PHYSICAL REVIEW D 91, 064003 (2015) Remnant symmetry, propagation, and evolution in fðTÞ gravity Pisin Chen* Leung Center for Cosmology and Particle Astrophysics and Graduate Institute of Astrophysics and Department of Physics, National Taiwan University, Taipei 10617, Taiwan and Kavli Institute for Particle Astrophysics and Cosmology, SLAC National Accelerator Laboratory, Stanford University, California 94305, USA † Keisuke Izumi Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan ‡ James M. Nester ,§ Department of Physics and Graduate Institute of Astronomy and Center for Mathematics and Theoretical Physics, National Central University, Chungli, 320, Taiwan and 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 (Received 7 January 2015; published 2 March 2015) It was recently argued that fðTÞ gravity could inherit “remnant symmetry” from the full Lorentz group, despite the fact that the theory is not locally Lorentz invariant. Confusion has arisen regarding the implication of this result for the previous works, which established that fðTÞ gravity is pathological due to superluminal propagation, local acausality, and nonunique time evolution. We clarify that the existence of the “remnant group” does not rid the theory of these various problems but, instead, strongly supports it. DOI: 10.1103/PhysRevD.91.064003 PACS numbers: 04.50.Kd, 95.30.Sf I. INTRODUCTION: fðTÞ GRAVITY AND The dynamical variable of TEGR as well as its fðTÞ REMNANT SYMMETRY extension [10] is the frame field (vierbein) feaðxÞg or, a General relativity (GR) is a geometric theory of gravity equivalently, its corresponding coframe field fe ðxÞg. The ∂ formulated on a Lorentzian manifold equipped with the vierbein is related to the coordinate vector fields f μg by μ ∂ a a μ Levi-Civita connection. This connection is torsionless and eaðxÞ¼ea ðxÞ μ and, similarly, e ðxÞ¼e μðxÞdx . The metric compatible; the gravitational field is completely vierbein eaðxÞ forms an orthonormal basis for the tangent described in terms of the Riemann curvature tensor. space TxM at each point x of a given spacetime manifold However, given a smooth manifold, one can equip it with ðM; gÞ. The metric tensor g is related to the vierbein field by other connections. If one chooses to use the Weitzenböck a b connection, then the geometry is flat—the connection being gμνðxÞ¼ηabe μðxÞe νðxÞ: ð1Þ curvature free (but still metric compatible). The gravita- tional field is now completely described in terms of the The Weitzenböck connection is defined by torsion tensor. Surprisingly, GR can be recast into the “teleparallel equivalent of GR” (TEGR or GR∥), which w ∇ ≔ a employs the Weitzenböck connection, a subject which has a XY ðXY Þea; ð2Þ large set of literature (see, e.g., [1–9]). a where Y ¼ Y ea. This means that we declare the vierbein w field to be teleparallel, i.e., covariantly constant: ∇ e ¼ 0. *[email protected] X a † [email protected] Equivalently, the connection coefficients are ‡ [email protected] § λ λ a Present address: Morningside Center of Mathematics, Acad- Γ νμ ¼ e a∂μe ν: ð3Þ emy of Mathematics and System Science, Chinese Academy of Sciences, 55 Zhongguancun Donglu, Haidian District, Beijing 100190, People’s Republic of China. It is then straightforward to show that this connection is ¶[email protected] curvatureless, but the torsion tensor is nonzero in general. 1550-7998=2015=91(6)=064003(6) 064003-1 © 2015 American Physical Society CHEN et al. PHYSICAL REVIEW D 91, 064003 (2015) Under a linear transformation of the bases feaðxÞg of the this could potentially explain the acceleration of the tangent vector field Universe [10]. It is well known that unlike TEGR, generalized theories 0 b b f T eaðxÞ → eaðxÞ¼La ðxÞebðxÞ; detðLa Þ ≠ 0; ð4Þ such as ð Þ are not locally Lorentz invariant [11].Of course, at a purely mathematical level, a given manifold the connection 1-form that can be parallelized admits infinitely many choices of vierbein. However, in a general teleparallel theory, such as w f T gravity, there exists a preferred frame compatible Γb θb ∇ Γb μ ð Þ aðxÞ¼h ; eai¼ μadx ð5Þ with the field equations. (This is the difference between kinematics and dynamics; the latter is determined by a transforms as Lagrangian.) While TEGR has local Lorentz symmetry and like GR has just 2 dynamical degrees of freedom, for all Γ0a −1 a Γd c −1 a c μb ¼ðL Þ d μcL b þðL Þ cL b;μ; ð6Þ nontrivial f’s it was thought that fðTÞ theory would be a preferred frame theory with 5 degrees of freedom and no where the comma in the subscript denotes the usual partial local Lorentz symmetry. The main message in a recent differentiation. work by Ferraro and Fiorini [12] is that this common belief We would like to emphasize that there is, in fact, a “ ” “ ” is, in fact, much more