Coordinate conditions and field equations for pure composite gravity
CCoordinate conditions and field equations for pure composite gravity
Hans Christian ¨Ottinger ∗ ETH Z¨urich, Department of Materials, CH-8093 Z¨urich, Switzerland (Dated: January 25, 2021)Whenever an alternative theory of gravity is formulated in a background Minkowski space, theconditions characterizing admissible coordinate systems, in which the alternative theory of gravitymay be applied, play an important role. We here propose Lorentz covariant coordinate condi-tions for the composite theory of pure gravity developed from the Yang-Mills theory based on theLorentz group, thereby completing this previously proposed higher derivative theory of gravity. Thephysically relevant static isotropic solutions are determined by various methods, the high-precisionpredictions of general relativity are reproduced, and an exact black-hole solution with mildly singularbehavior is found.
I. INTRODUCTION
Only two years after the discovery of Yang-Mills theo-ries [1], it has been recognized that that there is a strik-ing formal relationship between the Riemann curvaturetensor of general relativity and the field tensor of theYang-Mills theory based on the Lorentz group [2]. How-ever, developing this particular Yang-Mills theory into aconsistent and convincing theory of gravity is not at allstraightforward. The ideas of [2] have been found to be“unnatural” by Yang (see footnote 5 of [3]), whose workhas later been criticized massively in Chap. 19 of [4].Nevertheless, the pioneering work [2] may be consideredas the origin of what is now known as gauge gravitationtheory [5, 6].An obvious problem with the Yang-Mills theory basedon the Lorentz group is that it has the large number of 48degrees of freedom, half of which are physically relevant.One is faced with six four-vector fields satisfying second-order evolution equations. For the pure field theories,the physical degrees of freedom are essentially given bythe two transverse components of the four-vector fields,like in electrodynamics with its single vector potential.In view of this enormous number of degrees of freedomwe need an almost equally large number of constraints tokeep only a few degrees of freedom in a theory of grav-ity. In other words, we need a structured principle forselecting just a few ones among all the solutions of theYang-Mills theory based on the Lorentz group.A powerful selection principle can be implemented bymeans of the tool of composite theories [7, 8]. The basicidea is to write the gauge vector fields of the Yang-Millstheory in terms of fewer, more fundamental variables andtheir derivatives. The admission of derivatives in this so-called composition rule implies that the composite theoryinvolves higher than second derivatives. The power of thetool of composite theories results from the fact that, intheir Hamiltonian formulations [7, 8], the structure ofthe constraints providing the selection principle is highlytransparent. ∗ As the composite theory of gravity [9], just like theunderlying Yang-Mills theory, is formulated in a back-ground Minkowski space, the question arises how to char-acterize the “good” coordinate systems in which the the-ory may be applied. This characterization should beLorentz invariant, but not invariant under more generalcoordinate transformations, that is, it shares the formalproperties of coordinate conditions in general relativity.However, the unique solutions obtained from Einstein’sfield equations only after specifying coordinate condi-tions are all physically equivalent, whereas the coordi-nate conditions in composite gravity characterize phys-ically preferred systems. From a historical perspective,it is remarkable that Einstein in 1914 still believed thatthe metric should be completely determined by the fieldequations and, therefore, a generally covariant theory ofgravity was not desirable (see [10] for a detailed discus-sion). The important task of characterizing the preferredsystems in composite gravity is addressed in the presentpaper. Once it is solved, we can provide a canonicalHamiltonian formulation of composite theory of gravitybeyond the weak-field approximation [11] and we obtainthe static isotropic black-hole solution in a proper coor-dinate system.The structure of the paper is as follows. As a prepara-tory step, we present the various variables and relationsbetween them (Sec. II) and discuss their gauge transfor-mation behavior (Sec. III). A cornerstone of the devel-opment is the close relationship between the covariantderivatives associated (i) with a connection with torsionand (ii) with the Yang-Mills theory based on the Lorentzgroup. The core of the composite theory of gravity con-sists of the field equations presented for several sets ofvariables (Sec. IV) and the coordinate conditions charac-terizing the admissible coordinate systems (Sec. V). Asan application, we determine the static isotropic solutionsand provide the results for the high-precision tests ofgravity as well as an exact black-hole solution (Sec. VI).We finally offer a detailed summary of our results anddraw a number of conclusions (Sec. VII). A number ofdetailed results and arguments are provided in six ap-pendices. a r X i v : . [ g r- q c ] J a n II. VARIOUS VARIABLES AND RELATIONSBETWEEN THEM
For the understanding of composite theories, it is im-portant to introduce different kinds of variables and toclarify the relations between them. On the one hand,we have the metric tensors, tetrad variables, connectionsand curvature tensors familiar from general relativity andother theories of gravity. On the other hand, we have thegauge vectors and field tensors of the Yang-Mills theorybased on the Lorentz group.An important step is the decomposition of metric ten-sors in terms of tetrad or vierbein variables, g µν = η κλ b κµ b λν = b κµ b κν , (1)where η κλ = η κλ is the Minkowski metric with signa-ture ( − , + , + , +). Throughout this paper, the Minkowskimetric is used for raising or lowering space-time indices.For the inverses of the metric and the tetrad variables weintroduce the components ¯ g µν and ¯ b µκ . Note that theyare not obtained by raising or lowering indices of g µν and b κµ , respectively.Equation (1) may be regarded as the characterizationof metric tensors by symmetry and definiteness proper-ties. A general metric tensor may also be regarded as theresult of transforming the Minkowski metric. The decom-position of a metric g µν into tetrad variables b κµ is notunique. If we multiply b κµ from the left with