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Mathematical structure and physical content of composite in weak-field approximation

Author(s): Öttinger, Hans Christian

Publication Date: 2020-09-15

Permanent Link: https://doi.org/10.3929/ethz-b-000441983

Originally published in: Physical Review D 102(6), http://doi.org/10.1103/PhysRevD.102.064024

Rights / License: Creative Commons Attribution 4.0 International

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ETH Library PHYSICAL REVIEW D 102, 064024 (2020)

Mathematical structure and physical content of composite gravity in weak-field approximation

Hans Christian Öttinger * ETH Zürich, Department of Materials, CH-8093 Zürich, Switzerland

(Received 29 May 2020; accepted 17 August 2020; published 9 September 2020)

The natural constraints for the weak-field approximation to composite gravity, which is obtained by expressing the gauge vector fields of the Yang-Mills theory based on the Lorentz group in terms of tetrad variables and their derivatives, are analyzed in detail within a canonical Hamiltonian approach. Although this higher derivative theory involves a large number of fields, only a few degrees of freedom are left, which are recognized as selected stable solutions of the underlying Yang-Mills theory. The constraint structure suggests a consistent double coupling of matter to both Yang-Mills and tetrad fields, which results in a selection among the solutions of the Yang-Mills theory in the presence of properly chosen conserved currents. Scalar and tensorial coupling mechanisms are proposed, where the latter mechanism essentially reproduces linearized . In the weak-field approximation, geodesic particle motion in static isotropic gravitational fields is found for both coupling mechanisms. An important issue is the proper Lorentz covariant criterion for choosing a background Minkowski system for the composite theory of gravity.

DOI: 10.1103/PhysRevD.102.064024

I. INTRODUCTION composition rule expresses the corresponding gauge vector fields in terms of the tetrad or vierbein variables providing Einstein’s general theory of relativity may not be the a Cholesky-type decomposition of a metric. As a conse- final word on gravity. As beautiful and successful as it is, it quence of the composite structure of the proposed theory, seems to have serious problems both on very small and on it differs significantly from contentious previous attempts very large length scales. A problem on small length scales [6–8] to turn the Yang-Mills theory based on the Lorentz is signaled by 90 years of unwavering resistance of general group into a theory of gravity. relativity to quantization. A problem on the largest length Whereas the original formulation of composite gravity scales is indicated by the present search for “dark energy” in [2] was based on the Lagrangian framework, we here to explain the accelerated expansion of the universe within switch to the Hamiltonian approach. As a Hamiltonian general relativity. These problems provide the main moti- formulation separates time from space, it certainly cannot vation for continued research on alternative theories of provide the most elegant formulation of relativistic theo- gravity (see, for example, the broad review [1] of extended ries. However, the Hamiltonian framework has clear theories of gravity). advantages by offering a natural formulation of constraints A composite higher derivative theory of gravity has and a straightforward canonical quantization procedure. recently been proposed in [2]. The general idea of a For bringing constraints and quantization together, we here composite theory is to specify the variables of a “workhorse ” establish the constraints resulting from the composition rule theory in terms of more fundamental variables and their as second class constraints that can be treated via Dirac time derivatives [3,4]. The occurrence of time derivatives in – “ ” brackets [9 11], whereas gauge constraints can be handled the composition rule leads to a higher derivative theory, separately by BRST quantization (the acronym derives which is naturally tamed by the constraints resulting from from the names of the authors of the original papers the composition rule. For the composite theory of gravity [12,13]; see also [14,15]). Moreover, the Hamiltonian proposed in [2], the underlying workhorse theory is the approach provides the natural starting point for a gener- Yang-Mills theory [5] based on the Lorentz group, and the alization to dissipative systems. In particular, this approach allows us to formulate quantum master equations [16–19] *[email protected]; www.polyphys.mat.ethz.ch and to make gravity accessible to the robust framework of dissipative quantum field theory [20]. Published by the American Physical Society under the terms of The number of fields involved in the composite theory the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to of gravity is enormously large. Each Yang-Mills vector the author(s) and the published article’s title, journal citation, field has four components satisfying second-order differ- and DOI. ential equations so that, in the Hamiltonian approach, four

2470-0010=2020=102(6)=064024(18) 064024-1 Published by the American Physical Society HANS CHRISTIAN ÖTTINGER PHYS. REV. D 102, 064024 (2020)

2 3 μν additional conjugate momenta are required. For the Lorentz x , x are the spatial coordinates, and ημν ¼ η denotes the group, the six Yang-Mills vector fields associated with six Minkowski metric [with signature ð−; þ; þ; þÞ]. Greek infinitesimal generators (three rotations, three boosts) thus indices go from 0 to 3. The Minkowski metric, which is its result in 6 × 8 ¼ 48 fields. Gauge constraints eventually own inverse, is always used for raising or lowering reduce this number of degrees of freedom by a factor of 2 indices. Throughout this paper we set the speed (simply speaking, among the four components of a vector of light equal to unity (c ¼ 1). Assuming a background field, only the two transverse components carry physical Minkowski space comes with the advantage of offering a information). In addition, we consider 16 tetrad or vierbein clear understanding of energy, momentum, and their con- variables, again coming with conjugate momenta, so that servation laws. we deal with a total of 48 þ 32 ¼ 80 fields in our canonical Hamiltonian approach. Actually, this is not even the end of A. Tetrad variables and gauge vector fields the story as additional ghost fields would be introduced in κ the BRST approach for handling the gauge constraints. Our Standard tetrad or vierbein variables b μ result from a approach differs from the traditional Hamiltonian formu- Cholesky-type decomposition of a metric gμν, lation of general higher derivative theories developed by ¼ η κ λ ð Þ Ostrogradsky [21–23]. The Ostrogradsky framework gμν κλb μb ν; 1 would involve only 4 × 16 ¼ 64 fields, but would possess much less structure and fewer natural constraints [3]. A key which may also be interpreted as a coordinate transforma- task of the present paper is to elaborate in detail in the tion associated with a local set of base vectors. The context of the linearized theory that the constraints from the nonuniqueness of this decomposition is the source of the composition rule, together with the gauge constraints, gauge transformation behavior discussed in the next sub- reduce this enormous number of fields to just a few section. In the weak-field approximation, we write degrees of freedom, as expected for a theory of gravity. κ κ κλ ˆ Another important task of the present discussion of the b μ ¼ δ μ þ η hλμ; ð2Þ weak-field approximation is to provide guidance for the ˆ discussion of the fully nonlinear composite theory of where hλμ is assumed to be small so that we need to keep gravity. Understanding the structure of the constraints is only the lowest-order terms. In the rest of this paper, we use κ helpful also for proper coupling of the gravitational field to the symbol b μ for the linearized form (2) of the tetrad matter. Whereas the coupling of the Yang-Mills fields to variables rather than for the nonlinear version in Eq. (1) matter was considered previously [2], we here introduce a (with the exception of Appendix C, in which we consider properly matched additional coupling of the tetrad fields to the nonlinear static isotropic solution). It is convenient to ˆ matter. define the symmetric and antisymmetric parts of hμν, The structure of the paper is as follows. In a first step, we introduce the space of 80 fields for our canonical ˆ ˆ ˆ ˆ hμν ¼ hμν þ hνμ; ωμν ¼ hμν − hνμ: ð3Þ Hamiltonian formulation of linearized composite gravity, with special emphasis on gauge transformations and the In the weak-field approximation, we obtain the following implications of the composition rule (Sec. II). For the pure first-order expression for the metric (1): field theory in the absence of matter, we elaborate all evolution equations and constraints in detail, and we gμν ¼ ημν þ hμν: ð4Þ readily find the solutions for gravitational waves and static κ isotropic systems (Sec. III). We subsequently introduce a We denote the conjugate momenta associated with b μ μ double coupling mechanism for Yang-Mills and tetrad by pκ . Again, it is useful to introduce the symmetric and fields to matter into composite gravity. The modifications antisymmetric parts, resulting from the inclusion of matter are elaborated to obtain a complete theory of gravity that can be compared to h˜ μν ¼ pμν þ pνμ; ω˜ μν ¼ pμν − pνμ: ð5Þ linearized general relativity (Sec. IV). We finally summa- κ μ rize our results and draw a number of conclusions (Sec. V). The 32 fields b μ and pκ represent the canonical space The relation between the Lagrangian and Hamiltonian associated with the tetrad variables, which in turn character- approaches and some intermediate and additional results μν μν ize a metric. The fields h˜ and ω˜ may essentially be are provided in three appendixes. regarded as the conjugate momenta associated with hμν and ωμν, respectively (after properly accounting for normali- II. ARENA FOR COMPOSITE THEORY zation and symmetrization effects). For developing the composite theory of gravity, we The Hamiltonian description of a Yang-Mills theory is consider a fixed background Minkowski space where based on the four-vector fields Aaμ and their conjugates x0 ¼ ct is the product of the speed of light and time, x1, Eaμ, which are the generalizations of the vector potentials

