Injectivity Radius of Lorentzian Manifolds
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1. Em defined in the tan gent space or multi valued vector fields on the manifold This also induces a multi valued Riemannian metric gr as was explained before The following corollary immediately follows by repeating the proof of Theo rem 1 1 We note that the curvature covariant derivative bound imposed below is probably superfluous and could probably be removed by introducing a foli ation based on certain synchronous type coordinates as we did in Section 5 for Lorentzian manifolds On the other hand to the best of our knowledge this is the first injectivity radius estimate for pseudo Riemannian manifolds Corollary 9 2 Injectivity radius of pseudo Riemannian manifolds Let M g be a differentiable pseudo Riemannian n manifold with signature n1 N2 and let p M and T Ej E be a family of vectors in T M satisfying g Ei Ej 6ij Suppose that the exponential map exp is defined on Br 0 ro C TM and that Rm r lt tgs VRm r7 lt r on Br 0 ro 37 Then there exists a positive constant c n such that Inj M p T E Vol Br p c n ro n LA ro 9 where Br p r exp Br 0 r is the geodesic ball at p with radius r Proof Without loss of generality we assume r 1 In local coordinate system y let Ei Eig Pye ee i 1 m then grag Sap 2X EiaEig By the same computations as in the proof of Theorem 1 1 we obtain 1 VE lt ilr c n Igr 8rol lg2. It is useful to discuss first some basic definitions from Lorentzian geometry for which we can refer to the textbook by Penrose 20 Throughout this pa per M g is a connected and differentiable n 1 manifold endowed with a Lorentzian metric tensor g with signature To emphasize the role of the metric g or the point p we use any of the following notations 8X VY X Yg X Ve X Yy for the inner product of two vectors X Y at a point p M we sometimes also write X ie instead of g X X Recall that the tangent vectors X T M are called time like null or spacelike depending whether the norm gX X is negative zero or positive respectively Vectors that are time like or null are called causal The time like vectors form a cone with two connected components The manifold M g is said to be time orientable if we can select in a continuous way a half cone of time like vectors at every point p The choice of a specific orientation allows us to decompose the cone of time like vectors into future oriented and past oriented ones The set of all future oriented time like vectors at p and the corresponding bundle on M are denoted by T M and T M respec tively We also introduce the bundle T M consisting of elements of T M with unit length By definition a trip is a continuous curve y a b M made of finitely many future oriented time like geodesics We write p lt lt q if there exists a trip from p t
3. Theorem 1 1 Injectivity radius of Lorentzian manifolds Let M be a time orientable Lorentzian differentiable n 1 manifold Consider an observer p T consisting of a point p M and a reference future oriented time like unit vector T T M Assume that the exponential map exp is defined in the ball Br 0 ro C TM and the Riemann curvature satisfies a sup IRm lr S 1 6 y 0 r where the supremum is over the domain of definition of y and over every g geodesic y initiating from a vector in the Riemannian ball Br 0 ro C TyM Then there exists a constant c n depending only on the dimension n such that Inj M p T oe Vol Br p c n ro r n 1 0 To 1 7 This result should be compared with the injectivity radius estimate estab lished by Cheeger Gromov and Taylor 10 in Riemannian geometry Observe that the curvature assumption 1 6 can always be satisfied by suitably rescal ing the metric tensor It would be interesting to replace the volume term in the right hand side of 1 7 by Volg B p ro Finally in the last two sections of this paper we establish a volume compari son theorem for future cones and generalize our main theorem to the volume of a future cone Section 8 and we briefly discuss the regularity of the Lorentzian metric in harmonic like coordinates and present a generalization to pseudo riemannian manifolds Section 9 2 Preliminaries on Lorentzian geometry Basic definitions
4. IRmgy ly lt Cr on the ball Br p c e ro Step 3 Constructing geodesically convex functions Since the metrics gr and gn are comparable the volume ratio 1 ri 1 Volgy Bn p c ro is uniformly bounded above and below By applying the theory for Riemannian metrics in 10 the injectivity radius of the metric gy is bounded from below by c e ro Let u x dey p x 21 be the square of the distance function associated with the Riemannian metric gn which is a smooth function defined on the geodesic ball By p c e ro By the standard Hessian comparison theorem for Riemannian manifold we have 2 gnap lt VguaV gy pU lt 2 E 8N ag on the ball By p c e ro In terms of the original Lorentzian metric g the Hessian of the function u is ou 2 y y VaV pu Vona V gu pl Tap Tap Ta Since I gy Tln lt C sup S lt C by the estimate 5 4 and since also Vuly lt 2d on By p ro we conclude that 2 gnap 2 VaVgu 2 8N ag in the ball By p c e ro This completes the proof of Theorem 5 1 o 6 Injectivity radius of null cones We turn our attention now to null cones within foliated Lorentzian manifolds Our main result Theorem 6 1 below provides a lower bound for the null injec tivity radius under the main assumption that the exponential map is defined in some ball and the null conjugate radius is already controled Hence con trary to the presentation in Section 3 our m
5. It remains to consider the lift of the original geodesic loop y under the lifting the geodesics y1 72 are sent to two distinct line segments with respect to the vector space structure originating at the origin 0 which obviously do not intersect This is a contradiction and we conclude that in fact Inj gM p T 13 as announced This completes the proof of Theorem 3 1 5 Convex functions and convex neighborhoods We establish now the existence of convex functions and convex neighborhoods in M Let us recall first some basic definitions A function u is said to be geodesically convex if the composition of u with any geodesic is a convex function of one variable A set Q c Q is said to be relatively geodesically convex in Q if given any points p q Q and any geodesic segment y from p to q contained in Q one has y C Q A set Q is said to be geodesically convex in Q if Q is relatively geodesically convex in Q and in addition for any p q there exists a unique geodesic y connecting p and q and lying in Q We denote by dr the distance function associated with the reference Rie mannian metric gr Theorem 5 1 Existence of geodesically convex functions Let M g be a differ entiable n 1 manifold endowed with a Lorentzian metric g satisfying the regularity assumptions A1 A4 for some point p M and some future oriented unit time like vector field T and let gr be the reference Riemannian metric
6. J s satisfying J s Rm J s y s y 8 K0O 0 POlkr 1 we need to control its Riemannian length F s J r s as stated in 4 7 below Let 0 so be the largest subinterval of 0 12 4 in which the inequality J r lt 1 holds Using the equation satisfied by the Jacobi field and taking into account the curvature bound A3 we deduce that in the interval 0 so d se Vv Vy Dr 2 Vere Yy Vy D lt 2 Vo VelrlyIr Vy Jip 2 Ka EII Vyr 15 With 4 2 and the covariant derivative estimate in Lemma 3 2 we obtain d I y Jr lt 4K V Jlr 8 K2 4 4 We can next integrate 4 4 over an arbitrary interval 0 s c 0 so use the initial condition on the Jacobi field and obtain 2K gt AR 2K2 ax 1 1 8 lt Vy Jlr lt 1 e s 1 e e C Assuming that rz is small enough so that oa 1 e7 485 lt 1 2 and 2 e s 1 lt we infer that lt Vy Jlr lt 2 4 5 NI Re Hence using this inequality and Lemma 3 2 we find 4 lr lt 2 2K3 lt 1 so that F s J r s lt 2 2K3 s lt 2 2K3 ro 4 6 Further assuming that 2 2K3 r2 lt 1 we conclude that s rp Next we want to improve the rough estimate 4 6 Since d gV Dr Very Vy Dr Very Vy Dr then by substituting the previous estimates of J r s and V J r s and perform ing similar calculations as above we get EEA for some constant c3 gt 0 By integration this implies es lt Vyr ses and we arr
7. as an observer located at the point p This reference vector is necessary in order to define appropriate notions of conjugate and injectivity radii See Section 2 below for details Secondly we rely here on the elementary but essential observation that in the flat Riemannian and Lorentzian spaces geodesics are straight lines and therefore coincide Under our assumptions we will see that geodesics associ ated with the given Lorentzian metric are comparable to geodesics associated with the reference Riemannian metric On the other hand the curvature bound assumed on the Lorentzian metric implies in general no information on the curvature of the reference metric As we show below one of the main is sues is to guarantee the regularity of a foliation of the manifold by spacelike hypersurfaces Earlier work Let us briefly review some classical results from Riemannian geometry Let M g be a differentiable n manifold possibly with boundary endowed with a Riemannian metric g Throughout the present paper the manifolds and metrics are always assumed to be smooth Denote by B p r the corresponding geodesic ball centered at p M and with radius r gt 0 Suppose that at some point p M the unit ball B p 1 is compactly contained in M and that the Riemann curvature bound and the lower volume bound IRmgllL 80 1 lt K Vol B p 1 vo 1 1 hold for some constants K v gt 0 We use the standard notation L 1 lt
