The phase symmetry of general relativity
aa r X i v : . [ g r- q c ] F e b The phase symmetryof general relativity
Serhii Samokhvalov ∗ Dniprovsk State Technical University, Ukraine ***
Abstract
It is shown that the general relativity has a one-parameter compact symmetry and thissymmetry is analogous to the phase symmetry of quantum mechanics ∗ e - mail : [email protected] Introduction
The two main general properties of matter that were discovered in the creation of quantummechanics, namely, the discreteness (quantization) of certain physical quantities, as well as thewave nature of motion, are associated with the existence of fundamental compact symmetry -phase symmetry of quantum mechanics. From the point of view of the theory of symmetry, wecan say that the discovery of quantum-mechanical laws was the discovery of the fundamentalphase symmetry inherent in the laws of motion of matter.However, a certain compact symmetry is inherent in other wave phenomena, in particular theelectromagnetic field in the vacuum - the so-called Rainich-Heaviside symmetry [1]. The study ofthis symmetry (which by analogy with quantum mechanics we also called phase symmetry) wasperformed in [2], where several interesting properties of this symmetry were revealed. In partic-ular, using the fact that this symmetry is a consequence of the discrete symmetry of Maxwell’sequations in vacuum at the transformation of the Hodge duality for the electromagnetic fieldtensor, an explicitly phase-symmetric so-called ”vector Lagrangian” of electrodynamics simi-lar to Dirac’s Lagrangian was constructed. This Lagrangian has the interesting property thatthe quantity conserved due to phase symmetry (when applying the Neother theorem to thisLagrangian) is proportional to the energy-momentum tensor of the electromagnetic field.In this paper, it is shown that the gravitational field in vacuum is also characterized by acertain compact one-parameter symmetry, which is a generalization to the case of the gravita-tional field of Rainich-Heaviside symmetry of the electromagnetic field and which we will alsocall the phase symmetry. To reveal this symmetry, general relativity (GR) is considered in anorthogonal frame, which corresponds to the natural interpretation of the gravitational field asthe gauge field of the translation group [3]. It is shown that the energy-momentum tensor ofthe gravitational field is phase-symmetric. An explicitly phase-symmetric ”vector Lagrangian”of gravity is constructed and it is shown that similarly to the case of an electromagnetic fieldthe Noether current of such Lagrangian, which associated with phase symmetry, is proportionalto the energy-momentum tensor of the gravitational field.Greek indices belong to the coordinate basis, and Latin - to the frame basis. The transitionbetween them (indices replacement) is carried out using mutually inverted matrices h µm and h mµ . A point is placed in the place of the lowered or raised index (where this may lead tomisunderstanding). We first present the main provisions concerning the phase symmetry of classical electrodynamics[2], presenting them in a form convenient for our purposes generalization to the case of gravity.2he electromagnetic field strength tensor F µν is expressed in terms of potentials A µ as follows F µν = ∂ µ A ν − ∂ ν A µ where ∂ µ := ∂/∂x µ . Jacobi identity written for F µν ∂ ρ F µν + cicle ( ρµν ) = 0 (1)can be rewritten as: ∂ ν e F µν = 0 , (2)where e F µν := 12 ε µνρτ F ρτ (3)is the tensor, dual according to Hodge to the tensor F µν and ε µνρτ is absolutely antisymmetrictensor of Minkowski space. Equation (2) (or (1)), with three-dimensional reduction, decomposesinto the first pair of Maxwell equations.The second pair of Maxwell equations in four-dimensional form in the absence of sources ∂ ν F µν = 0 (4)are found from Lagrangian L M = − π F µν F µν (5)(here the speed of light c = 1).Infinitesimal transformations with parameter δϕδF µν = e F µν δϕ (6)and the transformations derived from it (taking into account formula (3)) δ e F µν = − F µν δϕ (7)keep the system of Maxwell equations in vacuum (2), (4) invariant, generalize the Rainich-Heaviside transformations and are called phase transformations of classical electrodynamics [2].At the phase transformations, the Lagrangian L M (5) obtains the term: δL M = − π F µν δF µν = − π F µν e F µν δϕ = − π ∂ µ A ν e F µν δϕ = − π [ ∂ µ ( A ν e F µν ) − ( A ν ∂ µ e F µν ] δϕ, which due to identity (2) (the first pair of Maxwell equations) is reduced to divergence. There-fore, the phase transformations (6) are the symmetry of electrodynamics generalized in the senseof [4] and preserves (due to (2)) second pair of Maxwell equations (4).3n [2] it was shown that there is a so-called ”vector Lagrangian” of electrodynamics L Dρ = l π ( e F ρν ∂ µ F νµ − F ρν ∂ µ e F νµ ) , (8)similar to Dirac’s Lagrangian ( l is length dimension constant), which has the following properties:- it is symmetric with respect to the transformation of the Hodge duality (3) and explicitlyphase-symmetric (invariant with respect to transformations (6), (7));- if we take the tensors F µν and e F µν as independent variables in it, then when they are varied,both pairs of Maxwell equations (2) and (4) are obtained as Lagrange equations;- phase transformations (6), (7) lead for Lagrangian L Dρ (8) to Noether current J µρ = − l π ( F σµ F σρ + e F σµ e F σρ ) = − l π ( F σµ F σρ − δ µρ F στ F στ ) (9)which coincides (with accuracy to multiplier l ) with the energy-momentum tensor of the elec-tromagnetic field. We note also the invariance of the tensor J µρ with respect to the dualitytransformation (3) and phase transformations (6), (7). Symmetry, similar to that considered above, is inherent in the theory of gravity, if it is presentedin the frame form [5], [3]. In this case, the potentials of the gravitational field are the componentsof (pseudo)orthonormal frame fields h mµ , and the strength of the gravitational field is describedby anholonomic coefficients F mµν = ∂ ν h mµ − ∂ µ h mν . In this section we discuss the basic relations of the theory of gravity in an orthogonal frame,which are used below.Due to the rotor structure of the tensor F mµν , the Jacobi identity also holds for it ∂ ρ F mµν + cicle ( ρµν ) = 0 , (10)which can be given the form: ∇ ν e F mµν = 1 h ∂ ν ( h e F mµν ) = 0 , (11)where ∇ ν is a covariant derivative in Riemannian space with metric g µν = h mµ h nν η mn , η mn is ametric of Minkowski space, h = det h mµ and e F mµν := 12 ε µνρτ F mρτ (12)4s the Hodge dual tensor to the gravitational field strength tensor F mµν . Here ε µνρτ is the ab-solutely antisymmetric tensor of four-dimensional Riemannian space in curvilinear coordinates.Identity (11) (or (10)) in a Riemannian space with a metric g µν is equivalent to the so-calledcyclic identity: R mρµν + cicle ( ρµν ) = 0and in the gravity theory plays a role similar to the first pair of equations of Maxwell’s electro-dynamics.Einstein equations in vacuum G µm := R µm − h µm R = 0derived from truncated Hilbert’s Lagrangian (M¨oller’s Lagrangian): L H = − R − ∇ σ R σ = 12 δ s pmn ω ms l ω lpn (13)(we assume that Einstein’s gravitational constant κ = 1). Here G µm is the Einstein tensor, R µm is the Ricci tensor, and R is the scalar curvature, δ s pmn := δ sm δ pn − δ pm δ sn - alternator, ω · nlk := 12 ( F · lnk + F · nlk − F · knl )- Ricci rotation coefficients and R p := ∇ σ h σp = F ssp = ω ssp .Using the gravitational field induction tensor (superpotential) B µνm := ω µ νm · − δ µνmp R p (14)allows us write down Lagrangian L H (13) in a form similar to the form of Maxwell’s Lagrangian L M (5): L H = 14 F mµν B µνm , (15)and Einstein equation in the form: G µm = −∇ ν B µνm − t µm = 0 , (16)where t µm = − h ∂ h µm ( hL H ) = B νµn F nνm − h µm L H (17)is the energy-momentum tensor of the gravitational field in the theory of gravity in an orthogonalframe [3], [4]. Einstein equations (16) are nonlinear, are much more complex than the equations of electrody-namics (4) (second pair of Maxwell equations) and are not invariant with respect to transfor-mations δF mµν · · = e F mµν δϕ F mµν through the tensor of its induction B µνm . First of all, note that B sνs = − R ν , whence it follows ω µ νm · = B µνm − δ µνmp B sps Thus, since F mµν = δ k lµν ω mk l , the result is: F mµν = δ k lµν ( B m · k l − δ mnk p η n l B sps ) . (18)Using this formula, Lagrangian L H (15) can be written in the form of a quadratic form withrespect to the gravitational field induction tensor: L H = 12 B m · n s B n sm − B m · m s B n sn . (19)Let us now consider the discrete transformation of the gravitational field induction tensor B µνm → b B µνm := e F · µνm , (20)which, according to (18), generates the following transformation of the strength tensor: F mµν → b F mµν = δ k lµν ( e F m · · k l − δ mnk p η n l e F · sps ) . (21)A unique property of the transformation (21), which is tested directly, is the fact that for a dualHodge tensor of the gravitational field strength, it leads to the transformation: e F · µνm → eb F · µνm = − B µνm , (22)which indicates that this transformation, as well as the transformation of the Hodge dualityin a four-dimensional pseudo-Riemannian space, has the properties of the imaginary unit, i.e.its repeated performance only leads to multiplication by -1. This is ensured the fact that acontinuous infinitesimal transformations with a parameter δϕδF mµν = b F mµν δϕ, (23)built on the basis of duality transformation (21), is compact, therefore, like the phase transfor-mations of quantum mechanics, associated with certain wave properties of the now gravitationalfield in the vacuum. The infinitesimal transformations (23) in terms of the dual tensors B µνm and e F · µνm are written as follows: δB µνm = e F · µνm δϕ, (24)6 e F · µνm = − B µνm δϕ (25)and in the finite version takes the form: B ′ µνm = B µνm cos ϕ + e F · µνm sin ϕ, e F ′ · µνm = − B µνm sin ϕ + e F · µνm cos ϕ. Property (22) occurs precisely because the gravitational field induction tensor is given byexpression (14), which follows from the truncated Hilbert’s Lagrangian (13) of general relativity.This gives grounds to call the quantities F mµν and b F mµν (as well as B µνm and e F µνm ) as GR-dualquantities , and transformation (21) (as well as transformations (20) and (22) agreed with them)as transformations of GR-duality .The following shows that transformations (24), (25), by virtue of identity (11), preserveEinstein equations in the frame variables (16), and therefore are the symmetry of Einstein’stheory of gravity in the (pseudo)orthonormal frame, which we will call the phase symmetryof general relativity . Indeed, due to the quadraticity of the Lagrangian L H with respect to B µνm (see formula (19)) and using formula (24), we find that during the phase transformationsLagrangian L H (15) obtains the term: δL H = 12 F Mµν δB µνm = 12 F mµν e F · µνm δϕ = − ∂ µ h mν e F · µνm δϕ = − h [ ∂ µ ( h e F · µmm ) − h mν ∂ µ ( h e F · µνm )] δϕ = [ ∇ µ e F · µmm − h mν ∇ µ e F · µνm ] δϕ. When performing identity (11) this term reduced to covariant divergence that ensures the in-variance of equations (16), which follow from the Lagrangian L H (Einstein equations in theorthogonal frame), at the phase transformations (24) (or (23)).The method of the above proof of the phase symmetry of Einstein equations shows that thearbitrary quadratic (relative to the strength tensor F mµν ) frame theory of gravity is invariantwith respect to transformations (24) with its induction tensor B µνm , which is determined bythe Lagrangian of the theory L = F mµν B µνm . The compactness of transformations (24), whichprovided by property (22) and therefore corresponds to a certain wave process, demonstratedby Einstein’s theory of gravity with Lagrangian (13), that distinguishes it from others. ButEinstein’s frame theory of gravity is unambiguously fixed among quadratic theories also by therequirement of local Lorentz invariance Λ g . The fact that the Λ g -invariant theory is also phase-symmetric probably has a deep basis, which may be related to the property of the absence ofvolume deviation in Einstein’s space-time during geodetic transfer [6].7 Consequences of phase symmetry of general relativity and”vector Lagrangian”
The above-proven invariance of Einstein equations with respect to phase transformations (24)under condition (11) when applied directly to equation (16) gives δG µm = −∇ ν δB µνm − δt µm = −∇ ν e F · µνm δϕ − δt µm = − δt µm = 0 , therefore, the energy-momentum tensor of the gravitational field t µm (17) is phase-invariant. Thisis easy to verify directly if the tensor t µm is represented as: t µm = 12 ( B nµs F snm + e F · nµs e B s · nm ) , (26)or in the variables e F · µνm and B µνm in the form: t µm = 14 ε mnkl ( B nµs e F skl − e F · nµs B skl · ) , (27)and apply transformations (24), (25). Really, δt µm = 14 ε mnkl ( δB nµs e F skl + B nµs δ e F skl − δ e F · nµs B skl · − e F · nµs δB skl · ) =14 ε mnkl ( e F · nµs e F skl − B nµs B skl · + B nµs B skl · − e F · nµs e F skl ) δϕ = 0 . Equations (11) in the frame theory of gravity play the role of the first pair of Maxwellequations in electrodynamics, but in the general case are not preserved at phase transformations.Really, ∇ ν δ e F · µνm = −∇ ν δB µνm δϕ = t µm δϕ. (28)The last equation in (28) is obtained under the condition that Einstein equations (16) hold.Therefore, the phase symmetry of the complete system of equations of the gravitational field(11) and (16) occurs only under the condition t µm = 0 . (29)In this case, the equations of the gravitational field become linear with respect to the variables e F · µνm and B µνm : ∇ ν e F · µνm = 0 , ∇ ν B µνm = 0 , (30)moreover, the transformation of the GR-duality (20), (22) translate these equations into oneanother, and thus become the symmetry of the complete system of gravitational equations (30).Note that the energy-momentum tensor of the gravitational field is also symmetric with respectto the transformation of duality (20), (22): b t µm = t µm , t µm is only a coordinate tensor (vec-tor) and in the case of Λ g -transformations associated with changes of reference frames [7] istransformed according to the non-tensor law [8]. Thus, the question arises under what condi-tions, given the Λ g -invariance of Einstein’s theory of gravity, we can find such Λ g -gauging (suchreference frame), for which t µm = 0 and, thus, there is a phase symmetry. For example, in [9]it is shown that condition (29) is satisfied in a free-falling reference frame in the Schwarzschildfield and it is assumed that in an arbitrary gravitational field in a free-falling reference frame theenergy-momentum tensor of the gravitational field must be zero due to the equivalence principle.If so, then the requirement of phase symmetry of gravity theory is equivalent to the requirementof choosing free-falling reference frames. This question, of course, requires further study.In the frame theory of gravity, as well as in electrodynamics, it is possible to construct a”vector Lagrangian” like Dirac’s Lagrangian: L Vρ = l e B m · ρ ν ∇ µ B νµm − F mρν ∇ µ e F · νµm ) , (31)which, like Lagrangian (8), has the properties:- it is symmetric with respect to the GR-duality transformation (21) and explicitly phase sym-metric (invariant with respect to transformations (24), (25));- if we take the tensor densities hB µνm and h e F · µνm as independent variables, then when theyare varied, from Lagrangian L Vρ (31) both tensor equations of the gravitational field (30) areobtained as Lagrange equations - cyclic identity and Einstein equation at the condition (29);- for the Lagrangian L Vρ , phase transformations (24), (25) lead to the Noether current J µρ = l B nµs F snρ + e F · nµs e B s · nρ ) = l t µm h mρ , (32)which (up to the multiplier l ) coincides with the energy-momentum tensor of the gravitationalfield (26), written in the coordinate frame. It is established that Einstein’s gravity theory in the orthogonal frame, like quantum mechanicsand electrodynamics, demonstrates (under certain conditions) the presence of compact phasesymmetry, and we found the duality transformation that provides this phenomenon.The conditions of phase symmetry of the gravitational field equations are considered andthe possible connection of the phase symmetry of gravity with the principle of equivalence isindicated. 9 phase-symmetric ”vector Lagrangian” of the gravitational field is constructed and it isshown that the consequence of its phase symmetry is the conservation law of energy-momentumof the gravitational field.However, a number of questions regarding phase symmetry in gravity remain unresolved.This is, first of all, the geometric nature of phase symmetry and its connection with the localLoretz invariance of the theory as well as with the principle of equivalence.In addition, the meaning of the so-called ”vector Lagrangian” in the theory of gravity remainsunclear, as well as its mysterious property, that the consequence of the phase symmetry ofthe ”vector Lagrangian” is the conservation of energy-momentum of the gravitational field,while for truncated Hilbert’s Lagrangian energy-momentum is conserved due to the generalizedtranslational invariance of the theory [4].All of these questions require further study, which may shed light on the connection betweengravity and quantum mechanics.
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