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By Noah Nissani

Based in the Noah Nissani and Elhanan Leibowitz's papers that appear in the bibliography.


Like-inertial global preferred frames, the non-rotating frames, are derived from the energy-momentum distribution by a variational procedure. It is shown that these global preferred frames, wherein the energy-momentum is globally conserved, share the experimental features of the inertial frames of Newton's mechanics and Einstein's Special Relativity.

1. Introduction

Newton's Mechanics (NM) and Einstein' Special Relativity (SR) recognize the existence of a preferred class of frames, the inertial frames, in which the laws of physics acquire a special form. Specially significant is the fact that it only is in this frames that the energy-momentum conservation takes the form of four continuity equations, derived from the ordinary divergenceless of the energy-momentum tensor. Namely, in these frames energy and momentum are four conserved quantities whose changes in a given volume are equal to their flux through the volume surface.

Numerous experimental facts support the existence of these frames wherein stars, pendulums and water buckets show special behavior. These frames are global in the sense that they extend throughout space and do not sensible change with time. Mach's philosophy intuited that despite all the frames are intrinsically equivalent, not all of them are equally related to the matter distribution actually existing in the universe. Therefore, the inertial frames are determined by the energy-momentum distribution throughout space-time.

NM and SR lack any relation to the energy-momentum distribution, which could allow for a theoretical determination of the inertial frames. The existence of a global preferred class of frames is, therefore, in the pre-general relativistic theories a well-founded experimental fact that lacks a theoretical explanation. On the other hand, General Relativity only recognizes a local class of like-inertial frames, the geodesic frames, in which the laws of physics take their special relativistic form. The numerous experimental facts that support the existence of global inertial frames are in GR totally neglected and Mach's principle remains a philosophical idea deprived of physical significance.

In SR the energy-momentum conservation implies four continuity equations for the energy and the three components of the momentum, which derive from the ordinary divergenceless of the energy-momentum tensor. This ordinary divergenceless is covariant with respect the Lorentz's transformation and hence valid in all inertial frames. However, it is clearly not valid in non-inertial frames. In GR the covariant divergenceless of the energy-momentum tensor implies very limited continuity equations only locally valid in the geodesic frames. Therefore, the energy-momentum tensor is assumed in GR only locally conserved in the geodesic systems of coordinates.

In an attempt to restore the global characteristic of energy-momentum conservation, Einstein and his continuators looked for a kind of energy excluded from the non-conserved tensor. This external kind of energy-momentum is intended to complement the energy-momentum tensor to a quantity with globally vanishing ordinary divergence. For philosophical reasons, non supported by experimental facts, this global energy-momentum conservation was required to hold in all systems of coordinates. However, mathematical reasons preclude the satisfaction by a symmetric tensor of a global ordinary divergencelessness condition valid in all coordinate systems. Hence, the reluctance to accept a physical law valid in a preferred system of coordinates leads paradoxically to acquiescence with a non-tensorial gravitational energy-momentum whose dependence on the coordinates annuls any apparent gain in covariance.

In previous papers [Nissani and Leibowitz 1988, 89, 89b, 90, 91, 92. Carmeli, Leibowitz and Nissani 1990] has been shown the existence of like-inertial general-relativistic globally preferred frames. In these frames the energy-momentum tensor is ordinary divergenceless and the stars, neglecting gravitational effects, move with constant velocity. Namely, they were shown fulfilling the experimental characteristics of the NM and SR inertial frames.

One of the consequences of the existence of these frames is that gravitational energy may be incorporated in the energy-momentum tensor, together with all the other members of the energy family. For this purpose a tensorial expression for the gravitational energy-momentum is needed. Such tensorial expression was proposed in reference [N. and L. 1991], and his consistency with experimental facts elaborated in [N. and L. 1992].

2. The Experimental Facts

NM and SR accept the existence of a preferred realization of their respective invariance group - the inertial frames. The existence of this preferred realization is supported by an abundance of terrestrial experiments and astronomical observations. As experimental facts they have to be taken into account by any physical theory concerned with coordinate systems. We will classify them, for the convenience of the ensuing discussion, in to the following six sets denoted EF 1 to 6:

EF 1) The local experiments that do not involve gravitation. They are satisfactorily explained by SR, and also by GR by means of the locally geodesic coordinates. In spite of their local character they point to the same frames where distant celestial bodies show special behavior.

EF 2) The local experiments that do involve gravitation, e.g. Newton's water bucket or Foucault's pendulum. They cannot be explained by means of the geodesic coordinates since in the absence of gravitation a pendulum is not expected to oscillate and the water will hardly enter the bucket. They also point to the frames wherein stars and galaxies are in constant velocity motion. Their explanation demands the existence of nongeodesic preferred frames.