8. nl lt c n on the ball Br 0 c n where fag F ag a minus sign for a lt n and a plus sign for a gt n In view of the computations in 13 Theorem 4 11 and Corollary 4 12 we deduce that l gl lt r c n where r y y Since dz Yo y ly yo we have for any point yo Br 0 c n Viptero Yor 2 Oap Sr on the ball Br 0 c n Since the metric gro plays the same role as gy cf the proof of Theorem 1 1 all arguments can be carried out and this completes the proof of the corollary o Acknowledgements The first author BLC was partially supported by Sun Yat Sen University via China France Russia collaboration grant No 34000 3275100 the Ecole Nor male Sup rieure de Paris the French Foreign Ministry and the Institut des Hautes Etudes Scientifiques IHES Bures sur Yvette The second author PLF was partially supported by the A N R Agence Nationale de la Recherche through the grant 06 2 134423 entitled Mathematical Methods in General Rel ativity MATH GR and by the Centre National de la Recherche Scientifique CNRS References 1 M T ANpERson Convergence and rigidity of manifolds under Ricci cur vature bounds Invent Math 102 1990 429 445 2 M T ANpERson On long time evolution in general relativity and ge ometrization of 3 manifolds Commun Math Phys 222 2001 533 567 38 3 M T ANDERSON Regularity for Lorentz metrics under curvature bound
9. 0d p for p M Given an arbitrary achronal and closed set S C M we define the future or past domains of dependence of S in M by D S p M every future resp past endless trip containing p meets S D S D S U D S Observe that domains of dependence are closed sets Next define the future or past Cauchy horizons H S p D S J p N D S 0 D S F D S H S H S U H S For instance the future Cauchy horizon is the future boundary of the future domain of dependence of S One can check that the Cauchy horizons are closed and achronal sets with D S H S US and 0D S H S Finally a future Cauchy hypersurface for M is defined as an achronal but not necessarily closed set S satisfying D S M For instance it is sufficient for S to be smooth achronal spacelike and such that every endless null geodesic meet M Reference metric As explained in the introduction one should not use the Lorentzian metric to compute the norm of a tensor since the Lorentzian norm may vanish even when the tensor does not This motivates the introduction of a reference Riemannian metric associated with a time like vector field as follows Let T be a future oriented time like unit vector field satisfying therefore Q T T 1 at every point p We refer to T as the reference vector field pre scribed on M Introduce a moving frame E a 0 1 n defined in M that is E is an or
10. Euclidian metric connecting q to p VO rE E te lo 24 This is a timelike curve for the Lorentzian metric g since 2s 2 R EF 0 Iy E n giji lt co c lt 0 0 which shows that Acill cI p t ro 0 t lt C ltl contains On the other hand we claim that the larger Euclidian cone A the null cone in other words Azc i EON PUT p t c1 ro 0 Indeed arguing by contradiction we suppose there exist a time fy c1 ro 0 and a point q Ag n Connected to p by a causal curve y y s with y 0 p After reparametrizing in time the curve is necessary we can assume that y t t x t for some t lt T lt 0 as long as the point y t lies in the coordinate system under consideration For this part of the curve at least we have Foe i dxi Mitac gee eye n Bi ae de which by 6 6 implies that A 2 lt C1 Co Therefore after integration we find x1 q AaS E lt VOo Cat lt Vaar lt ro Hence we can choose tj to the whole curve lies in the system of coordinates and is parametrized in the form y t t x t t to 0 Moreover we have Ix to lt VC1 Co ltol lt C1 ltol which contradicts our assumption q AS ie In conclusion we have localized the slices of the past null cone within annulus regions N P NE cA t c1 ro 0 c1 ltl C1 lel Step 2 The past null cone N p viewed as a graph with
11. Y s 0 y 0 e s 0 50 14 On the other hand to determine the length of y s with respect to the reference metric gr we proceed as follows 00 el Wry r0 7 6 2 Kerr 0 7 Or 2 Ver Yoyo Orl lt 2 Ver Vglr ly s It So by Lemma 3 2 ly s 2 lt 2 K y s and in consequence EOR K By integration of the above inequality and provided s is small enough so that 2s K y 0 r lt 1 we see that 5 v Olr lt Olr lt 21O 4 2 In view of 4 1 this implies oe E A Ole lt ly Ge lt 2e ly Ole 4 3 These inequalities hold for all s 0 1 2Ks y 0 r as long as y s Be p i2 In particular by restricting attention to geodesics whose initial vector has unit Euclidian length y 0 le 1 we see that y 0 72 C Be p iz where r2 ine 2 In turn this establishes that the exponential map at the point p is well defined on the ball Bg 0 r2 with a range included in the geodesic ball Be p i2 Step 2 Conjugate radius estimate Our second task is to determine a ball on which the exponential map is a local diffeomorphism and we therefore need to control the length of a Jacobi field along a geodesic Let y 0 r2 M be a g geodesic satisfying y 0 p and y 0 z 1 By the discussion in Step 1 we already know that the curve y lies in Bg p i2 and that maxse o r v s lr lt 2 7 Given an arbitrary Jacobi field along y J
12. associated with Then for any 0 1 there exists a positive constant ro depending only upon e the foliation bounds Kg K the curvature bound Kp the volume bound vo and the dimension of the manifold and there exists a smooth function u defined on Br p ro such that l e dr p lt u lt 1 e dr p 2 e gr lt Vu lt 2 gr Hence the function u above is equivalent to the Riemannian distance func tion from p and is geodesically convex for the Lorentzian metric In the proof 18 given below the function u is the Riemannian distance function associated with a new time like vector field denoted by N in the proof below The following corollary is immediate and provides us with a control of the radius of convexity which generalizes Whitehead theorem from Riemannian geometry 23 6 Corollary 5 2 Existence of geodesically convex neighborhoods Under the as sumptions of Theorem 5 1 for any 0 lt r lt ro there exists a set Q C Q which is geodesically convex in Br p 2ro and satisfies exp Br 0 r Q C exp Br 0 1 r Moreover one can always choose Q so that Br p r c Q C Br p 1 r where Br p r is the geodesic ball determined by the reference Riemannian metric Proof of Theorem 5 1 Step 1 Synchronous coordinate system Given gt 0 by applying the injectivity radius estimate in Theorem 3 1 to points near p we see that there exists a constant r depending on Ko K1
13. fact that r has no fixed point and the claim is proved Step 6 The pull back of the volume element of g is the same as the one of gr By combining this observation with our results in Steps 4 and 5 we find e n Vol Br 0 1 oe es Vol Br p c n which gives Vol Br p c n l gt c n Vol 610 D gt c n Vol Br p c n The proof of Theorem 1 1 is completed o 8 Volume comparison for future or past cones In Riemannian geometry under a Ricci curvature lower bound Bishop Gromov s volume comparison theorem allows one to compare the volume of small and large balls in a sharp and qualitative manner Let us return to Step 2 of Sec tion 5 where we introduced the index form associated with the synchronous coordinate system on time like geodesics By noticing that the index form is symmetric and that Jacobi fields minimize the index form in some sense we can extend the method of proof of the index comparison theorem However in a general Lorentzian manifold since the index form we needed without imposing a restriction on the geodesics is non symmetric we need to adapt the method of the index comparison theorem as follows Theorem 8 1 Volume comparison theorem for cones Let M g be a globally hyperbolic Lorentzian n 1 manifold Fix p M and a vector T TM with g T T 1 and suppose that the exponential map exp is defined on the ball Br 0 ro C T M determined by the reference inner product gr a
14. field defined along o such that TO 0 Vix V where V T M satisfies the orthogonality condition Vt V 0 Then we have W 1 V V V Vv V V V 2 V T p _ oo pa ae T Va V Va Vg R o V 0 V I V V 20 Recall that in the absence of conjugate points Jacobi fields minimize the index form I V V among all vector fields with fixed boundary values By applying a standard comparison technique from Riemannian geometry on the orthogonal space Vt on which the Lorentzian metric induces a Riemaniann metric we control the Hessian of t in terms of the curvature bound K2 K2 1 c e Ky 1 c e A g lvs HV OD s _ gl 6 3 pina Or ao eeie Since Vit 1 28 we deduce from 5 3 that Sii 98i 38i lt DE lt TE in the cone exp C4 ro 5 4 Combining 5 4 with the curvature formulas derived in Section 3 i e 1799kg Zu Ox 4 R Rint Rija a Ot OT Ot OT 1 ra o Rojil 5 Vi 5 81 V8 128i 1 38i 98 PE aaa pq Rojo 232 48 Or at we conclude that a C a lt Z on exp C4 ro 5 5 Finally relying on the formulas for the curvature of the reference Rieman nian metric gn we obtain C IRmevlv lt amp on exp C ro Observe that as could have been expected the upper bound blows up as one approach the point q which is the base point in our definition of the distance In particular this implies the following curvature bound near the point p
15. lift of o Uo through 0 and denote by p the end point of o Then it is clear that all the points p i 1 N are the pre images of p in By 0 1 2 We claim that they are distinct Indeed assuming that p pi forsomei j we would find Uo gt gone 8T 0r cn which gives Oj ao UaUg o UaUGg Oj 8r0 3c n 8r0 3c n 8r0 3c n This would imply o a ns and therefore p pj which is a contradiction In 8109 short this argument shows that the cancellation law holds for the homotopy class of not too long curves Step 5 Suppose that there exist two distinct g geodesics y 0 4 M and y2 0 12 M satisfying nO n0 p p POR p OR 1 and meeting at their endpoints that is y1 1 y2 l2 Then let h Iz and y yz U y 0 1 gt M Our aim is to prove that 1 gt c n Vol Br p c n 31 which will give us the desired injectivity radius From the loop y we define a map 7 Br 0 c n Br 0 2c n as follows for any y By 0 c n the point 7 y is the end point of the lift exp Oy Uy through the origin If one would have 7 y y then by the cancellation law established in Step 4 we would have y x 9 which is a contradiction So gT cn the map 7 has no fixed point Without loss of generality we assume that 1 lt c n Let N c n I be the largest integer less than c n I and let us use the notation 2y y o y etc Claim The classe
