Gravitational Energy-Momentum Tensor in General Relativity

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RELATIVISTIC GRAVITATIONAL ENERGY-MOMENTUM TENSOR

Gravitational Energy in the Surrounding of a Star - Fall Velocity
By Noah Nissani
Based in the Noah Nissani and Elhanan Leibowitz's papers that appear in the bibliography.

Abstract
The tensorial expression for the gravitational energy-momentum proposed by Nissani and Leibowitz [N & L, 91, 92] is thoroughly presented and used in calculating the integral gravitational energy in the surrounding of a star, and the fall velocity of a body in a gravitational field. In both cases the results coincide with Newton and Einstein's values up to the sixth and seventh digit, respectively.
1. Introduction
The existence in curved space-time of global like-inertial frames, the "non-rotating" frames, was shown in [Nissani and Leibowitz 1988, 89, 89b, 90, 91, 92. Carmeli, Leibowitz and Nissani 1990] and in "Like-Inertial Globally Preferred Frames in General Relativity" in this site. These frames are derived by a variational principle on the space-time integrals of the energy-momentum components. It is, therefore, in total accordance with Mach-Einstein's assumption, which asserts that the preferential characteristics of the inertial frames of Newton Mechanics (NM) and Special Relativity (SR) must be traced to their relation with the energy-momentum distribution.
As it was shown, these like-inertial frames of curved space-time satisfy all the experimental facts that characterized the inertial frames of flat space, including the global ordinary-divergencelessness of the energy-momentum tensor density. Therefore, the NM and SR energy-momentum conservation law, which is experimentally valid in the inertial frames alone, are rescue valid in the non-rotating frames of curved space-time.
Any attempt to extend the validity of this law to all coordinate frames, whichever be the background theory - NM, SR or General Relativity (GR), requires the introduction of spurious non-tensorial kinds of energy with strong coordinate dependence. Tremendous force fields would be required to maintain the whole universe performing its daily rotation around the earth, as they do in non-inertial coordinates rotating with the earth. Ad-hoc kinds of energy-moment would be required as source of these forces to maintain valid in every frame the energy-momentum conservation. These kinds of energy-momentum would vanish, or at least satisfy by themselves and ordinary-divergencelessness condition, and hence be inactive, in the inertial frames. Similarly, in General Relativity, the baseless requirement that the energy-momentum conservation law be valid in all coordinate system, led to an anomalous, non-localized, pseudo-tensorial gravitational energy-momentum. In the non-rotating frames, in which the covariant divergencelessness of the energy-momentum tensor becomes an ordinary divergencelessness, every pseudo-tensorial gravitational energy-momentum must vanish, or satisfy by itself an ordinary-divergencelessness condition.

2. Energy-Momentum Conservation
Since the experimental facts concerning energy-momentum conservation show it only holding in a preferred class of coordinates - the inertial frames of NM and SR, it is not justified the demand of its holding in all systems of coordinates. Furthermore, the existence in GR of global like-inertial frames wherein the covariant divergencelessness of the energy-momentum tensor becomes an ordinary divergencelessness, precludes the interchange of energy between any kind of energy included in this conserved tensor and any other excluded from it.
To exemplify the above assertion, let us assume, following [N and L, 88] (where may be found a more detailed exposition), a region of 3-space surrounded by a surface on which T vanishes. Between the times t1 and t2 this region traces a 4-volume D in space-time.

Applying the Gauss theorem one obtains, in a global non-rotating and local geodesic along the trajectory coordinate system, whose existence was shown in [N and L, 88 and 89], and where
    (1) that     (2) are four conserved integrals, which give the components, in the local-geodesic coordinates, of the energy-momentum affine 4-vector content of the 3-space region at two arbitrary times. Notice that, the above equalities are strictly valid, despite the finite extension of the 3-space region, since Eq.1 is valid throughout the region, when expressed in non-rotating coordinates.
Consider now a Weber antenna in the interior of the above otherwise totally empty 3-space region and a gravitational wave reaching it at a given moment between t1 and t2. If the gravitational energy-momentum carried by the wave is an anomalous kind of energy not included in the matter energy-momentum tensor T, then, this last vanishes on any Gauss surface surrounding the antenna, even while the gravitational wave transverses it. Therefore, as a consequence of the global ordinary-divergencelessness of the energy-momentum tensor density valid on the non-rotating coordinates, which in turn derives from the covariant-divergencelessness imposed on T by Einstein's equation, the energy-momentum of the antenna must remain strictly unaltered.
Therefore, as it was above asserted, no interchange can occur between two kinds of energy-momentum, one included and the other excluded from T. Hence, if gravitational energy-momentum interacts with normal tensorial kinds of energy, as it happens in the increasing velocity of a falling body, it must be tensorial and included in T.

