineTensor and DefineMathFunction NewDummy 5 16 NewDummy x converts all dummy indices occurring in the tensor ex pression x to computer generated dummy indices NewDummy x n or NewDummy x n1 n2 applies NewDummy to term n or to terms ni n2 of x For most bundles the generated dummy names are 8 REFERENCE LIST 76 of the form kn where k is the last index name associated with bundle and n is an integer For one dimensional bundles only the dummy name k itself is generated See also RenameDummy OrderDummy NewRule 6 2 NewRule is a DefineRule option NonNegative 2 3 2 4 2 6 Ricci recognizes NonNegative as a value for the Type option of DefineConstant DefineTensor and DefineMathFunction NonPositive 2 3 2 4 2 6 NonPositive is a value for the Type option of DefineConstant DefineTensor and DefineMathFunction NoOneDims NoOneDims is a value for the Mode option of AbsorbMetrics used inter nallv bv TensorSimplifv Norm Norm x is the norm of the tensor expression x It is automatically converted by Ricci to Sqrt Inner x Conjugate x See also Inner HodgeNorm Normal 7 2 Normal is a va
47. e Default is False or the value of Quiet if set See also DefineBundle UndefinelIndex UndefineBundle Declare 8 REFERENCE LIST 63 DefineMathFunction 2 5 2 6 DefineMathFunction f declares f to be a scalar valued function of one real or complex variable that can be used in tensor expressions If x is any rank 0 tensor expression then f x is also interpreted as a rank 0 tensor expression Ricci has built in definitions for the Mathematica functions Sin Cos Tan Sec Csc Cot and Log If you intend to use any other functions in tensor expressions you must call DefineMathFunction LIMITATION Currently Ricci has no provision for using mathematical functions depending on more than one variable in tensor expressions Options e Type gt type This can be a single keyword or a list of key words chosen from among Real Complex Imaginary Automatic Positive Negative NonPositive or NonNegative Real means f x is always real Imaginary means f x is always imagi nary Complex means f x can assume arbitrary complex val ues and Automatic means Conjugate f x flConjugatelx Positive Negative NonPositive and NonNegative all imply Real and mean that the function values always have the correspond ing attribute Default is Real e Quiet gt True or False Default is False or the value of Quiet if set See also Declare DefineRelation 6 1 DefineRelation tensor x condition defines a relation of the fo
48. e usual inner product multiplied by a numerical scale factor depending on WedgeConvention chosen so that if e are orthonormal 1 forms then the following k forms are orthonormal et A Me k aj L wl Ab 8 REFERENCE LIST 72 The arguments x and y must be alternating tensor expressions of the same rank In output form HodgeInner x y appears as lt lt x y gt gt See also Inner HodgeNorm Int ExtdStar HodgeNorm HodgeNorm x is the norm of the alternating tensor expression x with respect to the Hodge inner product It is automatically converted by Ricci to Sqrt HodgeInner x Conjugate x See also HodgeInner Norm Im 2 6 Ricci converts Im x to x Conjugate x 2I See also Conjugate Imaginary 2 3 2 4 Imaginary is a value for the Type option of DefineConstant DefineTensor and DefineMathFunction IndexOrderedQ 6 1 IndexOrderedQ indices returns True if indices are ordered cor rectly according to Ricci s index ordering rules first by name then by altitude lower before upper and False otherwise See also OrderDummy OrderCovD DefineRelation DefineRule IndexQ IndexQ i returns True if i is an index name and False otherwise See also DefineBundle DefineIndex Inner 3 5 If x and y are tensor expressions Ricci interprets Inner x y as the inner product of x and y taken with respect to the default metric s on the bun dle s with which x and y are associated If x and y have oppos
49. e a constant with respect to differentiation in all space variables The notation cl L i is always interpreted as a covariant derivative of c and thus 0 and Conjugate c is replaced by c if the constant is Real and by c if it is Imaginary Options e Type gt type This can be a single keyword or a list of key words chosen from among Complex Real Imaginary Positive Negative NonPositive NonNegative Integer Odd or Even De termines how the constant behaves under conjugation exponentia tion and logarithms Default is Real e Quiet gt True or False Default is False or the value of Quiet if set See also Conjugate UndefineConstant Declare Def ineIndex 2 1 2 2 DefineIndex i j k bundle causes the index names i j k to be associated with bundle e The first argument must be a symbol or list of symbols One or more of these may have already been defined as indices as long as they are associated with the same bundle as specified in this command e The second argument must be the name of a bundle Options e TeXFormat gt texformat Determines how the index appears in TeXForm output Default is the index name itself This should be specified only when one single index is being defined Note that as in all Mathematica strings backslashes must be typed as double backslashes thus to get an index name to appear as f in TEX output you would type TeXFormat gt beta e Quiet gt True or Fals
50. e default metric on g s bundle not by g To use g to raise indices you must explicitly multiply by components of Inverse g See also DefineBundle RiemannTensor ScalarCurv LeviCivitaConnection RiemannConvention 7 2 RiemannConvention is a DefineBundle option which specifies the index convention of the Riemannian curvature tensor Riemannian 7 2 MetricType gt Riemannianis a DefineBundle option for defining a Rie mannian tangent bundle RiemannSymmetries RiemannSymmetries is a value for the Svmmetries option of DefineTensor RiemannTensor 7 6 RiemannTensor gt name is a DefineBundle option which specifies a name to be used for the Riemannian curvature tensor for this bundle The default is Rm This option takes effect only when MetricType gt Riemannian has been specified RiemannTensor g represents the Riemannian curvature tensor of the arbitrary metric g Like the default Riemannian curvature tensor RiemannTensor g is a covariant 4 tensor When indices are inserted the components of the curvature tensor are computed in terms of covari ant derivatives of g and the curvature of the default metric on g s bundle which is assumed to be Riemannian NOTE If you insert any upper in dices they are considered to have been raised by the default metric on g s bundle not by g To use g to raise indices you must explicitly multiply by components of Inverse g See also DefineBundle LeviCivitaConnection RicciTe
51. e is used for any particular Riemannian tangent bundle either by setting the global variable RiemannConvention or by including the RiemannConvention option in the call to DefineBundle See DefineBundle in the Reference List Chapter 8 for details You may convert components of the Riemannian Ricci and scalar curvature tensors of a Riemannian tangent bundle to connection components Christof fel symbols and their derivatives by applying the Ricci rule CurvToConnRule 7 SPECIAL FEATURES 42 Connection components in turn can be converted to expressions involving di rectional derivatives of the metric components bv applving ConnToMetricRule Note however that the latter rule applies only to bundles that have been given the option CommutingFrame gt True since this is the only case in which the connection components can be expressed purely in terms of derivatives of the metric 7 3 Matrices and 2 tensors Because every bundle has a default metric any 2 tensor on a given bundle can be interpreted as a linear endomorphism of the bundle Ricci supports various matrix computations with such tensors The basic operation is matrix multiplication which is indicated by Dot For example if a and b are 2 tensors then a b or Dot a b represents their matrix product If v is a 1 tensor a v represents the matrix a acting on the vector v Because the metric is used automatically to raise and lower indices no distinction is made between row
52. e is when there is only one bundle the tangent bundle of the underlying manifold For example the command In 2 DefineBundle tangent n g i j k defines a real vector bundle called tangent whose dimension is n and whose metric is to be called g The index names i j and k refer to this bundle If any expression requires more than three tangent indices Ricci generates index names of the form k1 k2 k3 etc If the bundle is one dimensional there can only be one index name Every bundle must have a metric which is a nondegenerate inner product on the fibers By default Ricci assumes the metric is positive definite but you can override this assumption by adding the option PositiveDefinite gt False to your DefineBundle call The metric name you specify in your DefineBundle call is automatically defined by Ricci as a symmetric covariant 2 tensor associ ated with this bundle By default Ricci assumes all bundles to be real You may define a complex bundle by including the option Type gt Complex in your DefineBundle call For example suppose you also need a complex two dimensional bundle called fiber You can define it as follows In 3 DefineBundle fiber 2 h a b c Type gt Complex This specifies that fiber s metric is to be called h and that the index names a b and c refer to fiber The fifth argument in the example above Type gt Complex is a typical exam ple of a Ricci option Like options for
53. e represented by 0 tensors other objects such as vector fields differential forms metrics and curvatures are represented by higher rank tensors Tensors are created by the DefineTensor command For example to create a l tensor named alpha you could type In 6 DefineTensor alpha 1 This creates a real tensor of rank one which is associated with the current default tangent bundle usually the first bundle you defined All tensors are assumed by default to be covariant thus the tensor alpha defined above can be thought of as a covector field or 1 form To define a contravariant 1 tensor i e a vector field you could type 2 RICCI BASICS 14 In 7 DefineTensor v 1 Variance gt Contravariant To create a scalar function you simply define a tensor of rank 0 In 8 DefineTensor u 0 Type gt NonNegative Real The Type option of DefineTensor controls how the tensor behaves with respect to conjugation exponentiation and logarithms For a rank 0 tensor its value can be either a single keyword or a list of keywords as in the example above The complete list of allowable Type keywords for 0 tensors is Complex Real Imaginary Positive Negative NonPositive and NonNegative For higher rank tensors the Type option can be Real Imaginary or Complex The default is always Real If you include one of the keywords indicating positivity or negativity you may leave out Real which is assumed For higher rank
