Remnant Symmetry, Propagation and Evolution in f(T) Gravity
aa r X i v : . [ g r- q c ] M a r Remnant Symmetry, Propagation and Evolution in f ( T ) Gravity
Pisin Chen ∗ (1) Leung Center for Cosmology and Particle Astrophysics& Graduate Institute of Astrophysics & Department of Physics,National Taiwan University, Taipei 10617, Taiwan(2) Kavli Institute for Particle Astrophysics and Cosmology,SLAC National Accelerator Laboratory, Stanford University, CA 94305, U.S.A Keisuke Izumi † Leung Center for Cosmology and Particle Astrophysics,National Taiwan University, Taipei 10617, Taiwan
James M. Nester ‡ (1) Department of Physics & Graduate Institute of Astronomy& Center for Mathematics and Theoretical Physics,National Central University, Chungli, 320, Taiwan and(2) Leung Center for Cosmology and Particle Astrophysics,National Taiwan University, Taipei 10617, Taiwan § Yen Chin Ong ¶ Nordita, KTH Royal Institute of Technology and Stockholm University,Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
It was recently argued that f ( T ) gravity could inherit “remnant symmetry” from the full Lorentzgroup, despite the fact that the theory is not locally Lorentz invariant. Confusion has arisen re-garding the implication of this result for the previous works, which established that f ( T ) gravity ispathological due to superluminal propagation, local acausality, and non-unique time evolution. Weclarify that the existence of the “remnant group” does not rid the theory of these various problems,but instead strongly supports it. I. INTRODUCTION: f ( T ) GRAVITY ANDREMNANT SYMMETRY
General Relativity [GR] is a geometric theory of grav-ity formulated on a Lorentzian manifold equipped withthe Levi-Civita connection. This connection is torsion-less and metric compatible – the gravitational field iscompletely described in terms of the Riemann curva-ture tensor. However, given a smooth manifold one canequip it with other connections. If one chooses to use theWeitzenb¨ock connection, then the geometry is flat – theconnection being curvature-free [but still metric compat-ible]. The gravitational field is now completely describedin terms of the torsion tensor. Surprisingly GR can berecast into “teleparallel equivalent of GR” [TEGR, orGR k ] which employs the Weitzenb¨ock connection, a sub-ject which has a large set of literature [see, e.g., [1–9]].The dynamical variable of TEGR, as well as its f ( T ) ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Currently visiting the Morningside Center of Mathematics,Academy of Mathematics and System Science, Chinese Academyof Sciences, 55 Zhongguancun Donglu, Haidian District, Beijing100190, China. ¶ Electronic address: [email protected] extension [10], is the frame field [vierbein] { e a ( x ) } , orequivalently its corresponding co-frame field { e a ( x ) } .The vierbein is related to the coordinate vector fields { ∂ µ } by e a ( x ) = e µa ( x ) ∂ µ , and similarly e a ( x ) = e aµ ( x ) dx µ . The vierbein e a ( x ) forms an orthonormal ba-sis for the tangent space T x M at each point x of a givenspacetime manifold ( M, g ). The metric tensor g is relatedto the vierbein field by g µν ( x ) = η ab e aµ ( x ) e bν ( x ) . (1)The Weitzenb¨ock connection is defined by w ∇ X Y := ( XY a ) e a , (2)where Y = Y a e a . This means that we declare the vier-bein field to be teleparallel , i.e., covariantly constant: w ∇ X e a = 0. Equivalently, the connection coefficients areΓ λνµ = e λa ∂ µ e aν . (3)It is then straightforward to show that this connection iscurvature-less but the torsion tensor is nonzero in gen-eral.Under a linear transformation of the bases { e a ( x ) } ofthe tangent vector field e a ( x ) → e ′ a ( x ) = L ba ( x ) e b ( x ) , det( L ba ) = 0 , (4)the connection 1-formΓ ba ( x ) = h θ b , w ∇ e a i = Γ bµa dx µ , (5)transforms asΓ ′ aµb = ( L − ) ad Γ dµc L cb + ( L − ) ac L cb,µ , (6)where the comma in the subscript denotes the usual par-tial differentiation.We would like to emphasize that there is in fact a dif-ference between “parallelizable” and “teleparallel”. Aparallelizable manifold M means that there exists aglobal frame field on M [that is, the frame bundle F M has a global section]. For example, S