Aspects of nonrelativistic quantum gravity

Abstract

A nonrelativistic approach to quantum gravity is studied. At least for weak gravitational fields it should be a valid approximation. Such an approach can be used to point out problems and prospects inherent in a more exact theory of quantum gravity, yet to be discovered. Nonrelativistic quantum gravity, e.g., shows promise for prohibiting black holes altogether (which would eliminate singularities and also solve the black hole information paradox), gives gravitational radiation even in the spherically symmetric case, and supports non-locality (quantum entanglement). Its predictions should also be testable at length scales well above the "Planck scale", by high-precision experiments feasible with existing technology.Keywords: Quantum gravity; Black holes; Information paradox; TestableThe greatest fundamental challenge facing physics has for many years been to reconcile gravity with quantum physics. There have been numerous attempts to do so, but so far there is no established and experimentally/observationally tested theory of "quantum gravity", the two main contenders presently being string theory [1] and loop quantum gravity [2], with "outsiders" like twistor theory [3], noncommutative geometry [4], etc.The motivations for studying nonrelativistic quantum gravity, apart from the simple and well-defined mathematics, are:1) Quantum theory is supposed to be universal, i.e., it should be valid on all length scales and for all objects, as there in principle exists no size/charge/mass-limit to its applicability. In atomic physics the practical restriction comes about because there is a limit to arbitrarily large atomic nuclei as, i) the Coulomb force between protons is repulsive, eventually overpowering the strong nuclear force trying to hold the nucleus together, ii) the additional weak force makes neutron-rich nuclei decay before they grow too large. Also, the electric charge comes in both positive and negative, and as a result a big lump of matter is almost always electrically neutral 1 . Neither of these limitations are present in "pure" quantum gravity.2) For weak gravitational fields the nonrelativistic theory should be sufficient. The weak-field newtonian limit is even used for determining the constant κ in Einstein's field equations of general relativity G µν = κT µν . The nonrelativistic limit is also almost always sufficient for practical purposes in non-quantum gravity, except for a handful of extreme cases (notably black holes and the very early universe), although high-precision experiments in, e.g., the solar system can and do show deviations from the nonrelativistic theory, always in favor of general relativity [5].3) Even for strong gravitational fields the newtonian picture gives the same prediction as general relativity for the Schwarzschild radius of a spherically symmetric, nonrotating black hole, and correct order of magnitude results * Electronic address: c.johan.hansson@ltu.se 1 The same also applies for e.g. the strong force, as the three different color charges ("red", "green", "blue") always...

No-go of quantized general relativity

“…For very large n and l = l max = n − 1 (closely connected to Kepler's law, see [1]), we get that the average expectation value r , and the individual most probable distancer (where the probability density peaks) approach each other 2 , and…”

“…This is also the regime of conducted experimental tests, such as the gravitational interferometric COW-experiment (Colella-Overhauser-Werner) [10], and the experiment claiming to having "measured" a quantum gravitational state for the first time [11], [12], [13], [14]. See also the discussion of the latter experiment in [1]. These use neutrons (m i , m g ) near the surface of the earth, in the gravitational field of the whole earth (M ), resulting in enormous n 8 × 10 17 (b 0 10 −29 m) [1], giving no insight into, nor any test of, "true" quantum gravity, as this would require small to moderate n, far from the correspondence limit.…”

“…See also the discussion of the latter experiment in [1]. These use neutrons (m i , m g ) near the surface of the earth, in the gravitational field of the whole earth (M ), resulting in enormous n 8 × 10 17 (b 0 10 −29 m) [1], giving no insight into, nor any test of, "true" quantum gravity, as this would require small to moderate n, far from the correspondence limit. For neutrons near the earth surface, the relative uncertainty is ∆r r = ∆r r = r 2 − r 2 r = (2n) −1/2 7.9 × 10 −10 ,…”

Testing quantum gravity

“…However, as quantum theory is supposed to be universal -no maximum length built into its domain of applicability -a low-energy, large length-scale, formulation of the theory should still apply. A falsification of the low-energy limit, in the experimentally accessible weak-field regime, would also falsify the full theory of quantized gravity [1], hence making it possible to test, and potentially rule out, quantum gravity with existing or near-future technologies 1 . In fact, direct tests of the high-energy limit of general quantum gravity may never be possible.…”

“…2 The answer lies in macroscopic systems still obeying the rules and laws of quantum theory -in essence those described by macroscopic wavefunctions. For a free-falling, effectively two-body problem, it should then in principle be possible to measure, e.g., the resulting quantum gravitational excitation energies [1]. A positive result would show that the gravitational field is quantized, just like the quantized energy levels resulting from the Schrödinger equation for hydrogen is implicit proof of the quantization of the electromagnetic field.…”