any Lorentztransformation, the invariance of the Minkowski metricunder Lorentz transformations implies that we obtain an-other valid decomposition. This observation reveals theorigin of the underlying gauge symmetry of the compositetheory of gravity.The key role of the metric tensor in the present the-ory is the characterization of the momentum-velocity re-lation, so that it can be interpreted as an indication oftensorial properties of mass. While this is also the case ingeneral relativity, Einstein’s theory of gravity goes muchfurther in the geometric interpretation of the metric byassuming that it characterizes the underlying space-time.In contrast, the present theory is developed in an under-lying Minkowski space, which is the standard situationfor Yang-Mills theories.As a next step, we introduce the vector fields A ( κλ ) ρ in terms of the tetrad variables (the pair ( κ, λ ) of space-time indices should be considered as a label associatedwith the Lorentz group, ρ as a four-vector index), b κµ b λν A ( κλ ) ρ = 12 (cid:18) ∂g νρ ∂x µ − ∂g µρ ∂x ν (cid:19) + 12˜ g (cid:18) b κµ ∂b κν ∂x ρ − ∂b κµ ∂x ρ b κν (cid:19) . (2)From the Yang-Mills perspective, ˜ g is the coupling con-stant. From a metric viewpoint, ˜ g (cid:54) = 1 implies torsion(see Eq. (6) below). The antisymmetry of the right-handside of Eq. (2) in µ and ν leads, after resolving for A ( κλ ) ρ ,to antisymmetry in κ and λ . We have thus introduced a κ, λ ) (0 ,
1) (0 ,
2) (0 ,
3) (2 ,
3) (3 ,
1) (1 , a for the basevectors of the six-dimensional Lie algebra so(1 ,
3) and orderedpairs ( κ, λ ) of space-time indices. six vector fields associated with six pairs ( κ, λ ), or witha label a taking the values from 1 to 6 according to Ta-ble I. The pairs (0 , , ,
3) correspond to Lorentzboosts in the respective directions (involving also time)and the pairs (2 , , ,
2) correspond to rotationsin the respective planes, as can be recognized by analyz-ing the gauge transformation behavior of the fields A ( κλ ) ρ resulting from the freedom of acting with Lorentz trans-formations on b κµ (see Sec. III for details).Following standard procedures for Yang-Mills theories(see, e.g., Sect. 15.2 of [12], Chap. 15 of [13], or [14]), wecan introduce a field tensor in terms of the vector fields, F aµν = ∂A aν ∂x µ − ∂A aµ ∂x ν + ˜ gf bca A bµ A cν , (3)where f bca are the structure constants of the Lorentzgroup. A Lie algebra label, say a , can be raised or loweredby raising or lowering the indices in the pairs associatedwith a according to Table I. The structure constants canthen be specified as follows: f abc is 1 ( −
1) if ( a, b, c ) isan even (odd) permutation of (4 , , , , , , , ,
3) and 0 otherwise (see also Eq. (A3)).The definition (2) suggests the following general pas-sage from quantities labeled by a Lie algebra index to aquantity with space-time indices,˜ X µν = b κµ b λν X ( κλ ) . (4)One then gets a deep relation between covariant deriva-tives associated with metrics and connections on the onehand and covariant derivatives associated with a Yang-Mills theory based on the Lorentz group on the otherhand (for a proof of this fundamental relation based onthe structure of the Lorentz group, see Appendix A), ∂ ˜ X µν ∂x ρ − Γ σρµ ˜ X σν − Γ σρν ˜ X µσ = b κµ b λν (cid:20) ∂X ( κλ ) ∂x ρ + ˜ g f bc ( κλ ) A bρ X c (cid:21) , (5)where the connection Γ ρµν is given byΓ ρµν = 12 ¯ g ρσ (cid:20) ∂g σν ∂x µ + ˜ g (cid:18) ∂g µσ ∂x ν − ∂g µν ∂x σ (cid:19)(cid:21) = ¯ g ρσ ¯Γ σµν . (6)Unlike the Christoffel symbols obtained for ˜ g = 1, Γ ρµν isnot symmetric in µ and ν for ˜ g (cid:54) = 1. This lack of symme-try indicates the presence of torsion. Note, however, thatthe connection is metric-compatible for all ˜ g [15], that is, ∂g µν ∂x ρ − Γ σρµ g σν − Γ σρν g µσ = 0 , (7)which can be recast in the convenient form ∂g µν ∂x ρ = ¯Γ µρν + ¯Γ νρµ . (8)From the connection Γ ρµν , we can further construct theRiemann curvature tensor (see, e.g. [15] or [16]) R µνµ (cid:48) ν (cid:48) = ∂ Γ µµ (cid:48) ν ∂x ν (cid:48) − ∂ Γ µν (cid:48) ν ∂x µ (cid:48) + Γ σµ (cid:48) ν Γ µν (cid:48) σ − Γ σν (cid:48) ν Γ µµ (cid:48) σ . (9)In Appendix B, it is shown that the field tensor (3) canbe written in the alternative form˜ F µνµ (cid:48) ν (cid:48) =12 (cid:18) ∂ g νν (cid:48) ∂x µ ∂x µ (cid:48) − ∂ g νµ (cid:48) ∂x µ ∂x ν (cid:48) − ∂ g µν (cid:48) ∂x ν ∂x µ (cid:48) + ∂ g µµ (cid:48) ∂x ν ∂x ν (cid:48) (cid:19) + 1˜ g ¯ g ρσ (¯Γ ρµ (cid:48) µ ¯Γ σν (cid:48) ν − ¯Γ ρν (cid:48) µ ¯Γ σµ (cid:48) ν ) . (10)This explicit expression for ˜ F µνµ (cid:48) ν (cid:48) reveals its symmetryproperties: antisymmetry under µ ↔ ν and µ (cid:48) ↔ ν (cid:48) and,more surprisingly, symmetry under ( µν ) ↔ ( µ (cid:48) ν (cid:48) ). Acomparison between the expressions (9) and (10) yields aremarkable relationship between the Riemann curvaturetensor and the field tensor of the Yang-Mills theory basedon the Lorentz group,˜ g ¯ g µρ ˜ F ρνµ (cid:48) ν (cid:48) = R µνµ (cid:48) ν (cid:48) , (11)which holds for all values of the coupling constant ˜ g . III. GAUGE TRANSFORMATION BEHAVIOR
As a consequence of the decomposition (1), there ex-ists the gauge freedom of acting with a Lorentz transfor-mation from the left on the tetrad variables b κµ . In itsinfinitesimal version, this possibility corresponds to thetransformation δb κµ = ˜ g Λ ( κλ ) b λµ , (12)where Λ ( κλ ) is antisymmetric in κ and λ and can hencebe understood as Λ a according to Table I. For κ = 0,time is mixed with a spatial dependence in one of thecoordinate directions so that we deal with the respectiveLorentz boosts. If both κ = k and λ = l are both spatialindices, the antisymmetric matrix Λ ( κλ ) describes rota-tions in the corresponding ( k, l ) plane. For the inverse of b κµ , Eq. (12) implies δ ¯ b µκ = − ˜ g Λ ( κλ ) ¯ b µλ . (13)By using Eq. (12) in the composition rule (2), we ob-tain δA ( κλ ) ρ − ˜ g η κ (cid:48) λ (cid:48) (cid:104) A ( κ (cid:48) λ ) ρ Λ ( κλ (cid:48) ) − Λ ( κ (cid:48) λ ) A ( κλ (cid:48) ) ρ (cid:105) = ∂ Λ ( κλ ) ∂x ρ , (14) which, by means of Eq. (A1), can be written as δA aρ = ∂ Λ a ∂x ρ + ˜ gf bca A bρ Λ c . (15)This result demonstrates that the six vector fields A aρ indeed possess the proper gauge transformation behaviorfor the vector fields of the Yang-Mills theory based onthe Lorentz group. By means of the Jacobi identity forthe structure constants, f sba f cds + f sca f dbs + f sda f bcs = 0 , (16)we further obtain the gauge transformation behavior ofthe field tensor, δF aµν = ˜ gf bca F bµν Λ c . (17)Finally, we look at the gauge transformation behav-ior obtained for the Yang-Mills variables transformed ac-cording to Eq. (4). From Eqs. (12) and (14) we obtain δ ˜ A µνρ = b κµ b λν ∂ Λ ( κλ ) ∂x ρ . (18)As the metric is gauge invariant (gauge degrees of free-dom result only from its decomposition), the representa-tions (6) and (10) imply the gauge invariance properties δ Γ ρµν = δ ¯Γ σµν = 0 , (19)and δ ˜ F µνµ (cid:48) ν (cid:48) = 0 . (20) IV. FIELD EQUATIONS