is parallelizablebut S is not. Whether a manifold is parallelizable ornot depends on the topology but not on the connec-tion. In 4-dimensions, the necessary and sufficient con-dition for parallelizability is the vanishing of the secondStiefel-Whitney characteristic class. Teleparallel geome-try means one has a connection which is flat everywhere,i.e., has vanishing curvature. [A manifold with a telepar-allel connection is always parallelizable.]One then defines the contortion tensor, which is thedifference between the Weitzenb¨ock and Levi-Civita con-nections. In component form, it reads K µνρ = − (cid:16) T µνρ − T νµρ − T µνρ (cid:17) . (7)For convenience, one usually also defines the tensor S µνρ = 12 (cid:16) K µνρ + δ µρ T ανα − δ νρ T αµα (cid:17) . (8)In TEGR, the Teleparallel Lagrangian consists of theso-called “torsion scalar” T := S µνρ T ρµν . (9)It turns out that the “torsion scalar” only differs from theRicci scalar [obtained from the usual Levi-Civita connec-tion] by a boundary term: T = − R + div( · ), and so itencodes all the dynamics of GR. One could then pro-mote T to a function f ( T ), similar to how GR is gener-alized to f ( R ) gravity. For a general f , this would leadto a dynamical gravity theory that would have secondorder field equations [whereas f ( R ) gravity gives higherorder equations], with some kind of non-linear dynamicsthat differs from GR but nevertheless reduces to GR ina certain limit. The hope was that this could potentiallyexplain the acceleration of the universe [10].It is well-known that, unlike TEGR, generalized theo-ries such as f ( T ) are not locally Lorentz invariant [11].Of course, at a purely mathematical level, a given man-ifold that can be parallelized admits infinitely manychoices of vierbein. However, in a general teleparallel the-ory, such as f ( T ) gravity, there exists a preferred frame compatible with the field equations. [This is the dif-ference between kinematics and dynamics ; the latter isdetermined by a Lagrangian.] While TEGR has local Lorentz symmetry, and like GR has just 2 dynamical de-grees of freedom, for all non-trivial f ’s it was thoughtthat f ( T ) theory would be a preferred frame theory with5 degrees of freedom and no local Lorentz symmetry. Themain message in a recent work by Ferraro and Fiorini [12]is that this common belief is in fact much more subtle.They argued that, depending on the spacetime manifold, f ( T ) gravity may “inherit” some “remnant symmetry”from the full [orthochronous] Lorentz group, and there-fore there could exist more than one such preferred frame– even infinitely many.More precisely, Ferraro and Fiorini discovered a re-markable yet simple result that f ( T ) gravity is only in-variant under Lorentz transformations of the vierbeinsatisfying d ( ǫ abcd e a ∧ e b ∧ η de L cf ( L − ) fe,µ dx µ ) = 0 . (10)In other words, while this is fulfilled by a global Lorentztransformations, there are local Lorentz transformationsthat could also satisfy Eq.(10). The set of those lo-cal Lorentz transformations that satisfy Eq.(10), givena frame e a ( x ) [or equivalently the co-frame e a ( x )] thatsolves the field equations of f ( T ) gravity, is denoted by A ( e a ), and dubbed the “remnant group” [which can be,but is not necessarily, a group]. We will refer to the addi-tional symmetry embodied in the remnant group [in ad-dition to the global Lorentz transformation] as the “rem-nant symmetry”.Ferraro and Fiorini then asserted that the existence ofthe remnant group seems to be not consistent with theresults obtained by us in [13] and [14], in which we showedthat f ( T ) gravity is generically problematic – it allows su-perluminal propagation and local acausality [that is, tem-poral ordering is not well-defined even in an infinitesimalneighborhood]. The theory also suffers from non-uniquetime evolution, i.e., Cauchy problem is ill-defined. Thatis, given a full set of the Cauchy data, one cannot predictwhat will happen in the future with certainty. Note thatall these problems arise at the classical level. Ferraro andFiorini did not explain the reason they think their resultscontradict ours.In this work we wish to clarify that the existence of theremnant group does not, in fact, contradict our previousworks, but instead strongly supports it. II. COMMENTS ON PROPAGATION ANDEVOLUTION IN f ( T ) GRAVITY