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a TABLE I. Correspondence between label a for the base vectors in terms of six additional fields Λa. As the base vectors Tκλ of the six-dimensional Lie algebra soð1; 3Þ and ordered pairs of the Lie algebra defined in Eq. (6) are antisymmetric, ðκ˜ λ˜Þ ; of spacetime indices. only the antisymmetric part of bκλ is affected by gauge transformations, that is, a 123456 ðκ˜ λ˜Þ (0,1) (0,2) (0,3) (2,3) (3,1) (1,2) ; δhκλ ¼ 0; ð9Þ

whereas and the electric fields of electrodynamics, respectively. Whereas μ is the usual spacetime index, a labels the base a δωκλ ¼ −2g˜ΛaTκλ: ð10Þ vectors of the Lie algebra associated with the underlying Lie group. For the Lorentz group, which consists of the real The latter equation suggests that the six fields Λ can be 4 × 4 matrices that leave the Minkowski metric invariant, a chosen to make the six components of ωκλ equal to zero. the Lie algebra is six-dimensional. We here choose six We refer to this particular choice as the symmetric gauge. natural base vectors of the Lie algebra, three of which From Eqs. (7), (9), and (10), we further obtain generate the Lorentz boosts in the coordinate directions and the other three generate rotations around the coordinate ∂Λ δ ¼ a ð Þ axes. It is convenient to switch back and forth between the Aaμ μ ; 11 ∂x labels a ¼ 1; …; 6 for all six generators and the pairs (0,1), (0,2), (0,3) for the boosts in the respective directions which is the proper gauge transformation behavior for the (involving also time) and (2,3), (3,1), (1,2) for the gauge-vector fields of the linearized theory. This trans- rotations in the respective planes according to Table I.In formation rule implies gauge invariance of the combination particular, we can now write our base vectors of the Lie algebra as ∂ωκ˜ λ˜ 2gA˜ ˜ − ; ð12Þ ðκ˜ λÞμ ∂xμ a κ˜ λ˜ κ˜ λ˜ Tκλ ¼ δ λδ κ − δ κδ λ: ð6Þ which is obvious from the definition (7) and the gauge We finally need to specify the composition rule for invariance of hκλ. We moreover assume that all conjugate momenta are expressing the four-vector fields Aaμ in terms of the tetrad κ gauge invariant [cf. Eq. (49) of [15] ], fields b μ or, in view of Eq. (2) equivalently, the symmetric ˆ and antisymmetric parts hμν and ωμν, respectively, of hμν. δEaμ ¼ 0; ð13Þ For a ¼ðκ˜; λ˜Þ according to Table I, we postulate the simple composition rule and 1 ∂ ˜ ∂ 1 ∂ω hλμ hκμ˜ κ˜ λ˜ δ κλ ¼ 0 ð Þ Aaμ ¼ κ˜ − ˜ þ μ ; ð7Þ p : 14 2 ∂x ∂xλ 2g˜ ∂x It turns out below that the assumption (13) requires that where g˜ is a dimensionless coupling constant that controls ∂A μ=∂xμ must be a gauge invariant quantity. In view of the relative weight of the symmetric and antisymmetric a κ Eq. (11), this implies contributions to b μ. Only for g˜ ¼ 1 can the four-vector variables (7) be interpreted as a connection field [2]. Such ∂2Λ an interpretation would be essential for a closer relation to a ¼ 0 ð Þ ∂ ∂ μ : 15 general relativity. xμ x

In other words, the fields Λa generating gauge trans- B. Gauge transformations formations must become dynamic players and satisfy free As the Minkowski metric in the decomposition (1) is field equations. This idea is the basis of the BRST approach invariant under Lorentz transformations, the corresponding for handling gauge constraints. Moreover, we conclude κ transformation matrices can be applied to the factors b μ 2 without changing the metric. For infinitesimal Lorentz ∂ ωκλ μ ¼ 0; ð16Þ transformations, this implies the lowest-order gauge ∂xμ∂x transformation

which follows from the symmetric gauge ωκλ ¼ 0 and the δ ¼ −˜Λ a ð Þ bκλ g aTκλ; 8 gauge invariance of the left-hand side.

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C. Implications of composition rule and The composition rule (7) contains two types of equa- ∂ ∂ 1 ∂ω tions. If κ˜ or μ is equal to zero, it contains time derivatives h00 ¼ h0l − K 2 − 0l ð Þ ∂ ∂ 1 − Að0lÞl ˜ ∂ : 21 and hence implies time evolution equations for the tetrad t xl K g xl variables. Otherwise, the composition rule provides con- straints that must be satisfied at any time. From the expression for Að0lÞ0, we finally obtain The expressions for Að0kÞl þ Að0lÞk and AðklÞ0 lead to the ∂ω0l ∂h00 ∂hnn ∂hln unambiguous evolution equations (the Latin indices k, l ¼ 2gA˜ ð0 Þ0 þ g˜ ð1 − KÞ þ K − : ð22Þ ∂ l ∂ l ∂ l ∂ take the values 1,2,3) t x x xn ∂h 1 ∂h0 ∂h0 All the above evolution equations are gauge invariant. kl ¼ l þ k ¼ 0 ∂t 2 ∂xk ∂xl These evolution equations suggest that K could also be an appealing choice. 1 ∂ω ∂ω 0l 0k We now turn from the evolution equations to the þ Að0kÞl þ Að0lÞk − þ ð17Þ 2g˜ ∂xk ∂xl constraints implied by the composition rule. The obvious constraints are obtained by choosing only spatial indices in and Eq. (7), ∂ω ∂ ∂ kl h0l h0k ð Þ 1 ∂h ∂h 1 ∂ω ¼ 2gA˜ ð Þ0 − g˜ − : ð18Þ kl ¼ jl − jk þ kl ð Þ ∂ kl k l Aj : 23 t ∂x ∂x 2 ∂xk ∂xl 2g˜ ∂xj

The expression for Að0 Þ0 provides only the time derivative l Further constraints are obtained by considering Að0kÞl − ˜ þ ω of gh0l 0l, and there is no evolution equation for h00 Að0lÞk, whatsoever. Once we have made a decision about the evolution of h0μ, all evolution equations are fixed uniquely. ð0 Þ ð0 Þ 1 ∂h0 ∂h0 1 ∂ω0 ∂ω0 A k − A l ¼ l − k þ l − k : Choosing four conditions for h0μ is superficially remi- l k 2 ∂xk ∂xl 2g˜ ∂xk ∂xl niscent of imposing coordinate conditions for obtaining ð Þ unique solutions in general relativity, but the logical status 24 is entirely different. Whereas the coordinate conditions of general relativity have no influence on the physical In total, we have turned the composition rule for the 24 predictions of general relativity, in the canonical formu- components of the gauge vector fields and the 4 coordinate conditions (19) into the 16 evolution equations (17), (18), lation of composite gravity suitable conditions for h0μ are and (20)–(22) for the tetrad variables and the 9 þ 3 ¼ 12 used to characterize “good” or “valid” Minkowskian constraints (23) and (24). We refer to these constraints coordinate systems. If these conditions are Lorentz covar- resulting directly from the composition rule as the primary iant, we have no possibility of switching between different constraints of the composite theory. These primary con- types of conditions corresponding to different physical straints are not affected by coupling the gravitational field predictions. As obvious as these remarks may be, the to matter. However, the evolution equations for the tetrad proper appreciation of coordinate conditions in general variables should be expected to be changed by coupling relativity was a slow process, in which even Einstein could terms in the Hamiltonian. not easily detach himself from the idea of physically preferred coordinate systems [24]. An appealing set of Lorentz covariant conditions is III. PURE FIELD THEORY given by We are now ready to define the canonical Hamiltonian ∂ ∂ ν version of the composite theory of gravity in the weak-field hμν h ν approximation on the combined space of Yang-Mills and ¼ K μ ; ð19Þ ∂xν ∂x tetrad fields, following the general ideas developed in [4]. We first provide the Hamiltonian and then elaborate a ¼ 1 2 where K = corresponds to particularly convenient number of its implications. harmonic coordinates (in the linear approximation). We here adopt the conditions (19) as the tentative criteria for physically meaningful coordinates. They can be rewritten A. Hamiltonian as explicit time evolution equations, namely The Hamiltonian for the composite theory of pure gravity, ∂ ∂ ∂ ν h0l ¼ hln − h ν ð Þ K l 20 ¼ þ ð Þ ∂t ∂xn ∂x H HYM HYM=t; 25

