Symmetries from Locality. III. Massless Spin-2 Gravitons and Time Translations

PHYSICAL REVIEW D 102, 085007 (2020)

Symmetries from locality. III. Massless spin-2 gravitons and time translations

† Mark P. Hertzberg* and Jacob A. Litterer Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA

(Received 15 August 2020; accepted 27 September 2020; published 13 October 2020)

We relax the assumption of time translation and Lorentz boost symmetry in theories involving massless spin-2 gravitons, while maintaining a basic notion of locality that there is no instantaneous signaling at a distance. We project out longitudinal modes, leaving only 2 degrees of freedom of the graviton. Our previous work, which assumed time translation symmetry, found that the Lorentz boost symmetry is required to ensure locality at leading order. In this work, without assuming time translations or Lorentz boosts, we show that locality of the exchange action between matter sources demands that massless spin-2, at leading order, organizes into Einstein-Hilbert plus a Gauss-Bonnet term with a prefactor that is constrained to be a particular function of time; while in the matter sector we recover time translation and Lorentz boost symmetry. Finally, we comment on whether the time dependence of the Gauss-Bonnet prefactor may be forbidden by going to higher order in the analysis and we mention that other possibilities are anticipated if graviton mass terms are included.

DOI: 10.1103/PhysRevD.102.085007

I. INTRODUCTION largest of scales; so we shall not pursue this possibility here. In the second option (ii) one may (a) add a mass to the The modern picture of cosmology is remarkably suc- graviton (which takes the number of degrees of freedom cessful. However, it possesses a range of curious features; from two helicities to five polarizations). However, this this includes the need for dark matter, dark energy, coinci- possesses various strong coupling problems and no known dences, fine-tuning problems, and, more recently, possible UV completion (arguably due to problems with causality), tension with different measurements of the Hubble param- so we shall also ignore this possibility here. Or one may eter [1]. It has sometimes been suggested that one or more of (b) add other types of particles, especially spin-0 particles, these issues may be addressed by modifying or deforming which can potentially mediate long range forces if they are the underlying rules of gravitation, namely general relativity. extremely light. Such interactions are tightly constrained by But any theoretically sensible deformation of general solar system tests and are unlikely to solve the above relativity is difficult to come by. This is because there is a conceptual problems anyhow (one can just add scalars as theorem [2–4]: the unique local, unitary, Poincaré invari- part of the matter sector to act as dark matter of course); so ant theory of a species of massless spin-2 particles is, at we shall not pursue this possibility here either. large distances, general relativity. (Multiple gravitons with This leaves us with investigating option (iii), which has higher order interactions can lead to inconsistencies [5–7].) been investigated in the literature [8–20], and is the focus of Hence to escape the conclusion that we necessarily have this paper. In our previous papers [21–23] we considered general relativity in a local and unitary way, one must do deforming Lorentz boosts, while maintaining all other parts one (or more) of the following: (i) consider small distances, of the Poincaré symmetry, namely rotation symmetry, (ii) add extra degrees of freedom, (iii) deform Poincaré spatial translation symmetry, and time translation symmetry. symmetry. The first option (i) is not interesting to resolve In [22] we considered spin-1 particles (electromagnetism) puzzles in cosmology, which involve phenomena on the and in [23] we considered spin-2 particles (gravitation). By imposing a very basic notion of locality, that there is no *[email protected] instantaneous signaling, we showed that these particles must † [email protected] couple to conserved sources, associated with a symmetry. In the case of spin-1 we showed it must couple to a conserved Published by the American Physical Society under the terms of current, which requires the familiar Uð1Þ symmetry, and for the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to spin-2 we showed it must couple to a conserved 2-index the author(s) and the published article’s title, journal citation, symmetric current (energy-momentum tensor), which and DOI. Funded by SCOAP3. requires Lorentz boost symmetry. Since the latter recovers

2470-0010=2020=102(8)=085007(12) 085007-1 Published by the American Physical Society MARK P. HERTZBERG and JACOB A. LITTERER PHYS. REV. D 102, 085007 (2020)

Lorentz boosts and in turn the full Poincaré symmetry polarizations of the graviton. In each case we impose involving spin-2, then it is anticipated to recover the full locality, defined here as avoiding instantaneous signaling, structure of general relativity just from locality of spin-2. and find the same resulting theory. We believe this resulting For potential cosmological applications it is important to theory is valid regardless of the particular choice of how to move beyond the above assumptions. Most importantly, cut down the degrees of freedom. We find that we recover cosmology involves an expanding universe, i.e., a back- general relativity (GR) to the leading order we are working, ground space-time that breaks time translation invariance. plus an additional set of terms that involve a very specific While in the context of general relativity this is associated form of time translation symmetry violation. We then with spontaneous breakdown of time translations, there comment on what is anticipated to occur when working remains several mysteries as mentioned earlier. Therefore it to higher order. motivates exploring setups in which time translation symmetry is explicitly broken. This work is a first step II. THEORIES OF MASSLESS SPIN-2 in this direction. Our particular setup is as follows: we will assume, again, a preferred frame with rotation invariance A. Recap of Poincaré invariant massless spin-2 (which allows for a notion of particle spin) and translation Our goal in this work is to study and generalize theories invariance in space, but we will not assume Lorentz boosts of massless spin-2 particles. We will begin with the or time translation invariance. As a first investigation into standard theory of general relativity, defined around a flat this problem, we will assume the graviton is massless. This background, and then generalize this in a particular fashion may seem a very mild assumption since we are going to that we will describe in the next subsection. So it is useful project down to only the two polarizations of the spin-2 to recap the structure of general relativity (the unique local, graviton and we know that Poincaré symmetry plus unitary, Poincaré invariant theory of massless spin-2). To unitarity would dictate that it would need to be massless. study the leading order interactions, we expand the metric But since we are not assuming Poincaré symmetry in this gμν around a flat background as work, this argument is no longer valid. In fact in our previous paper [23] where we did not assume Lorentz gμν ¼ ημν þ κhμν; ð1Þ boosts, we allowed for mass terms for the graviton, but where ημν is the metric of Minkowski space-time and hμν is went on to show that locality required them to vanish. If we now give up both Lorentz boost and time translation the spin-2 field whose purpose is to describe interactions of invariance, we can again have mass terms for the graviton, spin-2 particles in a local way. To project out the unphysical degrees of freedom, one needs the gauge redundancy: and for the greatest generality we should only use some → ∂ α principle like locality to dictate what terms are allowed. hμν hμν þ ðμ νÞ. It is well known that this leads to the However, it turns out this issue is rather complicated. So, unique Poincaré invariant quadratic kinetic term for grav- for simplicity, we will assume the graviton is massless here, itons, which is the Einstein-Hilbert term expanded to and leave the inclusion of mass terms for future work. We quadratic order (up to field redefinitions and boundary terms) note that all current data is indeed consistent with our 1 L ηαβ∂ μν∂ − ημν∂ ð4Þ∂ ð4Þ assumption of a massless graviton [24]. However, we know kin ¼ 2 ð αh βhμν μh νh Þ that when expanding around a Friedmann-Robertson- ∂ μν∂ ð4Þ − ∂ μα∂ ν Walker (FRW) background space-time in standard general þ μh νh μh νhα; ð2Þ relativity, mass terms appear; so our work here will not be ð4Þ ð4Þ μν where h is the four-dimensional trace h ¼ η hμν. able to recover perturbations around FRW. Instead we view Furthermore, in order to maintain the gauge invariance at the present work as a solid starting point to explore the leading order, the graviton must couple to a symmetric explicit breakdown of time translations, albeit not the most 2-index conserved current, the energy-momentum tensor,ffiffiffiffiffiffiffiffiffiffiffiffiffiffi as general. To be clear, we are especially interested in explicit L − 1 κ μν κ κ ≡ p32π int ¼ 2 hμνT . Here the coupling is GN, breakdown, rather than spontaneous, which is already a ’ well studied problem. where GN is Newton s constant. To quadratic order in hμν,the L ð2Þ L L The structure of this paper is similar to the previous full action is GR ¼ kin þ int. papers [22,23], especially to the gravitation paper [23], and To leading order, i.e., ignoring the induced influence of μν μν we encourage readers to read these accompanying papers. gravitation on the matter sector itself, T obeys ∂μT ¼ 0. Since many details are already laid out there, our descrip- Since the graviton is massless and mediates long ranged tion here will be relatively brief, with a focus on the forces, it is highly nontrivial that it leads to local interactions. construction and primary results. Our paper is organized as (We note that just because a theory is Lorentz invariant does follows: We first recap the fact that standard general not imply locality; it is trivial to construct Lorentz invariant relativity is local. We then discuss a family of generaliza- theories that are nonlocal. For example, consider the Lag- tions that break time translations and Lorentz boosts. We rangian for a scalar field L ¼ð∂ϕÞ2=2 − ð∂ϕÞ4=Λ4 þ; then focus on classes of theories that cut down to two this leads to superluminality around nontrivial ϕ backgrounds

