Variational and Conservative Principles in Rosen’s Theory of Gravitation

I. Dobrescu and N. Ionescu-Pallas

Abstract

A variational Action Principle is introduced, in the Rosen’s Theory of Gravitation, in view of deriving field equations, motion equations and a canonical energy tensor. Using the constraint of metric invariance during the variational process along the trajectory, a certain relationship between the canonical tensor and the motion equations is established as a test for selfconsistency. Gravitostatics for spherical bodies is worked out, and the Equivalence Principle between gravitational mass and inertial mass does hold in a weak version (equality of masses, but not also of their space distributions).

Mathematics Subject Classification: 83CXX Key words: General Relativity, Bimetric Theories, Field Equations, Gravitation

The field and motion equations, as well as the canonical energy tensor, may by derived, in the case of Rosen’s theory of gravitation [1], from a certain Action Principle. Adopting the perfect fluid scheme for the Riemannian Matter tensor and specifying the field part of the Action as depending on Minkowskian quantities of definite variance, not exceeding the first order derivatives, the Action integral takes the form

A = \frac{1}{c} ∫ L\sqrt{−g} (d^{4}x) , (1)

where,

(2)  L = L_{m} + L_{f};
L_{m} = [(c^{2} + H) ρ − p]_{R} · K, K = \begin{matrix}√ & \overline{−γ} \\ √ & \overline{−g} \end{matrix};

L_{f} = \frac{c^{4}}{16πG}f (χ^{αβ}, γ_{αβ|λ}; h^{αβ}) .

Here , a subscript R stands for specifying that the labelled quantity is defined in a Riemannian space-time, whose metric is

(dS_{R})^{2} = γ_{μν}dx^{μ}dx^{ν}, (dS_{R}) > 0. (3)

Balkan Journal of Geometry and Its Applications, Vol.4, No.1, 1999, pp. 31-43 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Besides this, a flat space-time metric is adopted [2]

(dS_{M})^{2} = g_{μν}dx^{μ}dx^{ν}, (dS_{M}) > 0. (4)

For ensuring coherence of the whole variational process, not only the coordinates x^{α} and the signatures (time-like) should be the same for the two metrics, but also the metric tensor γ_{μν} of the Riemannian space-time must be considered as a tensor in M_{4} (Minkowskian space time) [3], [4]. Arbitrary coordinates in M_{4} are adopted (necessary for carrying out the variational calculations) and x^{0} = ct is taken as the time coordinate (with physical dimension of a length). So, the role of the two metrics is strongly dissymmetrized, γ_{μν} are some quantities preserving only the meaning of gravitational potentials, and the metric (dS_{R})^{2} turns out to be a simple mathematical artifact necessary to formulate the specific coupling of gravitational field to its sources. This bimetric philosophy (which restricts the main role of Riemannian metric to the motion equations) entitles us to treat the quantities γ_{μν} as true potentials, distinct from the metric functions g_{μν}. Now, performing the variations against γ_{μν}, in the action integral, we come to field equations; variation against g_{μν} delivers a canonical energy tensor, while the variation against the coordinates of a fluid particle delivers motion equations.

For preventing confusion with respect to raising and lowering of indices, we denote distinctly the co- and contra- variant aspects of the metric tensors, always keeping in mind the Minkowskian character of the involved quantities. So, we write

(5)  γ_{μλ}χ^{λν} = δ_{μ}^{ν}, g_{μλ}h^{λν} = δ_{μ}^{ν}
γ^{αβ} = γ_{μν}h^{μα}h^{νβ} ≠ χ^{αβ}, χ_{αβ} = χ^{μν}g_{μα}g_{νβ} ≠ γ_{αβ} .

The quantity K (in (2)) is a Minkowskian scalar, while the quantities ρ (mass density), p (pressure) and H (Helmoltz potential) are scalar in both M_{4} and R_{4}. H is defined as

H = ∫_{0}^{p(ρ)} \frac{dp}{ρ (p)}, (6)

this implying the existence of a reciprocal relationship between pressure and mass density p = p (ρ) , ρ = ρ (p) . (7)

Finally, as a rule,

(8)  γ = Det ∥γ_{μν}∥ , g = Det ∥g_{μν}∥
γ < 0, g < 0

The Minkowskian covariant derivatives are denoted by a vertical bar followed by a certain (Greek) subscript, or (equivalently) by a derivative symbol (D) followed by the same subscript. For instance

γ_{αβ|λ} ≡ D_{λ}γ_{αβ} ≡ γ_{αβ,λ} − G_{αλ}^{σ}γ_{σβ} − G_{βλ}^{σ}γ_{σα}, (9)

where

G_{μν}^{λ} = \frac{1}{2}h^{λσ} (g_{μσ,ν} + g_{νσ,μ} − g_{μν,σ}) (10)

are the Christoffel symbols of the Minkowski space time, and (,) stands for the usual partial derivative.

