What does it mean to unify physics?

My son likes physics, and knows that the “Holy Grail” of physics is to unify the models for the different types of physical forces into one. He knows, for example, that while quantum mechanics successfully describes the very small and relativity the very large, they can’t be used together to describe what goes on in a black hole or in a “big bang,” situations where tiny distances and huge masses coexist.

But he wanted to know what it actually means to unify two models. So I pulled a physics book off his shelf and rummaged around in the section on relativity for a formula I remembered seeing oh-so-many decades ago, and told him the following.

Imagine being out in space when a rocket zooming by at velocity v1 (relative to us) shoots a projectile in its direction of travel at velocity v2 (relative to the ship). Classical mechanics tells us that the velocity of the projectile relative to us is given by the equation

v = v1 + v2

But eventually it was discovered that nothing can go faster than the speed of light. The equation above works fine for low velocities, but fails when v1 and v2 are close to the speed of light (for which we use the symbol “c”), because v can never exceed c.

Einstein (well, let’s not forget Larmor and Lorentz) eventually gave us this equation, which fixes the problem:

v = ( v1 + v2 ) / ( 1 + v1*v2/c^2 )

When v1 and v2 are small relative to c, the bottom becomes very close to 1, so the old equation from classical mechanics gives an answer that is correct to a very high degree of accuracy. But when v1 and v2 are close to c, the bottom becomes significantly more than 1, enough to prevent v from reaching c. This equation “unified” classical mechanics with the fact that nothing can travel faster than light.

The same thing will happen in upcoming unifications. We have equations which very accurately describe the physical universe under certain conditions; whatever the equations of the unified model ultimately look like (and I have not the slightest clue about that), they will have to effectively reduce to the equations we have now under the right conditions. The new equations will be more general in that they will work under more (perhaps all) circumstances, because of terms which we don’t know about yet. These new terms will disappear when they need to (effectively giving us our old equations) and play a part when they need to (under circumstances that the current equations can’t handle).

He liked that explanation.

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