EF 3) The non-local observed facts such as the behavior of stars and galaxies. In an inertial frame, and when gravitational effects can be neglected, they are at rest or in motion at constant velocity. On the other hand in a "rotating" frame they perform an ordered simultaneous rotation. Observed from the earth they describe elliptic paths synchronized with the rotation of the earth around the sun. Confined to only local preferred coordinates we have no means to account for these facts that embrace the whole observable universe.

EF 4) Experiments that show the physical character of the preferred frames. The constancy of spectral shifts and measured angles indicates that the uniform velocity exhibited by the stars in these coordinates is a physical fact rather than a coordinate effect. The preferred frames appear as physically measurable coordinates that cannot be considered as mere labels.

The possibility to establish the preferred coordinates by measurement requires a time-independent metric. Continuous quickly variations of the metric as those caused by massive bodies in the proximity of their paths would affect the measurements. The existence of such time-independent metric relies, therefore, on the characteristics of the energy-momentum distribution. The slow velocities of stars and galaxies with respect to their neighbors characterize our universe as a quasi-static one. This allows the existence of coordinates with a nearly time-independent metric and orthogonal time coordinate. All this is specially true when dealing with the idealized configuration of an isolated body. Hence, according to the experimental results, the preferred measured coordinates are adaptable to the quasi-static characteristic of spacetime.

EF 5) Numerous local experiments and astronomical observations that support the energy-momentum conservation as holding true in the same frames wherein the special behavior of pendulums, water buckets and stars takes place. The uniform velocity of stars and galaxies is by itself a manifestation of this conservation law.

When gravitational effects are not negligible, the energy-momentum conservation demands a suitable definition of gravitational energy. All the attempts to extend the energy- momentum conservation to a law valid in all coordinate systems have failed in what concerns the physical nature of the ensuing gravitational energy.

EF 6) The Lorentz invariance of the Maxwell equations that implies the Lorentz group as a link between the preferred frames.

In the next section we define a class of general-relativistic preferred frames. We will do it by means of a variational demand on the components of the energy-momentum tensor. These preferred frames will be shown compatible with the above detailed six sets of experimental facts.

3. Mach's Principle as a Variational Principle on the Energy-Momentum Distribution.

In pursuit of a class of coordinate systems linked to the energy-momentum distribution, one is led to resort to a non-covariant requirement associated with the energy-momentum tensor. Accordingly, we will look for the coordinate systems wherein the integrals of the components of the energy-momentum tensor density over an arbitrary four dimensional volume of spacetime attain a stationary value.

To this purpose will define ten scalar actions as functionals of the energy-momentum tensor and four scalar functions of the coordinates:

eq1     (1)

By varying the scalar functions and requiring stationariness, one finds the following four Lagrange equations:

eq2     (2)

for the four scalar functions. Hence, eq.(2) define the class of coordinates wherein the integrals of the components of the energy-momentum tensor through an arbitrary four dimensional volume attain a stationary value. Notice that for an antisymmetric tensor, eq.(2) is satisfied for any arbitrary set of four scalar function, provided that the covariant divergenceless of the tensor vanishes, and for none otherwise. Furthermore, if the metric substitutes for the energy-momentum tensor, eq.(2) defines the harmonic coordinates [Fock 1964].

In the particular case of isolated massive points, as stars or galaxies at great distances from each other, and neglecting any gravitational contribution, the energy-momentum tensor may be written

eq3     (3)

where i runs over stars, j over the three spatial coordinates, M are the time components of covariant vectors that in the Schwarzschild coordinates of the respective star take the values of the Schwarzschild stellar masses, and U are the stellar four-vector velocities. Using (3) in (2) and putting

eq4     (4)

where V stands for the velocities of the stars in the selected coordinate systems that satisfy equation (2), and accepting the conservation of the stars' masses expressed by

eq4b     (4b)

one finds

eq5     (5)

Therefore, equation (2) defines, in strict general relativistic terms, the coordinate systems where the distant stars are at rest or in constant velocity motion. These coordinates have been named the nonrotating coordinates since they share with Newton's fixed stars frames, selected by Newton's water bucket, the most conspicuous characteristics.

Furthermore, denoting by g' and T' the values of the metric and energy-momentum tensors in the nonrotating coordinates, eq. (2) becomes

eq6     (6)

Namely, in the nonrotating coordinates the energy-momentum tensor density satisfies a global continuity equation. They satisfy, therefore, the experimental facts EF 3 and 5 of the preceding section that distinguish the inertial frames. Notice that if the energy-momentum tensor is replaced by the metric, eq.(6) becomes the deDonder condition of the harmonic coordinates.

In the next section it will be shown that the nonrotating coordinates can be specialized to be locally geodesic with respect to any given observer. It is precisely the existence of both, geodesic and nongeodesic nonrotating coordinates, that makes it possible, at least in principle, to explain the sets EF 1 and 2 of local experimental facts.

