Geodesic incompleteness and partially covariant gravity
aa r X i v : . [ g r- q c ] F e b Geodesic incompleteness and partiallycovariant gravity
Ignatios Antoniadis , ∗ and Spiros Cotsakis , † Laboratoire de Physique Th´eorique et Hautes Energies - LPTHESorbonne Universit´e, CNRS 4 Place Jussieu, 75005 Paris, France Institute for Theoretical Physics, KU LeuvenCelestijnenlaan 200D, B-3001 Leuven, Belgium Institute of Gravitation and Cosmology, RUDN Universityul. Miklukho-Maklaya 6, Moscow 117198, Russia Research Laboratory of Geometry,Dynamical Systems and Cosmology, University of the AegeanKarlovassi 83200, Samos, GreeceFebruary 2021 bstract We study the issue of length renormalization in the context of fully covariantgravity theories as well as non-relativistic ones such as Hoˇrava-Lifshitz gravity.The difference of their symmetry groups implies a relation between the lengths ofpaths in spacetime in the two types of theory. Provided that certain asymptoticconditions hold, this relation allows us to transfer analytic criteria for the standardspacetime length to be finite to the Perelman length to be likewise finite, andtherefore formulate conditions for geodesic incompleteness in partially covarianttheories. We also discuss implications of this result for the issue of singularities inthe context of such theories. ∗ [email protected] † [email protected] ontents Introduction
Non-relativistic theories of gravity have been very popular recently for a number ofdifferent reasons. In the direction of renormalizing general relativity, a prime example isprovided by the Hoˇrava-Lifshitz theory where a power-counting renormalizable theory isoffered by the drastic abandoning of Lorentz symmetry and introducing an anisotropicscaling between space and time [1, 2], [3, 4]. This has major implications for a number ofareas, especially in cosmology, where general relativity is partially abandoned and newdifficulties arise due to the presence of higher-order terms and the idea of scalar graviton,but also new light is shed to old issues like the possibility of having a scale invariantspectrum of cosmological perturbations even without inflation, a bouncing cosmology,or a possible non-chaotic evolution towards the singularity of a Bianchi IX cosmology[5]-[10].Due to the close relation of the Hoˇrava-Lifshitz equations to a generalized form of theRicci flow equation, new ways are possible for the construction of topological quantumgravity theories by utilizing special mathematical properties of the later [11]. Secondly,the Ricci flow as well as other non-relativistic geometric flows such as the mean-curvatureflow have long been known for their interesting properties especially with regard to theformation of singularities [12]-[16], in association with corresponding properties knownfor the Einstein flow [17]-[19], although a clear relation between the nature of singularitiesbetween different non-relativistic flows and general relativity is presently elusive. Thirdly,the importance of alternative geometries such as the Newton-Cartan (NC) geometry ingravitation through a non-relativistic expansion is useful in delineating more preciselythe relations of non-relativistic gravity and general relativity [20]-[22].A distinctive feature common to all the non-relativistic theories above is that theirsymmetry, or invariance , groups are not the full diffeomorphism group (as in the caseof a theory such as general relativity) but typically some proper subgroup of it. Thisresults in a different behaviour of the solutions (i.e., the space-time metric and variousfields) of a non-relativistic theory as opposed to those of general relativity, in particular4ith respect to their transformations by elements of their symmetry group (cf. [23],[24]. The solutions of the theory are of course invariant under the symmetry group, andconstitute the theory’s absolute elements . In general relativity, and in distinction withall the non-relativistic theories mentioned above, there are no absolute elements becausethe symmetry group coincides with the covariance group (GR is ‘generally covariant’).In a non-relativistic theory, we may of course write down a metric-solution of the fieldequations of the theory-which looks the same as in general relativity, however, the subtledifference in their symmetry groups mentioned above will reflect important changes inthe causality properties associated to the two metric structures.The purpose of this note is to study this point. In the next Section, we introduce thisdifficulty and in Section 3 we suggest a new approach for the study of causality in thetwo frameworks of a fully and a partially covariant theory. In Section 4, we establish theexistence of an asymptotic relation between the length functional for paths in space-timetypically used in a non-relativistic framework and the standard spacetime length. Thisrelation will allow us in Section 5 to find conditions for partially covariant flows to begeodesically incomplete from those for the spacetime length in fully covariant theories.We discuss our results in the last Section.