54. e rules BianchiRules FirstBianchiRule SecondBianchiRule or ContractedBianchiRules TensorSimplify calls CorrectAllVariances TensorExpand AbsorbMetrics PowerSimplify RenameDummy OrderDummy and CollectConstants Any of these commands can be used individually to provide more control over the simplification process 5 SIMPLIFVING AND TRANSFORMING EXPRESSIONS 30 5 2 SuperSimplifv SuperSimplify x does the same job as TensorSimplify x but works harder at renaming dummy index pairs to find the lexically smallest version of the expression If there are two or more dummy index pairs in any term of x that refer to the same bundle SuperSimplify tries renaming the dummy indices in all possible permutations and chooses the lexically smallest result If there are k dummy index pairs per term the time taken by SuperSimplifv is proportional to k while the time taken by TensorSimplify is proportional to k Thus SuperSimplify can be very slow especially when there are more than about 4 dummy pairs This command should be used sparingly only when you suspect that some terms are equal but TensorSimplify has not made them look the same 5 3 TensorExpand TensorExpand x expands products and positive integral powers in x just as Expand does but maintains correct dummy index conventions and does not expand constant factors TensorExpand is called automatically by TensorSimplify 5 4 AbsorbMetrics AbsorbMetrics x causes any metric
55. eS 26 26 28 28 29 29 29 29 30 30 30 30 30 31 31 32 32 33 33 34 34 34 34 34 36 36 38 CONTENTS 5 7 5 Connections torsion and curvature 44 7 6 Non default connections and metrics 0 0 46 8 Reference List 49 8 1 Ricci commands and functions 49 8 2 Global variables 2 0 020 020 0000 0000000000084 87 1 INTRODUCTION 6 1 Introduction 1 1 Overview Ricci is a Mathematica package for doing symbolic tensor computations that arise in differential geometry It supports e Tensor expressions with and without indices e The Einstein summation convention e Correct manipulation of dummv indices e Mathematical notation with upper and lower indices e Automatic calculation of covariant derivatives e Automatic application of tensor symmetries e Riemannian metrics and curvatures e Differential forms e Any number of vector bundles with user defined characteristics e Names of indices indicate which bundles they refer to e Complex bundles and tensors e Conjugation indicated by barred indices e Connections with and without torsion Ricci is named after Gregorio Ricci Curbastro 1853 1925 who invented the tensor calculus what M Spivak calls the debauch of indices This manual describes the capabilities and functions provided by Ricci To use Ricci and this manual you should be familiar with Mathematica and with the basic objects of differential
56. ed in input output and TEX forms for the default covariant basis vectors for this bundle The default is the name given in the BasisName option or Basis if BasisName is also omitted Basis covectors are alwavs represented internallv bv the name Basis 8 REFERENCE LIST 61 e MetricType gt Riemannian or Normal The default is Normal which means that the bundle s metric has no special properties If Riemannian is specified this bundle is assumed to be a Rie mannian tangent bundle and DefineBundle automatically defines the Riemannian Ricci and scalar curvature tensors Contrac tions of the Riemannian and Ricci curvatures are automatically converted to Ricci and scalar curvatures respectively To ap ply other identities use FirstBianchiRule SecondBianchiRule ContractedBianchiRules or BianchiRules When you specify MetricType gt Riemannian you may also give the RiemannTensor RicciTensor ScalarCurv and RiemannConvention options below e RiemannTensor gt name a name to be given to the Riemannian cur vature tensor for this bundle This tensor is automatically defined as a real covariant 4 tensor with Symmetries gt RiemannSymmetries This option takes effect only if MetricType gt Riemannian is spec ified The default is Rm e RicciTensor gt name a name to be given to the Ricci tensor for this bundle This tensor is automatically defined as a real covariant symmetric 2 tensor This option takes effect only if
57. es of connection forms Curv is a product tensor with rank 2 2 which is Alternating in its first two and last two indices Inserting the first two indices yields the matrix of curvature 2 forms associated with the default basis Inserting all four indices yields the coefficients of the curvature tensor See also Conn Curvature CurvToConnRule Curvature 7 5 Curvature cn is the curvature tensor associated to the connection cn Ordinarily cn should be an expression of the form Conn diff where diff is a 3 tensor expression representing the difference tensor between the default connection and cn See also Conn Curv CurvToConnRule CurvToConnRule 7 2 7 5 CurvToConnRule is a rule that converts components of curvature tensors including Riemann Ricci and scalar curvature tensors for Riemannian tangent bundles to expressions involving connection coefficients and their covariant derivatives It is a partial inverse for SecondStructureRule See also Curv Conn ConnToMetricRule SecondStructureRule Declare 2 1 2 3 2 4 Declare name options can be used to change certain options 8 REFERENCE LIST 59 for previously defined Ricci objects Declare name1 name2 options changes options for several names at once all names in the list are given the same options The first argument name must be the name of a bundle constant index tensor or mathematical function that was previ ously defined by a call to DefineBund
58. fore taking covariant derivatives Ricci automatically expands covariant derivatives of sums products and powers For example 4 DERIVATIVES 26 In 26 1 a L i1 b U i 2 i 2 Out 26 1 a b i In 271 L j i i k6 Out 27 2 a b ta b a b 1 J 1 j k6 4 3 Div Div x represents the divergence of the tensor expression x This is just the covariant derivative of x contracted on its last two indices If x is a k tensor Div x is a k 1 tensor Ricci assumes that the last index of x is associated with the underlying tangent bundle of x s bundle For example suppose a is a 2 tensor In 28 Divlal L i k8 Out 28 a i k8 Div is the formal adjoint of Del 4 4 Grad If x is any tensor expression Grad x is the gradient of x It is the same as Del x except that the last index is converted to contravariant by means of the metric If x is a scalar function 0 tensor then Grad x is a vector field 4 5 Laplacian Laplacian x is the covariant Laplacian of the tensor expression x There are two common conventions in use for the Laplacian and both are supported by Ricci Which convention is in effect is controlled by the value of the global variable LaplacianConvention When LaplacianConvention DivGrad the default Laplacian x is automatically replaced by Div Grad x If 4 DERIVATIVES 27 LaplacianConvention is set to PositiveSpectrum then Laplacian x is re placed by Div Gr
59. geometry manifolds vector bundles tensors connections and covariant derivatives Chapter 8 contains a complete reference list of all the Ricci commands functions and global variables Most of the important Ricci commands are described in some detail in the main text but there are a few things that are explained only in Chapter 8 I would like to express my gratitude to my collaborators on this project Dale Lear who wrote the first version of the software and contributed uncountably many expert design suggestions John Roth and Lee Nave who reworked some of the most difficult sections of code Jay Coskey and Pm Weizenbaum who con tributed invaluable editorial assistance with this manual the National Science Foundation and the University of Washington who provided generous financial support for the programming effort and all those mathematicians who tried out early versions of this software and contributed suggestions for improvement 1 INTRODUCTION 7 1 2 Obtaining and using Ricci Ricci requires Mathematica version 2 0 or greater It will also run with Math ematica 3 0 although it will not produce formatted StandardForm output Therefore when you use Ricci in a Mathematica 3 0 notebook you should change the default output format to OutputForm When you first load Ricci you ll get a message showing how to do this by choosing Default Output Format Type from the Cell menu The Ricci source file takes approximately 287K bytes of
60. h entry in the list specifies the bundle for the corresponding index slots Each entry can be either a single bundle name or a list of bundle names indicating a direct sum of bundles for that group of slots 8 REFERENCE LIST 67 Bundle gt Same is a special option for product tensors It means that indices from any bundle can be inserted in the first set of index slots but indices in the remaining slots must be from the same bundle or its conjugate This option is used internally by Ricci in defining the Basis tensor Bundle gt Automatic is another special form that can be used for product tensors It means that indices from any bundle can be in serted in the first set of index slots but indices in the remaining slots must come from the underlying tangent bundle of the first indices This option is used internally by Ricci in defining the Conn Tor and Curv tensors e Variance gt Covariant or Contravariant Specifies whether the tensor is covariant so that components have lower indices or con travariant upper indices This can be a list whose length is equal to the total rank of the tensor in which case each entry in the list specifies the variance of the corresponding index slot The default is Covariant Values may be abbreviated Co and Con e Quiet gt True or False Default is False or the value of Quiet if set See also UndefineTensor Declare Conjugate DefineBundle FirstBianchiRule BianchiRules Conn Tor Cur
61. ha beta appears as lt lt a b gt gt 3 PRODUCTS CONTRACTIONS AND SYMMETRIZATIONS 23 3 7 Int If v is a vector field and alpha is a differential form Int v alphal represents interior multiplication of v into alpha usually denoted ya or via with a nor malization factor depending on WedgeConvention chosen so that Int v is an antiderivation More generally if alpha and beta are any alternating tensor expressions with Rank alpha lt Rank beta then Int alpha beta represents the gen eralized interior product of alpha into beta sometimes denoted a V f Int alpha is defined as the adjoint with respect to the Hodge inner product of wedging with alpha on the left if alpha beta and gamma are alternating tensors of ranks a b and a b respectively then lt lt alpha 7 beta gamma gt gt lt lt beta Int gamma gt gt alpha 3 8 Alt If x is any tensor expression Alt x is the alternating part of x If x is already alternating then Alt x x Note that Alt expects its argument to be a pure or mixed tensor expression This means that you must apply Alt before you insert indices If Alt is applied to a component expression it does nothing because a component expression is considered to have rank 0 3 9 Sym If x is any tensor expression Sym x is the symmetric part of x If x is already symmetric then Sym x x Note that Sym expects its argument to be a pure or mixed tensor expression Th