16. m lt for the spaces of Lebesgue measurable functions Then according to Cheeger Gromov and Taylor 10 the injectivity radius Inj M p at the point p is bounded below by a positive constant i i1 K vo n Inj M p gt i 1 2 It should be noticed that this is a local statement for earlier work on the injectivity radius see 5 11 15 Moreover Jost and Karcher 16 relied on the regularity theory for elliptic operators and established the existence of coordinates in which the metric has optimal regularity and are defined ina ball with radius iy ip K vo n Precisely given gt Qand 0 lt y lt 1 there exist a positive constant C e y depending also upon K vo n and a system of harmonic coordinates defined in the geodesic ball B p iz in which the metric g is close to the Euclidian metric gg in these coordinates and has optimal regularity in the following sense e gps ese gr 7 1 3 r lldgllo amn lt Cle y 7 0 72 Here C and C are the spaces of continuous and H lder continuous functions respectively Harmonic coordinates are optimal 12 in the sense that if the metric is of class C in certain coordinates then it has at least the same regularity in harmonic coordinates The above results were later generalized by Anderson 1 and Petersen 22 who replaced the L curvature bound by an L curvature bound with m gt n 2 For instance one can take m 2 in dimension n 3 in
17. metric g T 1 of To 1 of FO _ 1 Ogi 00 ap ar hay a eT o Ti 4 7 Palgi Tk r 8 5 053s JE ij tij as well as the non trivial curvature terms 1 OiK OS Z OR x Riga Riga Fe ot ot t Jr ar Zp oa j 0 1 pq Sql 1 ay k 1 kq 14 a Roa Be ga 28 ae t 58 aTh G GP 1 m OF 28 1 23 1 Of af axa or 28 ae Bf axl 1 1 f Og OF Zij Ron 5 Spa Ya ap Sear T Sxl ar i 1 Of dl gj 1 af a pq pq A aN S Ro gal 38 ga alas o t za A po 8a 1 pee TE 280 1 pg OF 1 of 25 Ot 2f at at 2 Oxi DF ax and re FSi 1 yg Sin in 1 OF Mii 1 OF OF Rop 5 ViVif Sp 98 Sr oe af at ot 4f xi axl By applying the formulas above to both metrics g gr we estimate the Christoffel symbols as follows Recall that the difference I I ag can be regarded as a tensor field on M so that the following Riemannian norm squared is a scalar field on the manifold M Ver Vit gr q r Toz py Ce ryt z ry y STaa oe ge t We need also the expression of the Lie derivative of g along the vector field T By a direct computation from 3 2 we obtain Agi _ lon C pil Sij rg o 0 r8 oi Z ga T8 ij PEET 3 6 Lemma 3 2 Levi Cevita connection of the reference metric Suppose that g satisfies Assumptions A1 A2 Then the covariant derivative of the Lorentzian and Riemannian metrics are comparable precisely Ve Vir n
18. of this radius r where the exponential map is defined and has some good property We restrict attention to the geodesic ball Br p ro exp Br 0 ro recall that these sets depend upon the vector T given at p 27 As explained in the introduction by g parallel translating the vector T at p along a geodesic y from p we can define get a future oriented unit time like vector field T y defined along this geodesic To this vector field and the Lorentzian metric g we can associate a reference Riemannian metric gr along the geodesic In turn this allows us to compute the norm Rmglr of the Riemann curvature tensor along the geodesic Of course whenever two such geodesics y y meet away from p the corre sponding vectors T and T are generally distinct If we consider the family of all such geodesics we therefore obtain a generally multi valued vector field defined in the geodesic ball Br p ro We use the same letter T to denote this vector field In turn we can still compute the Riemann curvature norm Rm r by taking into account every value of T The key objective of the present section is the study of the geometry of the local covering exp Br 0 ro gt Br p ro and to compare the Lorentzian metric g defined on the manifold M with the reference Riemannian metrics gr As we will see in the proof below it will be convenient to pull the metric upstairs on the tangent space at p using the exponential map Indeed thi
19. r the restriction of exp to the ball By 0 7 c TM determined by the metric g at the point p is a diffeomorphism on its image The radius of injectivity at the point p is defined as the largest value of r such that the restriction exp By 0 7 is a global diffeomorphism In the Lorentzian case the exponential map is defined similarly but some care is needed in defining the notion of radius of injectivity First of all if the manifold is not geodesically complete which is a rather generic situation as illustrated by Penrose and Hawking s incompleteness theorems 14 the map exp need not be defined on the whole tangent space T M but only on a neighborhood of the origin in T M More importantly the Lorentzian norm of a non zero vector may well vanish consequently the radius of injectivity should not be defined directly from the Lorentzian metric g The definition below depends on the prescribed Riemannian metric grp at the point p Definition 2 1 The conjugate radius Conj M p T of an observer p T TIM is the largest radius r such that the exponential map exp is a local diffeomorphism from the Riemannian ball Br 0 r Bg 0 r C TpM to a neighborhood of p in the manifold M Similarly the injectivity radius Inj M p T of an observer p T T M is the largest radius r ssuch that the exponential map is a global diffeomorphism at every point of the ball By 0 r When a vector field T is prescribed on the manifold ra
20. short length for the metric gr By lifting the homotopy to the tangent space T B7 0 1 and by relying on the conjugate radius bound we reach a contradiction as was done in Section 4 In summary there exists a universal constant C n 1 c n depending only on the dimension such that the injectivity radius at each point y of Br 0 c n is bounded from below by 4c n Moreover by the Jacobian field estimate again we can prove the ball Br 0 c c T M defined by the Euclidean metric grp is covered by exp Br 0 3c n where Br 0 3c n c TyT M is a ball of radius 3c n defined by metric gry and any two points in Br 0 c n can be connected by a g geodesic which is totally contained in Br 0 2c n Further arguments are now required to arrive at the desired bound 7 1 Step 3 Riemannian metric gy induced on Br 0 2c n Consider a geodesic y satisfying y 0 0 and y 0 T and let us set y c n 2 q T de q de q 0 Then by following exactly the same arguments as in the main proof of Section 5 we construct a normal coordinate system of definite size such that g dt gi dx dx and gy dt gijdx dx and such that the corresponding reference Riemannian metric satisfies the following properties G eln gn lt gr lt 1 c n gn ii gy has bounded curvature lt C n see 5 5 and 30 iii for any fixed yo Br 0 c n the distance function d yo is strictly g convex on t
21. tensors defined on M as well as on submanifolds of M allowing us for instance to view L M gr as a Banach space In particular we will use later the L norm of a tensor field T on M restricted to an hypersurface IV All gr VAF AV gr x where dVs_ is the volume form induced on by the reference Riemannian metric The functional norm above depends upon the choice of the vector field T but another choice of T would give rise to an equivalent norm provided T remains in a fixed compact subset Observe in passing that the volume forms associated with the metrics g and gr coincide so that the spacetime integrals of functions in M g or M gr coincide for instance the volume Vol A and Vol A of a set A C M coincide Furthermore we observe that in order to define the reference metric gr at a given point p it suffices to prescribe a future oriented time like unit vector T at that point p only it is not necessary to prescribe a vector field In the situation where the reference metric need only be defined at the base point p we refer to T as the reference vector rather than vector field and we refer to p T TM as the observer at the point p This will be the standpoint adopted for our main result in Section 7 below Exponential map On a complete Riemannian manifold the exponential map exp TM gt M at some point p M is defined on the whole tangent space TM and is smooth For sufficiently small radius
22. the application to general relativity since time slices of Lorentzian 4 manifolds are Riemannian 3 manifolds It is only more recently that the same questions were tackled for Lorentzian n 1 manifolds M g Anderson 2 3 studied the long time evolution of solutions to the Einstein field equations and formulate several conjecture In particular assuming the Riemann curvature bound in some domain Q IRm llL 0 lt K 1 4 and other regularity conditions he investigated the existence of coordinates that are harmonic in each spacelike slice of a time foliation of M This work by Anderson motivated our investigation in the present paper On the other hand motivated by applications to general relativity and nonlinear wave equations using harmonic analysis tools Klainerman and Rod nianski 19 considered asymptotically flat spacetimes endowed with a time foliation and satisfying the L curvature bound IRm llz lt K 1 5 for every spacelike hypersurface To control the injectivity radius of past null cones they relied on their earlier work 17 18 on the conjugate radius of null cones in terms of Bell Robinson s energy and energy flux and derived in 19 a new estimate for the null cut locus radius We refer to these papers for further details and references on the Einstein equations Section 6 of the present paper is a prolongation of the work 19 Outline of this paper The present paper establishes four estimates