A good way to understand the number of dynamicaldegrees of freedom is from the Hamiltonian perspective.Following Dirac’s procedure we find primary constraints,introduce them into the Hamiltonian with Lagrange mul-tipliers, determine the multipliers if possible and find Indeed, they used a stronger expression “seems to discredit”. any additional secondary constraints. The constraintsare divided into two classes: first class are associatedwith gauge freedom, and second class related to non-dynamical variables. Here we are concerned with telepar-allel theories. The primary dynamic variable is the or-thonormal frame e aµ . Its conjugate momentum is P aµ .For the Hamiltonian analysis of teleparallel theories, seee.g., [6, 7, 15, 16].The Lagrangian never contains the time derivative of e a , consequently we always have the primary constraints P a . Preservation of these constraints lead to a set of 4secondary constraints referred to as the Hamiltonian andmomentum constraints. These 8 constraints are all firstclass, geometrically they generate spacetime diffeomor-phisms. We need not consider them any further.For the special subclass of theories of the form f ( T )there are 6 more primary constraints, P [ µν ] ≃ · − · − / , (11)where − − / − − / − − / f ( T ) gravity, as confirmed by the de-tailed work of Miao Li et al. [17], which was based on the Hamiltonian analysis of [18]. In other words, f ( T ) grav-ity generically propagates 5 degrees of freedom, i.e., thereare 3 additional degrees of freedom compared to standardGR. These extra degrees of freedom are very non-linearin nature. For example, they do not show up in the lin-ear perturbation of flat Friedmann-Lemaˆıtre-Robertson-Walker [FLRW] cosmological background [19].What about the option (ii) then? Case (ii) is “exotic”;it is some kind of geometry that is intermediate betweena metric and preferred frame theory. Our Hamiltoniananalysis identifies this possibility, which as we shall seebelow, is precisely one of the types that was found byFerraro and Fiorini . This is the reason why the word“generically” is crucial, for in [13] we found out that thenumber of physical degrees of freedom and the classes ofDirac constraints, can and do change depending on thevalues of the fields. That is, they are expected to be differ-ent on different background geometries . This is in fact, in agreement with the findings of Ferraro and Fiorini [12]that different geometries give rise to a different numberof “admissible frames”. We will further elaborate on thislater.However, precisely because of the possibility that fieldconfigurations can change the number of degrees of free-dom as well as the constraint structure of the theory,we expect anomalous propagation such as superluminalshock waves to arise. This is explained in detail alreadyin [13] in which we employed the well-understood PDEmethod of characteristics, pioneered by Cauchy and Ko-valevskaya. [See also Section 2 of [20] and the referencestherein for further explanation regarding this method].One has to be careful in distinguishing between thesymmetry of a theory and the symmetry of a particularsolution . For example, consider a complex scalar field ϕ with a simple potential: V = V ( ϕϕ ∗ ) + (1 − ϕϕ ∗ ) V ( ϕ ) . (12)Clearly only the V term in the potential has local U(1)symmetry because it is a function of ϕϕ ∗ , i.e., the [squareof the] absolute value of ϕ . On the other hand, V termdoes not because it depends on the explicit ϕ configu-ration. However, if the absolute value of ϕ is unity, i.e.if ϕϕ ∗ = 1, then the value of the potential V is invari-ant under U(1) transformation. Thus, for some specificfield configurations V has U(1) symmetry, but this is notthe symmetry of the theory for generic values of ϕ ! Thespecific values that “restores” U(1) symmetry to V hasphysical effect, it is the signal that the mode related toU(1) direction has become massless. Similarly in f ( T )gravity one expects that if the fields evolve such that Note that the determinant of the Poisson bracket matrix is apolynomial in the variables and their