What Hath General Relativity Wrought?

Even as far back as its inception, Einstein had grave misgivings about his field equations. For on the left-hand side there was a modified Ricci tensor that incorporated gravity in terms of the curvature of the manifold, while, on the right-hand side there was a physical tensor that dictated how energy and momentum would be distributed across spacetime. This ad-hoc mixture was not exclusive geometry on one side and physical fields on the other. The right-hand side could even be zero, which Einstein associated with the vacuum, while the left-hand side still could give solutions that could result in non-flat metrics, like the Schwarzschild solutions.

Einstein had hoped that all physical fields would have their own geometries, resulting in a unified theory, which in his time would have unified electromagnetism and gravitation. Such a unified theory would result in solutions representing matter free of all singularities. Three decades resulted in dismal failure.

Due to the coordinate freedom built into Einstein's equations, there is no notion of time. One must introduce the notion of time in order that the concept of 'initial data' has any meaning. This required a decomposition of 4 dimensional spacetime into 3+1 dimensions in order that Einstein's equations become amenable to numerical integration. However, there is nothing unique to the time slices that are taken so that there is no unique instantaneous direction that can be given.

Moreover, on account of nonlinearity, there is no rigorous solution to the two-body problem which Newtonian theory solved so easily. So where did the merger of binary black holes that numerical relativity claims there are and which LIGO has obtained their spectra? It seems like the holy grail of numerical relativity lies well beyond the reaches of the general theory of relativity.

In the beginning, it seemed like the assumption of geodesic motion was separate from the field equations. Yet, in 1927, Einstein and Grommer found their were consequences of said equations. How much unlike Maxwell's electromagnetic theory general relativity appeared to be. The Lorentz force could be written down independently of Maxwell's equations, and formed the bridge between field and particle motion. In fact, a gravitational field cannot be detected on a geodesic. Only geodesic deviation on a torsion free manifold where two neighboring geodesics either converge or diverge at constant velocity, or remain parallel, could be determined.

And general relativity gives no satisfactory explanation of gravitational radiation. Prior to 1960, it was admitted that radiation is bound to motion, and for retarded potentials, radiation could be made to vanish at will merely by a proper choice of the motion of the center of gravity. Since energy cannot be localized in general relativity, power balance equations involving a gravitational flux being balance by a decrease in energy made no sense. And only in the Newtonian approximation was there no difference between gravitational and inertial mass.

In the presence of gravitational waves, the law of conservation of energy involved a 'pseudo-tensor' which unlike a normal tensor, is made up of only first derivatives of the metric coefficients. As the name suggests, this pseudo-tensor could be made to vanish by a suitable coordinate transformation. This was emphasized over and over again in Leopold Infeld and Jerzy Plebariski's 1960 book Motion and Relativity where they address the question:

Can we find a reasonable coordinate system for which the gravitational radiation vanishes? In other words: is the gravitational radiation as defined here something that can be annihilated by a proper choice of a coordinate system, or is it something that has an absolute meaning?

Their conclusion was:

"gravitation radiation", or rather the flow of its energy, can be created or annihilated by a choice of a coordinate system. However, as we have shown, there exist reasonable coordinate systems in which "gravitational radiation" always vanishes.

Applying de Donder's condition on the perturbed metric, the linearized field equations lead immediately to the wave equation. To incorporate radiation, one had to go to a fifth order term in the perturbation, and consider the mixture timespace pseudo tensor components at eighth order. but only if an arbitrary function of time does not vanish!

Infeld sums up the results in the following terms:

What is the moral of this chapter? The results are indeed meagre and mostly of a negative character. They show that it is hardly possible to connect any physical meaning with the flux of energy and momentum tensor defined with the help of the pseudo-energy-momentum tensor. Indeed radiation can be annihilated by a proper choice of the coordinate system. On the other hand, if we use a coordinate system in which the flux of energy may exist, then it can be made whatever we like by the addition of proper harmonic functions starting with [the fifth order in the perturbation metric] as a function of time.

The big question is what in general relativity has changed in the intervening years to allow LIGO to measure the displacement of a freely falling body due to gravitational waves when their energy can be made to vanish by a mere coordinate transformation? Where do black holes emerge if not by extending the metric beyond its domain of validity? No, time and space do not swap roles inside the Schwarzschild radius for that radius determines the boundary of the space upon which the metric is defined.

Nothing has changed in the theory of general relativity that would warrant LIGO's claims. On the contrary, much has been forgotten, or conveniently swept underneath the proverbial carpet. Nothing has changed since Einstein uttered these words:

We do not have any satisfactory classical theory of radiation. Ritz understood this fact. He was an intelligent man....

How ironical this is since Ritz championed retarded potentials while Einstein considered both retarded and advanced potentials in their famous 1909 debate. The radiation he is talking about includes that of gravitational radiation, which he proposed and later dismissed. How times have changed!

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