Monday, July 29, 2019

Standard Model Point Particles In Classical General Relativity

One of the main reasons that we know we don't know everything about fundamental physics is that classical general relativity and the Standard Model of Particle Physics are mathematically inconsistent. The point particle problem isn't the only mathematical inconsistency between general relativity and the quantum mechanics of the Standard Model, but it is one of the most obvious ones.

The Schwarzschild Radius Of Fundamental Particles

In classical general relativity, any true point particle with a non-zero mass gives rise to a singularity, because it involves division a finite mass by zero radius and volume.

The Schwarzschild radius of a mass is distance from the center a black hole to the event horizon of a black hole, in the case of a mass of that amount in general relativity.

The Schwarzschild radius of an election's mass is about 1.353 * 10-57 meters v. the Planck length of 1.616 * 10-35 meters.

Schwarzschild radius is proportional to mass, so a top quark pole mass mass black hole would have a Schwarzschild radius of 4.58 * 10-55 meters (which is significant because the top quark is the heaviest fundamental particle in the Standard Model).

A black hole with a Plank length radius would have a mass of 3.85763 * 10−8 kg, which is 1.772 Planck masses. Note that a mass of 1 kg = 1.780 * 10−27 GeV/c2. So a Plank radius black hole would be roughly 1017 times more massive than the pole mass of a top quark.

The Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle says that uncertainty in position times uncertainty in momentum measured at the same time is always greater to or equal than the reduced Planck's constant divided by 2.

The Heisenberg uncertainty principle also states that uncertainty in amount of energy times uncertainty in time measured at the same time is always greater to or equal than reduced Planck's constant divided by 2.

The reduced Planck's constant is 6.582 * 10-16 eV * second/radian. Mass and energy are basically equivalent with an E=mc2 conversion factor for these purposes.


So, while you need some finite radius for a point particle in the Standard Model, for example, a la string theory, or some finite distance length in space-time to avoid the point particle problem that the pure point particle Standard Model causes in classical general relativity, that scale can be much less than the Planck scale. In particular, the necessary scale is deeply within a domain where mass and position cannot be well defined to sufficient precision in a single measurement.

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