derivatives. Generically ithas real roots. The rank cannot be constant if it admits genericsolutions [which include, but is not limited to, Minkowski andFLRW]. some extra symmetries emerge, there would be physicaleffects that accompany the changes in the number andtype of constraints [much of the effort in [13] and [14] isspent on showing that the superluminal propagations arenot simply due to gauge choice]. To be more specific, onecan consider a kinetic term for ϕ of the form:(1 − ϕϕ ∗ ) ∂ µ ϕ∂ µ ϕ ∗ . (13)This kinetic term does not have local U (1) symmetrybecause the derivative is not covariant. Generically thisterm is well-defined, but when the field configurations ap-proach the value such that ϕϕ ∗ = 1, the dynamical termvanishes. Under the local U (1) transformation, ϕϕ ∗ = 1still holds and the dynamical term remains naught. Thisis indeed a “remnant symmetry”. However, the differen-tial equations [the equations of motion] then behave verydifferently depending of the values of the fields. This isthe situation faced by f ( T ) gravity.In fact, the results of [12] actually support our results.As we recall, generically f ( T ) gravity has 5 degrees offreedom. However, for almost any f it is very likely thatthere would exist solutions where the Poisson bracket ma-trix has less than the generic rank. For any such solutionone or more of the generically second class constraintswill now be first class. These first class constraints willgenerate some local Lorentz transformations. Thus whatwe expect to see is that for each solution there is somesubgroup of the Lorentz group which acts as a local sym-metry gauge group. In other words, our analysis involv-ing rank changes already implicitly implied the same re-sult that Ferraro and Fiorini now discovered, using a dif-ferent, more straightforward analysis.The point is that the set A ( e a ) [which is generally not agroup] encodes all information about the change in rankof the Poisson bracket matrix – the size of this “group”[at each spacetime point] reflects the number of normallysecond class constraints, which have become first classfor a particular frame, i.e., the change in the rank of thePoisson bracket matrix. For a given function f ( T ), if A ( e a ) is empty for every solution to the equations, thenthe propagation has no problems. For a given f , and forall solution frames, if this set is an Abelian group of size independent of spacetime point , then we may have goodpropagating modes. However, if this set is an Abeliangroup with size varying from point to point, then we haveacausal propagation – which as we argued in [13], ariseas a consequences of the change in the number of degreesof freedom and the constraint structure of the theory.Most crucially, it should be noted that in [14], we con-structed an explicit example in which f ( T ) gravity and itsBrans-Dicke-generalization with scalar field suffers fromnon-unique evolution – starting with a perfectly homoge-neous and isotropic, flat FLRW universe, anisotropy cansuddenly emerge. By non-unique evolution, we do not simply mean the following: Given a spacetime geometry,a chosen tetrad field can evolve into another choice oftetrad, which corresponds to the same metric tensor [ifthere are multitudes of “admissible frames” this would not be a problem notwithstanding the discussion above, if physical observables only couple to the metric ]. In-stead we mean a stronger statement: even a geometry ,described by the metric tensor, can change drastically un-der the evolution, and such change cannot be predictedfrom initial data alone. This problem cannot be evadedeven if there are more than one “admissible frames” cor-responding to a fixed geometry. III. DISCUSSION