For obtaining the field equations and the energy canonical tensor, we resort to the following identity

\frac{1}{\sqrt{−g}} δ (\sqrt{−g}L) ≡ D_{λ}q^{λ} − \frac{1}{2}K (T_{αβ} + \frac{c^{4}}{8πG}E_{αβ}) δχ^{αβ} − \frac{1}{2}T_{αβ}δh^{αβ}, (11)

where

(12)  q^{λ} = P^{αβ∥λ}δγ_{αβ} + 2γ_{σν}P^{μν∥η}Ω_{ημ∥αβ}^{λσ}δh^{αβ},
P^{αβ∥λ} ≡ \frac{∂L_{f}}{∂γ_{aβ|λ}} ,
Ω_{μν∥αβ}^{λσ} ≡ \frac{1}{4} (δ_{β}^{λ}δ_{ν}^{σ}g_{μα} + δ_{α}^{λ}δ_{ν}^{σ}g_{μβ} + δ_{β}^{λ}δ_{μ}^{σ}g_{να}+

+δ_{α}^{λ}δ_{μ}^{σ}g_{νβ} − h^{λσ}g_{μα}g_{νβ} − h^{λσ}g_{μβ}g_{να}) ,

T_{αβ} = {(c^{2} + H) ρU_{α}U_{β} − pγ_{aβ}}_{R} ,
E_{αβ} ≡ R_{αβ} − \frac{1}{2}γ_{αβ} (χ^{μν}R_{μν}) ,
R_{αβ} = −\frac{1}{K} \{(\frac{∂f}{∂χ^{αβ}} − \frac{1}{2}γ_{αβ}χ^{μν} \frac{∂f}{∂χ^{μν}} ) +

+ (γ_{μα}γ_{νβ} − \frac{1}{2}γ_{αβ}γ_{μν}) D_{λ}p^{μν∥λ}\} ,

p^{αβ∥λ} = \frac{16πG}{c^{4}}P^{αβ∥λ},
T_{αβ} = D_{σ}Q_{αβ}^{σ} − 2 (\frac{∂L_{f}}{∂h^{αβ}} − \frac{1}{2}g_{αβ}L_{f}) ,
Q_{αβ}^{σ} = 4γ_{λζ}P^{μζ∥ν}Ω_{μν∥αβ}^{λσ}.

The (trivially) conservative tensor Ω is coming from variation against g_{αβ} of the Minkowskian affine connections [5].

Putting the variation of the action integral against χ^{αβ} to vanish, one obtains the field equations

R_{αβ} − \frac{1}{2}γ_{αβ} (χ^{μν}R_{μν}) = −8π \frac{G}{c^{4}} T_{αβ} . (13)

The denotation are intentional, so as to reach just the Einsteinian expression. However, R_{αβ} is not, necessarily, the Ricci curvature tensor, excepting the case when a certain special function is chosen for L_{f}. Even so, the tensor E_{αβ} (defined in (12)) has more in common with its Einsteinian homologous, because it satisfies the Einsteinian conservative condition ∇_{μ} (χ^{μα}χ^{νβ}E_{αβ}) = 0; (14)

justifying the statement that Rosen’s Theory is the most likely theory to the Einstein’s one. The result (14) may be proved in the following way. Performing the variation of the action integral against the coordinates of a fluid particle one obtains (just as in Einstein’s theory) [6] the geodetic type motion equation

δ ∫ (1 + \frac{1}{c^{2}} H) \sqrt{γ_{μν}dx^{μ}dx^{ν}} = 0. (15)

At the same time, the respective motion equations may be obtained by putting the Riemannian covariant divergence of the tensor T to vanish, accounting equally for the particle current conservation in both R_{4} and M_{4}. Accordingly, ∇_{μ} (χ^{μα}χ^{νβ}T_{αβ}) = 0 emerges as a consequence of the motion equation (15) and, through the result (13) one reach the result (14). To a certain extent, this may justify the interpretation of the bimetric theories as a natural extension of the Einstein’s Gravitation Theory. Einstein’s theory itself may be considered as a special bimetric theory for which the following identity does hold

∇_{μ} (χ^{μα}χ^{νβ}E_{αβ}) ≡ 0; (16)

Rosen’s gravitation theory may be obtained out of the general bimetric theory so far presented by specifying the function f

(17)  f = −\frac{1}{8}h^{αβ}W^{(λσ)(ρτ)}γ_{λσ|α}γ_{ρτ|β},
W^{(λσ)(ρτ)} ≡ (χ^{λρ}χ^{στ} + χ^{σρ}χ^{λτ} − χ^{λσ}χ^{ρτ} ) .