3. Tensorial Gravitational Energy
Having rescued in GR the global conservation of the covariant-divergenceless energy-momentum tensor as a law valid in a preferred class of coordinates, there is no more obstacle for the inclusion of a tensorial gravitational energy-momentum in this conserved tensor. With the gravitational energy-momentum included in the conserved tensor, the Einstein field equation must now take the form ,    (3) with the total energy-momentum tensor being now the sum of the matter and the gravitational tensors .    (4)
Assuming, as it must be expected, that the gravitational energy-momentum tensor be a functional of geometric elements, we define
    (4b) as the matter part of the Einstein's tensor, which my be now written: ,    (5) with .    (6)
Moller asserted [Moller,61, 66] that, to construct an expression of the gravitational energy-momentum, which could satisfy a minimal set of physical requirements, the metric alone is not sufficient, and more elements must be incorporated. Hence, in order to perform the above decomposition of the Einstein tensor, we must resort to other geometric elements than the metric and its derivatives.
Carmeli's SL(2 C) Gauge Theory of Gravitation presents an interesting example of decomposition of its tensor gauge field by means of four 2x2 vector-spinor complex matrices, which are the gauge potential of the theory [Carmeli 1982, Nissani 1984, Carmeli, Leibowitz and Nissani 1990]. Despite the fact that we will remain here in the classical mathematical frame of GR, our decomposition of the Einstein tensor resembles that of the Carmeli gauge field,
  . (7) Where the rotor and the commutator parts are tensors, and F and B are 2X2 spinor matrices. This gauge field relates to the the Riemann's curvature tensor by   , (8) with     (9)
Therefore, Carmeli's gauge field resembles the Ricci irreducible tensor, (irreducible by the habitual methods when written as a functional of the Christoffel symbols), in which two indices of Riemann's tensor are reduced by means of the above matrices. Its decomposition suggests the way we will decompose the Einstein's tensor following [N. and L.,1991].
Let us consider a tetrad of orthogonal and normalized vector fields fulfilling the orthonormality condition

    (10)
where the tetrad indices are raised and lowered with the Minkowski matrix. And the Ricci coefficients of rotation will be expressed by

    (11)
whith the antisymmetric relations

    (12)
that reduces the independent scalars to 24.
When written as a functional of the tensorial components of the Ricci coefficients of rotation,
  , (13)
the Ricci tensor decomposed, similarly to the Carmeli's gauge field tensor eq. (7), in two tensorial terms, a rotor and a commutator, of the Ricci coefficients:
  . (14)
We will assume now, and justify afterward this assumption by comparing with Newton and Einstein's results, that the commutator is associated with the matter and the covariant rotor with the gravitational energy-momentum, respectively:
   (15) and   . (16)
However, since all the following calculations will be performed assuming the vanishing of the total energy-momentum in the surrounding of a stellar body, the gravitational and the matter energy-momentum tensors will be of identical absolute value with opposed signs. Therefore, the interchange of the above definitions between the matter and the gravitational parts, would demand a mere change of sign in our calculations, and the results would remain unaltered. It would be in accordance with the Landau and Lifshitz association of the commutator of the affine connections appearing in the customary expression of the Ricci tensor, which vanishes in the local geodesic coordinates, with their proposed non-tensorial gravitational energy-momentum [Landau and Lifshitz, 51]. However, in our approach all kinds of energy-momentum are tensorial, and so too the rotor and commutator parts of the Ricci tensor, therefore, only future calculations performed in non-empty space could decide between the two possibilities.
In terms of the orthonormal tetrad we have:

