aa r X i v : . [ phy s i c s . g e n - ph ] O c t Energy momentum tensor in the nonsymmetric gravity
B. V. Gisin
IPO, Ha-Tannaim St. 9, Tel-Aviv 69209, Israel. E-mail: [email protected] (Dated:)General relativity is the theory with unclear energy momentum tensor. The proposed approachallows to construct the energy momentum tensor for relativity with non-symmetric metric. A con-sequence of the approach is confirmed in the nuclear physics and may be applicable to astrophysics
PACS numbers: 04.20.Cv, 04.20.Ex, 04.50.Kd, 04.40.Nr
INTRODUCTION
The theory of relativity allows the possibility of a non-symmetric metric tensor. A. Einstein, in his attempts tobuild out an Unified Field Theory, associated the anti-symmetric part of this tensor with electromagnetism [1].Afterwards it was found that the antisymmetric metrictensor may represent a generalized gravity [2] with a newforces and properties [3], [4].Presently, Einstein’s interpretation is not popularamong the physical community. This scepticism is ex-pressed in the phrase: ”Research in this direction ulti-mately proved fruitless; the desired unified field theorywas not found”. One of the problems is the transition toclassical electrodynamics. Another problem is the energymomentum tensor.However, the potential of Einstein’s interpretation isnot exhausted.The paper presents an approach, which on the onehand considers Einstein’s interpretation, on the otherhand is applicable to the generalized Moffat gravity. Thedistinguish feature of the approach is the construction ofthe energy momentum tensor so that the field equationare fulfilled due to Maxwell’s equation.In this way, in particular, the result obtained, thatthe current density squared corresponds to a mass den-sity. The paper demonstrates that this surprising resultis confirmed by experimental data.
FIELD EQUATIONSThe approach
Usually the field equations are governed by the prin-ciple of least action [1], [5]. The Lagrangian is formedfrom the metric tensor and the Ricci tensor. The mainargument in favor of the variational principle is the com-patibility of equations in the system. In this case theenergy momentum tensor (for brevity the energy tensor)must be postulated: ”Our problem now is introduce atensor T µν , of the second rank, whose structure we doknow but provisionally. . . ” [1].In the present approach this argument has no deci- sive significance. The energy tensor is constructed byinserting new terms so that the field equation turns intoequality due to Maxwell’s equation. Such a constructionmay consist of the inclusion of missing terms as well asthe compensation of redundant terms. All inserts in thetensor should be material, that is, they should disappearwhen the electric current goes to a zero.In a sense, the structure of Maxwell’s equation is op-timal for any field described by an antisymmetric tensorof the second rank. Therefore, for simplicity, we use herethe terminology of electrodynamics, in spite of the factthat this approach is applicable also to Moffat’s interpre-tation.In our approach we come back to the initial Einstein’sworks. We postulate that the covariant derivative of themetric tensor equals zero, and equate the Einstein tensorto energy tensor.Generally the symmetric g ( µν ) and antisymmetric g <µν> part of the metric tensor describes the gravita-tional and electromagnetic field. However, for a betterunderstanding of the role of the electromagnetic field andthe principle of constructing the energy tensor, prefer-ably to consider the case of absence of the gravitationalfield. In this case g ( µν ) = δ µν and the interval has the Eu-clidean form . However all manipulations with such a formof the metric tensor allows only the orthogonal coordi-nate transformations. Therefore we assume a very weakgravitational field, γ µν = g ( µν ) − δ µν , | γ µν | ≪ | g <µν> | ,which allows more general transformations. In addition,we assume also | g <µν> | ≪ . Using the assumptions, we expand parameters of thefield in power series in the electromagnetic tensor. Thisleads to arising symmetric terms in the energy tensor.The terms may be sources of the gravitational field. Thefield has the order of the electromagnetic field squared.For simplicity, we use the expansion including the sec-ond order. It is enough to demonstrate main features ofthe approach.The approach provides a simple algorithm of matchingthe field equation and Maxwell’s equation for every orderof the approximation.
