Proving the principle of equivalence using the gauge invariance approach

We have let’s say an event P.

We can describe this event using coordinate system/reference frame alphabar and
coordinate system/r.f. alpha.

We want to connect these 2 coordinate systems. So, we say that event P would be described by an observer travelling with coordinate system alphabar AS THOUGH
event P itself has moved to a new event P’ in coordinate system alpha.

This can be condensed into the equations:
x alpha(P’) = x alphabar(P)
x alpha(P’) = x alpha(P) + xi alpha

The bottomline is event P and event P’ are one and the same (as we recall from special relativity, events are objective, not subjective, entities).

In general relativity, all physical laws are derived from the metric.

Therefore, we want to see how the metric differs between r.f. alpha and r.f.
alphabar, so that we can see whether the physical laws we formulate in r.f. alpha
work the same way in r.f. alphabar (for e.g. F=ma in r.f. alpha won’t become
something like F=m^2.a^2 in r.f. alphabar). This is done via:

g(gamma,beta)(x alpha(P)) = g(gamma, beta)(x alphabar(P) – xi alpha)

where
g(gamma, beta) represent the physical laws
x alpha represent the reference frame
P is the event.

Using Taylor’s expansion and an identity to simplify, the spirit of the calculation
is that we want to get:
The difference between g(gammabar,betabar)(x alphabar(P)) and g(gamma,beta)(x
alpha(P’)).

*the observer travelling with r.f. alphabar should evaluate the physical laws using
the coordinate system of his r.f., that’s why we have to put g(gammabar,betabar)(x
alphabar(..)) for e.g. and NOT g(gammabar,betabar)(x alpha(..))
*x alpha(P) is NOT equal to x alphabar(P) so we want want to compare the evaluations
of the metric between that at x alphabar(P) and x alpha(P’) and NOT between that at
x alpha(P) and x alphabar(P)

When this difference is zerorized,
g(gammabar,betabar)(x alphabar(P)) = g(gamma,beta)(x alpha(P’)), this would mean that:

The physical laws evaluated by an observer travelling with coordinate system
alphabar for event P will not change their form, for e.g. F=ma will remain F=ma and
not change into some other form.

(for the time being, I don’t have LaTeX enabled yet on my site)

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~ by musafiremes on April 25, 2007.

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