    (17)
And for the Einstein's tensor we will have

    (18)
where parenthesis indicate symmetrization over the two indices.
The modified Einstein field equations for the 16 components of the tetrad will now take the form,
(19)
Therefore, the first of above equations is the tensorial expression of the gravitational energy-momentum that we must justify by showing its fitting with the Newton and/or Einstein values. It will be shown in the following sections, as it was done in [N. and L. 91, 92], that the above tensorial expression for the gravitational energy-momentum, gives with great approximation the Newton integral value of the gravitational energy in the surrounding of a star, and, by balancing gravitational with kinetic energy, the Einstein geodesic falling velocity.
The above set of equations (10) and (19) is invariant under global Lorentz transformations of the tetrads,
  , (20) while only the third of eqs. (19) remains invariant under a local (coordinate-dependent) Lorentz transformation. Therefore, given a solution of the set of eqs. (19), with the constrain of eq. (10), the general solution of the system will be given by applying to the tetrads a global Lorentz transformation. (For a more detailed treatment of the algebraic problem see [N & L, 91])
The compatibility in the general case of eqs. (19) and eq. (10) must not worry, since we are dealing with the physical laws of our particular universe as it was created by God, and revealed to us through experience. It is a universe composed by isolated massive bodies with nearly Schwarzschild metric in their surroundings.
A tetrad satisfying the above equations in a Schwarzschild background will be presented in the following section. The concordance of the results obtained whereby with the experimental facts, their implications concerning the nature of energy, and the new ways that this approach opens for future searching constitute its justification.

4. The Relativistic Splitting of the Newtonian Integral Energy
While in the Newton mathematical model of the physical universe, energy and volume are scalar values and , therefore, the spatial integral of the energy in a given inertial frame has a unique and well defined value, it split in three different relativistic values:
1) The conserved energy,
     (21),
which is the time component of an affine contravariant vector.
It is, clearly, a conserved quantity when T is the total energy-momentum, and the integral is performed in non-rotating coordinates. Though, we will call it the "conserved integral energy", even when it were a partial contribution to this conserved quantity from a partial energy-momentum tensor.
2) The Scalar Energy
     (22),
Where the integrand is a four-vector product of the energy-momentum tensor by an unitary time vector. In a static universe, as it seems nearly be the ours, and taking the unitary time vector normal to the hypersurface defined by the spatial static coordinates, it is a well-defined scalar parameter of the system.
3) The Time Component of an Affine Covariant Vector
     (23),
All the above three relativistic integral values of the energy-momentum content of a spatial three-volume, in which splits the Newtonian value, are invariant under static coordinates transformations.

5. Gravitational Energy-Momentum in the Surrounding of a Celestial Body.
Assume a stellar source of radius R, with total Schwarzschild mass M, , which is an integral energy of the type defined by eq. (23) [L and L 1975] that in our approach includes gravitational contribution.
In order to take advantage of Schwarzschild's solution, let us assume that the total energy-momentum (material plus gravitational) vanishes everywhere in the surrounding of the source, viz.,
, (24) what in the case of the sun, would imply to assume an amount of matter energy of only 1 millionth of the sun mass distributed in its surrounding. The metric, therefore, will be .   (25)
In this case a particular solution of eqs. (19) and (10) is given by
. (26)
Notice that in flat space-time (being M=0, or asymptotically when r tends to infinite) they will constitute a parallel tetrad with
  ,   (27) and therefore,      (28)
Replacing eq. (26) in (19) we obtain that the four non-vanishing components of the gravitational energy-momentum are:

     (29),
which, as it would be expected, vanish for M=0 and asymptotically for r tending to infinite.
By using eqs. (29) in the integrals (21) to (23) from the surface of the stellar body to infinity we obtain:
  , (30) for the conserved gravitational energy, whereas the amount of scalar gravitational energy is given by   ,    (31) and the contribution of the external gravitational energy to the Schwarzschild mass of the system results   .    (32)
For 2MG/R << 1, the above three relativistic expressions, in which splits the integral of Newton's gravitational energy, converge with this last