62. hs elas ete as eect are BD A See Pass Ba piu A AT Eto bale A ee dd ON a a Y 10 10 12 12 13 16 17 18 19 19 19 20 20 21 21 22 22 22 CONTENTS Simplifving and Transforming Expressions bil Tens rSimpliy a B A Es 5 2 SuperSimplifv e Hid Lensorbixpand s lo ia sieb a A ewe NB Gea E 5d AbsorbMetries 5 2 2 elec a ito LE ct i 5 5 RenameDummy co saz A a a 5 0 Order DUMMY 64 A Ws ae ai tl ee ee SE ee 9 1 CollectConstants i 2 202 505 ga eR KA aaa i ani ee Res 5 8 FactorConstants 00000000000 0004 5 9 SimplifyConstants o a a 5 10 BasisExpa d 2 275 24 ae aaa Ae eae a RE ee OS 5 11 BasisGather s 20 fae en bok pS Rb bee ee A 5 12 CovDExpand 0 0 000 ee ee 5 13 ProductExpand 0 00 000 eee a ee ee 5 14 PowerSimplifv 00 5 15 CorrectAllVariances 6 5 16 NewDummy alele Gt cd a SIT CommuteCovD r eE p a ae BRP ER A A A 58 Order Cov De si xa le cd Bi e bee Bee Bod Dele es 5 19 Cov DSimplify os en Se A a na i BA Di eb ok 5 20 LowerAllindices 0 5 21 TensorGancel L ick kote ln a e ae Boni Age ot ini Defining Relations Between Tensors 6 1 DefineRelation 0 0 0 00 eee ee ees 6 2 DetineRules cp h reos Wwe eh a ee PS Special Features 7 1 One dimensional bundles 000000085 7 2 Riemannian metrics 0 0 e 7 3 Matrices and 2 tensors e GA Product tensors i a il bini a e A AS Ar Pe Ew a
63. is means that you must apply Sym before you insert indices If Sym is applied to a component expression it does nothing because a component expression is considered to have rank 0 4 DERIVATIVES 24 4 Derivatives This chapter describes the differentiation operators provided bv Ricci The op erators Del Div Grad Extd ExtdStar and Lie below are Ricci primitives after some simplification thev are maintained unevaluated until vou insert in dices The rest of the operators described here are immediatelv transformed into expressions involving the primitive operators 4 1 Del The basic differentiation operator in Ricci is Del If x is a tensor expression Del x represents the total covariant derivative of x Del stands for the math ematical symbol V If x is a tensor of rank k Del x is a tensor of rank k 1 If x is a section of bundle b Del x is a section of the tensor product of b with its underlying co tangent bundle The extra index slot resulting from differentiation is always the last one and is always covariant When you insert indices into Del x you get components of the covariant derivative of x For example suppose a is a 2 tensor In 21 Del a Out 21 Del a In 221 L i L j L k Out 22 a ijk The operator Del can also be used to take the covariant derivative of a tensor with respect to a specific vector If v is a 1 tensor expression Del v x repre sents the covariant deriv
64. ite variance for example if x is a l form and y is a vector field then Inner x y represents the natural pairing between x and y The arguments x and y must have the same rank Ricci automatically expands inner products of sums and scalar multiples In output form Inner x y appears as lt x y gt See also Norm HodgeInner InsertIndices 2 6 If x is a tensor expression of rank k then InsertIndices x L i1 L ik causes the indices L i1 L ik to be inserted into the k index slots in order yielding a component expression In input form this can be abbreviated x L i1 Llikl Indices in each slot may be lower L i or upper U i If x has rank 0 then InsertIndices x or x converts it to a component expression by generating dummy indices as necessary 8 REFERENCE LIST 73 If one or more indices are inserted into a rank 0 expression then InsertIndices is automatically converted to CovD See also CovD Int 3 7 If x and y are alternating tensor expressions with Rank x lt Rank y Int x y represents the generalized interior product of x into y When x is a vector field Int x y is interior multiplication of x into y with a numerical factor depending on WedgeConvention In general Int x _ is the adjoint with respect to the Hodge inner product of wedging with x on the left if alpha beta and gamma are alternating tensors of ranks a b a b respectively then lt lt alpha beta gam
65. l order using the skew symmetry of the Lie bracket Lie derivatives of differential forms are not automatically expanded in terms of exterior derivatives and interior multiplication because there may be some 4 DERIVATIVES 28 situations in which vou do not want this transformation to take place Vou can cause them to be expanded bv applving the rule LieRule In 13 Lie v beta Out 13 Lie beta v In 14 LieRule Out 14 d Int beta Int d beta v v 5 SIMPLIFVING AND TRANSFORMING EXPRESSIONS 29 5 Simplifving and Transforming Expressions This chapter describes commands that can be used to simplifv tensor expres sions or to perform various transformations such as expanding covariant deriva tives in terms of ordinarv directional derivatives or vice versa The commands in this chapter that take onlv one argument can be applied only to selected terms of an expression For example TensorSimplify x n simplifies only term n leaving the rest of the expression unchanged TensorSimplify x n1 n2 n3 simplifies terms ni n2 n3 com bining terms if possible but leaving the rest of x unchanged The Reference List Chapter 8 describes which commands accept this syntax 5 1 TensorSimplifv The most general simplification command provided by Ricci is TensorSimplify TensorSimplify x attempts to put an indexed tensor expression x into a canonical form The rules applied by TensorSimplify may not always result i
66. lation tensor deletes the relation previously defined for tensor The argument must exactly match the first argument of the corresponding call to DefineRelation NOTE There is no UndefineRule function to remove the definition of a rule defined by DefineRule simply execute an assignment such as rulename or call DefineRule with the option NewRule gt True Option e Quiet gt True or False Default is False or the value of Quiet if set See also DefineRelation Undef ineTensor Undef ineTensor tensorname clears the definition of tensorname If you try to perform computations with previous expressions involving tensorname you may get unpredictable results Option e Quiet gt True or False Default is False or the value of Quiet if set See also DefineTensor Declare 8 REFERENCE LIST 88 UndefineTensorSymmetries UndefineTensorSymmetries name deletes the TensorSymmetry object created by DefineTensorSymmetries name UnderlyingTangentBundle UnderlyingTangentBundle x returns a list of bundles representing the underlying tangent bundle of the expression The tangent bundle is the direct sum of the bundles in the list It is assumed that all tensors used in a given expression have the same underlying tangent bundle Variance 2 4 Variance is a DefineTensor option The function Variance x returns a list of variances of the tensor expres sion x one for each index slot See also DefineTensor B
67. le DefineConstant DefineIndex DefineTensor or DefineMathFunction The allowable options depend on what type of object is given as the first argument the meanings of the options are the same as in the corresponding Define command Any options not listed explicitly in the call to Declare are left unchanged e Constant If name is a symbolic constant only the Type op tion is allowed It overrides any Type specification given when DefineConstant was called e Index If name is an index name the only allowable option is TeXFormat e Bundle If name is a bundle name the allowable op tions are FlatConnection ParallelFrame OrthonormalFrame PositiveDefinite CommutingFrame and TorsionFree e Tensor If name is a tensor name the allowable options are Type TeXFormat and Bundle e Math function If name is the name of a mathematical function the only allowable option is Type See also DefineBundle DefineConstant DefineIndex DefineTensor DefineMathFunction DefineBundle 2 1 7 1 DefineBundle name dim metric indices defines a bundle e name A symbol that uniquely identifies the bundle e dim The dimension of the bundle Can be a positive integer a symbolic constant or any constant expression e metric A name for the bundle s metric The metric is automatically defined by DefineBundle as a Symmetric or Hermitian 2 tensor It is assumed to be compatible with the bundle s default connection e indices A
68. led index slots it is first converted to a component expression before covariant derivatives are taken The Ricci functions that can be used to construct tensor expressions are de scribed in later chapters 2 7 Saving your work Ricci provides a function called RicciSave for saving all the definitions you have made during your current Mathematica session The command RicciSavel filename writes your definitions in the file called filename in Mathematica input form RicciSave also writes the current definitions of Ricci s predefined tensors Basis Curv Tor Conn and Kronecker in case you have defined any relations involv ing them and of its global variables those whose names start with Any 2 RICCI BASICS 19 previous contents of filename are erased You can read the definitions back in by typing lt lt filename 3 PRODUCTS CONTRACTIONS AND SYMMETRIZATIONS 20 3 Products Contractions and Symmetrizations This chapter describes the most important Ricci functions for specifying vari ous kinds of products contractions and symmetrizations of tensor expressions Although Ricci performs some automatic simplification on most of these func tions such as expanding linear combinations of arguments for the most part these functions are simply maintained unevaluated until you insert indices 3 1 TensorProduct The basic product operation for tensors is TensorProduct In input form this can be abbreviated TProd In 15