The metric tensor and Christoffel’s symbol
We use the normalized (dimensionless) metric tensor,electric current density and coordinates with x = it .The covariant derivative of vectors V µ and V µ is de-fined as V µ ; ν = V µ,ν − V σ Γ σµν , V µ ; ν = V µ,ν + V σ Γ µσν , (1)the semicolon and comma denotes the covariant and par-tial differentiation, respectively.We assume that g µν = δ µν + ϕ µν + γ µν . (2)where δ µν is the Kronecker delta, ϕ µν is the electromag-netic tensor.The covariant derivative of the metric tensor obeys theequation g µν ; σ ≡ g µν,σ − g κν Γ κµσ − g µκ Γ κσν = 0 , (3)Both the tensors g µν and g µν are connected by the rela-tion g σµ g σν = δ µν = g µσ g νσ (4)From this relation g µν is defined as the correspondingminor divided by the determinant g = | g µν | . The Christoffel symbol Γ σµν can be expressed as a func-tion of the metric tensor g µν , using the algebraic equa-tion (3) relative Γ κµσ and Γ κσν . However, in contrast tothe symmetric case, this expression is too sophisticated.Therefore we use successive approximations expandingΓ σµν in power series in ϕ µν and taken into account that γ µν can be presented as a sum of even power in ϕ .With help of Eq. (2) express ϕ νµ ; σ = ( g µν − g νµ ) ; σ and( g µν + g νµ ) ; σ in terms of the metric tensor and Christof-fel’s’ symbols. Then substitute the expression (2) andmake the cyclic interchange µ, ν, σ, we obtain the expan-sion, including the terms of the second orderΓ σµν = ϕ µν,σ + ϕ κν ϕ µσ,κ + ϕ µκ ϕ σν,κ ++ 12 ( γ σµ,ν + γ νσ,µ − γ µν,σ ) + . . . , (5) The Maxwell equation
In the Euclidean space the equation can be written asfollows ϕ { µν,σ } = 0 , ϕ µν,ν = j µ , (6)where the bracers mean the cyclic interchange µ, ν, σ . j µ is the current density j µ = ρ dx µ ds , (7) ρ is the charge density. In the Euclidean space co- and-contravariant vectors coincides.It is well know from quantum theory that the potentialis a more fundamental characteristic than components ofthe electromagnetic field. In the Euclidean space ϕ µν canbe expressed in terms of the potential as follows ϕ µν = A µ,ν − A ν,µ (8)Due to the expression (8) the first equation in (6) isfulfilled identically. But the Einstein tensor contain ϕ µν with two derivatives. Keeping it in mind, differentiatethe first equation of (6) in respect to x σ . Then, using thesecond equation, obtain ϕ µν,σσ − j µ,ν + j ν,µ = 0 . (9)This equation integrates both the field ϕ µν and matter j µ . We should expect appearance of this equation in thefirst approximation of the field equation [6] The Einstein tensor and field equation
Einstein’s tensor is equivalent to that in the generalrelatively G µν = R µν − g µν R. (10)where R µν is the Ricci tensor. R µν = Γ ρµν,ρ − Γ ρµρ,ν + Γ κµν Γ ρκρ − Γ κµρ Γ ρκν . (11)The covariant derivative of G µν equals zero. Equating G µν to the energy tensor G µν = T µν we obtain the fieldequation similarly to the symmetric theory. For conve-nience, the modified energy tensor T ∗ µν = T µν − g µν T, where T = g µν T µν , is in common use. Thus the fieldequation is as follows R µν = T ∗ µν . The first approximation
In the first approximationΓ σµν = ϕ µν,σ , R µν = ϕ µν,σσ − ϕ µσ,σν , (12)we find the field equation ϕ µν,σσ − ϕ µσ,σν = T ∗ µν . (13)A comparison with (9), in accordance with our ap-proach, allows to construct the energy tensor in the firstapproximation T ∗ µν = − j ν,µ , (14)This is an example of the inserted term.Eq. (13) easily can be rearranged as follows( ϕ µν,σ + ϕ σµ,ν + ϕ νσ,µ ) ,σ − ϕ νσ,µσ = − j ν,µ (15)Provided the condition (8), we obtain ϕ νσ,σ = j ν and theMaxwell equation (6). j ν,µ is a ’strange’ term since corresponding energy den-sity changes the sign together with electromagnetic field.Moreover its contribution in energy R j , dv vanishes bythe integration over all the spatial volume, in both theconditions: j µ = 0 at the boundary of integration andthe current conservation j µ,µ = 0. The second approximation
Using Eqs. (5), (6), it can be shown that the antisym-metric part of the Ricci tensor equals R <µν> = ϕ µν,σ j σ . (16)Obviously, the contribution in energy from this term alsoequal zero since ϕ ≡ . Accordingly to our approach,this term in the covariant form should be inserted in theenergy tensor T µν = − j ν ; µ + 2 j σ ϕ µν ; σ . (17)This is an example of the compensating term.It is noteworthy that the term j ν ; µ in the second ap-proximation exactly coincides with the term j α ϕ µν ; σ inthe first approximation. Therefore the inserted term in T µν is doubled.The symmetric part of the Ricci tensor in the secondapproximation is as follows R ( µν ) = j µ j ν + ( ϕ µκ ϕ σν ) ,κσ − ϕ σκ,µν ++ 12 ( γ σµ,νσ + γ νσ,µσ − γ µν,σσ − γ σσ,µν ) . (18)The term j µ j ν = ρ dx µ ds dx ν ds , (19)is of special interest. Its shape is similar to the mass termin the general relativity with symmetric metric, but therole of ρ plays the mass density. Since we use the dimen-sionless units the mass density should be proportional tothe gravitational constant, which is determined by com-paring the field equation with the Newtonian theory ofgravity.From Eq. (19) it follows that the charge densitysquared ρ corresponds to a mass density [7]. This