  .    (33)
For a stellar body with a ratio between R and Schwarzschild radius similar to the sun,
  ,   we obtain for the Newtonian value, calculated by balancing gravitational with kinetic energy,   ,   (34) and for the three relativistic integral values, respectively,   ,   (35)   ,    (36) and   ,    (37) which agree with the Newtonian value (34) up to the sixth digit. It is worthwhile noticing that, the deviation from the Newtonian value is of the same order than the splitting between the relativistic values themselves. And for a black hole, as it must be expected, we have   .   (38) Plotting the Newtonian and relativistic integral gravitational energy values per unity of the stellar mass M, against the stellar radius divided by the Schwarzschild radius, R/2MG, we have Fig. 3
The high agreement between the Newtonian integral gravitational value, obtained by balancing gravitational and kinetic energies, and based in the energy conservation law valid in the inertial frames, and the relativistic values in which it splits, gives a firm bases to the assumption that we can calculated the measured free fall velocity of a body, by balancing of relativistic kinetic energy with conserved integral gravitational energy, based in the relativistic energy conservation law valid in the non-rotating frames.
This assumption was found correct in [N&L 92], yet serious difficulties were found in the first attempts, which led to significant revelations on the nature of the energy content of the conserved energy momentum tensor, i.e., of the energy-momentum tensor appearing at the right side of the Einstein equation.

6. The General Relativistic Splitting of Time and Energy
Let us consider two possible ways of measuring general relativistic time and energy or, what is the same, two different general relativistic quantities, in which splits their Newtonian and special-relativistic counterpart.
1) The local time and energy. The first measured with identical clocks based in the frequency of photons emitted by atoms at repose in the non-rotating frame of the observer and placed in the vicinity of the measured event. And the second, by comparing with the energy of the same photons. Clearly, times measured this way at different points, in a gravitational field, can not be compared. It may expected the same difficulty when comparing energies measured this way, at different points along the path of a falling body, as we have to do when calculating its velocity by balancing kinetic with gravitational energy.
2) The universal time and energy, measured by comparing with the respective quantities of the same photon, but now emitted by an atom at repose somewhere in the non-rotating frame of the observer. The agreement between the resulting velocity of a falling body, obtained by using this second kind of energy when balancing gravitational and kinetic energies, with the Newtonian and Einstein velocities, shows that this is the kind of energy that must be considered as integrating the conserved energy-momentum tensor appearing in Einstein equation.

7. Falling Velocity of Matter in a Spherically Symmetric Gravitational Field
Following [N & L, 92] we return here to the spherical symmetric gravitational field of section 5, where the total energy-momentum tensot, including gravitational contribution, vanishes everywhere outside a stellar source. But now, we add to the previous configuration a test spherical layer of dust of infinitesimal thickness dr. Being the dust layer at rest at infinity, the three energy-moment integrals, eqs. (21) to (23), coincide and define its rest mass at infinity, dm. However, being the dust at rest at any other value r of the radial coordinate, these integrals split into six different values, depending on the kind of energy, local or universal, used in the integrand. Therefore, using in the integrand the universal kind of energy, in order to be able to compare energies at different points of the path, and the conserved integral energy, in order to balance kinetic with gravitational energies based in the global energy-momentum conservation law valid in non-rotating coordinates, we obtain:   .    (39) The above equation, where dV is the infinitesimal volume occupied by the dust, gives the value of the conserved integral universal rest mass of the dust at every point, which remains identical to the rest mass at infinity, dm.
While, the scalar integral, defined by eq. (22), of the universal rest mass of the dust, shows a gravitational redshift,
  .    (40) In the gravitational field of the system the dust layer will be falling with accelerated velocity and to the conserved universal rest mass will be added an increasing amount of conserved universal kinetic energy,   ,    (41) where the integrand varies along the path.
The contribution of the falling dust layer to the Schwarzschild mass parameter M, assumed being the time component of an affine covariant vector, defined by eq. (23), will be given by,
  ,    (42) and its scalar value, eq. (22),   ,    (43) where   .    (44)
By using eqs. (40) and (43), the ratio between the scalar energies of the dust layer at any value r of its path and of its rest mass will be given by
  ,    (45) being v the dust velocity measured im the local geodesic coordinates at rest in the non-rotating coordinates. Notice that the above relation, being a ratio between two energies, is independent of the kind of energy, local or universal, used in the respective integrands.
When the dust layer, falling from infinity, reaches the value r of the radial coordinate, the Schwarzschild mass inside this radius increases by dM, eq. (42), and the conserved gravitational energy outside this radius by:
  .    (46)
Assuming that the Schwarzschild coordinates are non-rotating, what will be sustained by the following results, we must have
  .    (47)
From eqs.(44),(45), (46) and (47) we now obtain for the velocity of the dust layer as it would be measured in the locally geodesic coordinates at rest in the non-rotating coordinates
  ,    (48) that shows the falling velocity as independent of the dust mass. It must be noticed that this result was obtained with total independence from Einstein assumption on the geodesic path followed by free bodies in a gravitational field. On the contrary, here the gravitational field was treated in a similar way to any other physical field. Developing the above expression of the dust velocity in powers of 2MG/r we obtain that for r>>2MG it results highly approximate to the geodesic Einstenian velocity,   .    (49)
Performing numerical calculations by means of the computer programs, whose sources can be download from this site, we find for the falling velocity of a body from infinity to the surface of a regular star with radius
  .    which agrees with the Einsteinian value,   ,    up to the seventh digit.
And comparing both velocities of a body falling from infinity to the surface of a black hole by plotting them for the five last Schwarzschild radii
  .