With the help of Eq. (5), the standard field equationsfor our Yang-Mills theory based on the Lorentz group(see, e.g., Sect. 15.2 of [12], Chap. 15 of [13], or [14]) canbe written in the manifestly gauge invariant form η µ (cid:48) µ (cid:48)(cid:48) (cid:32) ∂ ˜ F µνµ (cid:48)(cid:48) ν (cid:48) ∂x µ (cid:48) − Γ σµ (cid:48) µ ˜ F σνµ (cid:48)(cid:48) ν (cid:48) − Γ σµ (cid:48) ν ˜ F µσµ (cid:48)(cid:48) ν (cid:48) (cid:33) = 0 . (21)By means of Eq. (11), these field equations can be rewrit-ten in terms of the Riemann curvature tensor, η ρν (cid:48) (cid:18) ∂R µνµ (cid:48) ν (cid:48) ∂x ρ + Γ µρσ R σνµ (cid:48) ν (cid:48) − Γ σρν R µσµ (cid:48) ν (cid:48) (cid:19) = 0 . (22)In view of Eq. (9), this latter equation is entirely in termsof the variables Γ ρµν . The explicit form of the resultingequation is given in Appendix C. This observation offersthe option of the following two-step procedure: one firstdetermines the most general solution of the second-orderdifferential equations (C1) for Γ ρµν and then, in a post-processing step, one obtains the metric by solving thefirst-order differential equations (6). The post-processingstep selects those solutions Γ ρµν that can actually be ex-pressed in terms of the metric.Finally, we write the field equations directly as third-order differential equations for the metric. As the solu-tions of these third-order equations can be understoodin terms of selected solutions of the Yang-Mills theoryfound by post-processing, there is no reason to be con-cerned about the potential instabilities resulting fromhigher-order differential equations, known as Ostrograd-sky instabilities [17, 18]. Avoiding such instabilities is animportant topic, in particular, in alternative theories ofgravity [19–26]. We write all the third and second deriva-tives of the metric explicitly, whereas the first derivativesare conveniently combined into connection variables. Theresult is the following set of equations for the compositetheory of gravity obtained by expressing the gauge vec-tor fields of the Yang-Mills theory based on the Lorentzgroup in terms of the tetrad variables obtained by de-composing a metric,Ξ µνµ (cid:48) = 12 ∂∂x µ (cid:3) g µ (cid:48) ν − ∂ ∂x µ ∂x µ (cid:48) ∂g νρ ∂x ρ −
12 Γ σµ (cid:48) µ (cid:18) g (cid:3) g σν + ∂∂x ν ∂g σρ ∂x ρ − ∂∂x σ ∂g νρ ∂x ρ (cid:19) + η ρρ (cid:48) σρν (cid:18) ∂ g σρ (cid:48) ∂x µ ∂x µ (cid:48) − ∂ g µρ (cid:48) ∂x σ ∂x µ (cid:48) + 2 ∂ g µµ (cid:48) ∂x σ ∂x ρ (cid:48) − ∂ g σµ (cid:48) ∂x µ ∂x ρ (cid:48) − g ∂ g σµ ∂x µ (cid:48) ∂x ρ (cid:48) (cid:19) + η ρρ (cid:48) ˜ g (cid:20) Γ αµ (cid:48) µ (cid:18) αρ (cid:48) β + ¯Γ βρ (cid:48) α (cid:19) − Γ αρ (cid:48) µ ¯Γ αµ (cid:48) β (cid:21) Γ βρν − µ ↔ ν = 0 . (23)In view of the antisymmetry of Ξ µνµ (cid:48) implied by thelast line of the above equation, we can assume µ < ν so that Eq. (23) provides a total of 24 equations for theten components of the symmetric matrix g µν . If we wishto determine the time evolution of the metric from thethird-order differential equations (23), we need 30 ini-tial conditions for the matrix elements g µν and their firstand second time derivatives as well as expressions for thethird time derivatives.Closer inspection of the third-order terms in Eq. (23)reveals that the six equations Ξ mn = 0 for m ≤ n pro-vide the derivatives ∂ g mn /∂t , but that the remain-ing equations do not contain any information about ∂ g µ /∂t . Therefore, the remaining 18 equations con-stitute constraints for the initial conditions, and we arefaced with two tasks: (i) find equations for the time evo-lution of g µ , and (ii) show that the constraints are sat-isfied at all times if they hold initially (or count the ad-ditional constraints that need to be satisfied otherwise).It is not at all trivial to find the number of further con-straints arising from the dynamic invariance of the con-straints contained in Eq. (23). A controlled handling ofconstraints is more straightforward in a Hamiltonian set-ting. As the canonical Hamiltonian formulation has been elaborated only in the weak-field approximation [11], wesketch the generalizations required for the full, nonlin-ear theory of composite pure gravity in Appendix E. Asa conclusion, we expect (at least) four physical degreesof freedom remaining in the field equations (23) for g µν .Note that the Hamiltonian approach also provides thenatural starting point for a generalization to dissipativesystems. In particular, this approach allows us to for-mulate quantum master equations [27–30] and to makecomposite gravity accessible to the robust framework ofdissipative quantum field theory [31, 32].The issue of missing evolution equations is addressedin the subsequent section. As in the weak-field approxi-mation, coordinate conditions characterizing those coor-dinate systems in which the composite theory of gravitycan be applied provide the missing evolution equations. V. COORDINATE CONDITIONS
As we have assumed an underlying Minkowski spacefor developing composite gravity, we need to character-ize those coordinate systems in which the theory actuallyholds. These characteristic coordinate conditions shouldclearly be Lorentz covariant. Furthermore, the coordi-nate conditions should provide evolution equations for g µ because the field equations (23) determine the third-order time derivatives of g mn , but not of g µ . Therefore,the formulation of appropriate coordinate conditions isan important task. The status of coordinate conditionsin composite theory is very different from their status ingeneral relativity, where they have no influence on thephysical predictions.The coordinate conditions should be a set of fourLorentz covariant equations. An appealing form is givenby ∂g µρ ∂x ρ = ∂φ∂x µ , (24)where the potential φ is often assumed to be propor-tional to the trace of the metric. To eliminate the needof specifying a potential, we can write the second-orderintegrability conditions ∂∂x ν ∂g µρ ∂x ρ = ∂∂x µ ∂g νρ ∂x ρ . (25)After taking the derivatives with respect to x ν and sum-ming over ν , we arrive at the four Lorentz covariant co-ordinate conditions (cid:3) ∂g µρ ∂x ρ = K ∂∂x µ ∂ g