69. lue for the MetricType option of DefineBundle NoSymmetries 2 4 NoSymmetries is a value for the Symmetries option of DefineTensor Odd 47 2 3 Odd is a value for the Type option of DefineConstant OneDims OneDims is a value for the Mode option of CorrectAllVariances used internally by TensorSimplify OrderCovD 5 18 66 99 gt OrderCovD x puts all of the indices appearing after in the tensor expression x in order according to Ricci s index ordering rules first al phabetically by name then by altitude by adding appropriate curvature and torsion terms OrderCovD x n or OrderCovD x n1 n2 ap plies OrderCovD only to term n or to terms n1 n2 of x See also CommuteCovD CovDSimplify IndexOrderedQ 8 REFERENCE LIST 77 OrderDummy 5 6 OrderDummy x attempts to put the dummy indices occurring in the tensor expression x in a canonical form OrderDummy x n or OrderDummy x n1 n2 applies OrderDummy to term n or to terms ni n2 of x All pairs of dummy indices are ordered so that the lower member appears first whenever possible Then OrderDummy tries various permutations of the dummy index names in each term of x searching for the lexically smallest version of the expression among all equivalent versions Option e Method gt n specifies how hard OrderDummy should work to find the best possible version of the expression The allowable values are 0 1 and 2 The default is Method gt 1 which means tha
70. ma gt gt lt lt beta Int gamma gt gt alpha See also HodgeInner Wedge Dot WedgeConvention Integer 2 3 Ricci recognizes Integer as a value for the Type option of DefineConstant Inverse 7 3 If x is a 2 tensor expression Ricci interprets Inverse x as the matrix inverse of x See also Dot Kronecker Kronecker 7 3 Kronecker L i U j is the Kronecker delta symbol which is equal to 1 if i j and 0 otherwise Kronecker is a generic tensor representing the identity endomorphism of any bundle See also Dot Inverse L 2 2 2 4 L i represents a lower index i L i Bar is the internal representation for a lower barred index i it can be abbreviated in input form by LB i See also LB U UB LaplaceBeltrami 4 8 LaplaceBeltrami x is the Laplace Beltrami operator A dd d d applied to the differential form x It is automatically replaced by Extd ExtdStar x ExtdStar Extd x Option e Metric gt g asymmetric 2 tensor expression representing a metric to be used in place of the default metric for x s bundle which is assumed to be Riemannian See also Extd ExtdStar Laplacian 8 REFERENCE LIST 74 Laplacian 4 5 LB Laplacian x is the covariant Laplacian of the tensor expression x The sign convention of Laplacian is controlled by the global vari able LaplacianConvention When LaplacianConvention DivGrad the default Laplacian x is automatically replaced by Div Grad x When L
71. materials and or transmission John M Lee Department of Mathematics Box 354350 University of Washington Seattle WA 98195 4350 E mail lee math washington edu Web http www math washington edu lee CONTENTS Contents 1 Introduction VEL Overviews S49 4 dok sk g Ba ek BB pt a i kk 1 2 Obtaining and using Ricci 2 a 1 3 A brief look at Ricci Ricci Basics 251 Bundles oe si i a A RE e ee a 22 Indices ps ii a aj BB a 223 Constants os e E A EA 2A A E A e i ga bb h d A dh eee Bad AR a 2 5 Mathematical functions 6 2 6 Basic tensor expressions 20000 2T Savin yout WOTK Mei A elle n o Ta A a a woe dab aod a Products Contractions and Svmmetrizations 3 1 TensorProdutt sa eli a DS Be e Ge A Bi2 Wed Bers in ab rol O04 eed Pk a e Bee es 3 3 SvmmetricProduct 2 0 00200000002 2 ee BAY Dot e ee ale O Bt eet eo DA EA DID ANE eb AN MS oleae Ghee maine L he bout a 36 Hodgelnner e arts E y ALIA A ales Ba ae Re GP args do Sale Mate ARA tock ts By Lgl Bhs DA eae a B oe dea oh hl oe wo SS sot amp Bis Alte ix Sats a a Sie A Beales Stan a whe hr ty ee hi 8 DONA ON cn Tarde ate 2 he ue at eat ae add Oak 8 Derivatives AST 2D eli ud yh 92 Ante Mean ane oe bien So RE AE Bee es eke Sh 4 25 COND ye cee Reeth i GE B BE wee ae Bed DS Ass Div A Shy ja Matava KIM fiz ti ji Bob NN MAL AGP E AS GD ante 4 5 Laplaci ssibu oe tom ele ete Ab b Be i L ei woe AW BA G JERGA e
72. n Ordinarilv 7 SPECIAL FEATURES 48 the value of the Connection option should be an expression of the form Conn 4 diff where diff is a 3 tensor expression that represents the difference tensor between the given connection and the default connection For example suppose diff is a 3 tensor In 23 Della Connection gt Conn diff Conn diff Out 23 Del a In 24 L i L j L k k36 k37 Out 24 a a diff a diff ij jk k36 j i k i k37 j k Notice that the components are expanded in terms of covariant derivatives of a with respect to the default connection and components of diff You can also call CovD with the Connection option For example the following input expression results in the same output as Out 24 above In 25 CovD a L i L j L k Connection gt Conn diff Other operators that accept the Connection option are Div Grad and Laplacian On a Riemannian tangent bundle you can also specify that operations are to be computed with respect to a metric other than the default one This is done by adding the Metric option to the appropriate Ricci function For example suppose h is a symmetric 2 tensor To compute the covariant derivative of the tensor a with respect to the metric h you could use In 60 Del a Metric gt h h Out 60 Del a When you insert indices the components of this expression are computed in terms of covariant derivatives of a and h with respe
73. n on an n dimensional bundle the wedge product of all n basis elements is 1 n times the determinant function Thus if el e are basis covectors 1 eA Ae det n The other wedge product convention called Det is defined by arn p 249 aaa 8 pq where a is a p form and 8 a q form This convention is used for example in the texts by Spivak Boothby and Warner The name Det comes from the fact that under this convention if w are l forms and X are vectors wA Aw Xi Xi det w X In particular el A Ae det 3 3 SymmetricProduct In Ricci symmetric products are represented by ordinary multiplication Thus if a and b are l tensors then a b represents their symmetric product ax b a 2 tensor Similarly a 2 represents a x a In input form you can type a symmetric product just as you would an ordi nary product If you prefer for clarity you may indicate symmetric products explicitly using the SymmetricProduct function In 17 SymmetricProduct alpha beta Out 17 alpha beta In output form Ricci inserts explicit asterisks in products of tensors of rank greater than 0 to remind you that multiplication is being interpreted as a symmetric product The symmetric product is defined by a b Sym a 9 b where Sym is the projection onto the symmetric part of a tensor 3 4 Dot Dot is a Mathematica function that is given an additional interpretation by Ricci If
74. n the simplest looking expression but two component expressions that are math ematically equal will usually be identical after applying TensorSimplify TensorSimplify expands products and powers of tensors uses metrics to raise and lower indices tries to rename all dummy indices in a canonical order and collects all terms containing the same tensor factors but different con stant factors When there are two or more dummy index pairs in a single term TensorSimplify tries exchanging names of dummy indices in pairs and chooses the lexically smallest expression that results This algorithm may not always recognize terms that are equal after more complicated rearranging of dummy index names an alternative command SuperSimplify works harder to get a canonical expression at the cost of much slower execution TensorSimplify makes only those simplificatons that arise from the routine changing of dummy index names and the application of symmetries and alge braic rules that the user could easily check More complicated simplifications such as those that arise from the Bianchi identities or from commuting covariant derivatives are done only when you request them The basic reason for this dis tinction is so that you will not be confronted by a mystifyingly drastic and unx plainable simplification To reorder covariant derivatives that is the ones that come after use OrderCovD or CovDSimplify To apply Bianchi identities apply one of th
75. ne dimensional indices and indices that appear inside of differential op erators such as Del or Extd should be corrected This option exists mainly for internal use by TensorSimplify Default is A11 See also TensorSimplify LowerAllIndices AbsorbMetrics Covariant 2 4 CovD Covariant is a value for the Variance option of DefineTensor and may be abbreviated Co If an index slot is Covariant indices in that slot are lower by default See also DefineTensor 4 2 If x is a rank 0 tensor expression then CovD x L i L j is the component of the covariant derivative of x in the L i L j directions If x has any unfilled index slots then it is first converted to a component expression by generating dummy indices if necessary before the covariant derivative is computed The shorthand input form x L i L j is automatically converted to CovD x L i L j if x has rank 0 In output form covariant derivatives of a tensor are represented by subscripts following a semicolon Options for the explicit form CovD x L i L j1H only e Connection gt cn specifies that covariant derivatives are to be taken with respect to the connection cn instead of the default con nection Ordinarily cn should be an expression of the form Conn diff where diff is a 3 tensor expression representing the difference tensor between the default connection and cn e Metric gt g a symmetric 2 tensor expression indicating that the c
76. ngentBundle is a DefineBundle option The function TangentBundle x returns the tangent bundle list for the bundle x See also UnderlyingTangentBundle Tensor 2 4 Ricci s internal form for tensors is Tensor name i j 1k 1 where name is the tensor s name i j are the tensor indices each of the form L i or U il and k 1 are the indices resulting from co variant differentiation In input form this can be typed name i j k 1 Either set of indices in brackets can be omitted if it is empty See also DefineTensor UndefineTensor TensorCancel 5 21 TensorCancel x attempts to simplify each term of x by combin ing and cancelling common factors even when the factors have different names for their dummy indices TensorCancel x n or TensorCancel x n1 n2 applies TensorCancel only to term n or to terms n1 n2 of x See also TensorSimplify SuperSimplify TensorData TensorData name is a list containing data for the tensor name used internally by Ricci See also DefineTensor 8 REFERENCE LIST 84 TensorExpand 5 3 TensorExpand x expands products and positive integral powers in x just as Expand does but maintains correct dummv index conven tions and does not expand constant factors TensorExpand x n or TensorExpand x n1 n2 applies TensorExpand to term n or to terms n1 n2 of x If vou applv the Mathematica function Expand to an expression containing tensors with indices