064024-4 MATHEMATICAL STRUCTURE AND PHYSICAL CONTENT OF … PHYS. REV. D 102, 064024 (2020) consists of two contributions describing the workhorse ω˜ kl ¼ 0: ð31Þ theory and reproducing the evolution equations obtained from the composition rule, respectively. Our workhorse Only the external fluxes Jð0lÞμ can be nonvanishing in our theory is the linearized version of the Yang-Mills theory composite Yang-Mills theory of gravity. aμ based on the Lorentz group on the space ðAaμ;E Þ. The proper Hamiltonian is given by (see, e.g., Sec. 15.2 of [25], Chap. 15 of [26],or[15]; a derivation from the Yang-Mills B. Field equations Lagrangian is given in Appendix A) For the evolution of the conjugate momenta of the Z tetrad variables, we find the following results by means 1 ∂A ∂Aai ∂A ∂Aaj of Eq. (28): ¼ aμ þ ai − ai HYM E Eaμ j j 2 ∂x ∂xj ∂x ∂xi ∂h˜ kl ∂ðh˜ 0k − g˜ω˜ 0kÞ ∂ðh˜ 0l − g˜ω˜ 0lÞ ∂A ∂A 0 ¼ þ − a0 aj − aj a 3 ð Þ ∂ ∂ ∂ E E j d x: 26 t xl xk ∂xj ∂x ∂ðh˜ 0n − g˜ω˜ 0nÞ − 2Kδkl ; ð32Þ The Hamiltonian for coupling the Yang-Mills and tetrad ∂xn variables, ∂h˜ 00 ∂h˜ 0l ∂ω˜ 0l Z ¼ 2K þ 2g˜ð1 − KÞ ; ð33Þ ∂ ∂ l ∂ l ¼ _ κλ 3 t x x HYM=t bκλp d x Z ∂ ˜ 0l 1 ∂ ˜ ln 1 ∂ ˜ 00 1 ∂ ∂ω h h h hκλ ˜ κλ κλ κλ 3 ¼ þ ; ð34Þ ¼ h þ ω˜ d x; ð27Þ ∂t 2 ∂xn 2 ∂x 4 ∂t ∂t l ∂ω˜ 0l 1 ∂ ˜ ln 1 ∂ ˜ 00 is chosen such that the canonical evolution equations ¼ − h þ K h ð Þ n ; 35 ∂t 2g˜ ∂x 2g˜ 1 − K ∂xl ∂ δ ∂ κλ δ bκλ ¼ H p ¼ − H ð Þ κλ ; 28 and ∂t δp ∂t δbκλ ∂ω˜ kl reproduce the evolution equations (17), (18), and (20)–(22) ¼ 0 ð Þ ∂ : 36 for the tetrad variables. These evolution equations implied t by the composition rule and the coordinate conditions (19) Note that these equations for the conjugate momenta of the are of crucial importance for finding the Hamiltonian tetrad variables are independent of any other variables. The HYM=t, that is, for obtaining the complete canonical last of these evolution equations is consistent with our Hamiltonian formulation of the composite theory. κλ previous conclusion (31). According to the definition (29), We have introduced the variables p in a purely formal Eq. (35) can be rewritten as manner as the conjugate momenta of the tetrad variables. At this point, we can offer a physical interpretation. Note ∂Jð0lÞμ that, in view of the evolution equations of the tetrad ¼ 0; ð37Þ ∂xμ variables, the Hamiltonian HYM=t contains a contribution κλ that is bilinear in the variables p and the gauge vector which supports our interpretation of conjugate tetrad fields Aaμ. This contribution can be written in the form variables in terms of conserved fluxes. aμ −J Aaμ with the identifications The evolution equations for the Yang-Mills fields are obtained from (note the sign conventions) 1 1 ð0lÞ0 0l ð0lÞj ˜ lj K ˜ 00 lj −J ¼ g˜ω˜ ; −J ¼ h − h η ; ð29Þ aμ ∂A μ δ ∂ δ 2 2 1 − K a ¼ − H E ¼ H ð Þ aμ ; : 38 ∂t δE ∂t δAaμ and The resulting equations can be written in the following ð Þ0 ð Þ −J kl ¼ g˜ω˜ kl; −J kl j ¼ 0: ð30Þ form:

κλ ∂ a ∂ a The symmetric and antisymmetric parts of the variables p A0 a An ¼ −E0 þ ð39Þ play the role of external Yang-Mills fluxes. By requiring ∂t ∂xn Lorentz covariant fluxes, Eq. (30) immediately leads to the conclusion and

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a ∂A ∂Aa ð0 Þ ð0 Þ j ¼ −Ea þ 0 ð40Þ E l ¼ E k ; ð47Þ ∂t j ∂xj k l for the gauge vector fields, whereas their conjugate partners and the tertiary constraints are subsequently obtained as are governed by ∂ ðklÞ ∂ ð0jÞ ∂ ð0jÞ ∂ ∂ ðklÞ ∂ ðklÞ E0 El Ek An Aj ð0 Þ ð0 Þ − þ ¼ − ; ð48Þ ∂ l ∂ l j k l j n E0 En ð0lÞ ∂x ∂x ∂x ∂xn ∂x ∂x ¼ − − J0 ; ð41Þ ∂t ∂xn ð0lÞ ð0kÞ ðklÞ ∂E0 ∂E0 ∂En ðklÞ ðklÞ − ¼ : ð49Þ ∂E0 ∂En ∂x ∂x ∂x ¼ − ; ð42Þ k l n ∂t ∂xn The latter constraint has been simplified by means of the ð0 Þ ð0 Þ ∂E l ∂ ð0lÞ ∂2A l ∂2 ð0lÞ identity (45). Note that these tertiary constraints can be j ¼ − E0 − j þ An − ð0lÞ ð Þ j n j Jj ; 43 used to rewrite the evolution equations (42) and (44) as ∂t ∂x ∂x ∂xn ∂x ∂xn ð Þ ð0 Þ ð0 Þ ∂E kl ∂E k ∂E l and 0 ¼ 0 − 0 ð50Þ ∂t ∂xl ∂xk ðklÞ ð Þ 2 ðklÞ ð Þ ∂E ∂E kl ∂ A ∂2A kl j ¼ − 0 − j þ n ð Þ and j n j : 44 ∂t ∂x ∂x ∂xn ∂x ∂xn ðklÞ ð0 Þ ð0 Þ ∂E ∂E j ∂E j Note that these evolution equations are gauge invariant, j ¼ k − l : ð51Þ ∂ ∂ l ∂ k provided that Eq. (15) for Λa holds. These are the t x x linearized standard field equations for Yang-Mills fields, κλ ’ Up to this point, the variables p do not appear in the which are strongly reminiscent of Maxwell s equations of κλ electrodynamics. constraints. From now on, only the variables p occur in Equation (40), together with the representation (7), the constraints. In the next round, we find implies the useful identity ∂Jð0lÞj ∂Jð0kÞj − ¼ 0; ð52Þ ðlnÞ þ ðnkÞ þ ðklÞ ¼ 0 ð Þ ∂x ∂x Ek El En : 45 k l ð0 Þ0 ð0 Þ0 This identity remains valid when we later include matter ∂J l ∂J k − ¼ 0: ð53Þ [that is, it can more generally be derived from Eq. (B6)]. ∂xk ∂xl

C. Constraints As we assume that, in the absence of matter, the external fluxes (29) vanish, these last conditions are satisfied The primary constraints (23) and (24) must be valid at all trivially so that the hierarchy of constraints ends at this times. From the time derivative of the primary constraints point. we obtain secondary constraints, a further time derivative We have arrived at a total of 4 × 12 ¼ 48 constraints yields tertiary constraints, and so on. This iterative process, resulting from the composition rule, supplemented by the in which the required time derivatives are evaluated by three constraints (31) so that the total is 51. All these means of the evolution equations, is continued until no constraints are gauge invariant. This is a consequence of the further constraints arise. The crucial question is whether the fact that the composition rule is designed such that the four- iterative process stops before all degrees of freedom are vector fields Aaν possess the proper gauge transformation fixed by constraints. As the introduction revealed that, in behavior (11) and all the evolution equations are gauge the canonical Hamiltonian formulation of composite gra- invariant. We eventually argue in favor of the 16 constraints vity, we are dealing with 80 fields, we need around 75 pκλ ¼ 0 (or h˜ κλ ¼ ω˜ κλ ¼ 0), which would replace the 15 constraints to obtain an appropriate number of degrees of constraints (31), (52), (53) and actually increase the count freedom for a theory of gravity. by one. The secondary constraints obtained as the time deriva- In a Yang-Mills theory, half of the degrees of freedom tives of the primary constraints can be formulated nicely in can be eliminated by gauge constraints (roughly speaking, terms of Yang-Mills variables, the four-vector potentials have only transverse components,

ð0jÞ ð0jÞ no longitudinal or temporal ones). In our case, we have 24 ð Þ ∂A ∂A E kl ¼ l − k ; ð46Þ gauge constraints, which brings us to a total of 75 (or 76) j ∂xk ∂xl constraints for our 80 fields. It is quite remarkable that just

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2 2 a few of the 80 degrees of freedom survive, as we would ∂ hμν ∂ ¼ f ð Þ λ μ ν ; 57 expect for a theory of gravity. ∂xλ∂x ∂x ∂x