ρσ ∂x ρ ∂x σ , (26)actually with K = 1. Note that it is very appealing touse third-order equations as coordinate conditions be-cause we actually need only expressions for the thirdtime derivatives of g µ (stronger, first-order conditionsare needed for the Hamiltonian formulation; see Ap-pendix E). For K = 1, we would obtain such equa-tions for g m , but not for g . This is the reason whywe have introduced the factor K in Eq. (26). For any K (cid:54) = 1, we obtain the desired four evolution equationsfor g µ . Formally, we could stick to the first-order condi-tions (24), but then the potential φ would be describedby the second-order differential equations (cid:3) φ = K ∂ g ρσ ∂x ρ ∂x σ , (27)where suitable space-time boundary conditions would berequired. Note, however, that for K (cid:54) = 1, Eqs. (24) and(27) imply ∂ g ρσ ∂x ρ ∂x σ = 0 , (28)whereas Eq. (26) implies the weaker requirement (cid:3) ∂ g ρσ ∂x ρ ∂x σ = 0 . (29)The coordinate conditions (26) are an essential new in-gredient into the composite theory of gravity. Of course,these coordinate conditions take a particularly simpleform for K = 0, which is a possible choice. Alterna-tively, we could choose K = ˜ g/ (1 + ˜ g ) because we canthen express the coordinate conditions as ∂ ¯Γ µρσ ∂x ρ ∂x σ = 0 . (30)In the following, we leave the particular choice of K (cid:54) = 1open.From a structural point of view, the coordinate condi-tions (26) have the important advantage that they canbe implemented in exactly the same way as the gaugeconditions in Yang-Mills theories: one can add a term tothe Lagrangian that does not lead to any modification ofthe field equations, provided that the desired (coordinateor gauge) conditions are imposed as constraints. For thecoordinate conditions (26), the additional contributionto the Lagrangian is given in Appendix D. VI. STATIC ISOTROPIC SOLUTION
The study of static isotropic solutions of compositegravity is of great importance because these solutionsprovide the predictions for the high-precision tests of gen-eral relativity (deflection of light by the sun, anomalousprecession of the perihelion of Mercury, gravitational red-shift of spectral lines from white dwarf stars, travel timedelay for radar signals reflecting off other planets) andthe properties of black holes. Therefore, we here discussthese solutions in great detail. We assume that the static isotropic solutions are of thegeneral form, g µν = (cid:32) − β α δ mn + ξ x m x n r (cid:33) , (31)with inverse¯ g µν = (cid:32) − β δ mn α − ξα ( α + ξ ) x m x n r (cid:33) , (32)where α , β and ξ are functions of the single variable r = ( x + x + x ) / . The static isotropic metric (31) isgiven in terms of the three real-valued functions α , β and ξ . In the original work on the composite theory of grav-ity (see Sec. V of [9]), we had parametrized these threefunctions in terms of only two functions A and B : α = 1, β = B , and ξ = A −
1. This particular parametrizationcorresponds to standard quasi-Minkowskian coordinates.A problem with these quasi-Minkowskian coordinates isthat it is unclear how they can be generalized to full coor-dinate conditions for general metrics. The more generalform (31) of the metric is consistent with the coordinateconditions (26). In particular, we do not need to intro-duce a further function for characterizing the components g m . In general relativity, the form (31) of the metric(with g m = 0) can be achieved by shifting time by afunction depending on r (see Sec. 8.1 of [16]). Nonzero g m arise by Lorentz transformation of the metric (31)so that the form (31) belongs to a particularly simplesolution of coordinate conditions and field equations.The field equations (23) provide two third-order ordi-nary differential equations involving all three functions α , β and ξ . For K (cid:54) = 1, the coordinate conditions (26)lead to another third-order differential equation relating α and ξ , which is actually independent of K ; only for K = 1, no further condition arises. In the remainderof this section, we solve the three differential equationsfor our three unknown functions for K (cid:54) = 1 by variousmethods. A. Robertson expansion
The high-precision tests for theories of gravity dependon the behavior of the static isotropic solutions at largedistances. We therefore construct the so-called Robert-son expansion in terms of 1 /r . One obtains the followingresults, α = 1 + α r r + α r r + . . . , (33) ξ = ξ r r + . . . , (34)and β = 1 − r r + (cid:2) g − α + ξ ) (cid:3) r r + . . . , (35)where all higher terms indicated by . . . in these Robertsonexpansions are uniquely determined by the dimensionlessparameters α , α , ξ and the coupling constant ˜ g . How-ever, α and ξ are not independent but rather relatedby a cubic algebraic equation with a single real solutionestablishing a one-to-one relation between α and ξ (seeAppendix F). The parameter r with dimension of lengthis determined by the mass at the center creating the staticisotropic field, as can be shown by reproducing the limitof Newtonian gravity (see, e.g., Sec. 3.4 of [16]).An obvious strategy for finding the dimensionless pa-rameters is to make sure that the high-precision pre-dictions of general relativity are reproduced. This isachieved by choosing α + ξ = 2 , α = ˜ g. (36)Imposing a further relation between α and ξ is subtleas we have already established the cubic relationship be-tween these parameters given explicitly in Eq. (F1). Thisimplies that the first part of Eq. (36) can be satisfied onlyfor particular values of the coupling constant ˜ g . By us-ing Eq. (36) for eliminating α and ξ from Eq. (F1), weobtain the following equation for ˜ g ,(4 + 4˜ g − ˜ g − g )(2 − ˜ g ) = 0 . (37)Two of the roots of this polynomial equation of degreefour are real. In addition to the obvious root 2, implying α = 2 and ξ = 0, one finds the further real-valued root115 (cid:2) (1259 + 30 √ / + (1259 − √ / − (cid:3) , which is approximately equal to 1 . g composite gravity with the coordinate conditions (26)for K (cid:54) = 1 can reproduce the high-precision predictionsof general relativity. Note that the parameter α in theexpansions (33)–(35) remains undetermined as it is theonly term among the listed ones that does not affect thehigh-precision tests of gravity. B. Short-distance singularity
