Unifying Spacetime and the Forces

Let's start with a simple idea and expand on it. Imagine you have a total energy (E) and you subtract from that the energy of the strong, weak and electromagnetic forces. What do you have left? Gravitational energy.

Now we need something that represents the total energy. How about Schrodinger's Hamiltonian?

It's a good idea to express Schrodinger's Hamiltonian in terms of light speed squared (C^2). Why? Because light speed squared connects all the fundamental interactions (forces)--so we will set them all equal to C^2.

If we divide the Hamiltonian by mass (m) we set it equal to C^2. (C^2 = E/m.) Next, let's multiply each C^2 by time squared (t^2). Doing so gives us the spacetime metric. To make the terms equal, we should probably multiply each one by a coefficient K.

As you can see we've unified the forces and spacetime. What better way to spend a Sunday afternoon? The last equation, which is the spacetime metric, is just another way of saying, "Gravitational energy is equal to the total energy minus the other forces' energy."

In case you are curious, here are the variables: s is spacetime; z is the proton number; epsilon is the permittivity of free space; h-bar is Planck's constant; c is light speed; t is time; gij is the metric tensor. E is electricity; B is magnetism; I is current; p is charge density; Gn is Newton's constant; Gij is Einstein's tensor; Tij is the energy-stress tensor; Mh is the Higgs mass; g is the magnitude scaling constant; e is the natural exponent; m is the Yukawa particle mass; r is the particle radial distance; Fw is the weak force.

And let's not forget the equation for the Higgs field energy. The Greek letter there is the complex scalar field. H is its Hamiltonian (energy). The next term is kinetic energy and the last two terms are potential energy.

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.