D. Compact form of theory possibly after a minor redefinition of f. The goal of this subsection is to find a closed set of An interesting feature of these field equations is that differential equations for the tetrad variables. To reach this the function f, which results from integration, needs to be goal it is important to express all the Yang-Mills variables determined simultaneously with the solutions hμν.We arrive at a set of second-order differential equations because in terms of the tetrad variables. For the vector fields Aaμ, the desired expression is given by the composition rule (7). higher derivative equations play the role of integrability μ ˜ Their conjugates Ea can then be extracted from the conditions. The coupling constant g does not occur in these evolution equations (39) and (40) (see Appendix B for a equations. Possible antisymmetric contributions to the summary of the resulting expressions). tetrad variables are governed by the wave equations (16), As we have already recognized Jð0lÞμ ¼ 0 ¼ ω˜ kl, and all conjugate momenta of the tetrad variables must Eqs. (32)–(36) imply that all conjugate tetrad variables vanish according to Eq. (54). must be constant and can be assumed to be zero, E. Comparison to general relativity pκλ ¼ 0: ð54Þ Einstein’s field equation for pure gravity in the weak- field approximation to general relativity is given by a This is a very desirable condition for the natural canonical vanishing curvature tensor [see Eq. (A18)], Hamiltonian approach to composite theories. As the con- κλ jugate momenta p appear linearly in the Hamiltonian (27), 2 2 λ 2 λ 2 λ ∂ hμν ∂ h μ ∂ ∂ they lead to an unbounded Hamiltonian and consequently to − − h ν þ h λ ¼ 0 ð Þ λ λ ν μ λ μ ν : 58 the famous risk of instabilities in higher derivative theories ∂xλ∂x ∂x ∂x ∂x ∂x ∂x ∂x [21,22]. Avoiding such instabilities is an important topic, in particular, in alternative theories of gravity [27–34]. The It is important to note that the coordinates xμ in general constraints (54) provide the most obvious way of eliminat- relativity are not associated with an underlying Minkowski ing instabilities in the canonical Hamiltonian approach to space so that these field equations can be simplified by composite higher derivative theories, which differs from the suitable general coordinate transformations. If we impose usual Ostrogradsky approach [3,4]. Moreover, this con- the same coordinate conditions (19) as used in composite straint implies that we have to solve the original Yang-Mills gravity, the field equations (58) of linearized general equations without any modification. In other words, the relativity simplify to composite theory simply selects solutions of the Yang-Mills 2 2 λ theory based on the Lorentz group to obtain the composite ∂ hμν ∂ ¼ð2 − 1Þ h λ ð Þ theory of gravity. This insight provides a more direct λ K μ ν : 59 ∂xλ∂x ∂x ∂x argument for the stability of solutions. Note that the large number of constraints and the small number of remaining This equation coincides with Eq. (57) for composite gravity degrees of freedom indicates that the composite theory is λ for f ¼ð2K − 1Þh λ. It becomes particularly simple for highly selective. ¼ 1 2 From Eq. (44) we obtain harmonic coordinates with K = , which may be pic- tured as nearly Minkowskian (see, e.g., pp. 163 and 254 ∂2 ∂2 of [35]). hkl ¼ f ð Þ μ k l ; 55 As in general relativity, the solutions for the deviatoric ∂xμ∂x ∂x ∂x metric hμν in harmonic coordinates can assume all kinds of polarization states, including longitudinal and temporal where the unknown function f results from integration, and components. The actual polarization of gravitational waves similarly Eq. (42) gives depends on the nature of their source (typically binary 2 systems of two black holes, two neutron stars, or a black ∂ h0 ∂f0 l ¼ : ð56Þ hole and a neutron star during their in-spiral or merger ∂ ∂ μ ∂ l xμ x x phases). Equations (41) and (43) provide further integrability conditions that can be exploited in a similar manner. F. Static isotropic solution Consolidating all the results, we get the following compact To find the static isotropic solution for the weak-field formula summarizing the linear version of the composite approximation to composite gravity for the coordinate theory of pure gravity on the level of tetrad variables, conditions (19), we start from the general ansatz

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¼ β¯ð Þ ¼ α¯ð Þδ þ ξ¯ð Þxkxl ¼ ¼ 0 want to understand the motion of matter in a gravita- h00 r ;hkl r kl r 2 ;h0k hk0 ; r tional field. ð60Þ The most convenient options for describing matter are given by point particle mechanics or hydrodynamics. We 2 2 2 1=2 with r ¼ðx1 þ x2 þ x3Þ . The coordinate conditions (19) here consider a single point particle either generating a become gravitational field or moving in a gravitational field.

0 ¯ 0 ¯0 ¯ r½ð3K − 1Þα¯ − Kβ þðK − 1Þξ ¼2ξ: ð61Þ A. Particle in a gravitational field As a starting point for discussing particle motion in a A prime on a function of r indicates the derivative with weak gravitational field, we use the first-order expansion of respect to r. the standard Hamiltonian, We assume that also f in Eq. (57) is static and isotropic. With f ¼ fðrÞ, Eq. (57) leads to two equations, 1 ¼ γ − μν ð Þ Hm m pμpνh ; 66 2 00 0 2γm r β¯ þ 2rβ¯ ¼ 0 ð62Þ where m is the rest mass of the particle, hμν depends on the and j particle position x , the particle momentum is given by pj, 0 x x and we define −p0 ¼ p ¼ mγ, where k l ðr2ξ¯00 þ 2rξ¯0 − 6ξ¯ − r2f00 þ rf0Þ r2 2 1 ¼ δ ð 0 − 2ξ¯ − 2α¯ 00 − 2 α¯ 0Þ ð Þ p 2 kl rf r r ; 63 γ ¼ 1 þ ; ð67Þ m where each side of the latter equation must vanish separately. All these equations are of the equidimensional type; that is a function of the spatial components pj of the particle 0 is, in each term there are as many factors of r as there are momentum. When a higher-order definition of p is 0 ¼ derivatives with respect to r, suggesting simple power-law required, one should use p Hm, where Eq. (66) provides μν solutions. Equation (62) implies β¯ 0 ∝ r−2, and we hence the first-order result in h . The lowest-order energy- write momentum tensor is given by (see, e.g., Eq. (2.8.4) of [35]) δ ¯ r0 Hm pμpν 3 βðrÞ¼2 ; ð64Þ Tμν ¼ −2 ¼ δ ðx − xðtÞÞ r δhμν γm

dxμ dxν 3 where r0 is a constant length scale and a possible ¼ γm δ ðx − xðtÞÞ; ð68Þ additive constant has been omitted to obtain asymptotic dt dt α¯ Minkowskian behavior. Equation (61) suggests that has where the lowest-order result p ¼ mγdx =dt has been the same power-law decay, so that the right-hand side of j j used. The evolution equation dp =dt ¼ 0 for a free particle Eq. (63) implies rf0 ¼ 2ξ¯. Equation (61) provides a relation j in the absence of gravity leads to the result among prefactors, so that we can write j 2 ∂Tμν ∂Tμν dx pj ∂Tμν c¯ r0 c¯ − ð3c¯ − 2ÞK − 2K r0 ¼ − ¼ ð Þ ¯ j j ; 69 α¯ðrÞ¼ ; ξðrÞ¼ : ð65Þ ∂t ∂x dt p0 ∂x 1 − K r 1 − K2 r ν ¼ 0 Consistency with general relativity, which implies a from which, for , we obtain energy-momentum vanishing curvature tensor, requires c¯ ¼ 1. The condition conservation in the form c¯ ¼ 1 is not predicted by the weak-field approximation of μν ∂T pure composite gravity, but it arises naturally in the full, ¼ 0 ð Þ ∂ ν : 70 nonlinear theory or from a suitable coupling to matter (see x Sec. IV E below). By construction, the Hamiltonian (66) leads to geodesic motion in a weak field. The potential distortion of geodesic IV. COUPLING OF FIELD TO MATTER motion by further couplings between matter and field is explored in Sec. IV G below. Of course, we cannot really appreciate a theory of the gravitational field without coupling it to matter. On the one hand, we want to understand the gravitational field gen- B. Hamiltonian for coupling field and matter μν erated by matter, say for calculating the parameters c¯ and r0 The occurrence of h in the Hamiltonian (66) already in the solution given in Eq. (65). On the other hand, we implies a coupling of field and matter. It leads to geodesic

064024-8 MATHEMATICAL STRUCTURE AND PHYSICAL CONTENT OF … PHYS. REV. D 102, 064024 (2020) motion in the given field hμν, but it does not provide composite theory selects solutions from a pure Yang- meaningful field equations for determining gravitational Mills theory, which is a consequence of the vanishing fields. For that purpose we need to couple the Yang-Mills conjugate momenta pκλ of the tetrad variables established field to the energy-momentum tensor of matter. In in Eq. (54). In the presence of matter, it is more natural to Appendix A, the details of the coupling are discussed in select solutions of the Yang-Mills theory with suitable a Lagrangian setting, and the following Hamiltonian for the external fluxes, so that the conjugate momenta pκλ should coupling is obtained, no longer be expected to vanish. Use of the separate Z Hamiltonian (74) in addition to the previously suggested ¼ ð ðλnÞ j − ðλjÞ 0 − ð0lÞ j Þ 3 ð Þ coupling mechanism (71) allows us to find a consistently HYM=m F C λ E C λ E C l d x; 71 jn j j tuned coupling of both Yang-Mills and tetrad fields to matter. Note that the “general wisdom” about the possibil- with the field tensor of the linearized Yang-Mills theory, ities of coupling gravity to matter [36–38] is not beyond all