4. The Locally Geodesic Non-Rotating Coordinates

So far, we have not made use of the covariant divergencelessness of the energy-momentum tensor. We now have to resort to it to include in the same preferred coordinate systems, together with the global energy-momentum conservation and the special behavior of the distant stars (EF 3 and 5), the local special relativistic form of the laws of physics (EF 1). For it be possible to do this, the nonrotating coordinates have to include locally geodesic coordinates with respect to any given observer.

Note that in locally geodesic coordinates eq.(6) is equivalent to a covariant divergencelessness condition:

eq7     (7)

Therefore, the covariant conservation of the energy-momentum tensor is a necessary condition for the nonrotating coordinates to include geodesic coordinates with respect to any given observer. That this condition is also sufficient was illustrated in detail in reference [N. and L. 1989].

Namely, there is for any given observer a subclass of the geodesic coordinates wherein the energy-momentum tensor is globally conserved and the stars move with constant velocity. It is in these frames that the local experimental facts EF 1, EF 3 and EF 5 find their explanation. They will be referred to as the geodesic nonrotating coordinates.

It is easy to see from eq.(2) that the internal group of the nonrotating coordinates is defined by

eq8     (8)

It is a broad subgroup of the general mapping group that includes the Lorentz group. The nonrotating coordinates, defined by eq.(2), constitute its preferred representation.

The internal group of the geodesic nonrotating frames of a given observer is the subgroup of the group (8) made up by the transformations that are locally Lorentzian. This is in agreement with the fundamental role of the Lorentz transformation in the local experimental facts (EF 6). On the other hand, the non-locally Lorentzian transformations of the group link geodesic to non-geodesic nonrotating coordinates. The existence of the non-geodesic nonrotating coordinates, i.e., preferred frames where gravitational effects are present, makes possible the explanation of the experimental facts that do involve gravitation (EF 2).

5. Static Nonrotating Coordinates

As it was indicated in Section 2, there is strong experimental evidence of the (nearly) static characteristic of the inertial frames (EF 4). It is therefore natural to demand the existence of nonrotating coordinates that share this characteristic, when one deals with static configurations. Obviously, in a totally static universe one has to expect static solutions of eq.(2). But, clearly, we are not interested in the physics of a dead universe with an energy-momentum tensor totally independent of time. The question that arises is to what extent the nonrotating coordinates may continue to be assumed static for the study of the dynamics of test bodies in a static gravitational field. The answer seems to be that it rests on the appropriate definition of the gravitational energy-momentum and its inclusion in the conserved energy-momentum tensor.

To clarify the effect of the definition of gravitational energy-momentum on the physical properties of the nonrotating frames, consider the extreme case of a gravitational energy-momentum which is by definition everywhere null; or, what is for this case the same, a gravitational energy-momentum not included in the conserved energy-momentum tensor. Now, assume an isolated test body in an arbitrary gravitational field. The energy-momentum tensor in the vicinity of the test body would be, in this case, of the form described by eq.(3). Hence, if the conservation of the scalar rest mass of the body is assumed, eq.(5) holds true. Therefore, the test body would be at rest or with constant velocity with respect to the nonrotating coordinates. Namely, the nonrotating coordinates would escort any body in its motion. They by no means would exhibit the physical characteristics of the inertial frames.

Only a suitable expression for the gravitational energy-momentum as an integral part of the conserved energy-momentum tensor can lead to physically acceptable nonrotating coordinates. Such a tensorial expression for the gravitational energy-momentum, compatible with the existence of static nonrotating coordinates, was considered in [N. and L. 1992].

7. Remarks and Conclusions

A special class of coordinates, the nonrotating coordinates, is established by means of a variational principle imposed on the spacetime integrals of the components of the energy-momentum tensor. These preferred frames are therefore selected on account of their relation to the energy-momentum distribution in accordance with the Mach-Einstein assumption. Furthermore, they share with Newton's and Einstein's (special relativity) inertial frames their experimental properties:

1) They constitute the fixed-stars frames wherein stars and galaxies are at rest or in constant velocity motion.

2) They include locally-geodesic coordinates with respect any given observer. Hence, they serve as adequate "inertial" frames for the local experiments that do not involve gravitation.

3) They also include non-locally-geodesic frames making possible, at least in principle, the explanation of experimental facts that involve gravitation, e.g., the Newton water bucket and the Foucault pendulum experiments.

4) As it does in the inertial frames, the energy-momentum tensor satisfies in these coordinates a global conservation law.

5) In static spacetime, and with an appropriate tensorial definition of gravitational energy-momentum, they are adaptable to the static nature of spacetime.

6) The internal group of the geodesic nonrotating frames of a given observer is locally Lorentzian, in accordance with the central role that the Lorentz group play in the physical phenomena that do not involve gravitation.


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Relativistic Gravitational Energy=Momentum Tensor

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