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[25].) In fact it can be shown that if the sources were to In the Lorentz invariant case, these can be organized into a μ ν disobey the null energy condition Tμνn n ≥ 0,theninfact 4 × 4 symmetric matrix field hμν that transforms as a tensor the graviton could itself be superluminal around certain kinds under Lorentz transformations (up to gauge transforma- of background configurations. Instead if we assume the null tions). In the non-Lorentz invariant, but rotationally invari- energy condition, the theory of general relativity, with ant, case we can just refer to its components as standard leading order interactions, is in fact local. ≡ ϕ ≡ ψ As an important manifestation of this, we can consider h00 ;h0i ¼ hi0 i;hij ð4Þ the tree-level graviton exchange between a pair of matter (where ϕ and the Newtonian potential ϕ are related by μ 1 ð4Þ N sources. In harmonic (de Donder) gauge ∂μhν ¼ 2 ∂νh ij ϕ ¼ 2ϕN=κ). Also, the 3-trace is h ≡ δ hij. the equations of motion are known to become very simple To leading order, we need to form some appropriate □ − κ − 1 ð4Þη □ ∂2 − ∇2 hμν ¼ 2 ðTμν 2 T μνÞ, with ¼ t . One can generalization of the above dimension 4 operators that then readily check if the tree-level exchange action between describe the quadratic terms of Eq. (2). In this work we are sources is local. To do so, one first solves for the particular giving up both Lorentz boosts and time translations; this κ 1 ð4Þ solutions hμν ¼ − 2□ ðTμν − 2 T ημνÞ. The corresponding means there are many ways we can generalize the above tree-level graviton exchange action between sources is half theory of Poincaré invariant massless spin-2. A concrete 1 μν the interaction term − 4 κhμνT , giving the result generalization is the following: we can write out each of the terms of Eq. (2) in terms of its rotationally covariant 8L ψ ϕ ex Tij T T T00 T00 T0i T building blocks hij, i ¼ h0i, and ¼ h00. Then we can T − − 2T0 T00 : 3 κ2 ¼ ij □ 2 □ þ 2 □ i □ þ □ ð Þ insert a different function in front of every term as follows: _ Here we have broken up the energy-momentum tensor into L ¼ −2Aψ_ i∂jhij þ 2Bh∂iψ i − C∂iϕ∂ih þ D∂iϕ∂jhij a spatial tensor Tij, spatial vector T0i ¼ Ti0, and spatial − E∂ h∂ h − F ∂ ψ 2 G∂ h ∂ h H∂ ψ ∂ ψ ij i j ij ð i iÞ þ j ij k ik þ j i j i scalar T00 (along with T ≡ δ T ). This decomposition will ij I J K L be useful for comparison to our later analyses where we − _ 2 ∂ ∂ _ _ − ∂ ∂ L 2 h þ 2 ih ih þ 2 hijhij 2 khij khij þ int; break Lorentz symmetry explicitly. This exchange action is local, despite the presence of the inverse operators on the ð5Þ right-hand side. That is because the inverse operators are where the coefficients are allowed to a priori be arbitrary the wave operator, which is associated with retarded wave … propagation, as opposed to the inverse Laplacian, which is functions of time, AðtÞ;BðtÞ; ;LðtÞ. The Poincaré invari- associated with instantaneous action at a distance. Later ant theory of massless spin-2 is recovered by setting all when we deform several aspects of the Poincaré symmetry, these coefficients to 1. Although the above 12 kinetic terms we shall see that almost all deformations will in fact lead to are the full set of dimension 4 terms that describe the graviton in the Poincaré invariant case, it is not the full set inverse Laplacians and instantaneous action at a distance. allowed if we give up Lorentz and time translation So by imposing locality, we will be able to derive that symmetry as we are doing here. In particular one can also almost all deformations are forbidden, as we saw in our ∼ ∂ ϕ 2 ∼ϕ_ ∂ ψ previous pair of papers [22,23]. add a pair of terms ð i Þ and i i. However it is relatively simple to show (as we mention in [23]) that they lead to nonlocality, so they shall be ignored here. B. Spatially invariant massless spin-2 There are lower dimension quadratic terms that could be Let us make clear what symmetries we will explicitly included too. In particular one could include several dif- break and which remain. We assume a preferred frame that ferent dimension 2 mass terms and dimension 3 single- has rotation invariance and spatial translation invariance. derivative terms. In the time translationally invariant theory However, as in our previous pair of papers [22,23],wedo that we analyzed in Ref. [23], we addressed these possible not assume Lorentz boosts. Moreover, we now do not terms. In particular we allowed for mass terms, at least in assume time translations. Maintaining rotation will be part, and we mentioned that one could then perform field important in this work. In particular, this ensures we have redefinitions to remove the dimension 3 terms (generating a notion of particle spin and our goal will be to build an dimension 5 terms that are higher order and ignorable in interacting theory of spin-2 particles. Since we are inter- this analysis); we then showed that mass terms led to ested in maintaining an explicit notion of locality, we will nonlocality. Similarly, here we could include such lower use the formalism that makes locality manifest; namely we dimension terms. Indeed this is important to obtain per- can embed the spin-2 particles into fields. In principle we turbations around FRW in general relativity. However, for only need a 3 × 3 symmetric matrix to describe this, which simplicity, this will not be our focus here. Instead we will we call hij, which transforms as a tensor under rotations. focus on massless spin-2 and assume the leading quadratic But locality will in fact require us to also introduce a terms are dimension 4, as they are in general relativity. nondynamical scalar ϕ and a nondynamical vector ψ i. We suspect that if one sets the masses to zero then in fact