The expression of the tensor R_{μν} acquires the form

R_{μν} = \frac{1}{2K} (□γ_{μν} − χ^{λσ}γ_{μλ|α}γ_{νσ}^{|α}) , □ ≡ h^{αβ}D_{α}D_{β,} (18)

and the tensor T_{αβ} turns out to be

(19) T_{αβ} = \frac{c^{4}}{32πG} \{χ^{μλ}χ^{νσ} (γ_{μν|α}γ_{λσ|β} − \frac{1}{2}g_{αβ}γ_{μν|η}γ_{λσ}^{|η}) −

−2 [(ln K)_{,α} (ln K)_{,β} − \frac{1}{2}g_{αβ}h^{μν} (ln K)_{,μ} (ln K)_{,ν}] − P_{αβ}\} ,

(20) P_{αβ} ≡ (γ_{μβ|α}χ^{μν})_{|ν} + (γ_{μα|β}χ^{μν})_{|ν} + (γ_{μβ}^{|ν}χ_{α}^{μ})_{|ν} +

+ (γ_{μα}^{|ν}χ_{β}^{μ})_{|ν} − (γ_{μ|α}^{ν}χ_{β}^{μ})_{|ν} − (γ_{μ|β}^{ν}χ_{α}^{μ})_{|ν} − 2g_{αβ}□ ln K.

The contravariant derivative (...^{|α}) is lift via h^{αβ}.

There is a close analogy between the tensors T_{αβ} and T_{αβ} + \frac{c^{4}}{8πG}E_{αβ}, both being canonical energy tensors-the first one defined in M_{4} and the second one in R_{4}. Their conservative character is conditioned-through the intermediary of motion equationsby the semi-local connection (i. e. along the trajectory of a fluid particle) of the metrical functions variation to the variation of the fluid particle coordinates [7]

δh^{αβ} = D_{λ} (h^{λα}δx^{β} + h^{λβ}δx^{α}) , (21)

δχ^{αβ} = ∇_{λ} (χ^{λα}δx^{β} + χ^{λβ}δx^{α}) . (22)

Here, ∇_{λ} stands for the covariant derivative in R_{4}. The formulas (21), (22) are a specific feature of the bimetric doctrine, because they mean conservation, under variational changes, of both curved and flat metrics. Out of (22) and (11) we obtain the identity

(23) δ_{γ} (\sqrt{−g}L) ≡ \frac{∂}{∂x^{λ}} (\sqrt{−g}P^{αβ∥λ}δγ_{αβ}) − \frac{1}{2}\sqrt{−γ} (T_{αβ} + \frac{c^{4}}{8πG}E_{αβ}) δχ^{αβ}
≡ \frac{∂}{∂x^{λ}} (\sqrt{−g}P^{αβ∥λ}δγ_{αβ}) − \frac{∂}{∂x^{λ}} \{\sqrt{−γ} (δx^{σ}) [(\frac{T_{·σ}^{λ} + T_{σ}^{·λ}}{2} ) +
+ \frac{c^{4}}{8πG} (\frac{E_{·σ}^{λ} + E_{σ}^{·λ}}{2} )]\} + \sqrt{−γ} (δx^{σ}) ∇_{λ} [(\frac{T_{·σ}^{λ} + T_{σ}^{·λ}}{2} ) +
+ \frac{c^{4}}{8πG} (\frac{E_{·σ}^{λ} + E_{σ}^{·λ}}{2} )] ,

where T_{αβ} is a symmetrical tensor in both R_{4} and M_{4} and the index raising is made with the help of χ^{μν}. By asking the condition

δ_{γ}A = 0, (24)

we come to a (Riemannian) conservative tensor

τ_{αβ} ≡ T_{αβ} + \frac{c^{4}}{8πG}E_{αβ}, ∇_{μ} (χ^{μα}χ^{νβ}τ_{αβ}) = 0. (25)

On the other hand, taking in view that (in R_{4})