In this Section, we compare the Hoˇrava-Lifshitz theory to general relativity, especiallytheir causal structures, focusing on their covariance groups. This is a necessary first stepsince we shall import ideas and methods from general relativity to the framework of theformer theory. As discussed in the Introduction, what we describe below is valid notonly for the Hoˇrava-Lifshitz theory but also for any other non-relativistic theory basedon NC geometry, a well as on other geometric flows such as the Ricci flow.It is well known that in general relativity there is a way, pioneered by YvonneChoquet-Bruhat, to split the 4-metric, connection, and curvatures, in a Cauchy-adapted5rame (cf. [17] and refs. therein). Such a decomposition is very suitable for the studyof the initial value formulation of the theory (as well as its Hamiltonian (ADM) for-mulation). In this case, the evolution equations are expressed in terms of the spatial3-metric g ij and its extrinsic curvature K ij , but there are further gauge variables , thelapse N and the shift N i . These, together with the spatial metric, make up the 4-metric ( g ij , N, N i ), satisfying the Einstein equations and being invariant under the fulldiffeomorphism group, Diff( V ), of the spacetime manifold V .But in the Hoˇrava-Lifshitz theory, because of the restricted covariance of the fieldequations under (the subgroup of Diff( V )) FPDiff( V ) of foliation-preserving diffeomor-phisms, the functions N, N i are no longer gauge variables but are now promoted to fields on the 4-manifold V . The ‘4-metric’ ( g ij , N, N i ), solution of the Hoˇrava-Lifshitz fieldequations, has now a smaller invariance group than in general relativity. This has majorimplications for the causal structure of the theory. Below we briefly discuss some of theseimplications in the context of the Hoˇrava-Lifshitz theory.On the 4-manifold V = M t × [ t , ∞ ] , t ∈ R , we are given the following data: asmooth Riemannian metric g ij on the 3-manifold M t = M × { t } , a smooth function N ( t, x i ) defined on V , and a vector field N i ( t, x j ) tangent to the 3-manifold M . Aswe shall show below, the data N, N i , can play a role analogous to that of the lapsefunction and shift vector of the usual (3 + 1)-formalism in general relativity. A basicgeometric assumption of the Hoˇrava-Lifshitz gravity is the existence of a ‘book-keeping’line element form, ds = − N dt + g ij ( t, x ) (cid:0) dx i + N i dt (cid:1) (cid:0) dx j + N j dt (cid:1) . (1)It is usually assumed that N is a function of t only (the so-called ‘projectable’ case),but most importantly the form (1) considered as a solution of the Hoˇrava-Lifshitz fieldequations (identical to the Ricci flow equations when N = 1 , N i = 0),˙ g ij = − κ κ W N (cid:18) R ij − λ − Dλ − g ij R (cid:19) + ∇ i N j + ∇ j N i , (2)6s invariant under the action of the restricted group of foliation-preserving diffeomor-phisms t → ˜ t ( t ) , x → ˜ x ( t, x ), not of the full group of spacetime transformations [1], [2],[3], [5]. Despite the identical form of ds above to the standard ‘adapted-frame’ space-time decomposition of a metric as in general relativity (cf. e.g., [17]), (1) is not the usualproper time, and equipping the 4-manifold V with the form (1) does not make ( V , g HL )a spacetime in the general relativistic sense of the word.In particular, there is no null structure on ( V , g HL ) like in general relativity, nor arelativistic notion of causality or chronology, nor can there be the usual trichotomy oftimelike, null, spacelike, for vectors at any point p on V . These ‘peculiarities’ are alsoshared by other non-relativistic theories of gravity such as Newton-Cartan theories, aswell as geometric flows such as the Ricci flow, cf. [20], and [12], Section 14.1, respectively.In fact, in such theories, the classical metric (1) is just the leading term in an 1 /c