In this work we discuss why the existence of remnantsymmetry as shown by Ferraro and Fiorini does not con-tradict our previous works, which established the exis-tence of superluminal propagation, local acausality, andnon-unique time evolution in f ( T ) gravity [and its Brans-Dicke generalization]. In fact, these problems are closelyrelated to the remnant symmetry – while our Hamil-tonian analysis agrees with the results of Ferraro andFiorini, the same analysis also shows that there are se-rious dynamical difficulties, especially if one approachesa point where the rank of the Poisson bracket matrixchanges.We now conclude this work with some additional com-ments.In relation to the Hamiltonian analysis, the usual un-derstanding is that generically teleparallel Lagrangianshave no local frame gauge freedom. Beyond the relationsthat follow from diffeomorphism invariance [which areconnected with energy momentum] the equations satisfyno other differential identities. According to Noether’sSecond Theorem, a local gauge freedom means a differen-tial identity. In the Hamiltonian formulation of telepar-allel theories, generically the only first class constraintsare the Hamiltonian and momentum constraints. If thereare no other first class constraints, then there is no otherlocal gauge symmetries. It would be very interesting tosee how “remnant symmetries” fit into this scheme moreexplicitly.We now remark on the comment in [12] on the possibil-ity of constructing local inertial frames in f ( T ) gravity,and the hope that Zeeman’s theorem on R , [21] wouldthus ensure local causality. In the context of Zeeman’stheorem, the invariance group G of the Minkowski space-time [the orthochronous Lorentz group, the translationgroup and the dilatation group] induces the light conestructure of R , . This structure provides some causal-ity relation C which allows the definition of causalitygroup G c . In 4 dimensions, it turns out that G = G c .Indeed one could generalize the notion of Riemann nor-mal coordinates to other geometries defined by differentconnections, in particular the Weitzenb¨ock connection.This was accomplished in [22]. However, propagationis not defined at a point; it involves dynamical modesmoving from a spacetime point x to another [althoughthis distance could be in a ε -neighborhood of x ], andthis is problematic in f ( T ) gravity if the “size” of A ( e a )[and thus the remnant symmetry] changes from point topoint. Furthermore, even local causality is certainly notavoided. To see this, one should consider also the equa-tion that governs the propagation, derived in [13]: h f T M µνa αβb + 2 f T T S µνa S αβb i k µ k α ¯ e bβ = 0; (14)in which M µνa αβb := ∂S µνa ∂T bαβ . (15)Here the notation ¯ e bβ represents the change of the framein a certain direction, instead of the value of the frame. k µ denotes the normal to a characteristic hypersurfacewhich in this case could be timelike [signaling a tachyonicpropagation]. Here, the first term in the square bracketis the healthy propagation; it is the only term presents inthe case of TEGR, in which case f ( T ) = T ⇒ f T T ≡ local in nature. Thuswe see that even locally the characteristic cones in ageneric f ( T ) theory is not the same as TEGR, in par-ticular the cones depend also on the second term [in thesquare bracket], which generically differs from one space-time point to another.Lastly we would like to make a side remark that in [12],two classic works by Hayashi and Shirafuji were men-tioned [23, 24]. In particular, it was pointed out that [23]considered “restricted local invariance of this sort” in thecontext of “New General Relativity” [NGR] proposed in[24]. Indeed Hayashi and Shirafuji considered the possi-bility of a new type of geometry somewhere in betweenteleparallel and Riemannian one. It would have what wehave here called “remnant symmetry”, a preferred framedetermined up to a certain dynamically determined sub-group of the Lorentz group. It is worth mentioning that there has since been some criticisms by Kopczy´nski andfurther developments that came out of that [15, 25–27].