(26)  δ_{x}L_{R} = \frac{∂L_{R}}{∂ρ} δ_{x}ρ, δ_{x}ρ = ρU_{σ}U^{ν}∇_{ν} (δx^{σ}) − ∇_{σ} (ρδx^{σ}) ;
∇_{ν} (ρU^{ν}) = 0, U^{α} = \frac{dx^{α}}{dS} , dS = (γ_{μν}dx^{μ}dx^{ν})^{\frac{1}{2}} , \sqrt{−γ}L_{R} = \sqrt{−g}L ,

we have equally

(27)  δ_{x} (\sqrt{−g}L) = \frac{∂}{∂x^{ν}} \{\sqrt{−γ}ρ\frac{∂L_{R}}{∂ρ} δx^{σ} (U^{ν}U_{σ} − δ_{σ}^{ν})\} −
−  \sqrt{−γ} (δx^{σ}) ρ\frac{∂L_{R}}{∂ρ} γ_{σλ} \{\frac{d^{2}x^{λ}}{dS^{2}} + Γ_{αβ}^{λ}U^{α}U^{β}+
+  (U^{λ}U^{ν} − χ^{λν}) ∇_{ν} ln (\frac{1}{c^{2}} \frac{∂L_{R}}{∂ρ} )\} .

Now, taking in view that (onto the trajectory and with geodetic constraint (22))

δ_{γ}A + δ_{x}A = 0,

we come to the relevant result

∇_{λ} (T^{λμ} + \frac{c^{4}}{8πG}E^{λμ}) = ρ\frac{∂L_{R}}{∂ρ} ω^{μ}, (28)

where

ω^{μ} ≡ \frac{d^{2}x^{μ}}{dS^{2}} + Γ_{αβ}^{μ}U^{α}U^{β} + (U^{μ}U^{ν} − χ^{μν}) \frac{∂}{∂x^{ν}} ln (\frac{1}{c^{2}} \frac{∂L_{R}}{∂ρ} ) . (29)

But, equation (28) may be written (equivalently) as

∇_{λ} (T^{λμ} + \frac{c^{4}}{8πG}E^{λμ}) = −\frac{1}{m_{0}} ρχ^{μλ}Q_{λ}, (30)

where

(31)  Q_{λ} ≡ \frac{d}{dS} (\frac{∂L}{∂U^{λ}} )_{x} − (\frac{∂L}{∂x^{λ}} )_{U} ,
L ≡ −m_{0} \frac{∂L_{R}}{∂ρ} \sqrt{γ_{αβ}U^{α}U^{β}},
Q_{λ} ≡ −m_{0} \frac{∂L_{R}}{∂ρ} γ_{μλ}ω^{μ},

and it is transparent that Q_{λ} = 0 are the Euler-Lagrange equations of motion coming from the variational problem

δ ∫ LdS ≡ δ (−m_{0} ∫ \frac{∂L_{R}}{∂ρ} \sqrt{γ_{αβ}dx^{α}dx^{β}}) = 0. (32)

Asking the condition (25) in (28) we come to the geodetic motion equation (32). As the equation (32) may be recovered asking only ∇_{λ}T^{λμ} = 0, we reach the result (14), Q.E.D.

One of the specific features of the General Relativity Theory, namely, the derivation of the motion equations out of the field equations is to be found equally in Rosen’s Bimetric Theory. Indeed, by putting the tensor τ_{αβ} to vanish all over the 4-dimensional space-time, we get the field equations (13). At the same time, this condition entails the vanishing of the covariant derivative of the respective tensor as well (25), which, through the intermediary of (28) or (30), delivers the motion equations (ω^{μ} = 0 or, equivalently, Q_{λ} = 0).

An alternative way of calculating the total variation of action integral, along a fluid particle trajectory, against the particle coordinates variations, is to resort to the geodetic constraint (21) instead of (22). To notice that the condition (21) means the keeping of the Minkowskian metrics invariance during the variational process, while the condition (22) implies the same status for the Riemannian metrics. The putting on the same footing the two metrics may be considered as a basic feature of the bimetrism, regarded from the standpoint of the variational process. Thus, we can write

(33) δ_{g} (\sqrt{−g}L) + δ_{x} (\sqrt{−g}L) = \frac{∂}{∂x^{λ}} \{\sqrt{−g} [2γ_{σμ}P^{μν∥η} · Ω_{ην∥αβ}^{λσ}δh^{αβ}−
− \frac{1}{2} (T_{·α}^{λ} + T_{α}^{·λ}) δx^{α} +
+ (ρ\frac{∂L}{∂ρ})_{R} (U^{λ}U_{α} − δ_{α}^{λ}) δx^{α}]\} +

+ \sqrt{−g} (δx^{α}) \{D_{λ} (\frac{T_{·α}^{λ} + T_{α}^{·λ}}{2} ) −
− K (ρ\frac{∂L}{∂ρ})_{R} γ_{αλ}ω^{λ}\};

δ_{g}A + δ_{x}A = 0; (34)

D_{λ}T^{λμ} − K · (ρ\frac{∂L}{∂ρ})_{R} h^{αμ}γ_{αλ}ω^{λ} = 0; (35)

whence (accounting for the motion equations ω^{λ} = 0), we conclude about the Minkowskian conservativity of tensor T^{αβ}.