085007-3 MARK P. HERTZBERG and JACOB A. LITTERER PHYS. REV. D 102, 085007 (2020) the dimension 3 terms must vanish in order to maintain proportional to A_ and three additional terms in Eq. (10) locality, though we do not have a proof to present here. proportional to B_ , I_, K_ , as compared to GR in which We leave the important case of including time-dependent A ¼ B ¼¼L ¼ 1 are all constants. mass terms and time-dependent dimension 3 terms to The sources T μν are not obviously required to be future work. conserved here. As we did in [23], we parametrize the The leading order interaction allowed involves a single violation of source conservation by a scalar σ and vector wi graviton coupled to matter. Poincar´e invariance requires the μν as coupling to the energy-momentum tensor ∼hμνT ,aswe reviewed in the previous subsection. We generalize this to σ ≡∂ipi þ ρ_r ð11Þ 1 L − κ T μν ≡∂ τ _ int ¼ 2 hμν ; ð6Þ wi j ij þ pr;i; ð12Þ μν where T is a general source which does not a priori have where ρ_r ≡ ðA=DÞρ_ and p_ r;i ≡ ðA=HÞp_ r;i. In the GR limit μν any relationship to an energy-momentum tensor. In fact we have ∂μT ¼ 0, which means σ ¼ wi ¼ 0, but these since we do not a priori assume Lorentz boosts or time are generally nonzero when Lorentz and time translation translation invariance there does not a priori even exist a symmetry is abandoned. In fact we can use the classical conserved, not to mention symmetric, energy-momentum equations of motion to express these violations of con- tensor. We note that we will again specify the coupling as κ, servation in terms of the fields as which is useful for power counting. One could promote the coupling to be a function of time κ ¼ κðtÞ; however, AC κσ˜ ¼ðH − FÞ∇2∂ ψ þ B − ∇2h_ without loss of generality, we can absorb this time i i D dependence into the sources instead. Since we are only _ D _ A _ 2 assuming spatial rotation invariance, we again find it most þ A − A ∂i∂jhij − C∇ h ð13Þ convenient to decompose it into its scalar, vector, and D D tensor (under rotations) pieces, as κ˜ 2 − − AF ∂ ∂ ψ_ 2 _ ∂ ∂ ψ 00 0i i0 ij wi ¼ B A i j j þ B i j j T ≡ ρ; T ¼ T ≡ pi; T ≡ τij: ð7Þ H AB AB_ This class of theories of massless spin-2 is specified by þ − I ∂ ḧþ − I_ ∂ h_ 12 functions of time AðtÞ;BðtÞ; …;LðtÞ (in addition to the H i H i strength of gravity κ). As mentioned above, the Lorentz ̈ 2 2 AA invariant and time translationally invariant limit (general þðJ − EÞ∂i∇ h þðG − LÞ∇ ∂jhij − ∂jhij H relativity) is where all these quantities are set to 1. Hence 2 _ we have a large space of deformations from general A ̈ _ AA _ þ K − ∂jhij þ K − 2 ∂jhij relativity; namely 12 new functions of time. However, H H several of these functions can be eliminated by redefining 2 þðG − EÞ∂i∂j∂khjk þðD − CÞ∂i∇ ϕ: ð14Þ the fields ϕ, ψ i, and hij and exploiting the degeneracy in ρ τ meaning between the scalar sources and the trace of ij; This representation is useful because it shows explicitly that this leaves eight independent functions. σ and wi vanish in the GR limit A ¼ B ¼¼L ¼ 1, The classical equations of motion that follow from the since all terms on the right-hand side are zero, but are action (5) are nonzero otherwise. 2 κρ˜ ¼ −C∇ h þ D∂i∂jhij ð8Þ III. RESTRICTING TO 2 DEGREES OF FREEDOM ∂ κ˜ ∂ _ ∇2ψ − ∂ ∂ ψ − ∂ pi ¼ B ih þ H i F i j j ∂ ðA jhijÞð9Þ As is well known, in the Poincar´e invariant case the t massless unitary representations are individual helicities, ∂ ∂ ∂ and for spin-2 we need to keep both left- and right-handed κτ˜ ¼ 2δ ðB∂ ψ Þ − δ ðIh_Þþ ðKh_ Þ ij ij ∂t k k ij ∂t ∂t ij helicities to build a theory with local interactions. − A∂ ψ_ þ D∂ ∂ ϕ − E∂ ∂ h − Eδ ∂ ∂ h Furthermore, this uniquely specifies the free theory of ði jÞ i j i j ij k l kl Eq. (2) as it carries the gauge redundancy needed to cut − δ ∇2ϕ ∂ ∂ δ ∇2 − ∇2 → ∂ α C ij þ G k ðihjÞk þ J ij h L hij; down to these 2 degrees of freedom: hμν hμν þ ðμ νÞ. ð10Þ Then one can construct all the interactions, order by order in κ, to build up the full Einstein-Hilbert action. where κ˜ ≡ −κ=2. We can see breaking time transla- In contrast, it is sometimes suggested that the existence tion symmetry results in one additional term in Eq. (9) of the graviton can be derived by promoting the global