[The work by Chen, Nester and Yo [27] was the firstto call attention to the effect of non-linear constraintsand the relationship of changes in the rank of the Pois-son matrix with tachyonic characteristics]. Kopczy´nski’sobjection was in regard to one of the complications thatoccurs when one has some “local symmetry” for the grav-itational Lagrangian for a certain subclass of solutions.Briefly, if the LHS of the field equation has a local symme-try, then the RHS, i.e., the material energy momentumtensor must have the same symmetry. This could im-pose “unphysical” limitations on the matter sector. Forthe NGR theory, it was found that spin-1/2 Dirac fielddoes not give rise to any problem, but for a hypotheticalspin-3/2 field there is inconsistency [unless if one sub-scribes to non-minimal coupling to save the theory]. For f ( T ) gravity this may not be a serious problem, as onecould simply take the source energy-momentum tensorto be completely locally Lorentz invariant, as it is in GR.However this would mean that there is no material sourcefor the extra degrees of freedom.Despite the pathologies of f ( T ) gravity, the existenceof remnant symmetry is indeed interesting and may pro-vide further insights into the structure of teleparallelgravities, and perhaps modified gravity theories in gen-eral. Acknowledgments
The authors would like to thank Huan-Hsin Tseng fordiscussion. K.I. is supported by Taiwan National ScienceCouncil under Project No. NSC101-2811-M-002-103. [1] C. Moller, “Further Remarks on the Localization of theEnergy in the General Theory of Relativity”, AnnalsPhys. (1961) 118.[2] Y. M. Cho, “Einstein Lagrangian as the TranslationalYang-Mills Lagrangian”, Phys. Rev. D (1976) 2521.[3] F. W. Hehl, J. D. McCrea, E. W. Mielke, Y. Ne’eman,“Metric Affine Gauge Theory of Gravity: Field Equa-tions, Noether Identities, World Spinors, and Break-ing of Dilation Invariance”, Phys. Rept. (1995) 1,[gr-qc/9402012].[4] Y. Itin, “Coframe Teleparallel Models of Gravity: Ex-act Solutions”, Int. J. Mod. Phys. D (2001) 547,[gr-qc/9912013].[5] V. C. de Andrade, L. C. T. Guillen, J. G. Pereira, “Gravi-tational Energy Momentum Density in Teleparallel Grav-ity”, Phys. Rev. Lett. (2000) 4533, [gr-qc/0003100].[6] M. Blagojevi´c, M. Vasili´c, “Gauge Symmetries of theTeleparallel Theory of Gravity”, Class. Quant. Grav (2000) 3785, [arXiv:hep-th/0006080]. [7] M. Blagojevi´c, Gravitation and Gauge Symmetries , IoPPublishing, Bristol, 2002.[8] F. W. Hehl, “Gauge Theory of Gravity and Spacetime”,[1204.3672 [gr-qc]].[9] R. Aldrovandi, J. G. Pereira,
Teleparallel Gravity: AnIntroduction , Springer, Dordrecht, 2013.[10] E. V. Linder, “Einstein’s Other Gravity and the Acceler-ation of the Universe”, Phys. Rev. D (2010) 127301,[arXiv:1005.3039 [astro-ph.CO]].[11] B-J. Li, T. P. Sotiriou, J. D. Barrow, “f(T) Gravityand Local Lorentz Invariance”, Phys. Rev. D (2011)064035, [arXiv:1010.1041 [gr-qc]].[12] R. Ferraro, F. Fiorini, “The Remnant Group ofLocal Lorentz Transformations in f(T) Theories”,[arXiv:1412.3424 [gr-qc]].[13] Y. C. Ong, K. Izumi, J. M. Nester, P. Chen,“Problems with Propagation and Time Evolution inf(T) Gravity”, Phys. Rev. D (2013) 024019,[arXiv:1303.0993 [gr-qc]]. [14] K. Izumi, J-A. Gu, Y. C. Ong, “Acausality andNonunique Evolution in Generalized Teleparal-lel Gravity”, Phys. Rev. D (2014) 084025,[arXiv:1309.6461 [gr-qc]].[15] W-H. Cheng, D-C. Chern, J. M. Nester, “CanonicalAnalysis of the One-Parameter Teleparallel Theory”,Phys. Rev. D (1988) 2656.[16] J. M. Nester, “Positive Energy via the TeleparallelHamiltonian”, Int. J. Mod. Phys. A (1989) 1755.[17] M. Li, R-X. Miao, Y-G. Miao, “Degrees of Free-dom of f ( T ) Gravity”, JHEP (2011) 108,[arXiv:1105.5934 [hep-th]].[18] J. W. Maluf , J. F. da Rocha-Neto, “HamiltonianFormulation of General Relativity in the Telepar-allel Geometry”, Phys. Rev D (2001) 084014,[arXiv: gr-qc/0002059].[19] K. Izumi, Y. C. Ong, “Cosmological Perturbationin f(T) Gravity Revisited”, JCAP (2013) 029,[arXiv:1212.5774 [gr-qc]].[20] S. Deser, K. Izumi, Y. C. Ong, A. Waldron, “Problems of Massive Gravities”, Mod. Phys. Lett. A 30 (2015)1540006, [arXiv:1410.2289 [hep-th]].[21] E. C. Zeeman, “Causality Implies the Lorentz Group”,J. Math. Phys. (1964) 490.[22] J. M. Nester, “Normal Frames for General Connections”,Ann. Phys. (Berlin) (2010) 45.[23] K. Hayashi, T. Shirafuji, “Addendum to ‘New GeneralRelativity’ ”, Phys. Rev. D (1982) 3312.[24] K. Hayashi, T. Shirafuji, “New General Relativity”,Phys. Rev. D (1979) 3524.[25] W. Kopczy´nski, “Problems with Metric-Teleparallel The-ories of Gravitation”, J. Phys. A: Math. Gen. (1982)493.[26] J. M. Nester, “Is There Really a Problem with theTeleparallel Theory?”, Class. Quant. Grav. (1988)1003.[27] H. Chen, J. M. Nester, H.J. Yo, “Acausal PGT Modesand the Nonlinear Constraint Effect”, Acta Phys. Polon.B29