An unsatisfactory aspect of the tensor T^{αβ} is the presence in its structure of the second order derivatives of the potentials. These appear through the intermediary of the tensor P_{αβ} and cannot be put all in D’Alembert form, in order to be avoided by resorting to the field equations. Indeed, we can write

P_{αβ} = (χ_{β}^{μ}□γ_{μα} + χ_{α}^{μ}□γ_{μβ} − 2g_{αβ}□ ln K) + (γ_{μβ}^{|ν}χ_{α |ν}^{μ} + γ_{μα}^{|ν}χ_{β |ν}^{μ}) + Π_{αβ}, (36)

Π_{αβ} = (γ_{μβ|α}χ^{μν} + γ_{μα|β}χ^{μν} − γ_{μ |α}^{ν}χ_{β}^{μ} − γ_{μ |β}^{ν}χ_{α}^{μ})_{|ν} . (37)

If the tensor Π_{αβ} (which gathers all the non-D’Alembertian second order terms of T^{αβ}) would have an identically vanishing covariant divergence, any difficulty will no longer survive. Unfortunately, this is not the case, because we have instead

ξ^{α} ≡ Π_{|β}^{αβ} = (γ_{μ}^{α|ν}χ^{μβ} − γ_{μ}^{β|ν}χ^{μα})_{|νβ} ≠ 0 , (38)

ξ_{|α}^{α} ≡ 0. (39)

Further on , we shall prove that, however, the tensor T^{αβ} does agree with the equivalence principle in a weak version (i. e. equality between inertial and gravitational masses, but not between their spatial densities). In this respect, we need first to write down the field equations for a static spherically symmetric mass distribution. We denote

R_{μν} = \frac{1}{K} N_{μν}, N_{μν} = \frac{1}{2} (□γ_{μν} − ε_{μν}) , ε_{μν} = χ^{λσ}γ_{μλ|α}γ_{νσ}^{|α}, (40)

and write down the field equations in the form

N_{αβ} = −\frac{8πG}{c^{4}}K (T_{αβ} − \frac{1}{2}γ_{αβ}T )_{R} . (41)

Thereafter we adopt polar coordinates and a diagonal form for metrical tensor γ_{αβ}

(42)  x^{0} = ct, x^{1} = r, x^{2} = θ, x^{3} = φ;
γ_{00} = e^{+ν}, γ_{11} = −e^{+λ}, γ_{22} = −r^{2}e^{+σ}, γ_{33} = −r^{2} sin^{2} θe^{+σ},

and, after a long and tedious calculation, we obtain [8]

(43)  □e^{ν} = −△_{r}e^{ν}, −□e^{λ} = △_{r}e^{λ} − \frac{4}{r^{2}} (e^{λ} − e^{σ}) ,
−□ (r^{2}e^{σ}) = r^{2}△_{r}e^{σ} + 2 (e^{λ} − e^{σ}) ,
−□ (r^{2} sin^{2} θe^{σ}) = sin^{2} θ [r^{2}△_{r}e^{σ} + 2 (e^{λ} − e^{σ})];

(44)  ε_{00} = −ν^{′2}e^{ν}, ε_{11} = λ^{′2} + \frac{2}{r^{2}} e^{−σ} (e^{σ} − e^{λ})^{2} ,
ε_{22} = r^{2}σ^{′2}e^{σ} + e^{−λ} (e^{σ} − e^{λ})^{2} ,
ε_{33} = sin^{2} θ [r^{2}σ^{′2}e^{σ} + e^{−λ} (e^{σ} − e^{λ})^{2}];

(45)  N_{00} = −\frac{1}{2}e^{ν}△_{r}ν, N_{11} = \frac{1}{2}e^{λ} [△_{r}λ + \frac{4}{r^{2}} sinh (σ − λ)] ,
N_{22} = \frac{1}{2}r^{2}e^{σ} [△_{r}σ − \frac{2}{r^{2}} sinh (σ − λ)] , N_{33} = N_{22} sin^{2} θ .