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Poincar´e symmetry to the gauge symmetry of diffeomor- Perhaps the most basic way is to remove any longitudinal phisms. But this is incorrect, as any gauge symmetry can modes directly by imposing that they vanish as be trivially introduced by the Stueckelberg trick. Instead, the only known way to introduce the graviton within ∂iψ i ¼ 0; ∂jhij ¼ 0: ð15Þ the framework of effective field theory is to simply postulate it. Note that this choice is not merely a “gauge choice,” since Now in the present work, since we are giving up Lorentz the lack of Lorentz symmetry in the theory means there is boost and time translation invariance, but still studying no gauge redundancy in our field formalism, but a choice of massless spin-2, there is no longer an unambiguous theory. We will only need to work with the classical theory consequence for the number of degrees of freedom. in this analysis (which suffices to derive our basic con- However, we will once again just postulate the existence clusions) and so we can imagine the above is implemented of the two polarizations of the graviton in this work. We are by Lagrange multipliers. The corresponding classical motivated by the following: (i) we wish to make only small equations of motion are deformations to general relativity, while additional degrees of freedom could be viewed as large changes; (ii) all current κρ˜ ¼ −C∇2h ð16Þ evidence of gravitational waves is consistent with only 2 degrees of freedom [26]; and (iii) in the Lorentz invariant 2 κ˜p ¼ B∂ h_ þ H∇ ψ ð17Þ case additional degrees of freedom in massive gravity have i i i been argued to suffer strong coupling and causality ∂ ∂ problems [25,27–29]. κτ˜ −δ _ _ − ∇2 − ∂ ψ_ ij ¼ ij ∂ ðIhÞþ∂ ðKhijÞ L hij A ði jÞ Since the theory of Eq. (5) with prefactor functions t t − δ ∇2ϕ ∂ ∂ ϕ − ∂ ∂ δ ∇2 AðtÞ;BðtÞ; …;LðtÞ that are a priori allowed to be arbitrary C ij þ D i j E i jh þ J ij h: ð18Þ functions of time, there is no longer any gauge redundancy in the theory. This means that it can propagate more than Note the functions F and G no longer appear in the theory. 2 degrees of freedom. Hence to project down to only We can explicitly check on the status of source con- 2 degrees of freedom for the graviton, one has to impose servation in this theory. The scalar component of the usual μν some constraints on the fields and/or the prefactor func- conservation of sources ∂μT ¼ 0 is now tions. We consider two options, which encompass the most ∂ ρ general set of possibilities: (A) set ∂iψ i ¼ 0 ¼ ∂jhij; ∂ipi þ B ¼ 0 ð19Þ (B) fix parameters such that ∂iψ i and ∂jhij are directly ∂t C determined by the equations of motion. Both of these R examples may seem like a form of “gauge fixing,” but we so the quantity d3ρ=C is conserved, though in our choice emphasize that since there is no gauge redundancy of variables it is not simply the integral of the source ρ,but assumed, they are instead a choice of theory with different rescaled by a time-dependent coefficient in this conserva- physical predictions. In our previous work of Ref. [23] we tion equation. This implies that the matter sector must considered a third option as well. However, after enforcing involve some Abelian symmetry; though at this stage we locality, this third option was readily seen to be a special cannot identify its details as we would need more infor- case of the second, so we will not consider it here. mation about the matter sector (we will later see that Nevertheless, by demanding the tree-level exchange locality requires it to be related to a space-time symmetry). action is local, we find at the leading order to which we On the other hand the vector component of the usual μν are working that both cases (A) and (B) will collapse to the conservation of sources ∂μT ¼ 0 is rather complicated: same theory; namely GR with some additional terms. As in our previous work, we also do not a priori impose A AB ∂ ρ̈ E − J T μν ∂ τ þ p_ ¼ I − i þ ∂ ρ conservation laws on the sources . However, locality j ij H i H C∇2 C i will restrict its form tremendously, as we will describe. D − C AB B_ H_ þ ∂ ∇2ϕ þ − − I_ κ˜ i H B H A. Theory A: Transverse constraint ∂ ∂ ρ In this and the next section we describe two explicit ways i × : ð20Þ to cut down to 2 degrees of freedom in the above class of ∂t ∇2 C theories. This is analogous to the first two theories we studied in [23] (while the third theory studied in [23] was Enforcing locality.—The basic principle which we are found to be only a special case of the second theory once imposing is that of locality: no instantaneous signaling at a locality was imposed, so we will not repeat that analysis distance. However the previous equation will in general here). Moreover, we believe that our results are general and spoil this because there are several terms that involve apply to any such choice. inverse Laplacians. This means that even if the sources

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themselves (ρ, etc.) are initially local, they will evolve to 2 h τ ð∂ ∂ − δ ∇ Þτ 1 A ∂ ∂ ipj ensure sources (p , τ ) are nonlocally distributed in space. ij ¼ ij þ i j ij þ ð Þ i ij κ˜ □˜ 2□˜ ∇2 □˜ ∇2 ∂t H This would provide a means to instantaneously signal to a δ ∂ ∂ ρ distant observer by having the distant observer situated near ij 3J − 3E − L i j þ ðE − J þ LÞρ þ one of these effectively nonlocal sources and then dis- 2D 2D ∇2 turbing the source. Hence to maintain locality we need to δij ∂ ∂ ρ enforce that these terms vanish. This requires the following þ ðI − KÞ 2∇2 ∂t ∂t D set of conditions be satisfied: ∂ ∂ ∂ B ∂ ρ _ _ _ þ i j 2A B I H ∇4 ∂t H ∂t D AB ¼ IH; ¼ þ ;D¼ C; ð21Þ B I H 1 ∂ ∂ ρ þ ðK − 3IÞ ; ð27Þ where the first condition here arises from requiring the 2 ∂t ∂t D 2 coefficient of ∂iρ̈=∇ vanish, the second condition here 2 arises from requiring the coefficient of ∂iðρ=CÞ=∇ vanish, where we have defined a wave operator, and the third condition here arises from requiring the 2 ˜ 2 _ 2 coefficient of ∇ ϕ vanish [this final condition is needed □ ≡ K∂t þ K∂t − L∇ : ð28Þ because ϕ itself is highly nonlocal; see ahead to Eq. (26)]. With these three conditions enforced, the conservation This unusual wave operator merits some discussion. First, equation becomes local and nearly canonical. To make it though it involves functions of time through K ¼ KðtÞ and appear as canonical as possible it is useful to define a L ¼ LðtÞ it is still a linear operator, and therefore has a shifted tensor source as Green’s function, which lets us continue using its inverse to ’ − simply mean convolution with that Green s function. Or, E J more simply, whatever its functional form, this operator has τij ¼ τ˜ij þ δijρ: ð22Þ D a matrix representation, and the operations of matrix By then inserting into Eq. (20) with the above trio of multiplication and inversion are well defined. conditions, we then obtain The corresponding tree-level graviton-exchange action between a pair of sources is in fact 1=2 of the interaction A L L 2 −κ T μν 4 term ex ¼ int= ¼ hμν = . Then to determine ∂jτ˜ij þ p_ i ¼ 0: ð23Þ H under what conditions this exchange action is local, we So, in this theory, locality has enforced that the sources can now write the interaction Lagrangian, Eq. (6), only in terms of the sources by using the solutions (24)–(27) to obey almost standard conservation laws. (We note that τ˜ij eliminate the fields from the interaction Lagrangian. We couples linearly to the graviton, just as τij does.) Having ensured that the sources themselves are local, we can then integrate by parts (in the spatial integrals) in the can now move to another test of locality. Since the graviton action to replace all divergences using the conservation is massless (or even if it were very light) it will mediate a equations (19) and (23). Integrating by parts in time now long ranged force between matter sources. Hence it can introduces a great deal of complexity to the problem very easily lead to nonlocal interactions. One necessary because each term in the action is cubic in functions of □ condition to avoid this is that the exchange interaction time, and the operator in Eq. (28) also contains functions between a pair of matter sources, with no external grav- of time. The general nonlocal exchange action contains a itons, is local. (Later we will comment on additional great many unique terms. To study this in full generality ramifications that can emerge from considering external was only feasible with computer algebra software, such as gravitons.) So it is useful to have the inhomogeneous Mathematica. We will describe its general form and present particular solutions for the gravitational fields a few simple terms as concrete examples. Schematically, the exchange action now contains terms h −ρ of the form ¼ ð24Þ κ˜ D∇2 L ⫈ SD S SD S_ ex a1½t ða2½t Þþa3½t ða4½t Þ ψ ∂ ∂ ρ i pi B i ̈ ¼ þ ð25Þ þ a5½tSDða6½tSÞ; ð29Þ κ˜ H∇2 H ∇4 ∂t D where a ½t¼a ½AðtÞ;BðtÞ; …;LðtÞ are functions of ϕ −τ E þ L − 3J ρ n n the coefficients in the action and their time derivatives, κ˜ ¼ 2 ∇2 þ 2 2 ∇2 D D D D □˜ −1 ∇2 1 ∂ ∂ ρ ¼ ½ ; is some differential operator involving at 3 − least one power of the inverse wave operator and various þ ð I KÞ 4 ð26Þ 2D ∂t ∂t D∇ (positive or negative) powers of Laplacians, and S stands