expansion about zero ( c → ∞ ) [1, 2, 22]. For a given curve C : I ⊂ R → V on themanifold V , there can be no invariant definition of a notion of proper length for C using the form (1) in such an approximation, no notion of causal geodesics on ( V , g HL ),hence no obvious way to speak of the usual route through geodesic (in-)completeness tospacetime singularities, maximal curves, global hyperbolicity, etc.Hence, although there is an obvious interest in the further analysis and examination ofthe implications and classical meaning of non-relativistic theories with anisotropic scalingwhich also share desirable quantum properties [1, 2], given their restricted invariance (ofthe line element ds as in Eq. (1) and action) under only foliation-preserving diffeomor-phisms, there is an apparent difficulty to talk sensibly about dynamics and asymptoticproperties of spacetime fields near singularities in terms of standard spacetime notions.How are we to somehow import a generic notion of geodesic (in-)completeness into theseframeworks and therefore be able to compare non-relativistic gravity theories to the moreusual ones that allow a full spacetime interpretation?7 A new approach
Consider two points p, q ∈ V belonging to a suitable, globally hyperbolic region N of V with q in the future of p in the spacetime metric g , and introduce Minkowski normalcoordinates ( t, x i ) in the region N (that is ∂ t is timelike and future-pointing and thenull cone T p V is the set t − P ( x i ) = 0). We can follow standard arguments (cf. e.g.,[25], Proposition 7.2, [26], Lemma 6.7.2) and introduce Gaussian normal coordinates T, X, Y, Z on V , where T = ( t − P ( x i ) ) / , X = x /t, X = x /t, X = x /t . Thisis the so-called synchronous system in which the surfaces T = const. are spacelike whilethe curves X i = const. are timelike geodesics orthogonal to these. The metric g thentakes the standard form, ds GR = dT − g ij dX i dX j . (3)Setting p = C ( T ) , q = C ( T ), the spacetime length of any curve C connecting the points p, q ∈ N is given by (a dot denotes differentiation with respect to T ), l GR ( C ) = 1 T − T Z T T (cid:16) − g ij ˙ X i ˙ X j (cid:17) / dT, (4)and so the length is equal to 1 for the curves X i = const. (the 3-metric g ij is positive-definite). That is, the geodesic connecting the two points p, q has maximum length.We note that this process of obtaining maximal geodesic curves described above isnot available in the Hoˇrava-Lifshitz theory (at least in its standard ‘projectable’ versionwhere N = N ( t )). This is simply because it is not feasible to pass to synchronouscoordinates which have N depending on both t and the spatial coordinates x i as above(even a choice of coordinates which translates ds HL to the form (3) would not do, becausethe curves X i = const. are not geodesics on V ).However, we may proceed as follows. Given the resemblance of the Hoˇrava-Lifshitzfield equation to the generalized Ricci flow (2), it is not unreasonable to think of usingthe L -length function (the so-called Perelman length) as a possible means of address-ing geodesic problems (dependent on its second variation) in the framework of Hoˇrava-Lifshitz and other non-relativistic theories. We show below that the L -length function8s in fact asymptotically connected to the standard spacetime length of fully covarianttheories. One way to show this is to use a similar procedure to the length renormaliza-tion method for the so-called potentially infinite metrics. It is known that for the Ricciand other Riemannian curvature flows, this method can be used to derive the Perelman L -length (cf. e.g., Chapter 11 of [13]). What we need is a procedure that applies equally to both non-relativistic and fully co-variant metric theories. For this purpose, we introduce now the family of metrics g ξ which have the same generic form of both the fully covariant