77. ni n2 of x SuperSimplify works exactly the same way as TensorSimplify except that it calls OrderDummy with the option Method gt 2 so that all possible permutations of dummy index names are tried This is generally much slower for expressions having more than 4 dummy index pairs for an expression with k dummy index pairs per term the time taken by TensorSimplify is proportional to k while the time taken by SuperSimplify is proportional to k See also TensorSimplify OrderDummy TensorCancel Sym 3 9 Sym x represents the symmetrization of the tensor expression x If x is symmetric then Sym x x See also Alt SymmetricProduct Symmetric 2 4 Symmetric is a value for the Symmetries option of DefineTensor 8 REFERENCE LIST 83 SymmetricProduct 3 3 If x y and z are tensor expressions SymmetricProduct x y z or x y zorx y zrepresents their symmetric product Symmetric prod ucts are represented internally by ordinary multiplication In output form Ricci inserts explicit asterisks whenever non scalar tensor expression are multiplied together to remind the user that multiplication is being inter preted as symmetric product Mathematically the symmetric product is defined by axb Sym a b See also Times Power Sym ProductExpand SymmetricQ SymmetricQ x is True if x is asymmetric or Hermitian tensor expression and False otherwise Symmetries 2 4 Symmetries is a DefineTensor option TangentBundle 2 1 Ta
78. nsor ScalarCurv Same Bundle gt Same is a special DefineTensor option for product tensors ScalarCurv 7 6 ScalarCurv gt name is a DefineBundle option which specifies a name to be used for the scalar curvature for this bundle The default is Sc This option takes effect only when MetricType gt Riemannian has been specified 8 REFERENCE LIST 81 ScalarCurv g represents the scalar curvature of the arbitrary metric g When indices are inserted the scalar curvature is computed in terms of covariant derivatives of g and the curvature of the default metric on g s bundle which is assumed to be Riemannian See also RiemannTensor RicciTensor LeviCivitaConnection DefineBundle ScalarQ ScalarQ x is True if x is a rank 0 tensor expression with no free indices and False otherwise SecondBianchiRule T2 SecondBianchiRule is a rule that attempts to turn sums containing two differentiated Riemannian curvature tensors into a single term using the second Bianchi identity For example 2 Rm 2 Rm SecondBianchiRule i jkl m i jllm k gt 2 Rm ijmk l See also FirstBianchiRule ContractedBianchiRules BianchiRules DefineBundle SecondStructureRule 7 5 SecondStructureRule is a rule that implements the second structure equation for exterior derivatives of the generic connection forms For ex ample Extd Conn L i U j SecondStructureRule gt 1 j k j Curv Conn Conn 2 i i k See also FirstStruc
79. nto all unfilled index slots so it can be used to turn a scalar tensor expression into a component expression See also Basis BasisGather CovDExpand ProductExpand BasisGather 5 10 BasisGather x y attempts to recognize the basis expression for y in x and replaces it by y BasisGather x y1 y2 does the same thing for several expressions at once In effect BasisGather is a partial inverse for BasisExpand See also BasisExpand BasisName 2 4 BasisName gt name is an option for DefineBundle It specifies the name to be used in input and output form for the basis vectors for the bundle being defined 8 REFERENCE LIST 52 BianchiRules 7 2 BianchiRules L i L j L x1 is a rule that converts Riemannian cur vature tensors containing the indices L i L j L k in order to two term sums using the first and second Bianchi identities The three indices may be upper indices lower indices or a combination The first Bianchi rule is applied to any tensor with with Symmetries gt RiemannSymmetries the second is applied only to tensors that have been defined as Riemannian curvature tensors by DefineBundle EXAMPLES i Out 16 Rm el In 171 BianchiRules L j L k L i i Out 17 Rm Rm kj1 ljk m Out 20 Rm ijkl In 21 BianchiRules L x L 1 U m m m Out 21 Rm Rm ijk l ijl k See also DefineBundle FirstBianchiRule SecondBianchiRule ContractedBianchiRules Bundle 2
80. of index names associated with the appropriate bundle skipping those names that already appear in x as free indices When the list of index names is exhausted computer generated names of the form kn are used where k is the last index name in the list and n is an integer For one dimensional bundles n is omitted See also TensorSimplify OrderDummy NewDummy RicciSave 2 7 RicciSave filename causes the definitions of all symbols defined in Mathematica s current context along with Ricci s predefined tensors and global variables to be saved in Mathematica input form into the file filename The current context is usually Global and usually includes all symbols you have used in the current session The previous contents of filename are erased You can read the definitions back in by typing lt lt filename RicciTensor 7 6 RicciTensor gt name is a DefineBundle option which specifies a name to be used for the Ricci tensor for this bundle The default is Rc This option takes effect only when MetricType gt Riemannian has been spec ified RicciTensor g represents the Ricci curvature tensor of the arbitrary metric g When indices are inserted the components of the Ricci tensor are computed in terms of covariant derivatives of g and the curvature of 8 REFERENCE LIST 80 the default metric on g s bundle which is assumed to be Riemannian NOTE If you insert any upper indices they are considered to have been raised by th
81. orm hag hga in It is done this way because Ricci s index conventions require that all tensors be considered to be complex multilinear To obtain the usual sesquilinear form on a complex bundle V you apply the metric to a pair of vectors x y where x y V In Ricci s input form this inner product would be denoted by Inner x Conjugate y J For every bundle you define you must specify either explicitly or by default the name of the tangent bundle of the underlying manifold Ricci needs this information for example when generating indices for covariant derivatives By default the first bundle you define or the direct sum of this bundle and its conjugate if the bundle is complex is assumed to be the underlying tangent bundle for all subsequent bundles If you want to override this behavior you can either give a value to the global variable DefaultTangentBundle before defining bundles or explicitly include the TangentBundle option in each call to DefineBundle See the Reference List Chapter 8 for more details Another common type of bundle is a tangent bundle with a Riemannian met ric The DefineBundle option MetricType gt Riemannian specifies that the bundle is a Riemannian tangent bundle Riemannian metrics are described in Section 7 2 There are other options you can specify in the DefineBundle call such as FlatConnection TorsionFree OrthonormalFrame CommutingFrame PositiveDefinite and ParallelFrame If y
82. ou decide to change any of these 2 RICCI BASICS 12 options after the bundle has already been defined you can call Declare See the Reference List Chapter 8 for a complete description of Def ineBundle and Declare 2 2 Indices The components of tensors are represented by upper and lower indices Ricci adheres to the Einstein index conventions as follows For any vector bundle V sections of V itself are considered as contravariant l tensors while sections of the dual bundle V are considered as covariant 1 tensors The components of contravariant tensors have upper indices while the components of covariant tensors have lower indices Higher rank tensors may have both upper and lower indices ndices are distinguished both by their horizontal positions and by their vertical positions the altitude of each index indicates whether that index is co variant or contravariant Any index that appears twice in the same term must appear once as an upper index and once as a lower index and the term is under stood to be summed over that index Indices associated with one dimensional bundles are an exception to this rule see Section 7 1 on one dimensional bundles below In input form and in Ricci s internal representation indices have the form L namel for a lower index and U name for an upper index Barred indices for complex bundles only are typed in input form as LB name and UB name they are represented internally by L name Bar and
83. ovariant derivatives are to be taken with respect to the Levi Civita connection of g instead of the default metric for x s bundle which is assumed to be Riemannian See also Del InsertIndices CovDExpand CovDExpand 5 12 CovDExpand x converts all components of covariant derivatives in x to ordinary directional derivatives and connection coefficients CovDExpand x n or CovDExpand x n1 n2 applies CovDExpand 8 REFERENCE LIST 58 only to term n or to terms n1 n2 of x Directional derivatives with respect to basis elements are represented by Dell Basis L i For example ki v CovDExpand gt Del v Conn v Ig j i j i ki Ordinarv directional derivatives can be converted back to covariant deriva tives bv calling BasisExpand See also BasisExpand Del Conn CovDSimplify 5 19 Curv CovDSimplify x attempts to simplify x as much as possible by ordering all dummy indices including those that occur after the in component expressions CovDSimplify x n or CovDSimplify x n1 n2 applies CovDSimplify only to term n or to terms n1 n2 of x CovDSimplify first calls TensorSimplify then OrderCovD then TensorSimplify again See also OrderCovD CommuteCovD TensorSimplifv SuperSimplify 7 5 7 6 Curv is the generic curvature tensor associated with the default connection on any bundle It is generated when covariant derivatives are commuted and when SecondStructureRule is applied to derivativ
84. probably more efficient to use the other simplification commands individually 5 20 LowerAllIndices LowerAllIndices x converts all upper indices in x except those appearing on Basis covectors or metrics to lower indices by inserting metrics as needed 5 21 TensorCancel TensorCancel x attempts to simplify products and quotients of tensor expres sions by combining and canceling factors that are equal even though they have different dummy index names For example 5 SIMPLIFVING AND TRANSFORMING EXPRESSIONS 36 i 3 j 1 Out 37 1 u u u u si zj In 38 TensorCancel Out 38 1 u u yes 6 DEFINING RELATIONS BETWEEN TENSORS 37 6 Defining Relations Between Tensors Because the internal and external forms of tensors differ vou should never use the name of a tensor on the left hand side of an assignment or rule such as a b a b a gt b or a gt b Ricci provides two commands DefineRelation and DefineRule to create such assignments and rules This is a common source of error To emphasize NEVER USE THE NAME OF A TENSOR ON THE LEFT HAND SIDE OF AN ASSIGNMENT OR RULE USE DefineRelation OR DefineRule INSTEAD 6 1 DefineRelation If you want to cause one tensor always to be replaced by some other tensor expression or to be replaced only when certain indices are inserted you can use DefineRelation The basic syntax is shown by the following example In 39 DefineRelation a 2 b 1 Relation