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μν for any of the sources ρ, pi, τij. Note in the static limit independent. In fact building T out of the sources ρ, pi _ _ _ τ˜ ρ (A ¼ B ¼¼L ¼ 0), the second term would be a total and ij, we can easily rescale and pi by a constant to write ˜ μν derivative in the action, while the other two terms would ∂μT ¼ 0, so the theory is required to have a canonical not necessarily vanish. To enforce locality, for every conserved source. L distinct term in ex that contains any inverse Laplacians The new feature that departs from the Poincar´e invariant the functions an½t must be set to zero or a constant to theory is the functions E, J, and L which can be linear eliminate the nonlocal terms, making them identically zero functions of time, and must have the same time depend- or a total derivative as appropriate. In most cases each set of ence. Writing J ¼ J0 þ c4t etc., enforcing the conditions terms provides multiple possible constraint options to (31) leaves the exchange action eliminate the nonlocal parts, such as in the first term of 8L τ τ τ ρ τ ρ ρ 2 the schematic example, Eq. (29), where if this term were ex τ ij − f a1 g a2 − c3 Lpi 2 ¼ ij þ þ 2 2 pi nonlocal, either a1 or a2 could be set to zero to eliminate it. κ □ 2 □ 4c2 □ 8c □ c □ 2 1 Nevertheless, it turns out there is a unique set of conditions τ_ ∇2τ c3c4 ρ that eliminates all nonlocal terms from the exchange action. þ 2 þ c4 3 L c2 □ □ We emphasize that any terms in ex that have inverse ρ − 3 ρ_ Laplacians would give rise to instantaneous signaling of − c3c4 ρ ðL JÞ 2 c4 2 þ 2 one source to a distant source; which we wish to avoid in c2 □ 2□ this work. This would be in sharp contrast to all known 2 2 2 c3c ð3J − 5LÞ∇ ρ 2c c4 p_ physical theories (general relativity, electromagnetism, etc.) 4 ρ − 3 p i ; þ 2 2 □3 2 i □2 ð32Þ which avoid instantaneous signaling; and indeed avoid c2 c1 inverse Laplacians in the final exchange action. 3 − 18 − 3 2 − 11 2 Most of the sources can only couple to themselves, as where a1 ¼ L J, a2 ¼ JL J L , a3 ¼ ∼SS 3J − 5L, fρa1τg=□ ≡ ρða1τ=□Þþτða1ρ=□Þ, and the represented in Eq. (29) where each term . The only 2 ∼ρτ wave operator with K constant has become □ ≡ K∂t − possible mixed terms are , and turn out to be among the 2 least numerous. As a simple example, grouping nonlocal L∇ . This is now completely local and the first line has a terms ∼ρτ, the exchange action contains similar form to the GR action Eq. (3), but with coefficients a1 linear in time and a2 quadratic in time. We are left with τ ðI − KÞρ̈ Iρ̈ τ ρ K̈τ six unspecified parameters, c1, c2, c3, c4, J0 and L0. In the L ⊃ þ − ð30Þ 0 ρτ ex 2C □˜ ∇2 2C □˜ ∇2 2C □˜ ∇2 static case (c4 ¼ ) the two symmetric terms on the first line combine and the higher order □−1 terms vanish, in from which we find locality requires I ¼ K in order to agreement with our previous work on the static case. 2 remove the first term as it is proportional to 1=∇ . We also However, with c4 ≠ 0, we are left with a more general— — find K_ ¼ 0 is a necessary condition for locality, rendering explicitly time-dependent theory. I ¼ K ¼ const. Note, however, this does not completely Even in the static limit, it is not immediately obvious eliminate every term in Eq. (30), just the nonlocal ones; how Eq. (32) reduces to Eq. (3). As we remarked in [23],if integrating by parts in time generates local terms higher we rewrite the exchange action in terms of the conserved ρ → ρ order in □−1. The full set of necessary and sufficient sources and rescale ðc2=LÞ , in the static case we then conditions to eliminate all nonlocal terms from the tree- recover Eq. (3) exactly. However, this does not generalize level exchange action in the theory of this subsection are to the present case with LðtÞ a function of time. Expressing found to be the exchange action (32) in terms of the conserved source τ˜ij [from Eqs.(22) and (23)] we obtain

A ¼ B ¼ c1;C¼ D ¼ c2; 2 8L τ˜ τ˜ τ˜ fρ Lτ˜g ρ L ρ 2c3 Lp 2 ex τ˜ ij − − p i c1 κ2 ¼ ij □ 2 □ þ 2 □ þ 2 2 □ 2 i □ I ¼ K ¼ c3;H¼ c2 c2 c1 c3 τ_ ∇2τ ρ_ 2 2 ρ̈ _ _ _ c3c4 ρ ρ c3c4 ρ L c3c4 ρ 2EðtÞ¼JðtÞþLðtÞ; E ¼ J ¼ L ¼ c4; ð31Þ þ 2 þ c4 3 þ 2 2 þ 2 3 c2 □ □ c2 □ c2 □ 2 _ where c1, c2, c3, c4 are constants (independent of time); c1, − 2 c3c4 pi 2 pi 2 : ð33Þ c2, c3 are dimensionless, while c4 has units of inverse time. c1 □ We see that locality imposes that most of the coefficients obey the standard relations in the Poincar´e invariant theory In this form JðtÞ drops out, so there are really only five of general relativity. In particular, most of the coefficients constants here: c1, c2, c3, c4, and L0 (we may, as in the next are required to be constants. Moreover the conservation section, label L0 ≡ c5). Unlike in the static case, factors of t equations (19) and (23) are now simple because the (from L ¼ L0 þ c4t) can only appear next to source terms coefficients in those equations are required to be time inside the □−1. To appear to the left in any of these terms,

