Using the Young’s Modulus of spacetime itself calculated from the recent detection of gravitational waves, I’ve done some rough calculations to come up with a potential quantization of spacetime and a regularization cutoff for quantum field theory scattering calculations.
If we think about spacetime itself as a quantum field, then we can model it as a network of harmonic oscillators. To do this, we use Hooke’s law:
With k having units of N/m or kg/s2. The Young’s Modulus, Y, however, has units of kg/(m*s2). To get units of energy (Joules), we do the following calculation using the Planck length pl to calculate a (spherical) Planck volume:
This would represent a potential energy quantum of spacetime. This value is on order of 1056 times smaller than the Planck constant divided by Planck time. This either means something is incorrect with my strategy (most likely), or it could be the reason why spacetime appears to be continuous rather than quantized: an energy quantum of spacetime itself is 1056 times smaller than the energy quantum of other fields.
If we continue under the assumption that this energy calculation is correct, it can then be used to calculate a quantum of mass:
This so-called quantum of mass is smaller than the lower bound on what is thought to be able to generate a black hole, which is restricted by the Compton wavelength, and is thought to be the Planck mass of 2.176×10−8 kg.
But, to continue in my folly, I will assume that if what I’ve calculated is the quantum of mass, then I will use it to calculate the Schwarzchild radius for this mass quantum under the assumption that a black hole represents the maximum amount of information that can occupy a given space: if this previous calculation represents a quantum of mass (the lower bound on the mass of spacetime itself), then its Schwarzchild radius would represent the smallest volume that the smallest mass of spacetime could occupy, making it the quantum of spacetime.
And so, if this is the minimal length of spacetime – the quantum of spacetime – then that means that there cannot be a wavelength shorter than this, which puts an upper bound on energy. We can then calculate the frequency, energy, and momentum of a particle with maximum energy.
And so, with this calculation, the regularization cutoff for energy when calculating scattering events would be the Λ = 1.89×1010 GeV energy. This falls several orders of magnitude short of the ~1016 GeV estimated for grand unification gauge coupling of the strong, weak, and electromagnetic forces.
Of course, it’s most likely that my calculations are wrong and/or misguided and/or meaningless. However, in humoring my speculations, the reason for my maximum energy calculation being lower than that predicted for unified gauge coupling may be the loss of energy in the universe due to expansion: light becomes red-shifted as it travels long distances through the expanding universe, with the energy disappearing (loss of energy conservation) due to breaking of time translation symmetry from the expansion. Thus, it may be that the maximum energy allowable in a quantum of spacetime is decreasing over time: the unified gauge coupling of the three forces, which was the case just after the big bang, was possible then, but has since become impossible. Perhaps, on account of the expanding universe, the quantum of spacetime has increased in magnitude compared to the early universe, thus increasing the lower limit on allowable wavelengths, resulting in a reduced upper bound on energy.