Grid Metric Gravity as an Alternative to General Relativity

Grid Metric Gravity as an Alternative to General Relativity
Kathleen A. Rosser
(Submitted to viXra January 27, 2018)

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Cassini-Huygens Mission conducted tests of the Schwarzchild metric, a vacuum solution to Einstein’s Field Equations (–

General Relativity has been considered our most accurate theory of gravity for almost a century. Yet in all that time, a full understanding of the theory has not been achieved. Today many physicists question General Relativity (GR) for a variety of reasons. Prominent among them is the fact that all attempts to quantize GR have so far had limited success [1-3]. GR also does not account for anomalies in the galactic rotation curve, wherein the outer bodies of galaxies orbit with a speed approaching a constant at large r in accordance with the Tully-Fisher law [4,5]. Moreover, GR does not predict the apparent observed acceleration of universal expansion except with the reintroduction of Einstein’s cosmological constant Λ, which must be fine tuned in a way that is not fully explained [6-9]. As a result of these and other discrepancies, perhaps hundreds of theories of gravity have been proposed in recent decades [10-13]. Many of them are variants on GR involving changes to Einstein’s Field Equations [14-16], or the introduction of scalar [17-19] or vector potentials [20]. For reviews of these theories see references [21-26].

There are other reasons GR might be questioned. That Einstein’s Field Equations are incomplete without an independently postulated Equation of State is less than satisfying. Indeed, in many applications, the Equation of State (EoS) carries more information than the field equations themselves. Yet in actual practice, the EoS is often largely a product of conjecture, or tailored to give the desired result [27,28]. A complete theory of gravity independent of thermodynamics might be preferable from an aesthetic standpoint. Note also that most practical applications of GR utilize the Schwarzschild metric, a solution to Einstein’s Field Equations (EFE) for the vacuum. Yet the Schwarzschild solution does not explicitly require an EoS. This points to an inconsistency in the underlying principles of the theory.

A few researchers might also object to the predicted existence of black holes, invisible bodies in which all matter is confined within an event horizon at r = 2m, a so-called coordinate singularity. This prediction distinguishes GR from both Newtonian gravity and classical electromagnetism, for which singularities exist only at r = 0.

GR presents another problem, usually considered a mere inconvenience, but which may be interpreted as an actual contradiction. The metric , which is determined by solving EFE, is nonlinear. Thus, for example, there is no way to add two Schwarzschild metrics to obtain the space-time curvature for two masses. Yet mass is presumed to add linearly. Why might this be a contradiction? Consider two masses m1 and m2. Each has a metric such that the time component is of the form 1-2m/r. If the masses reside together at r = 0, the metric becomes 1-2(m1+m2)/r. Yet there is no simple mathematical operation on the two original metrics that gives the final result. This suggests that it is the masses that are fundamental physical quantities, not the metrics. Yet, according to EFE, the metric is the fundamental physical quantity.

Another feature of GR that is less than satisfactory relates to Einstein’s Field Equations. These equations have beensubject to theory-independent observational tests only in the case of the Schwarzschild metric, a vacuum solution. Notehowever that EFE simplify significantly in the vacuum. Have observational tests been conducted for the full EFE using a non-zero density distribution? Even if they have, the results would depend on the selected Equation of State, which is not known a priori, but is theory-dependent. Hence any such tests would not be conclusive. The Schwarzschild metric is thus the only feature of GR that has been verified observationally in a conclusive theory-independent way.

From this thinking there arises the conjecture that a complete theory of gravity can be constructed from Schwarzschild metrics alone. This idea is not new. A similar concept was proposed by Richard Lindquist and John Archibald Wheeler in 1957, and was called the black hole lattice theory. It is discussed in more detail in references [29,30], and is currently of interest in solving the cosmological back-reaction problem, a difficulty in the standard model arising from inhomogeneities in the mass density of the universe [31]. Here I refer to it as the Schwarzschild Grid Model, or SGM. The problem with SGM is again that the metrics cannot be directly combined, but must be “patched” together in a way not entirely rigorous according to the GR formalism. This implies that is not the fundamental physical quantity.

Below, I introduce a theory of gravity inspired by the SGM concept, but for which the fundamental physical quantity is the classical gravitational potential. The metric is regarded as a secondary quantity derived from the relevant potentials, and serves mainly to determine the particle Lagrangian. For a single mass, the formalism gives rise to a line element identical to the Schwarzschild line element up to order . Thus it is approximately consistent with all conclusive observational tests of GR. The theory introduced here utilizes the cosmic coincidence , where is the presumed mass of the universe and R the so-called radius of the visible universe, or co-moving horizon [32]. The new theory is linear in that the fundamental physical quantities, the potentials, add linearly. The line element depends on the metric in the usual way. In the theory developed here, the Lagrangian for a test particle depends on the metric just as in GR. Since all motion, as well as physically measurable observables such as time dilation and redshift [33], are determined by the metric, the theory qualifies as a full metric theory of gravity. Hence the Equivalence Principle is automatically satisfied. This theory will be called Grid Metric Gravity, or GMG. . . . READ MOREView PDF File (6 pages)>>