ðκλÞ ðκλÞ doubt [39] and, in the context of composite theories, this ðκλÞ ∂Aν ∂Aμ coupling can be even richer. Fμν ¼ − ; ð72Þ ∂xμ ∂xν For obtaining the composite theory of gravity in the presence of matter, we would like to add the Hamiltonians and the tensor HYM=m, Ht=m, and Hm introducing the coupling of field and ∘ matter to the Hamiltonian (25) of pure gravity. However, λ Cμν ¼ G1Tμν þ G2ημνT λ; ð73Þ there is a problem. With the help of Table II, we realize that the Hamiltonian (25) has dimension of length−1, and so ∘ where Tμν is the traceless part of the energy-momentum does the Hamiltonian HYM=m defined in Eq. (71). The tensor of matter defined in Eq. (A10) and the coefficients dimensions of the Hamiltonian Ht=m defined in Eq. (74) can G1 and G2 must have the same dimensions as Newton’s still be adjusted by the definition of Vκλ. However, the constant G (cf. Table II). The concrete values of G1, G2 can Hamiltonian Hm defined in Eq. (66) has dimensions of only be chosen once we have elaborated all the equations mass, which is what we actually expect for a Hamiltonian for gravitational fields coupled to matter. when using the speed of light as the unit for veloc- The Lagrangian associated with the Hamiltonian (71) for ities (c ¼ 1). the coupling of the Yang-Mills field to matter has pre- As the mismatch in dimensions can be regarded as an þ þ viously been proposed in Eqs. (51) and (52) of [2]. We here action factor, it seems natural to multiply HYM HYM=t ’ ℏ introduce an additional coupling of the tetrad field to HYM=m by Planck s constant . We do that implicitly by matter, using ℏ as the unit of action (ℏ ¼ 1), thus eliminating the Z dimensional mismatch. However, this choice of units þ κλ 3 implies that, in HYM HYM=t, we actually deal with the H ¼ Vκλp d x; ð74Þ t=m energy of gravitational field quanta, which is clearly not the most appropriate energy scale when we usually consider where, for the linearized theory, Vκλ can be assumed to be a problems involving gravity. We hence introduce a very Λ symmetric tensor to be constructed from the energy- small dimensionless parameter E to scale down the typical momentum tensor of matter (more precisely, the time energy associated with gravitationally interacting masses to derivative of Vκλ turns out to be a tensor; the defining the level of energies, equations and more insight into the role of the indices are provided in Sec. IV D). The idea behind this additional ¼ þ þ þ Λ ð þ Þ ð Þ H HYM HYM=t HYM=m E Ht=m Hm : 75 coupling is as follows. In the absence of matter, the In the Lagrangian formulation in Eq. (52) of [2], it can be Λ TABLE II. Dimensions of various quantities in terms of length recognized that E plays the role of a dimensionless (L) and mass (M) for c ¼ 1. cosmological constant in general relativity. We hence write Quantities Dimensions l 2 Λ ¼ p ð Þ κ E ; 76 gμν, b μ, hμν, ωμν, ΛE dimensionless D a −1 Aν , H L pffiffiffiffiffiffiffiffiffiffiffiffiffi a a a μν −2 pffiffiffiffi Eν , Bν , Fμν, R L where l ¼ ℏG=c3 ¼ G is the Planck length and D is κλ ˜ κλ κλ −3 p p , h , ω˜ L Λ the diameter of the observable universe. This parameter E Vκλ M −124 −3 can be estimated to be of the order of 10 . It is interesting Tμν L M −1 to note that even our formulation of classical gravity G, G1, G2 LM requires an action constant. A similar situation arises in

064024-9 HANS CHRISTIAN ÖTTINGER PHYS. REV. D 102, 064024 (2020) formulating the entropy of a classical ideal gas, indicating ∂ ðklÞ ðklÞ ∂2 ðklÞ 2 ðklÞ Ej ∂E0 Aj ∂ An that a deeper understanding of an ideal gas requires ¼ − − þ ∂t ∂xj ∂xn∂x ∂xj∂x quantum theory. The same conclusion may be true for a n n ∂C ∂C ∂ ∂ deeper understanding of gravity. þ jk − jl þ δ Cnl − δ Cnk ð Þ l k jk jl : 83 ∂x ∂x ∂xn ∂xn C. Modified field equations The fact that Cμν occurs in Eqs. (80) and (81) for the gauge In the presence of matter, the dynamic aspects of the vector fields underlines that the coupling of the stress tensor composition rule (7) are affected by the Hamiltonian H , t=m to the workhorse theory of composite gravity does not but not its static aspects. In other words, the primary happen via the usual flux mechanism for Yang-Mills constraints are unchanged, whereas the evolution equations theories. for the tetrad variables get modified. As Vκλ is assumed to μν The occurrence of h in Eq. (66) implies that the be symmetric, only the evolution equations (17), (20), and evolution equations (32)–(34) for the symmetrized con- (21) for hκλ get changed, jugate momenta h˜ κλ get modified, too. We find ∂hkl 1 ∂h0l ∂h0k ¼ þ þ Að0 Þ þ Að0 Þ ∂ ˜ kl ∂ð˜ 0k − ˜ω˜ 0kÞ ∂ð˜ 0l − ˜ω˜ 0lÞ ∂t 2 ∂xk ∂xl k l l k h ¼ h g þ h g ∂t ∂xl ∂xk 1 ∂ω0l ∂ω0k − þ þ 2Λ V ; ð77Þ ˜ 0n 0n 2˜ k l E kl ∂ðh − g˜ω˜ Þ g ∂x ∂x − 2Kδ þ 2Λ Tkl; ð84Þ kl ∂xn E ∂ ∂ ∂ ν h0l ¼ hln − h ν þ 2Λ ð Þ K l EV0l; 78 ˜ 00 ˜ 0l ˜ 0l ∂t ∂x ∂x ∂h ∂h ∂ω 00 n ¼ 2K þ 2g˜ð1 − KÞ þ 2Λ T ; ð85Þ ∂t ∂xl ∂xl E and and ∂ ∂ 1 ∂ω h00 ¼ h0l − K 2 − 0l þ 2Λ ð Þ Að0lÞl EV00: 79 ∂t ∂x 1 − K g˜ ∂x ∂h˜ 0l 1 ∂h˜ ln 1 ∂h˜ 00 l l ¼ þ þ 2Λ 0l ð Þ n ET : 86 ∂t 2 ∂x 2 ∂xl Equations (78) and (79) imply a tiny modification of the coordinate conditions (19). The occurrence of the energy-momentum tensor in The Hamiltonian HYM=m given in Eq. (71) depends only Eqs. (84) and (85) is a very important qualitative modifi- on the spatial components of the Yang-Mills fields, so that cation. As anticipated, the conjugate momenta of the the evolution equations (39), (41), and (42) for the temporal tetrad variables do not vanish in the presence of matter. components of the Yang-Mills fields remain unaffected. Λ Remember, however, that the dimensionless parameter E Equation (40) gets modified to is extremely small.

∂ ð0lÞ ð0lÞ Aj ð0lÞ ∂A0 ¼ −E þ − C þ δ C00 ð80Þ D. Modified constraints ∂t j ∂xj jl jl In the presence of matter, the primary constraints (23) and and (24) remain unchanged. The secondary constraints (46) change to ∂ ðklÞ ∂ ðklÞ Aj ðklÞ A0 ð0jÞ ð0jÞ ¼ −E þ þ δjkC0l − δjlC0k; ð81Þ ð Þ ∂A ∂A ∂V ∂V ∂t j ∂xj E kl ¼ l − k − Λ jl − jk j ∂xk ∂xl E ∂xk ∂xl whereas Eqs. (43) and (44) become þ δjkC0l − δjlC0k; ð87Þ

∂ ð0lÞ ð0lÞ ∂2 ð0lÞ 2 ð0lÞ E ∂E0 A ∂ An ð0 Þ whereas Eq. (47) becomes j ¼ − − j þ − J l ∂ ∂ j ∂ n∂ ∂ j∂ j t x x xn x xn ∂ ∂ ∂C 0 ∂ ð0lÞ ð0kÞ V0l V0k − j þ δ Cn0 ð Þ E − E ¼ Λ − : ð88Þ jl 82 k l E ∂ k ∂ l ∂xl ∂xn x x and The tertiary constraints (48) become

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∂ ðklÞ ∂ ð0jÞ ∂ ð0jÞ ∂2Vlν E0 El Ek ð0lÞν ¼ −Λ ð Þ − þ J E μ ; 96 ∂xj ∂xk ∂xl ∂xμ∂x ð Þ ð Þ ∂ ∂ kl ∂A kl ∂ ∂V ∂V ¼ An − j þ Λ jl − jk E where suitable initial and boundary conditions need to be ∂x ∂ j ∂xn ∂t ∂ k ∂ l ν n x x x imposed to find Vl . There is no need to choose any ∂T00 ∂T00 particular form of V00 because, according to Eq. (96), there þ δ G1 − δ G1 ; ð89Þ jk ∂xl jl ∂xk is no flux component associated with it. We hence assume V00 ¼ 0, unless there is any particular need to modify and Eq. (49) changes to Eq. (79). Note that ν is a four-vector index, whereas l is related to the labels of the Lie algebra (more precisely, l is ∂ ð0lÞ ∂ ð0kÞ ∂ ðklÞ ∂ ∂ μl ∂ μk E0 − E0 ¼ En þ Λ V − V ð Þ the label for the Lorentz boosts). Equations (94) and (95) E μ : 90 ∂xk ∂xl ∂xn ∂x ∂xk ∂xl can now be written as

Finally, the quaternary constraints (52) and (53) become ∂ ∂2Vlν ∂ ∂2Vlν ¼ lν ¼ − 0ν ð Þ μ T ; l μ T ; 97 ∂t ∂xμ∂x ∂x ∂xμ∂x ∂ ð0lÞj ∂ ð0kÞj ∂2 ∂ jl ∂ jk J − J ¼ −Λ V − V ð Þ E μ 91 ∂xk ∂xl ∂xμ∂x ∂xk ∂xl lν implying that V and Ht=m have dimensions of mass or ¼ 1 lν and energy (c ). As announced, V is determined by the energy-momentum tensor and vanishes in the absence of ∂ ð0lÞ0 ∂ ð0kÞ0 ∂2 ∂ 0l ∂ 0k matter. J − J ¼ −Λ V − V ð Þ ∂ ∂ E ∂ ∂ μ ∂ ∂ : 92 Note that the three derivatives in Eq. (97) are required to xk xl xμ x xk xl go from the level of lowest derivatives (tetrad variables) to the level of highest derivatives (conjugate tetrad variables), At this stage we have to make a proper choice of the κλ with the gauge vector fields and their conjugates in between functions V in the Hamiltonian in order to avoid further [compare, for example, Eqs. (77) and (84)]. Note that the constraints that would quickly make it impossible to find any different numbers of derivatives occurring in the various solutions to the entire set of constraints. For this purpose we fields are also reflected in the different powers of L−1 in added a coupling of matter to the tetrad variables in addition Table II. to the more obvious coupling to the Yang-Mills variables. In the presence of matter, the procedure for selecting As a first step, we want to identify further vanishing among the solutions of the Yang-Mills theory with external conjugate tetrad variables because, according to Eq. (29), fluxes extends the idea of composite theories. This selec- ω˜ 0l ˜ kl only the variables and h carry essential information. tion criterion should provide stability instead of the Careful inspection of the structure of the evolution equations vanishing conjugate momenta associated with the tetrad suggests the following choices of vanishing variables in variables for the composite theory of pure gravity. Again, addition to those given in Eq. (31), the selection is very restrictive so that the composite theory of gravity possesses only a few degrees of freedom. h˜ 00 ¼ 0; h˜ 0l − g˜ω˜ 0l ¼ 0: ð93Þ

