Why A Quantum Gravity Theory May Be Impractical

Finding that elusive graviton would be a boon to the Standard Model--but is a quantum theory of gravity practical?

According to the Wilkinson Microwave Anisotropy Probe, the ground-state energy density of the vacuum of space is E-10 joules per cubic meter.  This is essentially the energy of nothing.  Energy less than this is less than nothing.  For all intents and purposes, energies less than the vacuum don't exist.  For the graviton to exist, it should have an energy density greater than E-10.

Suppose we take two protons and place them a Bohr radius apart.  How much is the gravitational energy?  What is the gravitational energy density?

The formula used to measure gravitational energy is E = Gmm/r.  Where E is energy; G is Newton's constant (E-11); m is the proton mass (E-27) and r is the bohr radius (E-11).

E = (E-11)(E-27)(E-27)/E-11 = E-54 joules.

To determine the energy density, divide by the the volume, the cube of the Bohr radius:

E-54/E-33 = E-21 joules per cubic meter!

This is approximately 100 billion times less than the vacuum energy density!  No wonder the graviton has not been found.

Our gravitational energy formula suggests that we could reduce the radius to an infinitesimal size.  Surely we would get an astronomical amount of gravitational energy and gravitons would be as common as sand on on the beach.

There are a couple of problems with this strategy:  1.  Gmm/r can't be greater than (m^2c^4 + p^2c^2)^.5--the total energy of the protons.  2.  Most of that energy is due to the strong force and the quarks that make up the protons.  For quarks and protons to exist, they need the lion share of the available energy.

To get an energy density greater than the vacuum's there needs to be a lot more particles than just two protons.  According to the gravitational-energy formula, when you double the mass, you increase the gravitational energy four times.  The chances of finding a graviton increase exponentially as you add more mass.  Or does it?

When you add more mass, you add a haystack of particles for the graviton to hide in.  You might say that the graviton has its own uncertainty principle: the closer you get (quantum scale), the harder it is to find.  If you step back and look at the big picture (the mass of a double-star system) you can detect a very faint gravitational wave.  Such a wave has little or nothing to do with quantum physics, since you need huge masses just to get started.

These are some of the reasons why a quantum gravity theory may be impractical.

Update:  An elementary particle such as an electron at rest needs E-14 joules of energy to exist.  The gravitational energy of a proton, using the proton's classical radius, is (E-11)(E-54)/E-15 = E-50 joules.  This is not nearly enough energy to create an elementary particle such as the electron--so it appears the graviton does not exist at the quantum level, unless its energy requirement is a thousand billion trillion trillion times lower than the electron's.