Is Einsteins Equivalence theorem in General Relativity correct? It seems to me t

Question

Is Einsteins Equivalence theorem in General Relativity correct? It seems to me that it neglects the fact that gravitational acceleration depends upon separation distance squared, thus neglecting the effect of tidal forces.

For example, as I sit on earth, I experience the affect of earth's gravity; Although the acceleration on my head is slightly less than the acceleration acting on my feet. If I make the claim that my frame is equivalent to me being in a space ship traveling at g, doesn't that mean my whole body is accelerating at g uniformly? This is contrary to the previous statement though.

Explanation / Answer

The equivalence principle, as stated correctly by Einstein, says that these two situations are equivalent:

An uniformly accelerating observer in the absence of a gravitational field
A free falling observer in an uniform gravitational field

So, as you noted, this does not apply to the gravitational field of the Earth. Imagine you are in an elevator, free falling towards the Earth. You could let go of two pens - what you would see is that the two pens would come closer to each other, since each of them would be falling towards the centre of the Earth. You, as an observer, could then say with certainty that you are in a gravitational field.

In a hypothetical uniform gravitational field no experiment could reveal, whether you are in a gravitational field or not. Einstein took this thought experiment as a motivation for the development of General Relativity. In the mathematical construction of General Relativity, the equivalence principle does not play an important role.

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