085007-7 MARK P. HERTZBERG and JACOB A. LITTERER PHYS. REV. D 102, 085007 (2020) outside the □−1, requires some integration by parts in time, We again use these to write the exchange action only in _ −1 leaving only ∼L ¼ c4 outside the □ . Thus in the two terms of the sources, integrating by parts to replace σ terms symmetric in ρ and τ˜, in one term L comes with the ρ divergences using the definitions of q, , and wi. Once and in the other L comes with the τ˜. This means with LðtÞ a again, because of the cubic dependence on functions of function of time we cannot rescale ρ to make the first line time, integrating by parts to simplify under the fewest look more like Eq. (3) as we could in the static case. Our independent variables expands the total number of terms interpretation of the above result will come in Sec. IV. enormously. The general form is similar to that described in Eq. (29). Now that we have general σ and w [as opposed to the B. Theory B: Constraint from equations of motion i restricted form in Theory (A)], there are more possible In the above section we cut down to 2 degrees of freedom mixed terms than only ∼ρτ; there are also terms with τσ, of the graviton by imposing constraints directly on the ρσ, ρ∂iwi, τ∂iwi, and piwi. It is advantageous to first look fields in Eq. (15). A secondary approach is to impose some at the wiwi and σσ terms involving inverse Laplacians to conditions on the prefactor functions so that the equations see if locality requires some constraint that simplifies all of motion include constraints that cut down the degrees of these other terms involving σ and wi. Looking for nonlocal freedom directly. By studying the full theory with all 12 terms in w w and σσ, the exchange action contains … i i unknown functions AðtÞ; ;LðtÞ, and using the general equations for the nonconserved sources, Eq. (13) and (14), 1 σ 1 L ⊃ 2σ FKL 2 Gwi we can impose the constraints on functions ex ˜ 2 2 þ wi ˜ 2 ð39Þ □∇ Hðc1 − FKÞ □∇ G − L 2 A plus terms higher order in ∇−2 and derivatives of σ and w . A ¼ B ¼ c1;C¼ D ¼ c2;I¼ K ¼ : ð34Þ i H If σ and wi are general functions, then the theory is immediately nonlocal. To eliminate these terms, the only It is simple to show that this now fixes ∂ ψ and ∂ h by the i i i ij possible constraints on the coefficients AðtÞ; …;LðtÞ would equations of motion, leaving the theory with the desired impose some of these vanish [e.g., GðtÞ¼0], leading to an 2 degrees of freedom. In particular the classical equations overly trivial theory, which we decline to do. The most of motion enforce that they are given by the sources as direct way to avoid this problem is to impose that σ and wi σ 0 κσ˜ vanish, i.e., ¼ wi ¼ . In previous work [23] we allowed ∂ ψ for σ and w to themselves be given by derivatives of other i i ¼ − ∇2 ð35Þ i ðH FÞ local functions in order to maintain locality by having the Laplacians cancel out. But this only leads to ultralocal κ˜ 2E − G − J ∂ q ∂ h ¼ j þ w terms in the action, which were found to be associated with i ij ðG − LÞ∇2 2G þ J − 2E − L ∇2 j a completely decoupled, and hence irrelevant, sector. We ∂ σ_ _ _ ∂ σ will ignore this here as it does not represent a real − K j F − H AF j 2 þ 2 modification to GR. Instead by focusing on the important A ∇ F H HðH − FÞ ∇ case of only coupled sectors, we are forced by locality to − _ J E K impose σ ¼ wi ¼ 0 [we can also have wi ¼ðE − JÞδijρ=D, þ ∂jρ þ pj : ð36Þ D A but we can easily shift from τij to τ˜ij to eliminate this, as we did earlier in Eq. (22); so we can ignore that without loss of ∂ In the expression for ihij we have introduced the follow- generality]. ing function of the sources: Thus locality imposes that the sources are conserved and the theory is now similar in structure to Theory (A) of the _ A J − E K previous section, though the coefficients F and G are still q ≡∂ w − σ_ þ ∇2ρ − ρ_ j j present. The full set of conditions to make the exchange H D D F_ H_ AF K_ action of Theory (B) local are þ − þ σ ð37Þ F H HðH − FÞ A σ ¼ 0;wi ¼ 0;I¼ K ¼ c3; 2 which gives a nice form for the second derivative, E ¼ G ¼ J ¼ L ¼ c5 þ c4t; F ¼ H ¼ c1=c3: ð40Þ κ˜ q Again we see that the static case is recovered if c4 ¼ 0 in ∂ ∂ h ¼ : ð38Þ i j ij 2G − 2E þ J − L ∇2 agreement with Ref. [23]. For the time-dependent case with c4 ≠ 0, we have the conditions of Eq. (40). So we are left Enforcing locality.—As in the previous section, we can with five parameters of the theory, as in Theory (A). We now solve the equations of motion to obtain the inhomo- therefore recover the same local action Eq. (33) with the geneous solutions for the fields, given in the Appendix. same number of parameters. [We note that since F ¼ H and

