In the above video, Sabine Hossenfelder discusses one of the shortcomings of quantizing gravity. At high energies or short distances things go haywire and you get crazy big numbers or infinities. In this post I present one possible solution to this problem. It is not the only solution, and, only experiments will reveal which solution is correct, or, reveal that none are correct. Here is a list of variables we will be working with:
Let's start things off by taking two arbitrary masses (m',m) and creating a reduced-mass Schwarzschild radius:
At equation 2 below, we express the maximum energy of the gravitational field between the two masses. Notice at equation 3, if the radius between the two masses is taken to the zero limit, you don't end up with an infinity. Instead the maximum gravitational energy is conserved, and, said energy never exceeds the maximum energy available--which is always finite.
The equation at lines 2 and 3 can be written in a more familiar form of work (energy) equals force times distance (see 3a below):
If the distance (carrot r) is great, the gravitational force (mg) is small, but if the distance goes to zero, the force blows up to infinity, but ... the energy is conserved, since the infinite force is only along a zero distance.
The next step is to quantize what we have so far. Let's take equation 3a and use a scale factor (alpha). At equation 4, we multiply alpha by a time derivative of h-bar. That takes care of energy (E). On the right side of 4 we have alpha times the time derivative of momentum (p) times the distance. The time derivative of momentum is, of course, the force.
From 4 we derive Heisenberg's uncertainty principle for the singularity (reduced by the alpha factor):
What does equation 7 tell us? It indicates that if we know the exact position of a singularity, we are completely uncertain about its momentum, and vice versa.