The evolution equations for the conjugate tetrad variables E. Compact form of theory then reduce to the much simpler form As in Sec. III D, we would like to find a closed set of differential equations for the tetrad variables, but now in the ∂h˜ kl ∂h˜ kl presence of matter. Again we need to express all the Yang- ¼ 2Λ Tkl; ¼ −2Λ Tk0; ð94Þ ∂t E ∂xl E Mills variables in terms of the tetrad variables. Expressions for the vector fields Aaμ can be obtained from the evolution and equations (18), (22), (77) and the primary constraints (23). Their conjugates Eaμ can then be extracted from the ∂ω˜ 0l ∂ω˜ 0l original evolution equation (39) for the temporal compo- g˜ ¼ Λ T0l; g˜ ¼ −Λ T00: ð95Þ ∂t E ∂xl E nents and the modified Eqs. (80) and (81) for the spatial components of the gauge vector fields. For the convenience Note that the consistency between the two members of the reader, the explicit representations are listed in of each equation is guaranteed by energy-momentum Appendix B. By construction, these expressions satisfy conservation. the primary constraints identically. The quaternary constraints can now be satisfied if we We only need to consider the evolution equations for the κλ a construct V by solving the Poisson equations conjugate Yang-Mills fields Eμ (the higher constraints can

064024-11 HANS CHRISTIAN ÖTTINGER PHYS. REV. D 102, 064024 (2020) be verified in a straightforward manner). From Eq. (83) we The compact equation (103) has a remarkable similarity obtain with the linearized version of Einstein’s field equation (A6) with the curvature tensor (A18) in a harmonic coordinate 2 ∂ 1 ∂ h ∂C μ system, provided that we choose jk þ þ δ l l μ Cjk jk ∂x 2 ∂xμ∂x ∂xμ ¼ 8π ¼ 0 ð Þ 2 G1 G; G2 ; 104 ∂ 1 ∂ h ∂C μ ¼ jl þ þ δ k ð Þ k μ Cjl jl : 98 ∂x 2 ∂xμ∂x ∂xμ and f ¼ 0. The freedom of choosing the function f is the only leftover from the higher derivative nature of the theory. By using that the tensor Tμν in Eq. (73) satisfies the energy- It gives us the remarkable possibility to mimic the local momentum conservation (70), we obtain the following gauge degree of freedom associated with the general generalization of Eq. (55): coordinate transformations employed to achieve the one- parameter family of coordinate conditions (19), although 2 1 ∂ hkl 1 λ λ the composite theory is defined in Minkowski space. μ þ G1 Tkl − T ληkl þ 2G2T ληkl 2 ∂xμ∂x 2 F. Isotropic solution revisited 1 ∂2f ¼ ; ð99Þ As an application of our compact equations, we consider 2 ∂xk∂xl a mass M resting at the origin, which is represented by an where the function f results from integration of the third- energy-momentum tensor Tμν with only one nonvanishing 3 order equations. From Eq. (42) we obtain another integra- component, T00 ¼ Mδ ðxÞ. Equation (97) requires nonzero 0 bility condition, components Vl . A simple solution of this equation is found to be 2 2 ∂ 1 ∂ h0k ∂ 1 ∂ h0l þ G1T0 ¼ þ G1T0 : l ∂ l 2 ∂ ∂ μ k ∂ k 2 ∂ ∂ μ l 0 M x x xμ x x xμ x Vl ¼ ; ð105Þ 8π r ð100Þ which describes a purely orientational effect. The complete From Eqs. (41) and (82) we obtain after using Eqs. (70) list of conjugate tetrad variables is given by and (96) Λ M xl ˜ 0l ¼ ˜ω˜ 0l ¼ − E ω˜ kl ¼ ˜ kl ¼ ˜ 00 ¼ 0 ð Þ ∂ 1 ∂2 1 h g 3 ; h h : 106 h00 þ − λ η þ 2 λ η 4π r l μ G1 T00 T λ 00 G2T λ 00 ∂x 2 ∂xμ∂x 2 Note that the modification of the coordinate condition (78) ∂ 1 ∂2 h0l is extremely tiny, but independent of the distance from the ¼ μ þ G1T0l ð101Þ ∂t 2 ∂xμ∂x central mass. We now focus on the field equations (103) with the and parameter choices (104). Away from the origin, these equations have already been solved in Sec. III F.By ∂2 ∂ 1 hjl 1 λ λ integrating the simplified field equations μ þ G1 Tjl − T ληjl þ 2G2T ληjl ∂t 2 ∂xμ∂x 2 ∂2 ∂2ð − Þ 2 h00 hll f ∂ 1 ∂ h0 þ G1T00 ¼ 0; þ 3G1T00 ¼ 0; ð107Þ ¼ j þ ð Þ ∂x ∂x ∂x ∂x l μ G1T0j ; 102 n n n n ∂x 2 ∂xμ∂x

over a sphere around the origin and using hll − f ¼ lν ¯ respectively. Again, the choice (96) of V is of crucial 3ðα¯ þ ξÞ, we find r0 ¼ MG and c¯ ¼ 1 for the coefficients importance because it leads to further integrability con- in the solutions (64) and (65). More details about isotropic ditions. Equations (100)–(102) allow us to extend the solutions can be found in Appendix C. differential equation (99) to all components, G. Modified particle motion ∂2 1 hμν 1 λ λ þ G1 Tμν − T λημν þ 2G2T λημν For obtaining the motion of a particle with mass m 2 ∂ ∂ λ 2 xλ x in a gravitational field, it is convenient to divide the 1 ∂2f Hamiltonian (75) by Λ because the resulting equations ¼ ; ð103Þ E 2 ∂xμ∂xν then look more familiar. Whereas the variational problem of the Lagrangian approach is clearly unaffected by such a possibly after a minor modification of f. constant factor, it corresponds to a rescaling of the particle

064024-12 MATHEMATICAL STRUCTURE AND PHYSICAL CONTENT OF … PHYS. REV. D 102, 064024 (2020) momentum variables in the Hamiltonian formulation. that the theory would not admit any solutions. In addition to However, the particle trajectories remain unchanged. constraints associated with gauge degrees of freedom, there We assume that the influence of the Hamiltonian Ht=m is are constraints resulting from the composition rule. Quite negligibly small and only Hm and HYM=m contribute to the miraculously, we obtain exactly the right total number of particle motion. This assumption is justified by the extremely constraints. In the presence of matter, securing solutions by Λ ˜ κλ small factor E in h [see, for example, Eq. (106) for static avoiding too many constraints requires a consistently isotropic fields]. The Hamiltonian Ht=m might have an matched double coupling of matter to both Yang-Mills influence of the motion of mass only on cosmological length and tetrad fields. The possibility of finding a proper number and time scales. of natural constraints relieves the pressure to use smaller The resulting evolution equation for the particle momen- Lie groups like SU(2), which is behind the Ashtekar tum is given by variables proposed for a canonical approach to gravity in the context of dreibein variables [40,41]. ∘ μν dpj pμpν ∂ μν 2 μν λ Composite theories involve higher derivatives and ¼ h − ðG1R þ G2η R λÞ ; ð108Þ 2γ j Λ are hence prone to instability. For composite gravity, one dt m ∂x E would expect fourth-order differential equations. However, where we have used the expression (A21) for HYM=m, and the constraints lead to a very special feature of composite the evolution of the particle position is governed by higher derivative theories: they select solutions from a workhorse theory. For composite gravity this means that we 1 p p hkl dxj deal with selected solutions of the Yang-Mills theory based 1 þ h00 − k l 2 2γ2m2 dt on the Lorentz group. In the presence of matter, the Yang- ∘ Mills theory includes suitable external fluxes. This selec- 2G1 jμ pμ ¼ δjμ − jμ þ R tion effect guarantees the elimination of instabilities. As the h Λ γ E m selection is very restrictive, we hope that it also helps to ∘ kl ∘ λ eliminate potential problems associated with the noncom- 1 00 p p R R λ p þ R − k l þ j ð Þ pact nature of the Lorentz group (Yang-Mills theories are Λ G1 γ2 2 G2 γ2 γ : 109 E m m usually based for good reasons on compact Lie groups). As composite gravity provides selected solutions of a Yang- The factor in parentheses on the left-hand side of Eq. (109) Mills theory, it is much closer to the standard treatment of j j τ τ simply changes dx =dt into dx =d , where is the proper electroweak and strong interactions than general relativity. time of the particle moving in a gravitational field. As a consequence of the equivalence principle, gravity is For the static isotropic solution in the weak-field approxi- all about geometry. However, this remark does not imply mation, the curvature tensor vanishes. Equations (108) and that gravity must necessarily be interpreted as curvature in (109) then describe geodesic motion. However, this should spacetime [42]. The composite theory of gravity expresses not be taken for granted. For the fully nonlinear composite the Yang-Mills fields associated with the Lorentz group in theory of gravity, it has been shown in Appendix A of [2] that λ 00 terms of the tetrad fields associated with a spacetime only R λ and R vanish [however, that result was found in a metric. This metric is only used for expressing momenta standard quasi-Minkowskian coordinate system that does in terms of velocities and may hence be interpreted as an not satisfy the coordinate conditions (19)]. If one still wants anisotropy of mass. The metric has no effect on the measure to achieve geodesic motion, then one would have to choose used for the integrations in the Hamiltonian or Lagrangian, the scalar coupling of fields and matter through G2 rather which are performed in an underlying Minkowski space. than the tensorial coupling through G1. A more appealing Nevertheless, the particle motion in the field around a option is to search for coordinate conditions characterizing a central mass turns out to be geodesic. And nevertheless, the background Minkowski system that leads to a vanishing field equations for a tensorial coupling of the gravitational curvature tensor in matter-free space. field to matter are remarkably similar to general relativity in the weak-field approximation. In the nonlinear regime, V. SUMMARY AND CONCLUSIONS however, it might turn out to be necessary to use the scalar The main insight from this paper is this: A lot of things coupling to guarantee the geodesic motion of particles. could go wrong with composite gravity, but they do not. The canonical Hamiltonian formulation of the evolution The canonical Hamiltonian formulation of the composite equations of composite gravity in a large space is clearly theory of gravity obtained by expressing the gauge vector advantageous for quantization. The constraints resulting fields of the Yang-Mills theory based on the Lorentz group from the composition rule are found to be gauge invariant, in terms of tetrad or vierbein variables requires 80 fields, second class constraints. This suggests that, in the quan- not counting any ghost fields for handling gauge condi- tization process, they can be treated via Dirac brackets, and tions. A large number of constraints should arise, so that the gauge constraints can be treated independently with the gravity has only a few degrees of freedom, but not so many BRST procedure. Therefore, quantization of linearized

