Einstein, Gravitation and Singularities
Recently Lehmkuhl has posted a very interesting paper on Einstein’s development of General Relativity.
In words, General Relativty says:
(space-time geometry) = (a constant)(the energy-momentum tensor).
Einstein always felt that the left side of the equation was “made of fine marble” but the right side was “built of low-grade wood” since it did not seem to have the potential for eventually incorporating quantum phenomena.
As Lehmkuhl describes in considerable detail, Einstein’s immediate goal was to have the field equations of the space-time geometry lead to the equations of motion for objects (i.e., the geodesic equations) without the latter being introduced as a separate assumption. Einstein and Grommer published a paper in 1927 that accomplished that goal. Instead of using the energy-momentum tensor approach they only used the vacuum field equations (pure geometry) and represented the particles as curvature singularities (purely geometric objects) in the metric. It worked!
“[Einstein] argued that it is the first field theory in which the field equations and the equations of motion of matter do not have to be introduced as separate assumptions.”
Shortly thereafter Einstein wrote to Ehrenfest that “it is interesting that the field equations can determine the motion of singularities. I even think that this will one day determine the development of quantum theory, but the path to this is still in the dark.”
Discrete Scale Relativity [ https://arxiv.org/abs/physics/0701132 ], which attempts to take another step forward in Einstein’s program of “deepening” the symmetries and unities of physics, asserts that there are dominant curvature singularities at the centers of all fundamental objects in nature, including subatomic particles, stellar ultracompacts, and galaxies. DSR predicts that the dark matter is primarily composed of stellar-mass primordial black holes with singularities at their centers. DSR also requires that gravitation is the dominant interaction on all fundamental scales of nature. It accomplishes this by rejecting the absolute nature of the constant in the basic equation above. While the usual Newtonian constant G applies to macroscopic and stellar-scale phenomena, the constants on other fundamental scales of nature’s self-similar hierarchy are radically different. For example, the atomic scale G is 38 orders of magnitude larger than the stellar scale G. General Relativity is the same theory on each fundamental scale, but the strengths of the gravitational interactions are radically different when measured in our conventional units.
Of course, nature involves more than just singularities. Surrounding the curvature singularities are shells and disks of diffuse matter, and electromagnetic fields, that do all the wonderful things that matter can do.
The take-home lesson is this: perhaps it is useful to seriously consider DSR’s assertion that gravitation and curvature singularities are the fundamental scaffolding upon which the rest of nature, matter and quantum phenomena are constructed. Perhaps this would take us closer to removing the incompatibilities that have plagued General Relativity and Quantum Theory for decades.