23. x t x x i2 i2 x Be 0 11 for some K gt 0 depending only on Ko Ki K2 We then restrict attention to a smaller radius i2 i2 Ko Ki K2 lt i such that e Ki gt 0 and we pick up c gt 0 sufficiently large so that ee lt e Kin lt e Kin lt e In turn in view of Assumption A1 on the lapse function n and of the expression 3 3 of the reference Riemannian metric gr the above inequalities imply that the reference Riemannian metric gr is comparable to the n 1 dimensional Euclidean metric e Sag S Strap Se Oop x t x i2 i2 xX Be 0 i2 for some constant cy gt c depending upon c and Ko Introducing on the manifold the n 1 dimensional Euclidian metric E which we define in the constructed coordinates x and is of course indepen dent of the point on the manifold and the corresponding Euclidian metric ball Br p i2 we have established e SES 8T 4 lt e SE qE Belp i2 4 1 In the following we use the notation X for the Euclidian norm of a vector X Our first task is to determine the radius of a ball on which the exponential map is well defined This radius depends upon the reference vector field T Let y 0 so M be a geodesic associated with the Lorentzian metric g and satisfying y 0 p Assume that this geodesic is included in the Euclidian ball Be p i2 in which we have a good control of the metric gr Obviously we have 8
24. Injectivity Radius of Lorentzian Manifolds Bing Long Chen and Philippe G LeFlocht December 29 2006 Abstract Motivated by the application to spacetimes of general relativity we investigate the geometry and regularity of Lorentzian manifolds under cer tain curvature and volume bounds We establish several injectivity radius estimates at a point or on the past null cone of a point Our estimates are en tirely local and geometric and are formulated via a reference Riemannian metric that we canonically associate with a given observer p T where p is a point of the manifold and T is a future oriented time like unit vector prescribed at p The proofs are based on a generalization of arguments from Riemannian geometry We first establish estimates on the reference Riemannian metric and then express them in term of the Lorentzian met ric In the context of general relativity our estimates should be useful to investigate the regularity of spacetimes satisfying Einstein field equations 1 Introduction Aims of this paper The regularity and compactness of Riemannian manifolds under a priori bounds on geometric quantities such as curvature volume or diameter represent important issues in Riemannian geometry In particular the derivation of lower bounds on the injectivity radius of a Riemannian manifold and the construction of local coordinate charts in which the metric has optimal reg ularity are now well understood Moreover Cheeger Gro
25. K2 vo n such that for any q Br p 2ro the injectivity radius at q is 2rp at least and we can assume that en 8Tq lt exp gr lt e ST gr Br 0 ro Cc T M q E Brp 2ro Let y y s be the backward time like geodesic satisfying y 0 p and y 0 T and consider the past point q y ro 2 The future null cone at q with radius ro the orientation being determined by the vector field T is defined by Cy ro V T7M Vlen lt ro IV lt 0 V T gt 0 Observe that the gr length of y between p and q is approximatively ro 2 and that the norm y r is almost 1 while y q a 1 and y T gt 0 By the injectivity radius estimate in Theorem 3 1 the exponential map at q is a diffeomorphism from C ro onto its image which moreover contains the original point p Next introduce the set of vectors that are almost parallel to T V V es e V V Tq Calfo V TyM Vlg lt ro V T g gt 0 The notation c e gt 0 is used for constants that depend only on Ko Ki Kz vo and satisfy lim c e 0 We claim that there is constant c e gt 0 such that Br p c e ro C expy Cq r0 5 1 Actually we have Br p c e ro c Br q G c ro hence Br p c e ro C exp Br0 G c r0 19 Since the metrics gr and gr are comparable under the exponential map at q we see that geodesics o connecting q and points of Br p c e ro make an angle lt
26. Lrglt lt LK Ks 12 Proof In view of 3 5 the difference I I depends essentially upon the terms w and 28 which precisely appear in the expression of the Lie derivative 3 6 We omit the details o It is important to observe that the difference between the curvature tensors can not be similarly estimated and that this is one of the main difficulties to deal with in the present work For future reference we provide here the expressions of certain curvature coefficients of g and gr in terms of first order derivatives of the lapse function n and the induced metric g jx 1 OSir PS Ogu AS jx ep re 2 Aea A ae a SE ae Rijn Riva 4 4n2 ot oat ot ot 1 1 n 0g n Zij Roji Vsi z Vi 5 81 gt Glos t ax t da et Pip OS 1 On O8ij 1 An n shell Vn2 oP4I OS a a Raae a a ae oe o oa and 1 OSix Oj Ogu OSjx ees eee E Rra Rix TA t ot ot ot 1 1 n 0g n Zij RToj al i 5 8 a ala at ax at Vou P8iiy 1 pg Sip Big 1 An Ig 1 On On Rab UN Ce a GS ae Tae ag ne ee et a where Rin denotes the induced curvature tensor on the time slices L L 4 Derivation of the first injectivity radius estimate In this section we provide a proof of Theorem 3 1 Step 1 Radius of definition of the exponential map First of all we note that the injectivity radius of the Riemannian metric glz induced on the initial hypersurface Xp t 1 0 is controled as follows
27. Using Assumptions A3 and A4 we see that the Riemann curvature of the metric g z is bounded and the volume of the unit geodesic ball Volg By p 1 is bounded below Therefore according to 10 there exists a constant 1 i1 K2 vo such that the injectivity radius of glx at the point p is i at least Inj Xo p gt iy Moreover according to 16 we can also assume that i is sufficiently small so that given any gt 0 there exists a coordinates x defined in a ball with definite 13 size near p with x p 0 such that the metric glx is close to the n dimensional Euclidian metric ge 6 in these coordinates More precisely on the initial slice Xp we have Onis 9i 0 x eek lt e ij x1 Be 0 11 where we have set Bg 0 r la x lt ri c R The latter can be regarded as a subset of Xo by identifying a point with its coordinates and we also use the notation Bg p r for this Euclidian ball We can next introduce some coordinates x t x on the manifold by propagating the coordinates x chosen on Xo along the integral curve of the vector field T This construction allows us to cover the domain Q From Assumption A2 together with A1 and 3 6 we deduce that the induced metric on each slice of the foliation is comparable with the n dimensional Euclidian metric in some time interval 12 i2 that is e Ki ij lt gij x lt e Kin Oij
28. ain assumption see A3 below is not directly stated as a curvature bound However under additional assump tions it is known that the conjugate radius estimate can be deduced from an L curvature bound so that our result is entirely relevant for the applications Indeed in a series of fundamental papers 17 18 19 Klainerman and Rodnianski assumed onan L curvature bound and estimated the null conjugate and injectivity radii for Ricci flat Lorentzian 3 1 manifolds Our result in the present section is a continuation of the recent work 19 and covers a general class of Lorentzian manifolds with arbitrary dimension while our proof is local and geometric and so conceptually simple We use the terminology and notation introduced in Section 2 In particular a point p M and a reference vector field T are given and N denotes the past null cone in the tangent cone at p The null exponential map exp BN 0 r gt M is defined over a subset of this cone BZ 0 r Br 0 r A N5 and allows us to introduce the past null injectivity radius Null Inj M p T We also set BN p r exp B7 0 r We consider a domain Q c M containing some point p on a final slice Xo and foliated as Q Ju per 6 1 te 1 0 22 We assume that there exist positive constants Ko Ki K2 such that eX lt n lt e ino A1 lLrglrs Ki inQ A2 the null conjugate radius at p is r at least and the null exponential map satisfie
29. are here extended to the Lorentzian setting we analyze the properties of Jacobi fields and rely on volume comparison and homotopy arguments In our presentation see for instance our main result in Theorem 1 1 at the end of this introduction we emphasize the importance of having assumptions and estimates that are stated locally and geometrically and avoid direct use of coordinates When necessary coordinates should be constructed a posteriori once uniform bounds on the injectivity radius have been established Our motivation comes from general relativity where one of the most chal lenging problems is the formation and the structure of singularities in solutions to the Einstein field equations Relating curvature and volume bounds to the regularity of the manifold as we do in this paper is necessary before tackling an investigation of the geometric properties of singular spacetimes satisfying Einstein equations See for instance 2 3 for some background on this sub ject Two preliminary observations should be made First since the Lorentzian norm of a non zero tensor may vanish it is clear that only limited information would be gained from an assumption on the Lorentzian norm of the curvature tensor This justifies that we endow the Lorentzian manifold with a reference Riemannian metric denoted by gr below this metric is defined at a point p once we prescribe a future oriented time like unit vector T We refer to the pair p T
30. bounded slope We now obtain a Lipschitz continuous parametrization of the null cone For any fixed q An we consider the vertical curve passing through q y 1 m x q x q t c1ro 0 By Step 1 we know that there exists tT such that y t N p Moreover Tq is unique since N p is achronal and this defines a map F A gt N p lt c2 lt ciro such that F q y4 T4 It is obvious F c1ro 0 p 25 We claim that the map F is Lipschitz continuous with Lipschitz constant less than C1 as computed with the Euclidean metric E Namely by contradiction suppose that F q1 F q2 le gt C1 qi qzle for some q1 q2 Aon then by 6 7 in Step 1 F q1 would be chronologically related to F q2 and this would contradict the fact that N p is achronal Moreover from Step 1 it follows that FAS gt NP Be p ciro Step 3 Constructing an homotopy of curves on the null cone Suppose that y y2 are two past null geodesics from p satisfying y 0 720 Or 140 7 1 yi s1 72 S2 We claim that max sj s2 gt cro which will establish the desired injectivity bound by setting io cro We argue by contradiction and assume that max s1 52 lt cro Taking into account Assumption A2 and applying exactly the same arguments as in Step 1 of Section 4 we see that the gr lengths of the curves y1 y2 satisfy L yj gr lt sje s orn G 12 By Step 1 of the present proof we know t