metric g used earlier todiscuss the spacetime length l GR ( C ), as well as the foliation-preserving-diffeomorphisms-invariant metric g HL used earlier for the non-relativistic length function l HL ( C ). Fol-lowing [14], for the family of metrics g ξ we choose the lapse and shift functions suchthat, g ξ : ds ξ = − N ξ dt + g ij ( t, x ) dx i dx j , − N ξ ( t, x i ) = R ( g ij )( t, x i ) + ξ t < , N iξ = 0 , (5)with R being the scalar curvature of the 3-metric g ij ( t, x i ), and ξ a real negative constantlarge enough such that the above inequality for the scalar curvature is true.Then the standard spacetime length of any curve C : ( t , t ) → V with respect to themetrics g ξ is given by, l ξ ( C ) = Z t t (cid:16) − N ξ + g ij ˙ C i ˙ C j (cid:17) / dt = Z t t R + ξ t + (cid:12)(cid:12)(cid:12)(cid:12) dCdt (cid:12)(cid:12)(cid:12)(cid:12) g ( t ) ! / dt, (6)and this is well-defined provided the g ij -length of C is bounded from below. Then wewrite the integrand as (cid:18) | ξ | t (cid:19) / (1 + x ) / , x = 2 t | ξ | R + (cid:12)(cid:12)(cid:12)(cid:12) dCdt (cid:12)(cid:12)(cid:12)(cid:12) g ( t ) ! , (7)9nd expand (1 + x ) / keeping only up to the highest order non-trivial term in ξ . Wefind l ξ ( C ) = Z t t (cid:18) | ξ | t (cid:19) / + | ξ | − / √ √ t R + (cid:12)(cid:12)(cid:12)(cid:12) dCdt (cid:12)(cid:12)(cid:12)(cid:12) g ( t ) ! + O ( | ξ | − / ) ! dt, | ξ | > . (8)In this formula, of course, there are conditions - see below - for the spacetime length l ξ ( C )to be related to an extracted length corresponding to the second term in the integral,that is the Perelman length function l per ( C ) for the spacetime curve C , given by, l per ( C ) = Z t t √ t R + (cid:12)(cid:12)(cid:12)(cid:12) dCdt (cid:12)(cid:12)(cid:12)(cid:12) g ( t ) ! dt, (9)(the so called reduced length is l per ( C ) / √ t − √ t ), cf. [14]). We recall that in dis-tinction to the spacetime length which maximizes the length (proper time) between twopoints in spacetime, the Perelman length is a minimum along geodesics (cf. ([12])-([14])).However, since the first integral in the right-hand-side of (8) is finite, it follows that thespacetime length l ξ ( C ) is finite if and only if the Perelman length l per ( C ) is finite. Theconditions under which this is possible are given in the next Section.The fact that the Perelman length (9) may be extracted from Eq. (8) and so beconnected with the usual spacetime length l ξ ( C ) asymptotically for the family of metrics g ξ is very important. It shows that there may be a connection between the asymptoticnature of the fields near singularities met in various non-relativistic geometric flows suchas the Ricci flow [15], [16], and that near the ‘physical’ spacetime singularities defined asgeodesic incompleteness in general relativity. This is in fact the case, as we demonstratein the next section. It is well known that spacetime singularities and finite geodesic length as well as geodesiccompleteness and infinite such length may be described, not in the usual 4-dimensionalsense of the singularity theorems in general relativity, but through analytic criteria based10n conditions imposed on quantities appearing in the (3 + 1)-decomposition of spacetime,such as the extrinsic curvature of the time slices, [17], [27]. As we show below, thisapproach is particularly useful in discussing geodesic (in-)completeness in non-relativisticframeworks as well.There are two main results of this type giving sufficient conditions for causal geodesiccompleteness as well as of incompleteness, for the time interval of the form [ t , ∞ ), cf.[17], [27]. Because of the result we proved in the previous Section, we may examinewhether conditions proved for the spacetime length in Refs. [17], [27] may be true forthe Perelman length and non-relativistic (partially covariant) frameworks. Our timeintervals presently are necessarily finite of the form [ t , t ], not infinite like in the Refs.