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̈ G ¼ L here, it is compatible with Eqs. (35) and (36), since f ¼ −c4t: ð43Þ we also have σ ¼ wi ¼ 0. This is clearest seen by multi- plying both sides of Eq. (35) by (H − F) and Eq. (36) That is, the additional terms in Eq. (41) are nothing other by (G − L)]. than the Gauss-Bonnet action with the function fðtÞ determined by Eq. (43). IV. DISCUSSION If f is a constant, the action Eq. (42) is a surface term in four dimensions, so it is usually not included in GR even After enforcing locality, we see that both theories though this particular sum of curvature squared terms became identical, so we recover the same exchange action. satisfies the same symmetries as the Einstein-Hilbert That is, imposing all conditions in Eq. (40), the now local action. While we did not begin our analysis by explicitly tree-level exchange action of Theory (B) is exactly as in adding the Gauss-Bonnet action with a function of time Theory (A) after writing the exchange action in terms of the fðtÞ, these terms are contained in the general action of conserved sources, Eq. (33). We believe this is true for any Eq. (5) to quadratic order in hμν. After enforcing locality we particular construction that follows from the above class of recover Einstein-Gauss-Bonnet gravity with fðtÞ related to theories outlined in Sec. II (we can also have decoupled 2 sectors, as we elaborated on in [23], but this is of little the graviton speed vg ¼ LðtÞ by Eq. (43). ΔL interest here). Since GB is itself invariant under the linearized gauge transformations hμν → hμν ∂ μαν and both Theories (A) If we set the coefficient c4 ¼ 0 we recover general þ ð Þ ˜ μν relativity exactly (at this order). But with c4 ≠ 0 we have a and (B) end up with a conserved source, ∂μT ¼ 0, the theory of spin-2 with 2 degrees of freedom that explicitly full action of Eq. (41) enjoys the linearized gauge redun- breaks time translation invariance, while maintaining local- dancy of GR at this leading order. One can then gauge fix in ity in tree-level graviton exchange between conserved any desired fashion [e.g., as in Theory (A)] and thus obtain matter sources, and is therefore a deviation from general Eq. (33) for the tree-level exchange action between matter relativity. sources. So the general conceptual features of our final results are: A. Gauss-Bonnet the imposition of locality (avoiding instantaneous signal- Analyzing only the exchange action as we have in the ing) is so strong it forces our theory into a familiar gauge previous two sections lets us identify nonlocal interactions, redundant form, even though we did not assume gauge but obfuscates the structure of the theory. Returning to the redundancy as a starting point in Eq. (5). In particular, in full Lagrangian, after enforcing locality by Eq. (40) what addition to the quadratic Einstein-Hilbert term, which is we are left with is the Lagrangian of Eq. (5) with all well known to be gauge invariant, we have also found an allowed Gauss-Bonnet term with a prefactor that depends coefficients constant, plus a term ∝ c4 that breaks time on time. This in fact is the most general action at the order translation symmetry. If we set c4 ¼ 0, we recover time translations, and we can also absorb the constant coeffi- we are working (albeit with a particular function of time) cients by a redefinition of the fields, allowing us to recover which is gauge invariant. A general conceptual feature of T μν GR with its full Poincaré symmetry. In the case of interest, the sources will be discussed in the final subsection. with c4 ≠ 0, the theory breaks time translation symmetry and in turn the Lorentz boost symmetry of the full theory. B. Broken symmetries at this order For ease of notation and without loss of generality, we set The general local action we find for spin-2 with 2 degrees 1 all c1 ¼ c2 ¼ c3 ¼ c5 ¼ . This leaves only c4 as the of freedom, Eq. (41), has the linearized gauge invariance of adjustable parameter of the theory. We can then write GR, but the presence of the explicit function of time fðtÞ the full Lagrangian of the theory (5) as prevents time translation symmetry, and its relation to the 2 ̈ _ graviton speed by Eq. (43) [i.e., vg ¼ LðtÞ, f ¼ −L] ð2Þ 1 1 L L c4t ∂ h∂ h − ∂ h ∂ h prevents us from recovering the usual Lorentz boost ¼ GR þ 2 i i 2 k ij k ij invariance. We do not have a simple intuition as to why locality of spin-2 prefers to include the Gauss-Bonnet terms þ ∂ h ∂ h − ∂ h∂ h : ð41Þ j ij k ik i j ij in the action with a prefactor fðtÞ that can be at most cubic in time, when any function of time would maintain gauge Let us compare this to the Gauss-Bonnet action [30] with invariance at this order. an inserted prefactor function fðtÞ: In previous work [7] some of us found for any nontrivial ffiffiffiffiffiffi fðtÞ the theory of Eq. (41) allows for superluminal ΔL p− 2 − 4 μν μνρσ GB ¼ fðtÞ gðR RμνR þ RμνρσR Þ: ð42Þ gravitons. This is still true here, but in this work we demanded only a general notion of locality, that there is no At the leading order in hμν, to which we are working, and instantaneous signaling. So the theory is local in this sense, after performing some integrations by parts, we can identify though allows for gravitons with a time-dependent speed

085007-9 MARK P. HERTZBERG and JACOB A. LITTERER PHYS. REV. D 102, 085007 (2020) which can in general be superluminal, i.e., it can be faster Moreover, one could reasonably anticipate this as than the particle speeds of the matter sector, including follows: so far we have only allowed the external sources photons. to be part of the matter sector and we found that such a Nonzero c4 characterizes deviations from Poincar´e matter sector must exhibit time translations in order to invariance. To be compatible with observations the value maintain locality. In addition, however, one should also of c4 (which has units of inverse time) must be very small. consider the external sources to be gravitons. In this case, it As an example of a bound, the recent LIGO observations of seems plausible that a similar analysis would demand that merging neutron stars showed that the graviton’s speed the energy-momentum tensor of the graviton itself needs to matched the photon’s speed to an accuracy of jv − 1j ≲ be conserved, which would then demand time translations g 0 10−15 [31] (we set the speed of light to 1 here). Since in the graviton sector, giving c4 ¼ and the full Poincar´e the photons and gravitons traveled from this event for symmetry of general relativity. Another hint is the follow- t ∼ 108 years to Earth, we can translate this into the direct ing: we have remarked that locality has enforced that the final action of Eq. (41) is gauge invariant to this leading bound on the coefficient c4 of order. It is plausible that at higher order, locality will again −23 −1 demand gauge invariance; this would presumably require f jc4j ≲ jv − 1j=t ∼ 10 years : ð44Þ g to be a constant to recover the usual nonlinear diffeo- morphism invariance of general relativity. But we do not c4 a priori Since is dimensionful it is not obvious what its have a proof of these issues. expected value is. On the other hand, since this is such a An important direction for future work is to include mass small number in most fundamental units, it suggests that its terms and single derivative terms in the starting action. true value may very well be exactly zero. Although we could show that mass terms needed to vanish in order to maintain locality in the time translation invariant C. Derived symmetries and higher order case of [23] (and the single derivative terms could then be One of our primary findings is that locality requires the converted by field redefinitions into higher dimension ˜ μν operators), they may play an important role in the time- conservation of matter sources ∂μT ¼ 0. Since there is dependent case. This is especially important in that such only one nontrivially conserved 2-index tensor, then this terms are known to exist when expanding around an FRW T˜ μν μν must be the energy-momentum tensor ¼ T .Aswe background in ordinary general relativity. We leave this detailed at the end of [23], the existence of a conserved important issue for future work. symmetric energy-momentum tensor relies both on Lorentz and time translation invariance. Hence we have found that ACKNOWLEDGMENTS locality enforces those symmetries are obeyed by the matter sector. We would like to thank McCullen Sandora for very On the other hand, our final action allows for the helpful input. M. P. H. is supported in part by National breaking of time translations in the gravity sector. At the Science Foundation Grants No. PHY-1720332 and above level of analysis, this is not an inconsistency because No. PHY-2013953. we have only worked to leading order; in particular, we only need the matter sector to exhibit time translations for APPENDIX: EQUATIONS OF MOTION IN μν THEORY B (SEC. III B) ∂μT ¼ 0, and this is true so long as we ignore back- reaction from the graviton. Nevertheless this does suggest In this Appendix we report on the inhomogeneous that if one were to demand locality at higher order, then it is particular solutions to the equations of motion (8)–(10) possible that the time translation violation, as measured by with the conditions of Theory (B) to cut down to 2 degrees the parameter c4, would in fact have to vanish. of freedom, Eqs.(35) and (36):

−τ ∂ 3I − K ∂ ρ q E − 3J þ L ρ ϕ ¼ þ − þ 2C∇2 ∂t 2C∇4 ∂t C ð2G − 2E þ J − LÞ∇2 2C2 ∇2 2G − 4E þ 3J − L q 2A ∂ σ þ þ ðA1Þ 2G − 2E þ J − L 2C∇4 C ∂t ðH − FÞ∇4 p F ∂ σ A ∂ 1 K ∂ σ_ F_ H_ AF ∂ σ K_ ψ ¼ i þ i þ w − i þ − i þ p i H∇2 HðH − FÞ ∇4 H ∂t ðG − LÞ∇4 i A ∇2 F H HðH − FÞ ∇2 A i A J − E ∂ ρ A ∂ q þ 1 þ i − i ðA2Þ C G − L ∇4 G − L ∇6