But, as a result of the specific form of the tensor T_{αβ} in the perfect fluid scheme, we get the subsidiary condition among the components of the N_{αβ} tensor

N_{22} = r^{2}e^{σ−λ}N_{11} . (46)

The previous condition may be written, in explicit form as

△_{r} (σ − λ) − \frac{6}{r^{2}} sinh (σ − λ) = 0 . (47)

So, we come to the solution σ = λ . (48)

Now, by denoting

ν = −2\frac{ψ_{1}}{c^{2}} , λ = +2\frac{ψ_{2}}{c^{2}} , (49)

we reach the so called exponential metric

(dS)_{R}^{2} = e^{−2 \frac{ψ_{1}}{c^{2}}} (cdt)^{2} − e^{+2 \frac{ψ_{2}}{c^{2}}} \{(dr)^{2} + r^{2} [(dθ)^{2} + sin^{2} θ (dφ)^{2}]\} , (50)

where

(51)  △_{r}ψ_{1} = −4πGe^{\frac{3ψ_{2}−ψ_{1}}{c^{2}}} [ρ (1 + \frac{1}{c^{2}} H) + 2\frac{p}{c^{2}} ]_{R} .
△_{r}ψ_{2} = −4πGe^{\frac{3ψ_{2}−ψ_{1}}{c^{2}}} [ρ (1 + \frac{1}{c^{2}} H) − 2\frac{p}{c^{2}} ]_{R} .

Taking the source body to be a sphere of radius R, we define two masses, namely

M_{1} = ∫_{0}^{R} 4πr^{2}ρ_{1}dr , M_{2} = ∫_{0}^{R} 4πr^{2}ρ_{2}dr, (52) where the densities (ρ_{1}, ρ_{2}) are those from (51), i. e. △_{r}ψ_{1} (r) = −4πGρ_{1}, △_{r}ψ_{2} (r) = −4πGρ_{2}. To notice that, even outside the mass distribution, the metric (dS_{R})^{2} contains two potential functions, (unlike the simple exponential metric of Yilmaz and Rastall, which contains a single such function) [9], [10]

(dS_{R})^{2} = e^{−2 \frac{μ_{1}}{r}} (cdt)^{2} − e^{+2 \frac{μ_{2}}{r}} (dr⃗)^{2} , (53)

μ_{1} = \frac{GM_{1}}{c^{2}} , μ_{2} = \frac{GM_{2}}{c^{2}} , M_{1} ≠ M_{2} . (54)

The presence of two masses M_{1} and M_{2} in the static metric may be considered as a second unsatisfactory aspect of the Rosen’s bimetric theory of gravitation.

Coming back to the canonical energy tensor, and calculating the T_{00} component, we obtain, in the static case of a spherical source body

T_{00} = \frac{c^{4}}{32πG} [A + B − P_{00}] , (55)

(56)  A = −\frac{1}{2}χ^{λσ}χ^{μν}γ_{λμ|α}γ_{σν}^{|α} = \frac{2}{c^{4}} (3ψ_{2}^{′2} + ψ_{1}^{′2}) ,
B = (ln K)_{|α} (ln K)^{|α} = −\frac{1}{c^{4}} (3ψ′_{2} − ψ′_{1})^{2} ,
P_{00} = \frac{2}{c^{2}} △_{r} (3ψ_{2} + ψ_{1}) = −\frac{8πG}{c^{2}} (3ρ_{2} + ρ_{1}) .

We notice the elimination of the non-D’Alembertian terms of T_{00} in the static case. Thus

T_{00} = (\frac{3}{4}ρ_{2} + \frac{1}{4}ρ_{1}) c^{2} + \frac{1}{32πG} (ψ_{1}^{′2} + 6ψ′_{1}ψ′_{2} − 3ψ_{2}^{′2}) . (57)

Using the Eddington’s definition of the gravitational mass together with the correspondence principle between a relativistic theory of gravitation and the Newtonian theory, we come to the formula

M_{g} = lim_{r→∞} (\frac{1}{G}rψ_{1}) = ∫_{0}^{R} 4πr^{2}ρ_{1} (r) dr . (58)

At the same time, the inertial mass is defined as

M_{i} = \frac{1}{c^{2}} ∫_{0}^{∞} T_{00}4πr^{2}dr . (59)

The weak version of the equivalence principle between the two kinds of mass requires the vanishing of the following integral

I ≡ ∫_{0}^{∞} (\frac{1}{c^{2}} T_{00} − ρ_{1}) 4πr^{2}dr . (60)

Accounting for (57), the respective integral becomes

I = ∫_{0}^{∞} \{\frac{3}{4} (ρ_{2} − ρ_{1}) + \frac{1}{32πGc^{2}} (ψ_{1}^{′2} + 6ψ′_{1}ψ′_{2} − 3ψ_{2}^{′2})\} 4πr^{2}dr . (61)

But from (49) and (50), we obtain for the difference ρ_{2} − ρ_{1} the expression