pa-rameter, analogously to the mass density, is proportional to an electrodynamic constant. For this constant onlythe dimension is known.The term j µ j ν must be inserted in the energy tensor.Conveniently to divide the inserts into two parts. Thefirst part is the compensating term (19), which inserteduniquely. The second part is inserted as the same termbut with a constant C on the same basis as in the generaltheory of relativity.With this definitions T ∗ µν takes the form T ∗ µν = − j ν ; µ + 2 j α ϕ µν ; σ + (1 + C ) j µ j ν . (20)The gravitational field obeys field equation12 ( γ σµ,νσ + γ νσ,µσ − γ σσ,µν − γ µν,σσ )+ (21)+ ( ϕ µκ ϕ σν ) ,κσ − ϕ σκ,µν = Cj µ j ν . (22)In the general relativity a supplementary condition isimposed for simplification of the field equation. The pur-pose of the simplification is to cancel the first three termsin (21) γ σµ,νσ + γ νσ,µσ − γ σσ,µν = 0 . (23)This condition consists of 16 equations connecting 10functions γ µν , some of the equations coincide. Howeverthe number of equations may be reduced by symmetry.The same result can be obtained using the four conditionsΞ µν,ν = 0, whereΞ µν = γ µν − δ µν γ σσ . (24)The four conditions are interpreted as an ”analog of theLorentz condition” in electrodynamics.Under this condition the equation for the gravitationalfield is simplified − γ µν,σσ +( ϕ µκ ϕ σν ) ,κσ − ϕ σκ,µν = Cρ dx µ ds dx ν ds . (25)Up to now we considered the gravitational and electro-magnetic field as originated from one source.Assume, that the source of a gravitation field (not con-nected with an electromagnetic field) exists. We dividethe total mass of any charged object as well as nuclei orparticles on the proper mass and mass connected withthe charge.In that case instead the factor C in right part of Eq.(22) would be the factor ( γ + Cρ ) and the tensor γ µν is the tensor of total gravitational field originated fromboth the mass density γ and the mass density connectedwith the charge density Cρ .A symmetric part of the energy tensor in that case,exclusive of ( − j ν ; µ ) , would be T µν = T ∗ µν − g µν T ∗ T µν = [ γ + (1 + C ) ρ ] dx µ ds dx ν ds − . g µν (1 + C ) ρ . (26)In the approximation of low velocities and weak fields T ≈ γ + 12 (1 + C ) ρ (27)is the total mass density.Surprisingly the last term in the expression (27) canbe found in the semi-empirical mass formula. The Bethe–Weizs¨acker mass formula
The semi-empirical mass formula is used to approxi-mate the mass of an atomic nucleus m = Zm p + N m n − a ( A − Z ) A + E b ( A, Z ) , (28)where Z and N is the number of protons and neutrons, A = Z + N is the total number of nucleons, m p and m n are the rest mass of a proton and a neutron, respectively, E b the binding energy of the nucleus, a and E b are smalland determined empirically.The mass corresponding to ρ is defined by the integralover the volume of nucleus. In the liquid drop model ofnucleus approximately Z ρ dv ≈ Z A . (29)Eq. (28) can be changed to following form m = Zm pe + N m ne − a Z A + E b , (30)where m pe and m ne is the effective mass of the protonand neutron in nucleus m pe = m p + 3 a, m ne = m n − a. (31)The three first terms in the right part of Eq. (30)correspond to the integral of the term T from Eq. (26)over all the volume of nucleus Zm pe + N m ne − a Z A → Z [ γ + 12 (1 + C ) ρ ] dv. (32)Accordingly to Eq. (32), C < , moreover (1 + C ) < . Nevertheless, for any atomic nucleus the integral in (32)always is positive.However, if the charge density increases and becomeslarger than nuclear, gravity may be replaced by repulsionand energy may change the sign.
CONCLUSION
We have considered the general relativity with non-symmetric metric and proposed the approach for the con-struction of the energy momentum tensor.It was shown that the energy tensor can be build outfrom the inserted material terms so that the field equa-tion turns into the equality due to Maxwell’s equation.Distinctive feature of the approach is the start fromthe first approximation without the gravitational field.In the second approximation a source of gravitationalfield arises. This source is similar to that in the generalrelativity but with the mass density proportional to thecharge density squared.The term corresponding to the source finds surprisingconfirmation in nuclear physic. Another surprise is thenegative sing of this mass density.If such term is really applicable to nuclear physics thenincreasing the charge density may have far-reaching con-sequences from microphysics to astrophysics. However,for that we should use exact expression for the Christoffelsymbols as a function of the metric tensor.In astrophysics the charge density, which correspondsto the zeroth energy density, would determine the begin-ning of the Big Bang. [1] A. Einstein,
The Meaning of Relativity , University Press(1970), Fifth edition, Princeton, N. J.[2] J. W. Moffat,
Nonsymmetric Gravitational Theory , Phys.Lett. B 355 (3-4): 447–452 (1995).[3] R. T. Hammond,
New Spin on Einstein’s Non-SymmetricMetric Tensor , arXiv:1207.5170v1 [gr-qc] 21 Jul 2012.[4] N. J. Poplawski,
On the Nonsymmetric Purely AffineGravity , Modern Physics Letters A, vol. 22, No. 36 (2007)2701–2720.[5] H. Weyl,