064024-13 HANS CHRISTIAN ÖTTINGER PHYS. REV. D 102, 064024 (2020) Z a ν 1 ˜ 1 ∂Aμ ∂ composite gravity in the context of dissipative quantum ¼ − ð ðκ˜ λÞ þ κ˜ λ˜ Þð μν þ μν Þþ Aa 3 L Fμν Hμν Fðκ˜ λ˜Þ Hκ˜ λ˜ ν d x; field theory [20] seems to be straightforward. A compact 8 2 ∂xμ ∂x formulation of the equations for the metric is advantageous ð Þ for solving practical problems, even though these second- A3 order differential equations have some special features: a κ˜ λ˜ free function appears as a result of eliminating higher where the fourth-rank tensor Hμν is assumed to have the ˜ ðκ˜ λ˜Þ derivatives by integration; this function is reminiscent of same antisymmetries in κ˜, λ and μ, ν as Fμν .Wenow gauge degrees of freedom in general relativity. κ˜ λ˜ assume that Hμν is a linear function of the energy- The steps carried out here in great detail for the weak- momentum tensor of matter. The natural way of building field approximation should provide guidance for the proper a fourth-rank tensor with the required antisymmetries from canonical treatment of the fully nonlinear composite theory a symmetric second-rank tensor Cμν is of gravity proposed in [2]. Whereas many of the steps are straightforward and may actually be more transparent in the κ˜ λ˜ κ˜ λ˜ λ˜ κ˜ κ˜ λ˜ λ˜ κ˜ Hμν ¼ C μδ ν − C μδ ν − C νδ μ þ C νδ μ; ð Þ nonlinear setting (for example, true vector indices can be A4 recognized more easily), special attention must be paid to where we assume that the matter tensor Cμν is a linear the coordinate conditions that we want to use for character- izing appropriate Minkowskian coordinate systems (see combination of the trace-free and trace parts of the energy- Appendix C). It would be desirable to find coordinate momentum tensor Tμν. This assumption is motivated by the conditions for which a suitably defined curvature tensor equations of general relativity. ’ vanishes in empty space. Moreover, one needs to make a Einstein s (linearized) field equation is usually written in choice between coordinate conditions that are more in the the form spirit of general relativity or better matched to the 1 assumption of a background Minkowski metric. − λ η ¼ −8π ð Þ Rμν 2 R λ μν GTμν; A5

or in the alternative form APPENDIX A: FROM LAGRANGIAN TO HAMILTONIAN FOR COUPLING OF 1 ¼ −8π − λ η ð Þ FIELD TO MATTER Rμν G Tμν 2 T λ μν : A6 The goal of this Appendix is to derive the contribution to the Hamiltonian that expresses the coupling of the Yang- In Eq. (A5), the energy-momentum tensor Tμν on the Mills field for the Lorentz group to matter. We emanate right-hand side is divergence-free (conservation of energy from the following Lagrangian for a pure Yang-Mills and momentum), so that it has to be matched with the theory, divergence-free version of the curvature tensor Rμν on the left-hand side (Bianchi identity). Equation (A6) is obtained Z 1 1 ∂ a ∂ ν by means of the trace equation, a μν Aμ Aa 3 L ¼ − FμνFa þ ν d x; ðA1Þ 4 2 ∂xμ ∂x λ λ R λ ¼ 8πGT λ; ðA7Þ where, in the weak-field approximation, the field tensor is which follows by taking the trace of either version of given by Einstein’s field equation. A particularly useful form of Einstein’s field equation for our purposes is obtained by ∂ a ∂ a equating trace-free tensors rather than divergence-free a Aν Aμ Fμν ¼ − : ðA2Þ tensors, ∂xμ ∂xν ∘ ∘ ¼ −8π ð Þ The second contribution in Eq. (A1) represents a covariant Rμν GTμν; A8 gauge breaking term for removing degeneracies associated with gauge invariance (the particular form corresponds to with the trace-free tensors the convenient Feynman gauge). ∘ 1 For the Yang-Mills theory based on the Lorentz group ¼ − λ η ð Þ Rμν Rμν 4 R λ μν A9 we can replace summations over a by summations over κ˜, λ˜ ˜ according to Table I. If we sum over all pairs ðκ˜; λÞ and and assume antisymmetry in κ˜, λ˜ [cf. Eq. (7)], each term occurs ∘ 1 twice. We include the coupling of the Yang-Mills field to λ Tμν ¼ Tμν − T λημν: ðA10Þ matter by generalizing Eq. (A1) to 4

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Of course, Eq. (A8) now needs to be supplemented by This expression for HYM=m suggests that Eq. (A7) to reproduce the full content of Einstein’s field equations. The clear separation between trace-free and trace μ ðμλÞ R ν ¼ Fνλ ðA17Þ parts motivates our choice

∘ is an interesting tensor to look at. By using Eqs. (7) and λ Cμν ¼ G1Tμν þ G2ημνT λ; ðA11Þ (A2), we arrive at the following explicit representation in terms of the metric, in Eq. (A4), where the coefficients G1 and G2 must have the ’ 1 ∂2 μ ∂2 μ ∂2 λ ∂2 λ same dimensions as Newton s constant G (cf. Table II). μ ¼ h ν − h λ − h ν þ h λ ð Þ R ν λ ν λ ν ; A18 From the Lagrangian (A3), we obtain the conjugate 2 ∂xλ∂x ∂xλ∂x ∂xμ∂x ∂xμ∂x momenta which can be recognized as the Ricci curvature tensor in the δ ∂ a ∂ a ∂ a a L Aν A0 0 Aμ κ˜ λ˜ weak-field approximation [see, e.g., Eq. (7.6.2) of [35] or Eν ¼ − ¼ − þ þ δ ν − H : ðA12Þ _ ν ∂ ∂ ν ∂ 0ν Eq. (B5) of [2]; cf. also Eq. (58)]. By means of Eqs. (A17) δAa t x xμ and (A12), we can evaluate These equations can also be regarded as evolution equa- a μ ν ¼ ðλnÞ j − ðλjÞ 0 − ð0lÞ j tions for Aν (due to the gauge breaking term, even for R νC μ Fjn C λ Ej C λ Ej C l a ν ¼ 0). The evolution equations for Eν are given by − ð 0μ þ μjδ0 Þ ν ð Þ H0ν H0j ν C μ: A19 ∂ a δ Eν ¼ − L ∂t δAν If we define a a κ˜ λ˜ a a 0μ μ ∂ð þ Þ ∂ ∂Aμ ∂Aμ μ μ j 0 Fnν Hnν 0 R ν ¼ R ν þ H0ν þ H0 δ ν; ðA20Þ ¼ − − ν − δ ν : ðA13Þ j ∂xn ∂xμ ∂x ∂t the Hamiltonian (A16) can be rewritten as We can now evaluate the Hamiltonian H as the Legendre Z transform of L, ¼ Rμ ν 3 ð Þ HYM=m νC μd x: A21 ¼ H HYM Z For pure gravity, that is, in the absence of matter or for 1 ˜ 1 ˜ 1 þ ðκ˜ λÞ mn − ðκ˜ λÞ 0n þ κ˜ λ˜ mn 3 κ˜ λ˜ μ Fmn Hκ˜ λ˜ En Hκ˜ λ˜ HmnHκ˜ λ˜ d x; Hμν ¼ 0, the tensor R ν coincides with the curvature tensor 4 2 8 μ R ν. This simple direct coupling of curvature tensor and ðA14Þ energy-momentum tensor suggests that the developments of this Appendix are very natural and appealing. where HYM is given in Eq. (26). The contribution from Note that the arguments given in this Appendix are not restricted to the weak-field approximation (A2). 1 1 κ˜ λ˜ mn λ ν l n 0 0 Generalization to the full theory is straightforward. Even H H ˜ ¼ ðC νC λ þ C C − C 0C 0ÞðA15Þ 8 mn κ˜ λ 2 l n the formula (A17) can be generalized (for g˜ ¼ 1). describes a direct local self-interaction of matter. Such self- interactions are a well-known problem and should be APPENDIX B: REPRESENTATION OF analyzed within a careful renormalization procedure. We YANG-MILLS FIELDS IN TERMS OF here simply add the corresponding contribution to the TETRAD VARIABLES Lagrangian and thus eliminate it from the Hamiltonian. The remaining contribution characterizes the coupling between In addition to the modified composition rule for the field and matter, gauge vector fields resulting from the presence of matter, Z ∂ ∂ 1 hlμ h0μ 1 ∂ω0l 1 ðκ˜ λ˜Þ 1 ðκ˜ λ˜Þ 0 3 ¼ − þ − Λ ð Þ H ¼ F Hmn − E H n d x Að0lÞμ l μ EVlμ; B1 YM=m 4 mn κ˜ λ˜ 2 n κ˜ λ˜ 2 ∂t ∂x 2g˜ ∂x Z ðλnÞ j ðλjÞ 0 ð0lÞ j 3 1 ∂ ∂ 1 ∂ω ¼ ðF C λ − E C λ − E C Þd x: ðA16Þ hlμ hkμ kl jn j j l Að Þμ ¼ − þ ; ðB2Þ kl 2 ∂xk ∂xl 2g˜ ∂xμ