We next focus on singular behavior at small distances,which we expect to describe black holes. A glance at thefield equations (23) reveals that any fixed multiple of asolution is another solution of the field equations. Forthe “equidimensional” third-order differential equationsdetermining the functions α , β and ξ of r , we assume thefollowing form, α = c α r x , β = c β r x , ξ = c ξ r x , (38)with constants c α , c β , c ξ and an exponent x . We furtherassume that c α , c β , and x are different from zero. For ˜ g = 2, we then find that the field equations and coordinateconditions are equivalent to c ξ = 0 and x = 1. Forgeneral ˜ g , one can verify that the values x = 2˜ g , c ξ = 0 , (39)lead to a static isotropic solution of both field equationsand coordinate conditions. Of course, this solution isphysically unacceptable as a global solution because itdoes not converge to the Minkowski metric at large dis-tances. It does, however, characterize the asymptoticsingular behavior of physical solutions at short distances.The exponent x given in Eq. (39) speaks strongly infavor of choosing ˜ g = 2 (rather than an irrational value).We then obtain a solution decaying according to a 1 /r power law, the spatial part of which is a multiple of thethree-dimensional unit matrix. C. Numerical solution
After discussing the static isotropic solutions at largeand small distances from the center, we would now liketo consider their behavior over the entire range of r . Inparticular, we are interested in the influence of the so farundetermined parameter α in Eq. (33) on the behaviorof the solutions.To explore the full solutions, we solve the field equa-tions and coordinate conditions by numerical integration,starting from a large initial distance r i and then proceed-ing to smaller values of r . Assuming ˜ g = 2, the initialconditions at r i are given by the truncated third-orderexpansions α = 1 + 2 r r + α r r , (40) ξ = − α r r , (41) β = 1 − r r + 2 r r − r r . (42)These expressions do not only provide the values of thecoefficient functions at r i , but also their first and secondderivatives required for solving the third-order differen-tial equations for the functions α , β and ξ of r . Theactual numerical solution is performed with an implicitRunge-Kutta scheme of Mathematica.If r i is sufficiently large, that is, in the range of validityof the asymptotic solutions (40)-(42), the numerical solu-tions are expected to be independent of the choice of r i .This expectation is scrutinized in Figure 1. This figuredisplays the functions β and ξ for the values α = ± . r i = 50 and r i = 500, so that eachcurve for ξ actually consists of two overlapping curves and βξξ - - r / r FIG. 1. The functions β (dashed line) and ξ (continuous lines)characterizing the temporal and off-diagonal components ofthe isotropic metric (31) obtained from the composite theoryfor gravity for ˜ g = 2 and α = ± .
25. Positive and nega-tive values of ξ correspond to α = − .
25 and α = 0 . the anticipated independence of the results of r i is con-firmed. The result for β actually consists of four curves,which implies that ξ has remarkably little influence onthe function β until it touches the r axis.Figure 1 suggests that ξ diverges around the value r at which β touches the r axis (and numerical difficul-ties arise). According to Eqs. (38), (39), ξ must go tozero for small r . The real function ξ might actually endin a cusp singularity and develop a complex branch atsmaller r that reaches zero at r = 0 (see Sec. V C of[9]). Alternatively, ξ might jump from + ∞ to −∞ , orvice versa, to return as a real function to zero at r = 0,where it started at large r (this kind of behavior is foundfor the Schwarzschild solution of general relativity; seeSec. VI E). To avoid singularities at finite r we from nowon assume α = 0, for which ξ ( r ) is found to be identi-cally zero. Note that singularities would be much morealarming in the composite theory of gravity than in gen-eral relativity because they cannot be considered as ar-tifacts (“coordinate singularities”) removable by generalcoordinate transformations. D. An exact solution
As we have by now fixed the values of the coupling con-stant (˜ g = 2) and all the free parameters in the Robert-son expansions (33)-(35) ( α = 2, ξ = 0, α = 0), thereshould be a unique static isotropic solution, which is thecounterpart of the Schwarzschild solution in general rela-tivity. The Robertson expansions suggest that all highercoefficients α n , ξ n for n ≥ α consistsof only two terms and ξ vanishes identically, as alreadynoted in the numerical solutions. Then, a closed-form αβ r / r FIG. 2. The exact solutions (44) for the functions α and β characterizing the diagonal components of the isotropic metric(31) in the composite theory for gravity. expression for β can be found from the field equation4 r β = r (cid:16) r r (cid:17) β (cid:48) , (43)so that we arrive at the complete solution α = 1 + 2 r r , ξ = 0 , β = (cid:18) − (cid:114) r r (cid:19) . (44)These functions α and β are shown in Figure 2. Thepresent results are qualitatively similar to what wasfound in previous work on the composite theory of grav-ity for different coordinate conditions (see Fig. 1 of [9]).Note that β is non-negative, vanishes at r = (2 / r ,and that √ α ± √ β = 2, where the + sign holds for r ≥ (2 / r and the − sign for r ≤ (2 / r . The onlysingularities occur at the origin, and they are of the New-tonian 1 /r type. The most remarkable feature is that β reaches a local minimum at r = (2 / r , where β becomeszero. The observation that the proper time stands still atthis distance from the origin is the essence of black-holebehavior in the composite theory of gravity.An interesting consequence of β = 0 is revealed byconsidering the curvature scalar R = ¯ g νν (cid:48) R µνµν (cid:48) = ˜ g ¯ g µµ (cid:48) ¯ g νν (cid:48) ˜ F µνµ (cid:48) ν (cid:48) = ˜ g ¯ b µκ ¯ b νλ F ( κλ ) µν . (45)For the isotropic solution given in Eq. (44), we find R = 16 r r (cid:0) r r (cid:1) (cid:0)(cid:112) r r − (cid:1) , (46)which implies infinite curvature at r = (2 / r where β vanishes, and a change of sign at that point. This is animportant insight because, in the weak-field approxima-tion, the curvature scalar and tensor have been exploredfor the coupling of gravitational field and matter [11]. Ifwe want to keep geodesic motion of a mass point in agravitational field, however, the coupling should be donein terms of a scalar or tensor quantity that is given interms of second-derivatives of the metric and vanishes,at least for the static isotropic metric. In this context,the scalar identity (28) holding for the static isotropicsolution might be useful. A tensorial coupling could bebased on the following identity for the static isotropicsolution, ∂ g µν ∂x ρ ∂x ρ + 12 ¯ g ρρ (cid:48) ∂g µν ∂x ρ ∂g ρ (cid:48) σ ∂x σ = 2 r ( r + 2 r ) r η µν , (47)which implies that the trace-free part of the tensor onthe left-hand side vanishes. E. Comparison to Schwarzschild solution