[17], [27]. This is not a restriction, however. The simple choice of a new time parameter τ , given by τ = 1 t − t , (10)will transform the spacetime length integrals of the form R t t f ( t ) dt considered here inEq. (8), to the length integral, Z ∞ t − t g ( τ ) dτ, g ( τ ) = f (cid:18) t − τ (cid:19) τ , (11)and hence, the conditions found in [17], [27] will apply in the present context unaltered.Here we assume that f is continuous and unbounded on [ t , t ), but an analogous changeof time variable, τ = t − t , will transform an improper integral on ( t , t ] to one on[1 / ( t − t ) , ∞ ) for the function h ( τ ) = f (cid:0) t + τ (cid:1) τ .For causal geodesic completeness (that is, infinite geodesic length) the conditionsmay be phrased as follows (cf. [27] for definitions and complete proofs). Suppose thatthe spacetime ( V , g ) with metric given by (1) is such that the data N, N i , g ij are alluniformly bounded. Then if the spatial norms of the space gradients of the lapse, ( ∇ i N ) ,and of the extrinsic curvature, K ij K ij , are both bounded by time-dependent functions(which are integrable on ( t , ∞ )), then ( V , g ) is timelike and null future geodesicallycomplete. Equivalently, we may rephrase these conditions on the extrinsic curvature11sing its traceless part P ij = K ij − (1 / g ij K [27]. If the spatial norm of P , | P | g ij = K ij k ij − K , (12)is bounded on ( t , ∞ ), then the spacetime ( V , g ) is future timelike and null g-complete.(We may also rephrase these sufficient conditions for completeness as necessary ones for in completeness, that is when ( V , g ) is singular. Then K ij K ij must blow up (assumingthat the gradient of the lapse is bounded). Similarly, assuming that we have a singularspacetime, then we may use the traceless part to say that P ij must blow up somewhere.)Now we apply the above completeness results to the length function l ξ ( C ) given bythe first equality of Eq. (6), Under the same sufficient conditions the metric g ξ given by(5) will also be future geodesically complete. Namely, for an infinite length l ξ ( C ) (that is g ξ is future geodesically complete), we require that ( ∇ i N ξ ) and K ξ,ij K ijξ are uniformlybounded. However, if the length l ξ ( C ) given in Eq. (8) diverges (as it is necessary forcompleteness), we cannot separate the integral and extract the Perelman length (9) eventhough the integral of the first term converges on [ t , t ]. Therefore starting from aninfinite l ξ ( C ) length and looking for sufficient conditions for completeness is not usefulfor our purposes, as the situation becomes inconclusive.However, we may suppose that the length l ξ ( C ) is finite (i.e., geodesic incomplete-ness), and look for necessary conditions for incompleteness. This situation is explainedin great detail in Refs. [28, 29], and there are various cases to consider according towhich some of the quantities ( ∇ i N ξ ) or K ξ,ij K ijξ blow up or become discontinuous insome way. These will provide new criteria for the Perelman length to be finite. There-fore geodesic incompleteness is guaranteed in both frameworks, Ricci flow and generalrelativity.Further, we look for sufficient conditions for incompleteness: These are that ( ∇ i N ξ ) is integrable and that K ξ,ij v i v j ≥ k >
0, for all spatial vectors v , cf. Ref. [17]. Theseconditions will also give sufficient criteria for the Perelman length to be finite in thiscontext. 12 Discussion
