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Sunday, July 3, 2016

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.

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