The Schwarzschild solution of general relativity in har-monic coordinates is given by (see, e.g., Eq. (8.2.15) of[16]) α = (cid:16) r r (cid:17) , ξ = r + r r − r r r , β = r − r r + r . (48)We here compare this solution to the static isotropic so-lution (44) of composite gravity.The functions α in Eqs. (44) and (48) differ by the term r /r . This term does not matter for the high-precisiontests. Whereas the singularity at the origin is 1 /r forcomposite gravity, it is 1 /r for general relativity. Thisobservation goes nicely with the exponent x in Eqs. (38),(39), for ˜ g = 2 and ˜ g = 1, respectively, where generalrelativity corresponds to the torsion-free case ˜ g = 1.Whereas ξ vanishes in composite gravity, it has a sin-gularity at r = r for the Schwarzschild solution, witha jump from + ∞ for r = r +0 to −∞ for r = r − . Whilethis may be considered as a coordinate singularity in gen-eral relativity, this would not be possible for a theory inMinkowski space. For the high-precision tests, the ab-sence of a 1 /r contribution to ξ is crucial.Also β is remarkably different for the two solutions.Whereas β is non-negative in composite gravity, itchanges sign at r for the Schwarzschild solution. Al-though the two solutions look so different, their truncatedthird-order expansions (42) coincide. The coincidence ofthese expansions to order 1 /r is crucial for satisfying thehigh-precision tests. VII. SUMMARY AND CONCLUSIONS
Yang-Mills theories are formulated on a backgroundMinkowski space, and so is the composite theory of grav-ity that selects a small subset of solutions from the Yang-Mills theory based on the Lorentz group. Such theoriesare covariant under Lorentz transformations but, unlike general relativity, not under general coordinate transfor-mations. Therefore, it is important to characterize thecoordinate systems, in which the composite theory ofgravity should be valid, by coordinate conditions. Wehere propose the Lorentz covariant third-order equations(26) for the metric as appealing coordinate conditionsthat nicely supplement the third-oder differential equa-tions for the composite theory of gravity. Their alter-native formulation in Eqs. (24) and (27) shows that weessentially introduce a potential for the divergence of themetric, where the potential itself satisfies a second-orderdifferential equation.In the original work on composite gravity [9], nogeneral coordinate conditions were given. The staticisotropic solution was determined for quasi-Minkowskiancoordinates, which are defined only for solutions of thisparticular type and do not satisfy the new coordinateconditions. Also the coordinate conditions previouslyused in the complete Hamiltonian formulation of the lin-earized theory, or weak-field approximation, of compositegravity [11] differ from the present proposal. Therefore,previous results are qualitatively similar but quantita-tively different from our previous results. The coordinateconditions (26) complete the nonlinear theory of purecomposite gravity proposed in [9].The field equations for pure composite gravity can beexpressed in a number of different ways. One option is tosolve the field equations of the Yang-Mills theory basedon the Lorentz group and, in a post-processing step, se-lect those solutions that can be properly expressed interms of the derivatives of the tetrad variables obtainedby decomposing the metric. Alternatively, one can in-troduce a gauge-invariant connection with torsion andformulate second-order differential equations entirely interms of those. One is then interested in the solutions forthe connection that can be properly expressed in termsof first derivatives of the metric. A final possibility isto write third-order evolution equations directly for themetric.In the various formulations of the field equations, itis difficult to count the number of degrees of freedom ofcomposite gravity. This difficulty is a consequence of theprimary constraints arising from the composition rule ofcomposite theories and serving as a selection principle forthe relevant solutions of the underlying Yang-Mills the-ory. A canonical Hamiltonian formulation on the com-bined spaces of tetrad and Yang-Mills variables providesthe most structured form of both field equations and co-ordinate conditions. This formulation suggests that com-posite gravity has four degrees of freedom (whereas theYang-Mills theory based on the Lorentz group has 24 de-grees of freedom). The Hamiltonian formulation suggeststhat we deal with two types of constraints: (i) constraintsresulting from the composition rule and (ii) gauge con-straints. As the former can be handled by Dirac brackets[33–35] and the latter by the BRST methodology (theacronym derives from the names of the authors of theoriginal papers [36, 37]; see also [14, 38]), the path toquantization of composite gravity is clear. This is a majoradvantage of an approach starting from the class of Yang-Mills theories, which so successfully describe electro-weakand strong interactions and for which quantization is per-fectly understood, and imposing Dirac-type constraints.In addition, this background reveals why composite theo-ries, although they are higher derivative theories, are notprone to Ostrogradsky instabilities.The fact that just a few degrees of freedom of theYang-Mills theory based on the Lorentz group survivein the composite theory of gravity is also reflected inits static isotropic solutions. Its Robertson expansionhas two free dimensionless parameters in addition to theYang-Mills coupling constant. For reproducing the high-precison predictios of general relativity, one of the freeparameters and the coupling constant (˜ g = 2) need tobe fixed. The remaining dimensionless parameter canbe chosen to avoid singularities at finite distances fromthe origin. A closed-form solution for the static isotropicmetric, which plays the same role in composite gravityas the Schwarzschild solution in general relativity, hasbeen found. The solution displays a 1 /r singularity atthe origin but remains finite at all finite values of r . Theonly remarkable feature is g = 0 at a particular distancefrom the origin, which is of the order of the Schwarzschildradius; for all other values of r , we have g < ACKNOWLEDGMENTS
I am grateful for the opportunity to do this work duringmy sabbatical at the
Collegium Helveticum in Z¨urich.