85. pter AbsorbMetrics 5 4 AbsorbMetrics x simplifies x by eliminating any metrics that are con tracted with other tensors and using them instead to raise or lower in dices AbsorbMetrics x n or AbsorbMetrics x n1 n2 applies AbsorbMetrics only to term n or to terms n1 n2 of x Option e Mode gt All or NoOneDims NoOneDims indicates that metrics asso ciated with one dimensional bundles should not be absorbed unless they are paired with other metrics This option exists mainly for internal use by TensorSimplify so that it can control the order in which different metrics are absorbed Default is A11 See also TensorSimplify LowerAllIndices CorrectAllVariances Alt 3 8 Alt x is the alternating part of the tensor expression x If x is alternating then Alt x x Alt is also a value for the global variable WedgeConvention See also Sym Wedge WedgeConvention Alternating 2 4 Alternating is a value for the Symmetries option of DefineTensor Any 2 1 Any can be used as a value for the Bundle option of the DefineTensor command Bundle gt Any means that indices from any bundle can be inserted 8 REFERENCE LIST 51 Bar 2 2 2 4 Bar is an internal symbol used by Ricci for representing conjugates of tensors and indices The internal form for a barred index is L i Bar or U i Bar In input form these can be abbreviated LB i and UB i The internal form for the conjugate of a ten sor is Tensor name
86. respect to conjugation exponentiation and logarithms After a constant has been defined you can change its Type option by calling Declare as in the following example In 12 Declarel d Type gt NonNegative Real In the Ricci package any expression that does not contain explicit tensors is assumed to be a constant for the purposes of covariant differentiation so in some cases it is not necessary to call DefineConstant However if you attempt to insert indices into a symbol c that has not been defined as either a constant or a tensor you will get output that looks like c L i L j indicating that Ricci does not know how to interpret c For this reason as well as to tell Ricci what type of constant c is it is always a good idea to define your constants explicitly Note that constants are not given Mathematica s Constant attribute for the following reason Some future version of Ricci may allow tensors depending on parameters such as t in this case t will be considered a constant from the point of view of covariant differentiation but may well be non constant in expressions such as D f t t You can test whether a tensor expression is constant or not by applying the Ricci function ConstantQ 2 4 Tensors The most important objects that Ricci uses in calculations are tensors Ricci can handle tensors of any rank and associated with any vector bundle or direct sum or tensor product of vector bundles Scalar functions ar
87. rivatives that require component expressions You may insert indices into any pure or mixed tensor expression thereby con verting it to a component expression by typing the indices in brackets after the expression as in x L i U j which is a shorthand input form for the explicit Ricci function InsertIndices x L i L j For most tensor expressions the number of indices inserted must be equal to the rank of the tensor expression The only exception is product tensors described in Section 3 below For example if alpha and beta are 1 tensors their tensor product indicated by TensorProduct alpha beta is a 2 tensor so you can insert two indices 2 RICCI BASICS 18 In 12 TensorProduct alpha betal L i1 L j11 Out 12 alpha beta ij If x is a scalar rank 0 expression it can be converted to a component expres sion by inserting zero indices as in InsertIndices x or x or by typing BasisExpand x For example Inner alpha beta represents the inner prod uct of alpha and beta In 13 Inner alpha beta BasisExpand ki Out 13 alpha beta k1 If you type indices in brackets after a component expression one in which all index slots are already filled you get components of covariant derivatives Thus In 14 13 L j ki ki Out 14 alpha beta alpha beta 53 k1 ki j If vou tvpe indices in brackets after a scalar rank 0 expression that is not a component expression i e has unfil
88. rm tensor x condition associated with the tensor s name The first argument tensor must be a single tensor name with or without indices and can be written in Ricci s input form If index names used in tensor are associated with bundles then the relation is applied only when indices from those bundles appear in those positions The condition argument is optional if present it should be a true false condition that must be satisfied before the relation will be applied It may involve the tensor name or indices that appear in tensor For example if i is associated with bundle a and p with bundle b DefineRelation tl L i1 L p 1 0 8 REFERENCE LIST 64 specifies that t is to be replaced by O whenever its first index is associated with a and its second with b In general the relation substitutes x suitably modified in place of any expression that matches tensor after insertion of indices renaming of indices covariant differentiation conju gation or raising or lowering of indices DefineRelation has the Ho1dA11 attribute so its arguments will not be evaluated when the command is executed This means you must write out tensor x and condition in full rather than referring to a variable name or a preceding output line such as Out 10 so that DefineRelation can properly process the indices that appear in the arguments Option e Quiet gt True or False Default is False or the value of Quiet if set
89. rt a component expression for a tensor back to pure tensor form BasisGather allows vou to do this For example we can undo the effect of BasisExpand in Out 34 above In 35 BasisGather 34 a Out 35 2 a The second argument to BasisGather can in principle be any tensor expression although BasisGather may not recognize the basis expansion of complicated expressions It works best if you apply TensorExpand or TensorSimplify before applying BasisGather You can gather several tensor expressions at once by putting a list of expressions in the second argument 5 12 CovDExpand CovDExpand x expands all covariant derivatives in x into ordinary direc tional derivatives and connection components Directional derivatives of com ponents of tensors are represented by expressions like Dell Basis L i a L j L k 1 For example 5 SIMPLIFVING AND TRANSFORMING EXPRESSIONS 33 Out 35 a jk i In 36 CovDExpand k3 k4 Out 36 Del a l a Conn a Conn i jk k3 k j i j ka k i Here Conn represents the coefficients of the default connection with respect to the default basis See Section 7 5 Connections torsion and curvature for more information 5 13 ProductExpand ProductExpand x expands symmetric products and wedge products of 1 tensors that occur in x and rewrites them in terms of tensor products For example assuming WedgeConvention Alt you would get the following re sult i j Out 5 Basis 7
90. s See the Reference List Chapter 8 for details OrderDummy is called automatically by TensorSimplify 5 7 CollectConstants CollectConstants x groups together terms in the tensor expression x hav ing the same tensor factors but different constant factors and performs some simplification on the constant factors CollectConstants is called automatically by TensorSimplify 5 8 FactorConstants FactorConstants x applies the Mathematica function Factor to the constant factor in each term of x 5 9 SimplifyConstants SimplifyConstants x applies the Mathematica function Simplify to the con stant factor in each term of x 5 10 BasisExpand If x is any tensor expression BasisExpand x expands all tensors in x into component expressions multiplied by the default basis vectors and covectors For example suppose a is a 2 tensor 5 SIMPLIFVING AND TRANSFORMING EXPRESSIONS 32 In 34 BasisExpand 2 al ki k2 2a Basis X Basis ki k2 Out 34 Because of the limitations of the index notation there is no provision for ex panding tensors automaticallv with respect to anv basis other than the default one vou can usuallv accomplish the same thing bv using DefineRule to create transformation rules between the default basis and other bases If x is a scalar rank 0 expression BasisExpand x causes all index slots to be filled thus converting x to a component expression 5 11 BasisGather Sometimes vou mav want to conve
91. sents an upper index i U i Bar is the internal representation for an upper barred index i it can be abbreviated in input form by UB i See also UB L LB UB 2 2 2 4 UB i is the input form for an upper barred index i It is converted automatically to the internal form U i Bar See also U L LB Bar UndefineBundle 2 1 UndefineBundle bundle clears the definition of bundle its metric and all its indices If bundle is a Riemannian tangent bundle it also clears the definitions of its Riemann Ricci and scalar curvature tensors If you try to perform computations with previous expressions involving bundle s indices you may get unpredictable results Option e Quiet gt True or False Default is False or the value of Quiet if set See also DefineBundle UndefineIndex Declare 8 REFERENCE LIST 87 UndefineConstant UndefineConstant c removes c s definition as a constant Option e Quiet gt True or False Default is False or the value of Quiet if set See also DefineConstant Declare Undefinelndex Undef ineIndex i or UndefineIndex i j k removes the association of indices with their bundles If you try to perform computations with previous expressions involving these indices you may get unpredictable results Option e Quiet gt True or False Default is False or the value of Quiet if set See also DefineIndex DefineBundle UndefineBundle Declare UndefineRelation 6 1 UndefineRe
92. single tensor with or without indices then the rule is also applied to expres sions that match lhs after insertion of indices covariant differentiation or conjugation DefineRule has the Ho1dA11 attribute so its arguments will not be evalu ated when the command is executed This means you must write out 1hs rhs and condition in full rather than referring to a variable name or a preceding output line such as Out 101 so that DefineRule can properly process the indices that appear in the arguments Options e Quiet gt True or False Default is False or the value of Quiet if set e NewRule True or False Default is False See also DefineRelation DefineTensor 2 4 7 4 DefineTensor name rank defines a tensor e name A symbol that uniquely identifies the tensor e rank Rank of the tensor For ordinary tensors this must be a non negative integer For product tensors this is a list of positive integers Options e Symmetries gt sym Symmetries of the tensor For ordinary ten sors the allowed values are Symmetric Alternating or Skew Hermitian SkewHermitian RiemannSymmetries NoSymmetries or the name of an arbitrary symmetry defined by calling DefineTensorSymmetries For a product tensor the value of the Symmetries option must be either NoSymmetries or a list of symme tries whose length is the same as that of the list of ranks each symme try on the list can be Symmetric Alternating or Skew Hermiti