ðλnÞ Note that the fields Fjn contain only spatial derivatives of we have the representation of their conjugate momenta a the spatial components of Aμ, and no time derivatives. obtained from the evolution equations (39), (80), and (81):

064024-15 HANS CHRISTIAN ÖTTINGER PHYS. REV. D 102, 064024 (2020) 1 ∂ ∂h μ ∂h0μ ∂V μ 1 ¼ l − − Λ l ð Þ Y ¼ ; ðC3Þ Eð0lÞ0 l E ; B3 2 2 ∂xμ ∂t ∂x ∂xμ r ðg˜ þ r=r0Þ 1 ∂ ∂h μ ∂h μ with a free parameter r0, which is closely related to the ¼ l − k ð Þ EðklÞ0 k l ; B4 2 ∂xμ ∂x ∂x Schwarzschild radius. As a next step, one should choose the form of the static ∂2 ∂2 2 2 isotropic metric to be used in the composition rule for the 1 h0j hlj ∂ h00 ∂ h0l Eð0 Þ ¼ − − þ gauge vector field (C1). It is well known from general l j 2 ∂xl∂t ∂t2 ∂xj∂xl ∂xj∂t relativity that the proper form of the isotropic metric depends ∂ ∂ Vlj Vl0 on the choice of coordinates [see, e.g., Eq. (8.1.3) of [35] ]. þ C − δ C00 þ Λ − ; ðB5Þ jl jl E ∂t ∂xj The choice of coordinates does not matter in general relativity, where general coordinate transformations are and possible, but it does matter in the composite theory of gravity, where only Lorentz transformations in the back- ∂2 ∂2 2 2 1 hkj hlj ∂ h0k ∂ h0l ground Minkowski space are allowed. We here compare Eð Þ ¼ − − þ kl j 2 ∂xl∂t ∂xk∂t ∂xj∂xl ∂xj∂xk standard quasi-Minkowskian coordinates (see Sec. 8.1 of þ δ − δ ð Þ [35]) and harmonic coordinates. jkC0l jlC0k: B6 In the previous work [2], we used standard quasi- Minkowskian coordinates for associating a metric with the Yang-Mills solution (C2), (C3). An important con- clusion was that g˜ should approach 0 to reproduce the high- APPENDIX C: STATIC ISOTROPIC SOLUTION precision predictions of general relativity and that particu- IN HARMONIC COORDINATES larly nice black hole solutions result when 0 is approached Static isotropic solutions play an important role in the from below. However, these conclusions depend on the theory of gravity. They are the starting point (i) for many of assumption that composite gravity can be applied mean- the predictions that have been tested with high precision ingfully in standard quasi-Minkowskian coordinates (as and (ii) for the theory of black holes. We here offer a few their name might suggest). remarks on the role of coordinate conditions in the fully For comparison, we here consider harmonic coordinates, nonlinear composite theory of gravity for static isotropic which can be defined in more general situations and may be solutions. regarded as nearly Minkowskian (see, e.g., pp. 163 and 254 We start with the static isotropic solutions of the Yang- of [35]). We assume the following form of a static isotropic Mills theory, from which the solutions of the composite metric [see, e.g., Eq. (8.1.3) of [35] ], which is sufficiently theory are then selected. We assume that these solutions are general for imposing harmonic coordinate conditions: of the form −β 0 a ¼ ð Þ a l ð Þ gμν ¼ ; ðC4Þ Aν Y r Tlνx ; C1 0 αδ þ ξ xmxn mn r2 2 2 2 1=2 with r ¼ðx1 þ x2 þ x3Þ . More explicitly, by means of a with inverse the definition (6), the 24 components of Aν can be listed in matrix form, 1 − β 0 0 1 ¯μν ¼ ð Þ 000 g δ ξ : C5 x1 x2 x3 0 mn − xmxn B C α αðαþξÞ r2 000 0 −x3 x2 a B C Aν ¼ YB C; ðC2Þ @ 000 x3 0 −x1 A In our previous work based on standard quasi- α ¼ 1 β ¼ 000−x2 x1 0 Minkowskian coordinates, we assumed , B, and ξ ¼ A − 1 [2]. We are now interested in solutions where the index a of the Lie algebra (see Table I) labels the gμν of the nonlinear theory that satisfy the harmonic columns and the spacetime index ν labels the rows. The coordinate conditions function Y has to be determined from the Yang-Mills equations for the gauge vector field (C1). The field ∂gρν 1 ∂gμν g¯μν ¼ g¯μν : ðC6Þ equations for gauge vector fields of this form and their ∂xμ 2 ∂xρ solutions have been discussed in Sec. V of [2]. Most remarkable is the closed-form solution of the fully non- According to these conditions, the functions in Eq. (C4) are linear equations, related by the differential equation

064024-16 MATHEMATICAL STRUCTURE AND PHYSICAL CONTENT OF … PHYS. REV. D 102, 064024 (2020) 0 1 ξ0 α þ 2ξ α0 4ξ β0 p1ffiffi 0 − þ ¼ ; ðC7Þ β α þ ξ α þ ξ α α β ¯ μ B C r b κ ¼ @   A: ðC12Þ δ 1 1 0 pmkffiffi þ pffiffiffiffiffiffi − pffiffi xmxk α α 2 where a prime on a function of r indicates the derivative αþξ r with respect to r. The Schwarzschild solution of general relativity in The nonlinear decomposition rule harmonic coordinates is given by [see, e.g., Eq. (8.2.15) of [35] ] 1 ∂gμν ∂gμ0ν 0 a ¼ ¯ μ − ¯ μ AaνT κλ 2 b κ μ0 ∂ μ b λ 2 2 ∂x x r0 r − r0 r þ r0 r0 α ¼ 1 þ ; β ¼ ; ξ ¼ : ðC8Þ 1 ∂ κ0 þ − 2 b μ ¯ μ ¯ μ r r r0 r r0 r þ ðb κηκ0λ − b ληκ0κÞðC13Þ 2g˜ ∂xν One can easily verify that the functions given in Eq. (C8) indeed satisfy Eq. (C7). For the Schwarzschild solution, we leads to two equivalent representations of Y, moreover have ðα þ ξÞβ ¼ 1. Note that the harmonic coordinate conditions (C6) are rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi! 1 ξ − rα0 1 1 α 1 α þ ξ Lorentz covariant. The same would be true for the follow- Y ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 − − ing class of simpler coordinate conditions: 2r2 ðα þ ξÞα gr˜ 2 2 α þ ξ 2 α ð Þ ∂gρν ∂gμν C14 ημν ¼ Kημν ; ðC9Þ ∂xμ ∂xρ and which leads to 1 β0 0 0 0 2ξ Y ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðC15Þ ð3K − 1Þα þ Kβ þðK − 1Þξ ¼ : ðC10Þ 2r ðα þ ξÞβ r

As pointed out before, in composite gravity, solutions for For harmonic coordinates, we find the second-order different coordinate conditions are not equivalent. The Robertson expansions condition (C6) is very much inspired by the thinking of general relativity. Once the decision in favor of a back- 2 2 2 2 4 2 α ¼ 1 þ r0 − ˜ r0 β ¼ 1 − r0 þ ˜ r0 ξ ¼ r0 ground Minkowski space has been made, Eq. (C9) may g 2 ; g 2 ; 2 : actually be the more appropriate choice. r r r r r Our construction of Yang-Mills fields is based on the ðC16Þ symmetric tetrad variables obtained by factorizing the metric (C4), Similar expansions can be obtained for the coordinate pffiffiffi conditions (C9), provided that K ¼ 1=2. By matching the β 0 κ terms that contribute to the high-precision predictions of b μ ¼ pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi ; ðC11Þ 0 αδ þð α þ ξ − αÞ xkxm km r2 general relativity with the expansions of the Schwarzschild solution (C8), we find g˜ ¼ 2 from the second-order with inverse expansion of β.

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