31. c e with y q at the point q as measured by the metric gr4 By reducing the constant c e if necessary the claim is proved Let t be the Lorentzian distance from q it is defined on exp C ro and is a smooth function on exp C r0 p Using the claim 5 1 we deduce that t is smooth in the ball Br p c e ro and satisfies 5 c 6 ra lt t lt e e ro in the ball Br p c e ro 5 2 It is clear also that Vt 1 V71 Vt 0 We now introduce a new foliation Let z be coordinates on the level set hypersurface t t p and by following the integral curves of the unit time like vector field N Vt let us construct coordinates z with zp t in which the Lorentzian metric g takes the simple form NS g dz gijdz dz Let gn n be a new reference Riemannian metric based on the vector field N By Lemma 3 2 using the equation satisfied by future g geodesics we see that log o z lt K3 ro Recall that we allow r to depend upon This inequality shows that the vector field N makes an angle lt c e with T everywhere on exp Cy ro From this we conclude that the two metrics are comparable 1 c e er lt en lt 1 c e gr inthe cone exp Ca ro Step 2 Hessian comparison theorem and curvature bound for the reference metric gy Since p exp Cq 70 let o 0 t p M be the future time like geodesic connecting q to p and let V be the Jacobi
32. cture of space time Cam bridge Univ Press 1973 15 E Hentze Aann H KARCHER A general comparison theorem with appli cations to volume estimates for submanifolds Ann Sci Ecole Norm Sup 11 1978 451 470 16 J Jost anD H Karcuer Geometrische Methoden zur Gewinnung von a priori Schranken f r harmonische Abbildungen Manuscripta Math 40 1982 27 77 17 S KLAINERMAN AND I RopNIANsKI Ricci defects of microlocalized Einstein metrics J Hyperbolic Differ Equa 1 2004 85 113 18 S KLAINERMAN AND I Ropniansxi Rough solutions of the Einstein vacuum equations Ann of Math 161 2005 1143 1193 39 19 S KLaAINERMAN AND I Ropniansxt On the radius of injectivity of null hypersurfaces J Amer Math Soc to appear 20 R Penrose Techniques of differential topology in relativity CBMS NSF Region Conf Series Apli Math Vol 7 1972 21 S Peters Convergence of Riemannian manifolds Compositio Math 62 1987 3 16 22 P PETERSEN Convergence theorems in Riemannian geometry in Com parison Geometry Berkeley CA 1992 93 MSRI Publ 30 Cambridge Univ Press 1997 pp 167 202 23 J H C WuiTEHEAD Convex regions in the geometry of paths Quart J Math Oxford 3 1932 33 42 40
33. d y r c E r with constant in r scalars c and X c y 0 r 1 We used here that by definition y is g parallel transported Let V a r E r be a Jacobi field along a radial geodesic y y r with V 0 0 and V 0 r 1 Then the Jacobi equation takes the form a r Ea REg Ey Es r c a r pe ax a lt dat a lt 22 ax a we obtain V r r lt e and thus V r 7 lt e 1 By substituting this result into the above formulas the estimate can be im proved again Indeed by computing and estimating the second order deriva tive 4 Ya aAa as we did for the Jacobi field estimate of Section 4 we can check that Since r C n r lt gt laal2 lt e 1 along the geodesic Denote by go gr the Lorentzian and the Riemannian metrics at the origin 0 which are nothing but the metrics at the point p and let ay y be Cartesian coordinates on Br 0 1 with a Zep g0 0 Nag Assuming that the radius under consideration is sufficiently small so that 1 C n yl lt 1 we conclude from the Jacobi field estimate that the exponential map is non degenerate and that the metric along the geodesic are comparable In turn since this is true for every radial geodesic we can define the pull back of the metric to the tangent space and the conclusion hold in the whole ball B7 0 1 that is 1 C n lyl gro lt gry lt 1 C n lyl gro y Br 0 1 7 3 By cons
34. e metric To establish Theorem 3 1 it is convenient to introduce coordinates on Q chosen as follows Fix arbitrarily some coordinates xf on the initial slice Xp Then transport these coordinates to the whole of Q along the integral curves of the vector field T This construction generates coordinates x on Q such that x and the vector 0 dt is orthogonal to 0 oxi so that the Lorentzian metric takes the form g n d gidx dx 3 2 where n is the lapse function and gj is the Riemannian metric induced on the slices The reference Riemannian metric in the domain Q then takes the form QT wdt gijdx dx 3 3 and the Riemannian norm of a vector X has the explicit form gr X X n X X XIX We want to control the discrepancy between the reference Riemannian met ric gr and the original Lorentzian metric g as measured in the connections V and Vz and the curvature tensors Rm and Rm Clearly these estimates should involve the constants arising in A1 A4 Consider the general class of metrics i fd gidx dxi 3 4 which allows us to recover both the Lorentzian f n and the Riemannian f n metrics In view of the expressions of the Christoffel symbols and the Riemann curvature nee a e ap 2 ax x axe a Ba tO se Hh Cy cee ab TC p t PA Rigo Ort Oxf Vanlgs Tp Poor Rapys ERa Rag Rayo 11 we compute explicitly the Christoffel symbols associated with the
35. ery slice of the foliation We will prove Theorem 3 1 Injectivity radius of foliated manifolds Let M be a differentiable manifold endowed with a Lorentzian metric g satisfying the regularity assumptions A1 A4 at some point p and for some foliation 3 1 Then there exists a positive constant ig depending only upon the foliation bounds Ko K the curvature bound Ko the volume bound vo and the dimension of the manifold such that the injectivity radius at p satisfies Inj M p T gt ip 10 The following section is devoted to the proof of this theorem Observe that the conditions A1 A4 are local about one point of the manifold and are stated in purely geometric terms requiring no particular choice of coordinates Of course the conclusion of Theorem 3 1 hold globally in M if the assumptions A1 A4 hold also globally at every point of the manifold Our assumptions do depend on the choice of the time like vector field T but the dependence of the constants arising in A1 A4 should not be essential however it is conceivable that when applying this theorem in a specific situation a quantitatively sharper estimate would be obtained with a choice of an almost Killing field that is a field T corresponding to a small Lie derivative rg Later in Section 7 a more general approach is presented in which the vector field T is constructed from a single vector prescribed at the point p Basic estimates on the referenc
36. h we now extend to a globally hyperbolic Lorentzian manifold Let A be the star shaped domain with respect to 0 in T M such that exp AN Br 0 ro is a diffeomorphism on its image and the image of dA N Br 0 ro is set of cut locus in Br p ro Let x4 be the characteristic function of A Since s x s is decreasing in s we see that x P x is also decreasing in s Now we get two functions on Br 0 r0 whose quotient is decreasing along radial geodesics Observe that M is globally hyperbolic so any point in FC p ro is connected to p by a maximizing time like geodesic This also implies that the integration of xap over Br 0 s gives the volume Vol Br p s Then by integrating xap and ox over Br 0 s and after a simple calculation we deduce that Vol FC p s Volx B s is decreasing in s The case of the ratio Vol FCz p s Volx B s is similar The proof of the theorem iscompleted o We are now in a position to prove Corollary 8 2 Injectivity radius based on the volume of a future cone Let M be a manifold satisfying the assumptions in Theorem 1 1 and assumed to be globally hyperbolic and let T TyM be a reference vector Let X be a subset in the unit sphere S included in the future cone N If Vol FCz p ro 2 vo gt 0 then the inequality Inj M p T ro 2 c Z 0 holds where FCz p ro exp FC ro with FCp ro 0 lt VIr lt ro T Vyr lt 0 V lt 0 ex VIr 7 and the c
37. hat the Euclidean lengths of y y2 satisfy Liven lt Sro j 1 2 In particular y1 y2 C N p N Be p cro and we can thus concatenate the curve y y2 and obtain y yz U y 0 s1 s2 gt N p N Br p ro Since FA ar D N p N Be p ro there exists a smooth family of curves de 0 51 2 gt N p such that 01 Y Oo p de 0 oe s1 82 p g 0 1 Specifically we choose oe s F eF y s where the multiplication by e is defined by relying on the linear structure of A3 Br 0 ctr Equivalently by setting x s x y s we have the explicit formula de s F c10 X 8 ex s 26 It is clear that the Euclidean and gr lengths of d satisfy L oe 8E lt e l Cy L y gz lt C478 ro 3 5 8 L oe gr lt cy Ivy By Assumption A3 on the null conjugate radius we can lift to the null cone of the tangent space T M the continuous family of loops o and we obtain a continuous family of curves o defined on 0 s1 s2 such that 6 0 0 LG erp lt ro Observe that the property L G grp lt Cc ro lt ro guarantees the existence of this continuous lift By continuity all of the curves o are loops containing 0 As observed earlier in the proof for the case of bounded curvature o1 consists of two distinct segments which clearly can not form a closed loop and we have reached a contradiction o 7 Injectivity radius of an observer in a Lore
38. he Jacobi field defined on 0 s1 and satisfying V 0 0 and V s1 va Clearly Vo s s y and the vectors V and V V are orthogonal to y for alli gt 1 Consider the Jacobian of the exponential map s J dexp ig which is given by 1 OAV AA Vi p s 7 2n jay oY 2 s y 0 A V5 0 A VOl Denote by px s the corresponding quantity in the simply connected Lorentzian n 1 manifold with constant curvature K2 Define the index form L X Y iy V X Vy Y g Rmg y X y Y ds 0 where X Y are vector fields along y and Rm y X y Y Rm y X y Yg Observe that I is symmetric in X Y It is easy to see ds S S1 2n log p Y Vj s1 Vis ViVi VI 7 i 1 Let E s be the parallel transport of v along y Since there are no conjugate points along y the Jacobi field minimizes the index form among all vector fields 34 with fixed boundary values This is the same as in Riemannian geometry The reason is that the length of time like geodesic without conjugate points is locally maximizing among all nearby time like curves with the same end points Let V s S F s then ae Vi lt I Vi Vi and sinh s sinh s ds Ei E K is 8 on ani sinh s1 Tams 2 Rm Ei yE Ka s1 sinh s z 9 sinhs sinhs 28 ico y yy n Kz ds lt 0 The following is a simple but very important observation due to Gromov whic