subtle. They argued that, depending difference between parallelizable and teleparallel. A on the spacetime manifold, fðTÞ gravity may “inherit” parallelizable manifold M means that there exists a global some “remnant symmetry” from the full (orthochronous) frame field on M (that is, the frame bundle FM has a global 3 2 Lorentz group, and, therefore, there could exist more than section). For example, S is parallelizable but S is not. one such preferred frame—even infinitely many. Whether a manifold is parallelizable or not depends on the More precisely, Ferraro and Fiorini discovered a remark- topology but not on the connection. In four dimensions, able yet simple result that fðTÞ gravity is only invariant the necessary and sufficient condition for parallelizability under Lorentz transformations of the vierbein satisfying is the vanishing of the second Stiefel-Whitney character- istic class. Teleparallel geometry means one has a con- ϵ a ∧ b ∧ ηde c −1 f μ 0 dð abcde e L fðL Þ e;μdx Þ¼ : ð10Þ nection which is flat everywhere, i.e., has vanishing curvature. (A manifold with a teleparallel connection is In other words, while this is fulfilled by a global Lorentz always parallelizable.) transformations, there are local Lorentz transformations One then defines the contortion tensor, which is that could also satisfy Eq. (10). The set of those local the difference between the Weitzenböck and Levi-Civita Lorentz transformations that satisfy Eq. (10), given a frame connections. In component form, it reads a eaðxÞ (or, equivalently, the coframe e ðxÞ) that solves the field equations of f T gravity, is denoted by A ea 1 ð Þ ð Þ μν μν νμ μν “ ” K ρ ¼ − ðT ρ − T ρ − Tρ Þ: ð7Þ and dubbed the remnant group (which can be, but is 2 not necessarily, a group). We will refer to the additional symmetry embodied in the remnant group (in addition For convenience, one usually also defines the tensor to the global Lorentz transformation) as the remnant 1 symmetry. μν μν δμ αν − δν αμ Sρ ¼ 2 ðK ρ þ ρT α ρT αÞ: ð8Þ Ferraro and Fiorini then asserted that the existence of the remnant group seems to be not consistent with1 the results In TEGR, the teleparallel Lagrangian consists of the so- obtained by us in [13,14], in which we showed that fðTÞ — called “torsion scalar” gravity is generically problematic it allows superluminal propagation and local acausality (that is, temporal ordering μν ρ T ≔ Sρ T μν: ð9Þ is not well defined even in an infinitesimal neighborhood). The theory also suffers from nonunique time evolution; i.e., It turns out that the torsion scalar only differs from the Ricci the Cauchy problem is ill-defined. That is, given a full set scalar (obtained from the usual Levi-Civita connection) by of the Cauchy data, one cannot predict what will happen in a boundary term: T ¼ −R þ divð·Þ, and so it encodes all the the future with certainty. Note that all these problems arise dynamics of GR. One could then promote T to a function at the classical level. Ferraro and Fiorini did not explain the fðTÞ, similar to how GR is generalized to fðRÞ gravity. For reason why they think their results contradict ours. a general f, this would lead to a dynamical gravity theory In this work, we wish to clarify that the existence of the that would have second order field equations (whereas remnant group does not, in fact, contradict our previous fðRÞ gravity gives higher order equations), with some kind works but, instead, strongly supports it. of nonlinear dynamics that differs from GR but, never- theless, reduces to GR in a certain limit. The hope was that 1Indeed, they used a stronger expression, “seems to discredit.” 064003-2 REMNANT SYMMETRY, PROPAGATION, AND EVOLUTION … PHYSICAL REVIEW D 91, 064003 (2015) II. COMMENTS ON PROPAGATION AND where # denotes “the number of.” This gives for the EVOLUTION IN fðTÞ GRAVITY possibilities (i), (ii), and (iii), respectively, (i) ½2ð16Þ − 2ð8Þ − 6=2 ¼ 5, A good way to understand the number of dynamical (ii) ½2ð16Þ − 2ð10Þ − 4=2 ¼ 4, degrees of freedom is from the Hamiltonian perspective. (iii) ½2ð16Þ − 2ð14Þ − 0=2 ¼ 2. Following Dirac’s procedure, we find primary constraints, The possibility (iii) corresponds to TEGR, while (i) is the introduce them into the Hamiltonian with Lagrange multi- generic case for fðTÞ gravity, as confirmed by the detailed pliers, determine the multipliers if possible, and find any work of Li et al. [17], which was based on the Hamiltonian additional secondary constraints. The constraints are di- analysis of [18]. In other words, fðTÞ gravity generically vided into two classes: first class are associated with gauge propagates 5 degrees of freedom; i.e., there are 3 additional freedom, and second class are related to nondynamical degrees of freedom compared to standard GR.