ρ_{2} − ρ_{1} = −4\frac{p_{R}}{c^{2}} e^{\frac{3ψ_{2}−ψ_{1}}{c^{2}}} (62)

so that the integral I may be thought to the form

I = −\frac{1}{c^{2}} ∫_{0}^{∞} \{3p_{R}e^{\frac{3ψ_{2}−ψ_{1}}{c^{2}}} − \frac{1}{32πG} (ψ_{1}^{′2} + 6ψ′_{1}ψ′_{2} − 3ψ_{2}^{′2})\} 4πr^{2}dr . (63)

In the Newtonian approximation, we can write

p_{R} = p_{N} + O (\frac{1}{c^{2}} ) , ψ_{1} = ψ_{N} + O (\frac{1}{c^{2}} ) , ψ_{2} = ψ_{N} + O (\frac{1}{c^{2}} ) , (64)

and the integral (63) becomes

I = −\frac{1}{c^{2}} ∫_{0}^{∞} (3p_{N} − \frac{ψ_{N}^{′2}}{8πG}) 4πr^{2}dr + O (\frac{1}{c^{4}} ) . (65)

The equilibrium pressure p and the gravitational potential ψ are expressed, in the nonrelativistic approximation, by the formulas

p_{N} = − ∫_{r}^{R} ρψ′dr, △_{r}ψ_{N} = −4πGρ_{N}, (66)

ρ_{N} = \{ \begin{matrix}ρ_{N} (r) r ∈ (0, R) \\ 0 r ∈ (R, ∞) \end{matrix} p_{N} = \{ \begin{matrix}p_{N} (r) r ∈ (0, R) \\ 0 r ∈ (R, ∞) \end{matrix} (67)

so that the integral we want to estimate may be transformed in a suitable way

I = −\frac{1}{c^{2}} ∫_{0}^{R} (3p_{N} − \frac{1}{2}ρ_{N}ψ_{N}) 4πr^{2}dr + O (\frac{1}{c^{4}} ) . (68)

Now, inserting p_{N} from (66) in (68) and working out the necessary calculations we come to the result we look for

∫_{0}^{R} (3p_{N} − \frac{1}{2}ρ_{N}ψ_{N}) 4πr^{2}dr ≡ 0, → I = O (\frac{1}{c^{4}} ) , (69)

id est

M_{i} − M_{g} = O (\frac{1}{c^{4}} ) . (70)

This is the quantitative expression of the statement made by Nathan Rosen (without proof), that his theory does comply with the equivalence principle.

Conclusions

The variational formulation of the Rosen’s Theory of Gravitation in the perfect fluid scheme and the derivation of the energy tensor are entirely original. The proof of the equivalence principle between the inertial mass and the gravitational mass is also given here for the first time. In the original Rosen’s work what is proved, in connection with this principle, is the equivalence between the inertial force and the gravitational force. The similarity between the Einstein’s Theory and the Rosen’s one is given in our paper, pointing out the bimetric feature of the two theories. Finally, a clear difference between the Rosen’s exponential metric and the similar metrics from the literature is put into evidence. Acknowledgement. The authors would like to thank to the reviewer and editors of BJGA for several helpful comments on our preliminary version of this paper. One of the authors (I. D.) is obliged to Professor Ioan Gottlieb for his constant moral assistance and useful advice. Thanks are also conveyed to Phys. Dr. Marius Piso, Chief of the Gravitation Research Laboratory for his encouragement during the elaboration of this work and to Miss Sylvia Onofrei for her interest in this subject in the initial phase of the work.

Appendix General static metric of spherical symmetry

Solving eq (47) for σ ≠ λ one obtains

u (ρ) = 0.987 606 7 × 10^{0}ρ^{2} + 2.675 775 0 × 10^{−2}ρ^{6}
+ 1.204 552 6 × 10^{−3}ρ^{10} + 6.421 986 8 × 10^{−5}ρ^{14}
+ 3.718 234 4 × 10^{−6}ρ^{18} + 2.260 255 4 × 10^{−7}ρ^{22}
+ 1.418 832 9 × 10^{−8}ρ^{26} + 9.111 626 1 × 10^{−10}ρ^{30}
0 < ρ ≤ 1 ,
u (ρ) = 1.000 000 0 × 10^{0}ρ^{−3} + 1.515 151 5 × 10^{−2}ρ^{−9}
+ 4.679 144 4 × 10^{−4}ρ^{−15} + 1.707 923 6 × 10^{−5}ρ^{−21}
+ 6.725 845 2 × 10^{−7}ρ^{−27} + 2.766 368 7 × 10^{−8}ρ^{−33}
+ 1.170 356 0 × 10^{−9}ρ^{−39} + 5.050 235 9 × 10^{−11}ρ^{−45}
1 ≤ ρ < ∞,

where u = σ − λ and ρ = r/r_{0}, r_{0} standing for an arbitrary constant with physical dimension of a length. The metric acquires the form