8. "Subtle is the Lord but malicious it isn't". (Einstein)
Results that agree up to the sixth and seventh digit with Newtonian and Einstenian values, respectively, are attained by means of a tensorial expression of the gravitational energy-momentum, incorporated to the conserved energy-momentum tensor as a normal and equal-rights member of the family. They include the integral gravitational energy in the surrounding of an ordinary star, and the fall velocity of a body. This last calculated by balancing gravitational and kinetic energy.
The above results clearly show that this new approach, which, at the same time that respects the fundamentals of General Relativity, introduces significative innovations in the interpretation of the meaning and nature of the energy-momentum appearing in their equations, deserves attention.
The treatment exposed here of the classical relativistic gravitational field, in a similar way to any other field of forces, together with the new assumption concerned the nature of the conserved energy-momentum, opens a window of hope to its possible introduction in the family of the relativistic quantum fields.

REFERENCES
Carmeli, M., Classical Fields, Wiley-Interscience, N.Y. (1982).
Carmeli, M., Leibowitz, E. and Nissani, N.(1990) Gravitation: SL(2,C) Gauge Theory and Conservation Laws, World Scientific, Singapore, p. 77.
Einstein, A. and Grommer, J. (1927) Sitzungsber. Preuss. Acad. Wiss., Phys. Math. K 1, 2.
Fock, V.,(1964) The Theory of Space Time and Gravitation, Pergamon Press, Oxford, p. 369.
Infeld, L. and Schild, A. (1949) Rev. Mod. Phys. 21, 408.
Landau, L.D. and Lifshitz, E.M. (1975) The Classical Theory of Fields, Pergamon Press, Oxford, p.303.
Moller, C., Ann. Phys. (N.Y.) 12 (1961) 118.
Moller, C., Mat. Fys. Medd. Dan. Vid. Selsk. 35 Num.3 (1966)
Nissani, N., SL(2,C) Gauge Theory of Gravitation: Conservation Laws, Physics Reports, V.109, N.2 (1984).
Nissani, N. and Leibowitz, E. (1988) Physics Letters A, 126, 447.
Nissani , N. and Leibowitz, E.(1989). International Journal of Theoretical Physics, Vol. 28, No. 2, 235.
Nissani, N. and Leibowitz, E. (1989b). Proceedings of the Fifth Marcel Grossmann Meeting on General Relativity, D.G. Blair and M.J. Buckingham editors, R. Rufini series editor, World Scientific, Singapore, Vol. A, p. 415.
Nissani, N. and Leibowitz, E. (1990) Proceedings of the Third Canadian Conference on General Relativity and Relativistic Astrophysics, World Scientific, Singapore.
Nissani, N. and Leibowitz, E. (1991) Int.J.Th.Physics, Vol.30, No.6, 837.
Nissani, N. and Leibowitz, E. (1992) Int.J.Th.Physics, Vol.31, No.12, 2065.

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