Appendix A: Relation between covariant derivatives
The reformulation of equations for the Yang-Mills the-ory based on the Lorentz group in the metric language isbased on the identity f bc ( κλ ) B b C c = η κ (cid:48) λ (cid:48) (cid:104) B ( κ (cid:48) λ ) C ( κλ (cid:48) ) − C ( κ (cid:48) λ ) B ( κλ (cid:48) ) (cid:105) , (A1)which, in view of the definition (4), can be rewritten inthe alternative form b κµ b λν f bc ( κλ ) B b C c = ¯ g ρσ (cid:16) ˜ B ρµ ˜ C σν + ˜ B ρν ˜ C µσ (cid:17) . (A2) These remarkably simple identities follow from the formof the structure constants of the Lorentz group. Afterwriting the structure constants in the following explicitform (see Table I for the index conventions), f abc = η κ a λ c η κ b λ a η κ c λ b − η κ a λ b η κ b λ c η κ c λ a + η κ a κ b (cid:0) η κ c λ a η λ b λ c − η κ c λ b η λ a λ c (cid:1) + η κ a κ c (cid:0) η κ b λ c η λ a λ b − η κ b λ a η λ b λ c (cid:1) + η κ b κ c (cid:0) η κ a λ b η λ a λ c − η κ a λ c η λ a λ b (cid:1) , (A3)the result (A1) is obtained by straightforward calcula-tion.We can now use Eq. (A2) to evaluate the right-handside of Eq. (5), b κµ b λν (cid:20) ∂X ( κλ ) ∂x ρ (cid:48) + ˜ g f bc ( κλ ) A bρ (cid:48) X c (cid:21) = ∂ ˜ X µν ∂x ρ (cid:48) + ¯ g ρσ (cid:20) (cid:18) ˜ g ˜ A ρµρ (cid:48) − b κρ ∂b κµ ∂x ρ (cid:48) (cid:19) ˜ X σν + (cid:18) ˜ g ˜ A ρνρ (cid:48) − b κρ ∂b κν ∂x ρ (cid:48) (cid:19) ˜ X µσ (cid:21) . (A4)By using the composition rule (2) we recover the fun-damental relationship (5) with the definition (6) of theconnection following from¯Γ µρν = b κµ ∂b κν ∂x ρ − ˜ g ˜ A µνρ . (A5) Appendix B: Alternative expression for field tensor
From the definitions (3) and (4) and the fundamentalrelations (5) and (A2), we obtain˜ F µνµ (cid:48) ν (cid:48) = ∂ ˜ A µνν (cid:48) ∂x µ (cid:48) − Γ σµ (cid:48) µ ˜ A σνν (cid:48) + Γ σµ (cid:48) ν ˜ A σµν (cid:48) − ∂ ˜ A µνµ (cid:48) ∂x ν (cid:48) + Γ σν (cid:48) µ ˜ A σνµ (cid:48) − Γ σν (cid:48) ν ˜ A σµµ (cid:48) − ˜ g ¯ g ρσ (cid:16) ˜ A ρµµ (cid:48) ˜ A σνν (cid:48) − ˜ A ρνµ (cid:48) ˜ A σµν (cid:48) (cid:17) . (B1)By means of Eq. (A5), we obtain˜ g (cid:32) ∂ ˜ A µνν (cid:48) ∂x µ (cid:48) − ∂ ˜ A µνµ (cid:48) ∂x ν (cid:48) (cid:33) = ∂ ¯Γ µµ (cid:48) ν ∂x ν (cid:48) − ∂ ¯Γ µν (cid:48) ν ∂x µ (cid:48) + ∂b κµ ∂x µ (cid:48) ∂b κν ∂x ν (cid:48) − ∂b κµ ∂x ν (cid:48) ∂b κν ∂x µ (cid:48) , (B2)and, again Eq. (A5), gives ∂b κµ ∂x µ (cid:48) ∂b κν ∂x ν (cid:48) = ¯ g ρσ (¯Γ ρµ (cid:48) µ + ˜ g ˜ A ρµµ (cid:48) )(¯Γ σν (cid:48) ν + ˜ g ˜ A σνν (cid:48) ) . (B3)By combining Eqs. (B1)–(B3), we finally arrive at˜ F µνµ (cid:48) ν (cid:48) = 1˜ g (cid:18) ∂ ¯Γ µµ (cid:48) ν ∂x ν (cid:48) − ∂ ¯Γ µν (cid:48) ν ∂x µ (cid:48) + ¯Γ σµ (cid:48) µ Γ σν (cid:48) ν − ¯Γ σν (cid:48) µ Γ σµ (cid:48) ν (cid:19) . (B4)This expression for the field tensor coincides with the onegiven in Eq. (10) when the definition (6) of the connectionis used.0 Appendix C: Field equation for connection
By inserting the expression (9) for the Riemann curva-ture tensor in terms of the connection, the field equation(22) for the composite theory of gravity can be written asa second-order differential equation for the connection, ∂ Γ µµ (cid:48) ν ∂x ρ ∂x ρ − ∂ Γ µρν ∂x ρ ∂x µ (cid:48) + η ρρ (cid:48) (cid:20) Γ σµ (cid:48) ν ∂ Γ µρσ ∂x ρ (cid:48) − Γ µµ (cid:48) σ ∂ Γ σρν ∂x ρ (cid:48) + Γ µρσ (cid:18) ∂ Γ σµ (cid:48) ν ∂x ρ (cid:48) − ∂ Γ σρ (cid:48) ν ∂x µ (cid:48) (cid:19) − Γ σρν (cid:18) ∂ Γ µµ (cid:48) σ ∂x ρ (cid:48) − ∂ Γ µρ (cid:48) σ ∂x µ (cid:48) (cid:19) + Γ µρ (cid:48) σ Γ σρσ (cid:48) Γ σ (cid:48) µ (cid:48) ν + Γ µµ (cid:48) σ Γ σρσ (cid:48) Γ σ (cid:48) ρ (cid:48) ν − µρσ Γ σµ (cid:48) σ (cid:48) Γ σ (cid:48) ρ (cid:48) ν (cid:21) = 0 . (C1)Note that η ρρ (cid:48) occurs rather than ¯ g ρρ (cid:48) , so that there isno need to know the metric for solving this equation. Appendix D: Modified Lagrangian
The Lagrangian for a pure Yang-Mills theory, includinga covariant but gauge breaking term for removing degen-eracies associated with gauge invariance (the particularform corresponds to the convenient Feynman gauge), isgiven by L = − (cid:90) (cid:18) F aµν F µνa + 12 ∂A aµ ∂x µ ∂A νa ∂x ν (cid:19) d x. (D1)We propose to add the further term L cc = 12 (cid:90) (cid:18) ∂ g µν ∂x µ ∂x σ ∂ g ρν ∂x ρ ∂x σ − K ∂ g µν ∂x µ ∂x ν ∂ g ρσ ∂x ρ ∂x σ (cid:19) d x, (D2)implying the functional derivative δL cc δg µν = ∂∂x µ (cid:18) (cid:3) ∂g ρν ∂x ρ − K ∂∂x ν ∂ g ρσ ∂x ρ ∂x σ (cid:19) , (D3)which vanishes upon imposing the coordinate conditions(26) as constraints. If the gauge conditions and the coor-dinate conditions are imposed as constraints, the abovemodifications of the Lagrangian for the pure Yang-Millstheory have no effect on the field equations. Appendix E: Hamiltonian formulation