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1 ∂ ∂ τ δ E þ L − J ∂ ∂ LðL − 3JÞ − GðJ þ LÞþEðG þ 3LÞ h ¼ τ þ i j − δ þ ij ρ þ i j ρ ij □˜ ij ∇2 ij 2 2 C 2∇2 CðG − LÞ ∂ ∂ ∂ ∂ J − E ρ Kρ_ δ 2G − 2E þ 3J − L q þ i j 2K 1 − − þ ij ∇4 ∂t ∂t G − L C C 2 2G − 2E þ J − L ∇2 ∂ ∂ Gð2G − 2E þ JÞþ3LðG − 2E þ JÞ − L2 ∂ ∂ ∂2 Kð4E − 3G − 2J þ LÞq þ i j q þ i j 2∇4 ðG − LÞð2G − 2E þ J − LÞ ∇6 ∂t2 ðG − LÞð2G − 2E þ J − LÞ 2∂ ∂ ∂ Fσ GKσ_ F_ H_ AFGσ ∂ σ þ i j A þ − − − A ∇4 ∂t HðH − FÞ AðG − LÞ F H HðH − FÞðG − LÞ ∂t H − F _ _ 2∂ ∂ ∂ ∂ 1 −Kσ_ F H AFσ ∂ i ∂ pj þ i j K þ − þ A ð Þ ∇6 ∂t ∂t G − L A F H HðH − FÞ ∇2 ∂t H 2 2 _ _ A ∂ i ∂ 1 K G∂ i K þ ð w þ p − ð w þ p ; ðA3Þ ∇4 ∂t2 G − L jÞ A jÞ ðG − LÞ∇2 jÞ A jÞ

˜ 2 where as above □ ≡∂tðK∂tÞ − L∇ , and we can take the trace to obtain −ρ q h ¼ þ : ðA4Þ C∇2 ð2G − 2E þ J − LÞ∇4

[1] L. Verde, T. Treu, and A. G. Riess, Tensions between the [13] P. Horava, Quantum gravity at a Lifshitz point, Phys. Rev. D early and the late Universe, Nat. Astron. 3, 891 (2019). 79, 084008 (2009). [2] S. Weinberg, Photons and gravitons in S matrix theory: [14] J. Khoury, G. E. J. Miller, and A. J. Tolley, On the origin of Derivation of charge conservation and equality of gravita- gravitational Lorentz covariance, Classical Quantum Grav- tional and inertial mass, Phys. Rev. 135, B1049 (1964). ity 31, 135011 (2014). [3] S. Weinberg, Photons and gravitons in perturbation theory: [15] J. Collins, A. Perez, and D. Sudarsky, Lorentz invariance Derivation of Maxwell’s and Einstein’s equations, Phys. violation and its role in quantum gravity phenomenology, in Rev. 138, B988 (1965). Approaches to Quantum Gravity, edited by D. Oriti [4] S. Deser, Self-interaction and gauge invariance, Gen. (Cambridge University Press, Cambridge, 2006), p. 528–547. Relativ. Gravit. 1, 9 (1970). [16] C. Charmousis, G. Niz, A. Padilla, and P. M. Saffin, Strong [5] R. M. Wald, Spin-2 fields and general covariance, Phys. coupling in Horava gravity, J. High Energy Phys. 08 (2009) Rev. D 33, 3613 (1986). 070. [6] M. P. Hertzberg, Constraints on gravitation from causality [17] A. Papazoglou and T. P. Sotiriou, Strong coupling in exten- and quantum consistency, Adv. High Energy Phys. 2018, ded Horava-Lifshitz gravity, Phys. Lett. B 685, 197 (2010). 2657325 (2018). [18] T. Griffin, K. T. Grosvenor, C. M. Melby-Thompson, and [7] M. P. Hertzberg and M. Sandora, General relativity from Z. Yan, Quantization of Horava gravity in 2 þ 1 dimensions, causality, J. High Energy Phys. 09 (2017) 119. J. High Energy Phys. 06 (2017) 004. [8] G. Amelino-Camelia, Relativity in space-times with short [19] J. W. Moffat, Superluminary universe: A possible solution distance structure governed by an observer independent to the initial value problem in cosmology, Int. J. Mod. Phys. (Planckian) length scale, Int. J. Mod. Phys. D 11, 35 (2002). D 02, 351 (1993). [9] J. Magueijo and L. Smolin, Lorentz Invariance with [20] E. Pajer, D. Stefanyszyn, and J. Supeł, The boostless an Invariant Energy Scale, Phys. Rev. Lett. 88, 190403 bootstrap: Amplitudes without Lorentz boosts, arXiv:2007 (2002). .00027. [10] V. A. Kostelecky, Gravity, Lorentz violation, and the stan- [21] M. P. Hertzberg and M. Sandora, Special relativity from soft dard model, Phys. Rev. D 69, 105009 (2004). gravitons, Phys. Rev. D 96, 084048 (2017). [11] J. Collins, A. Perez, D. Sudarsky, L. Urrutia, and H. [22] M. P. Hertzberg and J. A. Litterer, Symmetries from locality. Vucetich, Lorentz Invariance and Quantum Gravity: An I. Electromagnetism and charge conservation, Phys. Rev. D Additional Fine-Tuning Problem? Phys. Rev. Lett. 93, 102, 025022 (2020). 191301 (2004). [23] M. P. Hertzberg, J. A. Litterer, and M. Sandora, Symmetries [12] D. Mattingly, Modern tests of Lorentz invariance, Living from locality. II. Gravitation and Lorentz boosts, Phys. Rev. Rev. Relativity 8, 5 (2005). D 102, 025023 (2020).

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[24] M. Tanabashi et al. (Particle Data Group), Review of [28] S. Deser and A. Waldron, Acausality of Massive Gravity, particle physics, Phys. Rev. D 98, 030001 (2018). Phys. Rev. Lett. 110, 111101 (2013). [25] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and [29] S. Deser, M. Sandora, A. Waldron, and G. Zahariade, R. Rattazzi, Causality, analyticity and an IR obstruction to Covariant constraints for generic massive gravity and analysis UV completion, J. High Energy Phys. 10 (2006) 014. of its characteristics, Phys. Rev. D 90, 104043 (2014). [26] B. P. Abbott et al. (LIGO Scientific and Virgo Collabora- [30] D. Lovelock, The Einstein tensor and its generalizations, tions), Observation of Gravitational Waves from a Binary J. Math. Phys. (N.Y.) 12, 498 (1971). Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016). [31] B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi-GBM, [27] N. Arkani-Hamed, H. Georgi, and M. D. Schwartz, Effec- and INTEGRAL Collaborations), Gravitational waves and tive field theory for massive gravitons and gravity in theory gamma-rays from a binary neutron star merger: GW170817 space, Ann. Phys. (Amsterdam) 305, 96 (2003). and GRB 170817A, Astrophys. J. Lett. 848, L13 (2017).

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