93. t dummy index names are interchanged in pairs until the lexically smallest version of the expression is found this is used by TensorSimplify Method gt 2 causes OrderDummy to try all possible permutations of the dummy index names this is used by SuperSimplify Method gt 0 means don t try interchanging names at all If n is the number of dummy index pairs in the expression the time taken by Method gt 1 is proportional to n while the time taken by Method gt 2 is proportional to n This can be very slow especially if there are more than 4 dummy pairs per term See also TensorSimplify SuperSimplify RenameDummy NewDummv ParallelFrame 2 1 ParallelFrame is a DefineBundle option Plus 2 6 Ricci transforms expressions of the form a b L i into Insertindicesl a b Llil Any number of upper and or lower indices can be inserted in this way provided that the number of indices is consistent with the rank of a b Positive 2 3 2 4 2 6 Ricci recognizes Positive as a value for the Type option of DefineConstant DefineTensor and DefineMathFunction PositiveDefinite 2 1 PositiveDefiniteis a DefineBundle option PositiveSpectrum 4 5 PositiveSpectrum is a value for the global variable LaplacianConvention 8 REFERENCE LIST 78 Power 2 6 If x is a tensor expression and p is a positive integer Ricci interprets x p as the p th symmetric power of x with itself Ricci transforms expressions of the form x p i
94. tensors you can specify tensor symmetries using the Symmetries option For example to define a symmetric covariant 2 tensor named h type In 9 DefineTensor h 2 Symmetries gt Symmetric Other common values for the Symmetries option are Alternating or equiva lently Skew and NoSymmetries the default To associate a tensor with a bundle other than the default tangent bundle use the Bundle option In 10 DefineTensor omega 2 Bundle gt fiber Type gt Complex Variance gt Contravariant Covariant This specifies that omega is a complex 2 tensor on the bundle named fiber which is contravariant in the first index and covariant in the second If the value of the Variance option is a list as in the example above the length of the list must be equal to the rank of the tensor and each entry specifies the variance of the corresponding index position The Bundle option may also be typed as a list of bundle names this means that the tensor is associated with the direct sum of all the bundles in the list Any time you insert an index into a tensor that does not belong to one of the bundles with which the tensor is associated you get 0 After a tensor has been defined you can change certain of its options such as Type and Bundle by calling Declare For example to change the 0 tensor u defined above to be a complex valued function you could type In 11 Declarel u Type gt Complex 2 RICCI BASICS 15
95. the bundle being defined CollectConstants 5 7 CollectConstants x groups together terms in the tensor expres sion x having the same tensor factors but different constant fac tors CollectConstants x n or CollectConstants x n1 n2 applies CollectConstants only to term n or to terms n1 n2 of x The constant factors are simplified somewhat if a constant fac tor is not too complicated LeafCount c lt 50 Ricci factors the constant For larger constant expressions it applies the Mathemat ica function Together See also TensorSimplify FactorConstants SimplifyConstants ConstantFactor TensorFactor CommuteCovD 5 17 CommuteCovD x L i L j 1 changes all adjacent occurrences of in dices L i L j after the to L j L i by adding appropriate curva ture and torsion terms The second and third arguments may be upper or lower indices See also OrderCovD CovDSimplify 8 REFERENCE LIST 54 CommutingFrame 2 1 CommutingFrame is a DefineBundle option Default is True CompatibilitvRule 7 5 CompatibilityRule is a rule that transforms derivatives of metric com ponents into connection coefficients using the fact that the default con nection is compatible with the metric which is expressed bv d gij Conni Conn i See also FirstStructureRule SecondStructureRule StructureRules Complex 2 1 2 3 2 4 2 6 Ricci recognizes Complex as a value for the Type option of DefineConstant DefineTensor DefineB
96. the same length as the list of ranks Each symmetry applies to the corre 7 SPECIAL FEATURES 44 sponding set of index slots The only other acceptable form for product tensors is Symmetries gt NoSymmetries which is the default One common use of product tensors is in defining arbitrary connections and their curvatures In fact the built in tensors Conn and Curv are defined as product tensors For example the following command defines a rank 4 tensor whose first two index slots refer to the sum of fiber and its conjugate and whose last two refer to tangent It is skew symmetric in the last two indices with no symmetries in the first two It is contravariant in the second index and covariant in the other three In 62 DefineTensorl omega 12 2 Bundle gt fiber Conjugate fiber tangent Symmetries gt NoSymmetries Skew Variance gt Co Con Co Co When you insert indices into this tensor you may either insert the first two indices only or insert all four at once There is no provision for inserting only the last two indices When the first two indices are inserted the result is a skew symmetric covariant 2 tensor associated with the tangent bundle For example assuming index names a b c are associated with fiber In 5 omega L a U b b Out 5 omega a In 6 BasisExpand b k6 k7 Out 6 omega Basis 7 Basis a K6 k7 Another way to use a product tensor is to represent a section of Hom a
97. them in terms of tensor prod ucts ProductExpand x n or ProductExpand x n1 n2 applies ProductExpand only to term n or to terms n1 n2 of x See also BasisExpand Sym Alt SymmetricProduct Wedge Quiet Quiet is an option for some of the defining and undefining commands in the Ricci package The option Quiet gt True silences the usual messages printed by these commands The default is the value of the global variable Quiet which is initially False 8 REFERENCE LIST 79 Rank Re Real Rank x returns the rank of the tensor expression x If t is a tensor without indices Rank t is the total rank as specified in the call to DefineTensor If t is a tensor with all index slots filled Rank t 0 Ift is a product tensor with some but not all index slots filled then Rank t is the total rank of t minus the number of filled slots For any other tensor expression x Rank x depends on the meaning of the expression See also DefineTensor 2 6 Ricci converts Re x to x Conjugate x 2 See also Conjugate 2 1 2 3 2 4 2 6 Ricci recognizes Real as a value for the Type option of DefineBundle DefineConstant DefineTensor and DefineMathFunction RenameDummy 5 5 RenameDummy x changes the names of dummy indices in x to standard names RenameDummy x n or RenameDummy x n1 n2 applies RenameDummy only to term n or to terms n1 n2 of x RenameDummy chooses dummy names in alphabetical order from the list
98. thout the name of the basis vector For example Dell Basis L i x appears as Del x i Options e Connection gt cn specifies that the covariant derivative is to be taken with respect to the connection cn instead of the default con nection Ordinarily cn should be an expression of the form Conn diff where diff is a 3 tensor expression representing the difference tensor between the default connection and cn e Metric gt g a symmetric 2 tensor expression indicating that the covariant derivative is to be taken with respect to the Levi Civita connection of g instead of the default metric for x s bundle which is assumed to be Riemannian See also Grad CovD Extd Div Laplacian CovDExpand Det 7 3 If x is a 2 tensor expression Ricci interprets Det x as the determinant of x using the metric to convert x to a Contravariant Covariant tensor if necessary Det is also a value for the global variable WedgeConvention See also Dot WedgeConvention Dimension Dimension bundle returns the dimension of bundle See also DefineBundle Div 4 3 Div x is the divergence of the tensor expression x which is the covariant derivative of x contracted on its last two indices If x is a k tensor Div x is a k 1 tensor Ricci assumes without checking that the last index of x is associated with the underlying tangent bundle of x s bundle Div is the formal adjoint of Del Options 8 REFERENCE LIST 69
99. tor x returns the product of all the constant factors in x which should be a monomial See also TensorFactor CollectConstants ConstantQ 2 3 ConstantQ x returns True if there are no explicit tensors in x and False otherwise See also DefineConstant ContractedBianchiRules 7 2 ContractedBianchiRules is a rule that simplifies contracted covariant derivatives of the Riemannian and Ricci curvature tensors using con tracted versions of the second Bianchi identity For example assuming RiemannConvention SecondUp which is the default 1 Out 3 Rm ijkl In 4 ContractedBianchiRules Out 4 Rc Re ik 3j jk i Similarly j Out 5 Re nen er In 6 ContractedBianchiRules 1 Out 6 Sc 2 sd See also BianchiRules FirstBianchiRule SecondBianchiRule Contravariant 2 4 Contravariant is a value for the Variance option of DefineTensor and may be abbreviated Con If an index slot is Contravariant indices in that slot are upper by default See also DefineTensor CorrectAllVariances 5 15 CorrectAllVariances x changes the variances upper to lower or 8 REFERENCE LIST 57 lower to upper of indices in x whose variances are not cor rect for their positions bv inserting appropriate metric coefficients CorrectAllVariances x n or CorrectAllVariances x n1 n2 applies CorrectAllVariances only to term n or to terms n1 n2 of x Option e Mode gt All or OneDims OneDims indicates that only o