39. he ball Br 0 2c n and more precisely 2 c n gw Ved yo 2 e n gy on Br 0 2c n for any yo Br 0 c n The Hessian of the distance function defined by the Riemannian metric gy is naturally computed using the covariant derivatives defined by the Lorentzian metric g Step 4 Suppose that p1 py are distinct pre images of p in the ball B7 0 c n We claim that any p Br p c n has at least N distinct pre images in Br 0 1 and refer to this property as a lower semi continuity property Generalizing the terminology in 10 we use the notation a a b when two ST curves a b defined on M and with the same endpoints are homotopic through a family of curves whose lift have gro lengths lt A Relying on the lift and the linear structure we see that for any curve starting from p with after lifting through 0 gro length A lt 1 there exists a unique g geodesic y lt with the same end points as defined on M such that G ae ye This fact establishes a 8T 0r one to one correspondence between the following three concepts i equivalence class of curves through p with gro lengths lt 3c n ii radial geodesic segments of gr lengths lt 3c n and iii points in the ball By 0 3c n c T M Let o be a g geodesic connecting p to p in Br p c n Observe that the images of the lines Op by the exponential map o exp Opi are distinct geodesic loops through p Denote by g the
40. hen the spatial dimension is n 3 and the manifold is Ricci flat according to Klainerman and Rodnianski 17 18 Assumption A3 is a consequence of the following L curvature bound IRmellt2 _ r lt K 6 2 for some constant K gt 0 Assumption A4 concerns the metric on the initial hypersurface and is only slightly stronger than the volume bound A4 Furthermore according to Anderson 1 and Petersen 22 the property A4 is also a consequence of the curvature bound IRmolln x_ser lt K 6 3 for m gt n 2 and some constant K gt 0 and a volume lower bound at every scale pn Volgi By p 1 vo r 0 ro 6 4 In summary by combining Theorem 6 1 above with the results in 19 16 we conclude 23 Corollary 6 2 Einstein field equations of general relativity Let M gQ be a Lorentzian 3 1 manifold satisfying the vacuum Einstein equation Ric 0 6 5 Suppose that near some point p M there exists a foliation Q of the form 6 1 satisfying Assumptions A1 A2 and such that the L curvature assumption 6 2 holds on the initial spacelike hypersurface X_ Then there exists a positive constant ig depending only upon the foliation bounds Ko K and the curvature bound K such that the null injectivity radius satisfies Null Inj M p T 2 to Proof of Theorem 6 1 Step 1 Localization of the past null cone N p between two flat null cones Assumption A3 provides us with a bound o
41. ive at the following lower bound for the norm of the Jacobi field KVD ess a F s gt Ver gt z gt e 4s for some c4 gt 0 On the other hand using again the above estimates we have d 1 SE lt Verny J Dor Ko F ee C3 2 22 C5 s e s Ka 2 2K3 s lt e for some constant c5 gt 0 This leads to the upper bound F s lt es 16 In summary we have established that the norm of the Jacobi field is com parable with s e 4s lt F s lt e s s 0 r2 4 7 By the definition of Jacobi fields these inequalities are equivalent to controling the differential of the exponential map that is for s 0 r2 es Wr lt Idexp syo W Ir lt g IWlr We conclude that the pull back of the reference metric to the tangent space at p satisfies e grp lt exp gr lt e Srp in the ball Br 0 r2 C TM 4 8 In particular since the conjugate radius of the Lorentzian metric is precisely defined from the reference Riemannian metric these inequalities show that the conjugate radius of the exponential map is r2 at least Step 3 Injectivity radius estimate We are now in a position to establish that Inj M p T 2 r3 r2e 4 We argue by contradiction and assume that y 0 s gt M and y2 0 s2 M are two distinct g geodesics satisfying max s S2 lt r3 and y 0 y0 p ly O lr Iy30 lr 1 y s y2 82 q We will reach a contradiction and this wil
42. l establish that the injectivity radius is greater or equal to r3 as can be checked by using the fact that the exponential map is at least a local diffeomorphism By Step 1 we know that 71 72 C Be p 2e 73 By concatenating these two curves we construct a geodesic loop containing p y y3 U y1 0 51 82 gt Be p 2e 13 which need not be smooth at p or q Since y is contained in the image of the ball Br p r2 under the exponential map we can define an homotopy of y with the origin x 0 by setting in the coordinates constructed earlier l s ey s e 0 1 The curves I 0 1 s2 gt Bg p 2e r3 satisfy Te 0 T s1 52 p Tol0 1 p Ti y Moreover we have I s e lt 2 lt 2e and thus I s r lt 2e In particular the gr lengths computed with the reference metric of the loops Te are less than Le gr lt 2e r3 17 Since the exponential map is a local diffeomorphism from the ball Br 0 r2 T M to the manifold and in view of the estimate 4 8 on the exponential map it follows that all the loops T can be lifted to the ball Br 0 rz in the tangent space with the same origin 0 Consequently we obtain a continuous family of curves T 0 s1 s2 gt T M satisfying T 0 0 ee 0 1 At this juncture we observe that since T s1 s2 for e 0 1 all cover the same point p and since the curve To is trivial and the family is continuous T s1 5 0 0 1
43. mov s theory pro vides geometric conditions for the strong compactness of sequences of mani folds and has become a central tool in Riemannian geometry See for instance 1 4 5 7 8 9 16 21 22 Department of Mathematics Sun Yet Sen University 510275 Guang Zhou P R of China E mail mcscb mail sysu edu cn tLaboratoire Jacques Louis Lions amp Centre National de la Recherche Scientifique Universit de Paris 6 4 place Jussieu 75252 Paris France E mail lefloch ann jussieu fr 2000 AMS Subject Classification 53C50 83C05 53C12 53C22 Key Words Lorentzian geometry Riemannian geometry general relativity Einstein equation injec tivity radius Jacobi field null cone comparison geometry Our objective in the present paper is to present some extension of these clas sical techniques and results to Lorentzian manifolds Recall that a Lorentzian metric is not positive definite but has signature Motivated by re cent work by Anderson 2 and Klainerman and Rodnianski 19 we derive here several injectivity radius estimates for Lorentzian manifolds satisfying certain curvature and volume bounds That is we provide sharp lower bounds on the size of the geodesic ball around one point within which the exponential map is a global diffeomorphism and therefore we obtain sharp control of the manifold geometry Our proofs rely on arguments that are known to be flexible and effi cient in Riemannian geometry and
44. n harmonic coordinate system x with respect to the Riemannian metric gy such that X x lt 1 and for every 0 lt y lt 1 Ign ap Oapl lt C1 1 lOgn lt 1 c n lOgnler lt 1 c n y In the construction of harmonic coordinates we may also assume that lx Z lero lt c n Since V gln lt 1 c n and that in these coordinates V lt 1 c n we have 0g lt 1 c n Finally to estimate the metric we write gag Naglp lt c1 7 and l gl lt 1 c n and we conclude that gag Magl lt ame c n The proof is completed o Pseudo Riemannian manifolds Finally we would like discuss pseudo Riemannian manifolds M g also re ferred to as semi Riemannian manifolds Consider a differentiable manifold M endowed with a symmetric non degenerate covariant 2 tensor g We assume that the signature of g is n1 n2 that is n negative signs and nz positive signs Riemannian and Lorentzian manifolds are special cases of pseudo Riemannian manifolds Fix p M and an orthonormal family T consisting of n vectors E1 E2 En TM such that E Ej g 6 Based on this family we can define a reference inner product gr on T M by generalizing our construction in the Lorentzian case and by using this inner product we can then define the ball Br 0 s C T M By parallel translating F E2 En along radial geodesics from the origin in T M we obtain vector fields E1 E2
45. n the null conjugate radius we need to control the null cut locus radius We proceed as in Section 4 and introduce coordinates near the point p such that x p 0 Precisely relying on Assumptions A1 A2 and A4 we determine the coordinates x x so that x tand the spatial coordinates x are transported via the gradient of the function t from the coordinates prescribed on the initial slice L_ Then the Lorentzian metric reads g n dt Qij dx dx and satisfies for some Co C gt 0 Te 1 G lt n lt Co G ij lt Qij lt C1 Oyj 6 6 for all r lt t lt 0 and x x lt ro and in these coordinates the reference Riemannian metric gr is comparable to the n 1 dimensional Euclidian metric gg dt dx dx 1 T amp E lt gr lt C1 ge 6 7 1 Denote by Bg q r the Euclidean ball with center q and radius r Note that these inequalities holds within a neighborhood of p in Q The forthcoming bounds will hold in a neighborhood of the past null cone only To simplify the notation we set 1 1 ay C a Co A In each time slice of parameter value t a we introduce the n dimensional Euclidian ball with radius b Aen ar ane ae e Co which is centered around the point p with coordinates a 0 0 We also define Ay Nic ar similarly For any point q ina slice satisfying ro lt to lt 0 and x q x q lt c t we consider the line for the