In this note we have considered possible relations between fully covariant theories andnon-relativistic ones, especially with regard to their causality structures and the issue ofsingularities (in the sense of geodesic incompleteness). Since two such theories are verydifferent due to the existence of ‘absolute elements’, it is not obvious how their causalstructures could be related if at all. This is also an issue that may arise when one asks ina non-relativistic context whether or not some property deduced from a special solutionhas possibly a more general (i.e., ‘generic’) meaning or significance.The main result we reported in this work is that if the spacetime length of pathsis finite, then the Perelman length is also finite asymptotically if certain conditionshold. This suggests a possible relation between the spacetime singularities of generalrelativity associated with geodesic incompleteness, and those of non-relativistic flowssuch as the Ricci flow, the Hoˇrava-Lifshitz theory, or in the context of Newton-Cartangeometry. We believe that this relation extends without much change in the setting ofany diffeomorphism invariant theory vs. a non-relativistic one.It is therefore possible that, with certain adjustments, the proofs of key results inglobal causality theory in general relativity may be applicable in the wider context of non-relativistic gravity. However, the exact formulation of something like this is at presenttotally unknown. Conversely, some techniques in Ricci flow (also applicable in principleto other partially covariant flows) may become useful in situations where the spacetimelength is finite and a possible extension of the solution is necessary. Unfortunately, ourpresent results cannot deduce an infinite Perelman length from geodesic completeness(infinite spacetime length). However, it follows from our incompleteness result about thefiniteness of the spacetime length, that if we know that the Perelman length is infinite,then the spacetime length will also be infinite. Therefore geodesic completeness in apartially covariant theory would imply standard spacetime geodesic completeness underthe other conditions considered in this paper. In the Ricci flow, an infinite Perelmanlength is obtained by the process of surgery at specific times, the singular times. This13aises the intriguing question whether one could similarly continue the Einstein flowusing surgery at the singularities in general relativity. We hope to return to this problemin the future.
Acknowledgments
Work partially performed by I.A. as International professor of the Francqui Foundation,Belgium. This paper is dedicated to the fond memory of our beloved friend and colleague,the late Ioannis Bakas. Giannis was not only a brilliant mathematical physicist, but alsoa extremely kind, generous, pleasant, and honest person. He had a very wide knowledgeand deep interest in both theoretical physics and pure mathematics, as this is clearlytestified by his scientific work. He took a very active part in early discussions of thisproject, and had he lived to this day, this paper would have been a better one.14 eferences [1] P. Horava, JHEP 03 (2009) 020; e-Print: 0812.4287 [hep-th][2] P. Horava, Phys.Rev.D 79 (2009) 084008; e-Print: 0901.3775 [hep-th][3] I. Bakas, J. Phys. Conf. Ser. 283 (2011) 012004, arXiv:1009.6173.[4] I. Bakas, Fortsch.Phys. 59 (2011) 937; e-Print: 1103.5693 [hep-th][5] I. Bakas, F. Bourliot, D. Lust, M. Petropoulos, Class. Quant. Grav. 27 (2010) 045013,arXiv:0911.2665.[6] S. Mukohyama, Class.Quant.Grav. 27 (2010) 223101; e-Print: 1007.5199 [hep-th][7] R. Brandenberger, Phys.Rev.D 80 (2009) 043516; e-Print: 0904.2835 [hep-th][8] S. Nojiri, S. D. Odintsov, Phys.Rept. 505 (2011) 59-144; e-Print: 1011.0544 [gr-qc][9] T. Clifton, P. G. Ferreira, A. Padilla, C. Skordis, Phys.Rept. 513 (2012) 1-189; e-Print: 1106.2476 [astro-ph.CO][10] I. Antoniadis, P. O. Mazur and E. Mottola, JCAP (2012), 024; e-Print: 1103.4164[gr-qc].[11] A. Frenkel, P. Horava, S. Randall, Topological Quantum Gravity of the Ricci Flow;e-Print: 2010.15369 [hep-th][12] J. Morgan and G. Tian, Ricci flow and the Poincar´e conjecture , (AMS, 2007).[13] B. Chow, P. Lu and L. Ni,
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