100. tureRule CompatibilityRule StructureRules Curv Conn CurvToConnRule SecondUp 7 2 SecondUp is a value for the RiemannConvention option of DefineBundle SimplifyConstants 5 9 SimplifyConstants x applies the Mathematica function Simplify to the constant factor in each term of the tensor expression x 8 REFERENCE LIST 82 SimplifyConstants x n or SimplifyConstants x n1 n2 ap plies SimplifyConstants only to term n or terms n1 n2 of x See also CollectConstants FactorConstants ConstantFactor Skew 2 4 Skew is a synonym for Alternating SkewHermitian SkewHermitian is a value for the Symmetries option of DefineTensor SkewQ SkewQ x is True if x is an alternating or skew Hermitian tensor expres sion and False otherwise StructureRules 7 5 StructureRules is the union of CompatibilityRule FirstStructureRule and SecondStructureRule Summation If a and b contain indices then Ricci transforms a b p internally to Summation a b p to prevent the expression from being expanded to ap b p Summation is not printed in output form instead Summation a b p appears as if it were a b p See also Power PowerSimplify SuperSimplify 5 1 5 2 SuperSimplify x attempts to put the tensor expression x into a canonical form so that two expressions that are equal are usu ally identical after applying SuperSimplify SuperSimplify x n or SuperSimplify x n1 n2 applies SuperSimplify only to term n or to terms
101. ules e Type gt type For rank 0 tensors this can be a single keyword or a list of keywords chosen from among Complex Real Imaginary Positive Negative NonPositive or NonNegative For higher rank tensors only Real Complex or Imaginary is allowed De termines how the tensor behaves under conjugation exponentiation and logarithms The default is always Real e TeXFormat gt texformat Determines how the tensor appears in TeXForm output Default is the tensor s name Note that as in all Mathematica strings backslashes must be typed as double back slashes thus to get a tensor s name to appear as f in TEX output you would type TeXFormat gt beta e Bundle gt bundle The bundle with which the tensor is associated This can be a list meaning the tensor is associated with the direct sum of all the bundles in the list If a complex bundle is named and the Type option is not Complex the tensor is automatically associated with the direct sum of the bundle and its conjugate Bundle gt Any indicates that the tensor can accept indices from any bundle The default value for the Bundle option is DefaultTangentBundle If the tensor is given an index associated with a bundle not in this list the result is 0 For a product tensor the Bundle option can be either a single bundle meaning that all index positions are associated with the same bundle or a list whose length is the same length as the list of ranks Eac
102. undle and DefineMathFunction Con 2 4 Con is an abbreviation for Contravariant Conjugate 2 1 2 2 2 4 2 6 The Ricci package modifies the Mathematica function Conjugate to han dle tensor expressions and indices The behavior of a tensor constant index bundle or mathematical function under conjugation is determined by the Type option specified when the object is defined In input form you indicate the conjugate of tensor t by typing Conjugatelt you may follow this with one or more sets of indices in brackets as in Conjugatelt L i L j You indicate the conjugate of an index L i or U i by typing LB i or UB i To indicate the conjugate of bundle b type Conjugate b You can compute the conju gate of any tensor expression x by typing Conjugate x In output form conjugates of tensors bundles and indices appear as barred symbols The behavior of tensors and indices under conjugation is determined by the Type options specified when the tensor and its associated bundles are defined For example assume a b and c are 2 tensors of types Real Complex and Imaginary respectively Assume i is an index associated with a real bundle and p is an index associated with a complex bundle 8 REFERENCE LIST 55 Conn Conjugatela l a _ ip ip Conjugate b b _ ip ip Conjugate c c _ ip ip See also Re Im DefineTensor DefineConstant DefineBundle Declare 7 5 7 6 Conn is the name used for the generic connection
103. undles VectorFieldQ VectorFieldQ x returns True if x is a contravariant l tensor expression and False otherwise Wedge 3 2 Wedge x y z represents the wedge or exterior product of x y and z The arguments of Wedge must be alternating tensor expressions They need not be covariant tensors however Ricci handles wedge products of covariant contravariant and mixed tensors all together using the metric to raise and lower indices as needed Ricci automatically expands wedge products of sums and scalar multiples and arranges factors in lexical order by inserting appropriate signs The interpretation of Wedge in terms of tensor products is determined by the global variable WedgeConvention In output form wedge products are indicated with a caret x7 y7 Zz See also WedgeConvention Alt TensorProduct ProductExpand 8 2 Global variables This section describes Ricci s global variables these are symbols that can be set by the user to control the way Ricci evaluates certain expressions For example to change the value of WedgeConvention to Det just execute the assignment WedgeConvention Det 8 REFERENCE LIST 89 The variables described below all begin with and are all saved automatically if you issue the RicciSave command DefaultTangentBundle 2 1 The global variable DefaultTangentBundle can be set by the user to a bundle or list of bundles It is used by DefineBundle as the de fault value for the TangentBundle
104. v Basis DefineTensorSymmetries DefineTensorSymmetries Del DefineTensorSymmetries name perm1 sgni permN sgnN 1 defines a TensorSymmetry object that can be used in DefineTensor Here permi permN must be lists that are nontrivial permutations of 1 2 k and sgni sgnN must be constants ordinarily 1 A tensor with this symmetry has the property that if any of the permutations permi permN is applied to its indices the value of the tensor is multiplied by the corresponding constant sgn1 sgnN Ricci automatically tries all non trivial iterates of each specified permutation until the indices are as close as possible to lexical order However Ricci does not try composing different permutations in the list See also TensorSvmmetrv DefineTensor UndefineTensorSymmetries 4 1 Del x is the total covariant derivative of the tensor expression x If x is a k tensor then Del x is a k 1 tensor The last index slot of Del x is the one generated by differentiation it is always covariant and is associated with the underlying tangent bundle of the bundle s of x If x is a scalar function 0 tensor then Del x is automatically replaced by Extd x 8 REFERENCE LIST 68 Del v x is the covariant derivative of x in the direction v v must be a l tensor expression In output form Del v x appears as Del x v Covariant derivatives with respect to contravariant basis vectors are displayed wi
105. v of two terms even when their dummy indices have different names For example 1 INTRODUCTION 9 In 11 9 10 ij k Qutliil alpha beta g t alpha beta i j k In 12 TensorSimplify i Out 12 2 alpha beta i You can take a covariant derivative of this last expression just by putting another index in brackets after it In 13 L j i i 2 alpha beta alpha beta L i j Out 13 The remainder of this manual will introduce you more thoroughly to the capa bilities of Ricci 2 RICCI BASICS 10 2 Ricci Basics There are four kinds of objects used by Ricci bundles indices constants math ematical functions and tensors This chapter describes these objects and then describes how to construct simple tensor expressions The last section in the chapter describes how to save your work under Ricci 2 1 Bundles In Ricci the tensors you manipulate are considered to be sections of one or more vector bundles Before defining tensors you must define each bundle you will use by calling the DefineBundle command When you define a bundle you must specify the bundle s name its dimension the name of its metric and the index names you will use to refer to the bundle You may also specify either explicitly or by default whether the bundle is real or complex the name of the tangent bundle of its underlying manifold and various properties of the bundle s default metric connection and frame The simplest cas
106. version of e is to be replaced by v when the first two indices of e are contracted with each other Note that you do not use underscores in the dummy index names for DefineRelation as you would in defining a relation directly with Mathematica The replacement takes place regardless of the actual names of the indices or whether they are upper or lower provided the corresponding indices refer to the same bundles as i and j For example In 431 e U j L j L k Out 43 v k In general the relation created by DefineRelation recognizes any equivalent relations resulting from raising and lowering indices changing index names conjugation and inserting extra indices provided that all indices refer to the same bundles as the corresponding indices that were used in the call to DefineRelation It does not recognize relations resulting from symmetries however you must define all such equivalent relations explicitly You can specify that the relation should be applied only under certain conditions by giving DefineRelation a third argument which is a True False condition that must be satisfied before the relation is applied The condition may refer to the index names used in the first argument For example suppose u is a scalar function on a Riemannian tangent bundle and you would like the first and second covariant derivatives of u always to be placed in lexical order this is not done automatically by Ricci You could define the following
107. with indices Ricci automatically converts it to TensorExpand See also TensorExpand Extd 4 6 Extd x is the exterior derivative of x which must be an alternating covariant tensor expression In output form Extd x appears as d x Ricci automatically expands exterior derivatives of sums scalar multiples powers and wedge products See also Del ExtdStar LaplaceBeltrami ExtdStar 4 7 ExtdStar x is the adjoint of the operator Extd applied to x which must be an alternating covariant tensor expression If x is a k form then 8 REFERENCE LIST 70 ExtdStar x is a k 1 form In output form ExtdStar x appears as shown here In 26 ExtdStar x Out 261 d x Option e Metric gt g asymmetric 2 tensor expression representing a metric to be used in place of the default metric for x s bundle which is assumed to be Riemannian See also Extd LaplaceBeltrami Div FactorConstants 5 8 FactorConstants x applies the Mathematica function Factor to the constant factor in each term of the tensor expression x FactorConstants x n or FactorConstants x ni n2 applies FactorConstants only to term n or to terms n1 n2 of x See also CollectConstants SimplifyConstants ConstantFactor TensorFactor FirstBianchiRule 7 2 FirstBianchiRule is a rule that attempts to turn sums containing two 4 tensors with Symmetries gt RiemannSymmetries into a single term using the first Bianchi identity For example

A Mathematica package for doing tensor calculations in differential

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