(dS)^{2} = e^{− \frac{2}{c^{2}} ψ_{1}} (cdt)^{2} − e^{+ \frac{2}{c^{2}} ψ_{2}} ·
· \{e^{− \frac{2}{3} u} · \frac{(r · dr)^{2}}{r^{2}} + e^{+ \frac{1}{3} u} \frac{(r × dr)^{2}}{r^{2}} \} .

For a constant mass density inside the spherical source, we have

ψ_{1} = \frac{GM}{r} , ψ_{2} = \frac{GM}{r} (1 − \frac{4}{5} \frac{GM}{c^{2}R} ) , r > R .

The equation

\frac{d^{2}u}{dρ^{2}} + \frac{2}{ρ} \frac{du}{dρ} − \frac{6}{ρ^{2}} sinh u = 0

is fulfilled within a relative error of 10^{−6} and u (ρ) is continuous in the point ρ = 1. No prescription is made in the framework of Rosen’s bimetric theory for determining r_{0}. Fock’s gauge does not apply because ψ_{1} ≠ ψ_{2}. However, if r_{0} has the magnitude order of GM/c^{2}, then the gravitational tests are not changed by the presence of the function u (ρ) .

The following empirical rules concerning u (ρ) may be established

u (ρ) = ∑_{n=1}^{n=∞} a_{n}ρ^{4n−2}
a_{1} = 0.987 606 7
\frac{a_{n+1}}{a_{n}} = (7.339 333 7 − 6.720 578 5\frac{1}{n} + 2.090 597 6\frac{1}{n^{2}} ) × 10^{−2}
0 ≤ ρ ≤ 1,

u (ρ) = ∑_{n=1}^{n=∞} b_{n}ρ^{−6n+3}
b_{1} = 1
\frac{b_{n+1}}{b_{n}} = (4.829 979 3 − 3.652 148 1\frac{1}{n} + 0.337 320 3\frac{1}{n^{2}} ) × 10^{−2}
1 ≤ ρ < ∞ .

For checking the efficiency of these approximations, we calculated

u_{1−0} = u_{1+0} = 1.015 637 2
△u_{1−0} = 7.196 849 4; △u_{1+0} = 7.196 847 7
(\frac{6}{ρ^{2}} sinh u)_{1∓0} = 7.196 851 9 .

For determining the coefficients of the empirical formulas a_{n+1}/a_{n}, b_{n+1}/b_{n}, the first few terms of the series expansions of u (ρ) were directly calculated

a_{2} = \frac{1}{66}a_{1}^{3}, a_{3} = \frac{7}{14 960}a_{1}^{5}, a_{4} = \frac{181}{10 597 664}a_{1}^{7}
b_{2} = \frac{1}{36}b_{1}^{3}, b_{3} = \frac{1}{780}b_{1}^{5}, b_{4} = \frac{281}{4 009 824}b_{1}^{7} .

References

[1] N. Rosen, A Bimetric Theory of Gravitation, GRG Vol. 4, No. 6 (1973), pp. 435-447; A Theory of Gravitation, Ann. Phys. 84, 455 (1974).

[2] N. Rosen, General Relativity and Flat Space, Phys. Rev. 57, 147 (1940).

[3] A. Papapetrou, Einstein’s Theory of Gravitation and Flat Space, Proc. Roy. Irish Acad., A52, 11 (1948).

[4] N. Rosen, Flat Space Metric in General Relativity Theory, Ann. Phys. 22, 1 (1963).

[5] N. Ionescu-Pallas, Vectorial Theory of Gravitation II, Romanian Rep. Phys. Vol. 45, No. 1, (1993), pp. 19-52.

[6] V. A Fock, The Theory of Space, Time and Gravitation, Pergamon Press, Oxford, 1958.

[7] Sir A. S. Eddington, The Mathematical Theory of Relativity, Cambridge University Press, 1960.

[8] I. Dobrescu, N. Ionescu-Pallas, Communication at the Conference on Theoretical Physics in Bistritza, Romania, June 1998

[9] H. Yilmaz, New Aproach to General Relativity, Phys. Rev. 111, 1417 (1958).

[10] P. Rastall, An Improved Theory of Gravitation, Canad. J. of Phys. 46, 19, 2155 (1968)

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