For the weak-field approximation of composite grav-ity, a canonical Hamiltonian formulation with a detailedanalysis of all constraints has been given in [11]. Wehere sketch how that approach can be generalized to afull, nonlinear theory of pure gravity selected from theYang-Mills theory based on the Lorentz group.The underlying space of the Hamiltonian formulationconsists of the tetrad variables b κµ and the gauge vector fields A aν associated with the Lorentz group as configura-tional variables, together with their conjugate momenta p κµ and E aν (where E aj = F aj and E a = ∂A aµ /∂x µ )[8, 11]. This space consists of 80 fields, but massive con-straints arise from the composition rule and gauge in-variance so that, in the end, the composite theory of puregravity turns out to possess only four degrees of freedom.The generalization of the Hamiltonian (25)–(27) of [11]is obtained by introducing the Hamiltonian for the full,nonlinear version of Yang-Mills theory, H pure = (cid:90) (cid:20) E aµ E aµ + 14 F aij F aij − E a ∂A aj ∂x j − E aj (cid:18) ∂A a ∂x j + ˜ gf bca A bj A c (cid:19) + ˙ b κµ p κµ (cid:21) d x, (E1)where the functional form of the 16 time derivatives ˙ b κµ in terms of the configurational variables b κµ and A aν isobtained from 12 components of the composition rule(2) and the four coordinate conditions (24) (the poten-tial φ is assumed to be a functional of g µν ). For puregravity without external sources, we can impose the 16constraints p κµ = 0 so that the composite theory con-sists of selected solutions of the Yang-Mills theory basedon the Lorentz group [8, 11]. The terms involving E a in the Hamiltonian (E1) are associated with the gaugebreaking term in the Lagrangian (D1). Of course, thisHamiltonian implies the canonical evolution equations forthe entire set of 80 fields.The generalization of the weak-field approximation be-comes particularly simple if we introduce the followingvariables eliminating the nonlinear effects of the couplingconstant, ˘ A µνρ = ˜ A µνρ − g Ω µν/ρ , (E2)with Ω µν/ρ = b κµ ∂b κν ∂x ρ − ∂b κµ ∂x ρ b κν , (E3)and ˘ E µν = ˜ E µν − η ρρ (cid:48) g (cid:16) Γ σρν ¯Γ µρ (cid:48) σ − Γ σρµ ¯Γ νρ (cid:48) σ (cid:17) , (E4)˘ E µνj = ˜ E µνj − g (cid:18) Γ σjµ ¯Γ σ ν − Γ σjν ¯Γ σ µ (cid:19) , (E5)as further modifications of the variables ˜ A µνρ and ˜ E µνρ defined in Eq. (4). For example, the composition rule (2)takes the linear form˘ A µνρ = 12 (cid:18) ∂g νρ ∂x µ − ∂g µρ ∂x ν (cid:19) , (E6)which corresponds to Eq. (7) of [11] in the symmetricgauge and includes 12 primary constraints. Also the evo-lution equations for ˘ A µνρ and hence also the 12 secondary1constraints keep the same form as in the linearized the-ory (cf. Eqs. (39), (40) and (46), (47) of [11]). The 12tertiary constraints can be obtained by acting with theoperator (cid:3) on the primary constraints. The invarianceof the tertiary constraints follows from p κµ = 0. In orderto verify the above statements, one needs the identity ∂ Ω µν/ρ ∂x ρ − η ρρ (cid:48) (cid:0) Γ σρµ Ω σν/ρ (cid:48) + Γ σρν Ω µσ/ρ (cid:48) (cid:1) = 0 , (E7)which is the counterpart of Eq. (16) of [11] and can beinferred from the gauge invariance of the left-hand sideof Eq. (E7). Finally, the 24 evolution equations for ˘ E µνρ correspond to the field equations given in various formsin Sec. IV.As the structure of the Hamiltonian and the con-straints for the full, nonlinear theory is so similar (mostlyeven formally identical) to the case of the linear weak-field approximation, we expect the same count of 24 +3 ·
12 + 16 = 76 constraints for 2 · (16 + 24) = 80variables. Half of the 24 constraints associated withgauge invariance result from the gauge conditions E a = ∂A aµ /∂x µ = 0, which establish a relationship betweenthe (unphysical) temporal and longitudinal modes ofthe four-vector potentials. The above arguments sug-gest that pure composite gravity possesses (at least) fourphysical degrees of freedom, just as in the thoroughlyelaborated special case of the weak-field approximation[11]. Appendix F: A cubic equation
The coefficients α and ξ in the Robertson expansions(33), (34) are related by the following cubic equation,10 ξ + 10˜ gξ (4 α + 5 α ξ + 2 ξ ) − g (cid:2) ξ − ( α + ξ )(8 α + 9 α ξ − ξ ) (cid:3) − g (cid:2) α + ξ ) (cid:3) (3 α + 2 ξ )+ ˜ g ( α + ξ )(36 + 11( α + ξ ) ) = 0 . (F1)Its only real solution for α in terms of ξ is given by α = (cid:104)(cid:16) w + (cid:113) w − w (cid:17) / + w (cid:16) w + (cid:113) w − w (cid:17) − / − ξ (40 + 85˜ g − g + 33˜ g ) (cid:105) / (cid:2) g (40 − g + 11˜ g ) (cid:3) , (F2)with w = 36˜ g (cid:0) − g + 190˜ g − g (cid:1) + 5 (cid:0)
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