46. ntzian manifold Main result We are now a in a position to discuss and prove Theorem 1 1 stated in the introduction As we have seen in the proof of the previous section once the injectivity radius is controled one can construct a foliation satisfying certain good properties On the other hand the concept of injectivity radius is clearly independent of any prescribed foliation As this is more natural we will now present a general result which avoids to assume a priori the existence of a foliation This will be achieved by relying on purely geometric and intrinsic quantities and constructing coordinates adapted to the geometry Such a result is conceptually very important in the applications The result and proof in this section should be viewed as a Lorentzian generalization of Cheeger Gromov and Taylor s technique 10 originally developed for Riemannian manifolds Let M g be a differentiable n 1 manifold endowed with a Lorentzian metric tensor g and consider a point p M and a vector T T M with gy T T 1 That is we now fix a single observer located at the point p As explained in Section 2 the vector T induces an inner product gr r on the tangent space T M We assume that the exponential map exp is defined in some ball Br 0 ro C T M determined by this inner product which is of course always true in a sufficiently small ball Controling the geometry at the point p precisely amounts to estimating the size
47. o q A causal trip is defined similarly except that the geodesics may be causal instead of time like and we write p lt q if there exists a causal trip from p toq The set J p q e M p lt lt q is called the chronological future of the point p and J p q e M q lt lt p is called the chronological past The causal future and past are defined similarly by replacing lt lt by lt The future or past sets of a set S C M are defined by 5 Jp POs Ep pes pes and one easily checks that J S are open but that J S need not be closed in general A future set F C M by definition has the form F J S for some set S C M Similarly a past set satisfies F J7 S for some S A set is called achronal if no two points are connected by a time like trip Observe that a set can be spacelike at every point without being achronal and that an achronal set can be null at some or even at every point A set B C M is called an achronal boundary if it is the boundary of a future set that is B 0J S J S J S for some S c M One can check that given a non empty achronal boundary the manifold can be partitioned as M PUB U F where B is the boundary of both F and P and moreover any trip from p P to q F meets B at a unique point Observe also that any achronal boundary is a Lipschitz continuous n manifold For instance in Section 6 below we will be interested in the geometry of past null cones that is the sets
48. on the radius of injectivity of Lorentzian manifolds which hold either in a neighborhood of a point or on the past null cone at a point Our assumptions are formulated within a geodesic ball or within a null cone and possibly apply in a ball with arbitrary size as long as our curvature and volume assumptions hold All assumptions and statements are local and geometric An outline of the paper is as follows In Section 2 we begin with basic material from Lorentzian geometry and we introduce the notions of reference metric and exponential map for Lorentzian manifolds In Section 3 we state our first estimate Theorem 3 1 below for a class of manifolds that have bounded curvature and admit a time foliation by slices with bounded extrinsic curvature In Section 4 we provide a proof of this first estimate and we introduce a technique that will be used in variants throughout this paper we combine two main ingredients sharp estimates for Jacobi fields along geodesics and an homotopy argument based on contracting a possible loop to two linear segments In Section 5 our second main result Theorem 5 1 shows under the same assumptions the existence of convex functions distance functions and convex neighborhoods this result leads us to a lower bound of the convexity radius In Section 6 our third estimate Theorem 6 1 covers the case of null cones under the assumption that the manifold has L bounded curvature on every spacelike slice this p
49. onstant c X depends only on the distance measured by T of to the null cone Proof First we recall there is a constant C X depending only on the distance of to the null cone such that Ric y y C Z ly for any time like geodesic y with y 0 By the volume comparison theorem for future cone established in Theorem 8 1 we have Vol FCz p c n ro Vol FCz p ro gt CX 35 and combining this result with Theorem 1 1 the corollary follows o 9 Final remarks Regularity of Lorentzian metrics Following the strategy proposed in the present paper we now transfer to the Lorentzian metric the regularity available on a reference Riemannian metric Clearly the regularity obtained in this manner depends on the way the refer ence Riemannian metric is constructed The interest of our approach below is to provide a simple derivation using harmonic like coordinates for the Rie mannian metric we see immediately that the Lorentzian metric has uniformly bounded first order derivatives For the optimal regularity achievable with Lorentzian metrics we refer to Anderson 3 Proposition 9 1 Regularity in harmonic like coordinates Under the assump tions and notation of Theorem 1 1 define Vol Br p c n ro n 1 ro 0 r c n r where c n is the constant determined in this theorem Then for any gt 0 there exist a constant c n with lim 9c1 n 0 and a coordinate
50. rovides a generalization and an alternative proof to the result by Klainerman and Rodnianski in 19 Next in Section 7 we establish our principal and fourth result stated in Theorem 1 1 below which provides an injectivity radius bound under the mild assumption that the exponential map exp is defined in some ball and the curvature Rm is bounded Most importantly this is a general result that does not require a time foliation of the manifold but solely a single reference future oriented time like unit vector T at the base point p This is very natural in the context of general relativity and p T is interpreted as an observer at the point p Given an observer p T we can define the ball Br 0 r c T M with radius r determined by the reference Riemannian inner product at p and we can also define the geodesic ball Br p r exp Br 0 r In turn the radius of injectivity Inj M p T is defined as the largest radius r such that the exponential map is a diffeomorphism from Br 0 r onto Br p r Let us then consider an arbitrary geodesic y y s initiating at p and let us g parallel transport the vector T along this geodesic defining therefore a vector field T along this geodesic only At every point of y we introduce the reference metric gr and compute the curvature norm Rm 7 This allows us to express the curvature bound For the convenience of the reader we state here our main result and refer to Section 7 for further details
51. s e gry IBN 0 r0 S exp sT laxo n S e gryp IBY 0 70 A3 and finally there exists a coordinate system on the initial slice L_ such that the metric g z is comparable to the n dimensional Euclidian metric gf in these coordinates eer lt glz se ge in Briel ro A4 We refer to K as the effective conjugate radius constant Theorem 6 1 Injectivity radius of null cones Let M be a differentiable n 1 manifold endowed with a Lorentzian metric g satisfying the regularity assumptions A1 A2 A3 and A4 at some point p and for some foliation 3 1 Then there exists a positive constant ig depending only upon the foliation bounds K Ky the null conjugate radius ro the effective conjugate radius constant Ky and the dimension n such that the null injectivity radius of the metric g at p satisfies Null Inj gM p T io It is interesting to compare the assumptions above with the ones in Section 3 Assumptions A1 and A2 are concerned with the property of the foliation and were already required in Section 3 Assumption A3 should be viewed as a weaker version of the L curvature condition A3 Recall that under the assumptions of Theorem 3 1 which included a curvature bound an analogue of A3 valid in the whole of Q was already established in 4 8 It is expected that A3 is still valid when the curvature in every spacelike slice is solely bounded in some L space Indeed at least w
52. s Jour Math Phys 44 2003 2994 3012 4 A Besse Einstein manifolds Ergebenisse de Math Series 3 Springer Ver lag 1987 5 J CHEEGER Finiteness theorems for Riemannian manifolds Amer J Math 92 1970 61 94 6 J CHEEGER AND D Esin Comparison theorems in Riemannian geometry North Holland Amsterdam Oxford American Elsevier Pub New York 1975 7 J CHEEGER AND M Gromov Collapsing Riemannian manifolds while keep ing their curvature bounded I J Diff Geom 23 1986 309 346 8 J CHEEGER AND M Gromov Collapsing Riemannian manifolds while keep ing their curvature bounded II J Diff Geom 32 1990 269 298 9 J CHEEGER K Fuxaya AND M Gromoy Nilpotent structures and invariant metrics on collapsed manifolds J Amer Math Soc 5 1992 327 372 10 J CHEEcER M Gromov anp M Taytor Finite propagation speed ker nel estimates for functions of the Laplace operator and the geometry of complete Riemannian manifolds J Differential Geom 17 1982 15 53 11 S Y CuEnc P Li ann S T Yau Heat equations on minimal submanifolds and their applications Amer J Math 106 1984 1033 1065 12 D M DeTurck ann J L Kazpan Some regularity theorems in Riemannian geometry Ann Sci Ecole Norm Sup 14 1981 249 260 13 R S Hamitron A compactness property for solution of the Ricci flow Amer J Math 117 1995 545 572 14 S Hawkinec ann G F Exus The large scale stru
53. s y 2y Ny are distinct homotopy classes for the relation groc n If this were not true then by the cancellation law we would have jy za 0 groen for some 1 lt j lt N We already know that all T is defined from Bz7 0 c n to Br 0 c n for i lt j Since for any y Br 0 c n we have exp Oy U jy ae exp Oy which implies that ni id We use here the notation n Then we define a function u Br 0 c n gt R by Tly Thy ete u y O y O ry A 0 niy Since Ti id it is easy to see u 7t y u y for any y Br 0 c n That is to say u is Ty invariant By Step 3 u is strictly g geodesically convex on Br 0 c n More precisely since for any g geodesic 0 59 gt Br 0 c n 7 are still g geodesics in Br 0 c n and 2 2 n LEE V7dz 0 E 8 E S VCO drh EOLA EO gt 9 amp s s gt 0 Observe that 3 2 u lB 0 2 ja c n c n any gt ew f and D u 0 lt SOE so the minimum of function u over Br 0 c 1 is only achieved at at an interior point say yo Br 0 c n Then by 7 invariance of u we have u 7y o u yo lt jce n 2 and this implies T yo Br 0 c n By the injectivity radius estimate at yo T M Q there exists a g geodesic connecting yo to 7 Yo 32 which is contained in Br 0 2c n By using the strong g geodesic convexity of u we conclude that 7 Yo yo This contradicts the
54. s will be possible once we will have estimated the conjugate radius in Step 1 of the proof below and will know that the exponential map is non degenerate on Br 0 ro Pulling back the Lorentzian metric g on M by the exponential map we get a Lorentzian metric g exp g defined in the tangent space on the ball Br 0 ro We use the same letter g to denote this metric Then the geometry in the tangent space is particularly simple since the g geodesics on M passing through p are radial straightline in Bzr 0 ro A third view point could be adopted by restricting attention within the cut locus from the point p and by imposing the curvature assumption within the cut locus only We are in a position to prove the main result of the present paper that was stated in Theorem 1 1 Proof of Theorem 1 1 After scaling we may assume that r 1 and so we need to show Inj M p T 2 c n Vol Br p c n 7 1 Step 1 Estimates for the metric gr and its covariant derivative Let Ey T E En be an orthonormal frame at the origin in T M for the Lorentzian metric g By g parallel transporting this basis along along a radial geodesic y y r satisfying y 0 0 Iy 0 lr 1 we get an orthonormal frame defined along the geodesic We use the same letters E to denote these vector fields Since d Z Ea Ep 0 we infer that IE E 1 along the geodesic 28 The same argument also implies YOr YOr aL WOk b Ok L 7 2 an
55. system x satisfying x p 0 and defined for all x x x lt 1 r such that lap Nagl lt c1 n 9 1 rosal lt cx t ey where Nag is the Minkowski metric in these coordinates Proof By scaling we may assume r1 1 By Step 1 in the proof of Theorem 1 1 we know that the Riemannian metric gr is equivalent to the Riemannian metric gto on Br 0 4c n By considering a lift and using again the results in Step 1 this implies Br p e n Br q 3c n q Brp e n Applying the same argument as in Theorem 1 1 we deduce that the injectivity radius of any point in Br p c n is bounded from below by c n As in Step 3 in the proof of Theorem 1 1 or in Step 2 of Section 5 we see that there exists a synchronous coordinate system y T y of definite size around p such that the metrics g dt gj dy dy and gn dT g dy dy the Riemannian metric constructed therein satisfy the following properties on the geodesic ball Br p c n a l1 c n gn lt gr lt 1 ce n gn b gn has bounded curvature lt 1 c n 36 c Itl 1V tly lt 1 c n Ir In particular this implies V gly lt 1 c n Since the volume Vol Br p c n is bounded from below it follows from 10 that the injectivity radius of gn at p is bounded from below by c n By the theorem in 16 on the existence of harmonic coordinates for any small gt 0 there exists a
56. t p Suppose also that the Ricci curvature satisfies on Br p ro Ric V V n Kp Vie for all time like vector fields V Then for any 0 lt r lt s lt ro the inequality Vol FC p r _ Volx B r Vol FC p s Volx B s holds with FC p r exp FC p r and FC P 1 0 lt IVlgra lt tor IVE lt 0 T Vero lt 0 33 and Volx B r is the volume of the ball with radius r B r analogous to Br p r M in the simply connected Lorentzian n 1 manifold with constant curvature K that is Rapys K Saygps Sad 8py More generally if X is a subset in unit sphere S such that IVI lt 0 T V lt 0 forall V amp then the inequality Vol FCz p r _ Volx B r Vol FCz p s Volx B s holds with FCx p r exp FCx p r and V FCs p r V FC p r v z This result will be used shortly to control the injectivity radius of null cones butis also of independent interest For definiteness we state the result for future cones Proof Let y 0 so M be a future oriented time like geodesic satisfying y 0 p and y 0 1 We are going to use the standard technique to compute the rate of change of the volume element along y Given s 0 89 assume that any point in 0 s is neither a conjugate point nor a cut point with respect to p Let vo y s1 v1 V2 Un be an orthonormal basis at y s1 with respect to 84 s Let also Va be t
57. the Riemannian metric gr and compute the norms of tensors We begin with a set of assumptions encompassing a large class of Lorentzian manifolds with L bounded curvature and we state our first injectivity estimate in Theorem 3 1 below The forthcoming sections will be devoted to further generalizations and variants of this result We fix a point p M and assume that a domain Q C M containing p is foliated by spacelike hypersurfaces with future oriented time like unit normal T OS E 3 1 te 1 1 The positive coefficient n is defined by the relation 2 nT or Bis d 0 m 3 3e a In the context of general relativity n is the proper time of an observer mov ing orthogonally to the hypersurfaces and is called the lapse function The geometry of the foliation is determined by this function n together with the Lie derivative Lrg The latter is nothing but the second fundamental form or extrinsic curvature of the slices 1 embedded in the manifold M We always assume that the geodesic ball Bs p 1 C Xo determined by the induced metric glz is compactly contained in Xo We introduce the following assumptions eM lt n lt e ino A1 lLrglr lt Ki in Q A2 IRmgir lt K gt in Q A3 Volgi Bs p 1 2 vo A4 where Ko Ki K2 and vo are positive constants Observe that Assumption A4 is a condition on the initial slice only together with the other assumptions it actually implies a lower volume bound for ev
58. ther than a vector at the point p we use the notation Inj M p T instead of Inj M p Tp The radii Conj gM p T and Inj gM p T are essentially independent of the choice of the reference vector as long as it remains in a fixed compact subset of T M We also need the notion of injectivity radius for null cones Given a point p Manda reference vector T T M we consider the past null cone at p N X T M 8p X X 0 g T X 20 which is defined a subset of the tangent space at p Denote by BY 0 7 By 0 1 Bg 0 7 A N5 the intersection of the Riemannian grp ball with radius r and the past null cone and by N p OF p the past null cone at p Consider now the restriction of exp to the past null cone denoted by exp BN O r c N gt N p CM which we refer to as the null exponential map Definition 2 2 The past null conjugate radius Null Conj M p T ofan observer p T T M is the largest radius r such that the null exponential map exp is a local diffeomorphism from the punctured Riemannian ball B 0 r 0 c T M to a neighborhood of p in the past null cone The null injectivity radius Null Inj M p T of an observer p T T M is defined similarly by requiring the map exp to be a global diffeomorphism 3 Lorentzian manifold endowed with a reference vector field A first injectivity radius estimate From now on we fix a reference vector field T which allows us to define
59. thonormal basis of the tangent space at every point and consists of the vector eo T supplemented with n spacelike unit vectors e j 1 7 Denoting by E the corresponding dual frame the Lorentzian metric tensor takes the form 8 Nap E EP where nag is the Minkowski metric This decomposition suggests to consider the Riemannian version obtained by switching the minus sign in noo 1 into a plus sign that is gr ap E Q EP where dag is the Euclidian metric Clearly gr is a positive definite metric it is referred to as the reference Riemannian metric associated with the frame Ey For every p M since T is time like the restriction of the metric gp to the orthogonal complement T C TpM is positive definite and the reference metric can be computed as follows if V aT V and W bT W with V W T then grp V W ab g V W In the following we use the notation STV W VY Wyry 8ra V IVR for vectors the norm of tensors is defined and denoted similarly In contrast with the Lorentzian norm the Riemannian norm A r of a tensor A at a point p M vanishes if and only if the tensor is zero at p Moreover as long as T remains in a compact subset of the bundle of half cone T M the norms associated with different reference vectors are equivalent The reference Riemannian metric also allows one to define the functional norms for Lebesgue and Sobolev spaces of
60. truction of the metric gr we have Vo Vg VT T schematically and VT 0 0 and it is useful to control the covariant derivative too To this end write the radial vector field as a4 B ade mae eF with 2 2 1 as stated already in 7 2 Using that VTI Va TEV gT grey ge and computing the derivative of IVT along radial geodesics we find d q NT lt C n VT 2 V2 VT VT r By using that 0 a 1a y MgO Sha hope a eN 29 we obtain ae or oy and therefore thanks to the curvature assumption 1 V V o T See a T R ye Oj oy r T d 2 FYT lt FIV Tp Cl IVT Ip C IVT r This implies the following bound for the covariant derivative IVTIr y lt C n ly lyl lt 1 C n 7 4 which also provides a bound for the difference Vg Vg Step 2 Estimate of the injectivity radius of g on Br 0 c n Since the curvature on Br 0 1 is bounded and that V VelF lt C n 1 c n on the ball Br 0 c n we can follow the argument in Section 4 and bound from below the conjugate radius for any point in the ball By 0 3c n 4 Next given any point y By 0 c n 2 let y and y2 be two geodesics which meet at their end points and have short length with respect to the metric gr or gro By using the linear structure on B7 0 1 a subset of the vector space T M we can construct an homotopy of the loop y U y3 to the origin such that each curve have also
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