Notes on Relativity and Cosmology - Physics Department, UCSB

7.3. THE EQUIVALENCE PRINCIPLE 179
To understand Einstein’s answer, let’s consider a tiny box of spacetime from
our diagram above.
cε
ε
This acceleration was
matched to g
For simplicity, consider a ‘square’ box of height ǫ and width cǫ. This square
should contain the event at which we matched the “gravitational field g” to the
acceleration of the rocket.
In this context, Einstein’s proposal was that
Errors in dimensionless quantities like angles, v/c, and boost parameters
should be proportional to ǫ 2 .
Let us motivate this proposal through the idea that the equivalence principle
should work “as well as it possibly can.” Suppose for example that the gravitational
field is really constant, meaning that static observers at any position
measure the same gravitational field g. We then have the following issue: when
we match this gravitational field to an accelerating rocket in flat spacetime, do
we choose a rocket with α top = g or one with α bottom = g? Recall that any rigid
rocket will have a different acceleration at the top than it does at the bottom.
So, what we mean by saying that the equivalence principle should work ‘as well
as it possibly can’ is that it should predict any quantity that does not depend
on whether we match α = g at the top or at the bottom, but it will not directly
predict any quantity that would depend on this choice.
To see how this translates to the ǫ 2 criterion above, let us consider a slightly
simpler setting where we have only two freely falling observers. Again, we
will study such observers inside a small box of spacetime of dimensions δx = cǫ,
δt = ǫ. Let’s assume that they are located on opposite sides of the box, separated
by a distance δx.
In general, we have seen that two freely falling observers will accelerate relative
to each other. Let us write a Taylor’s series expansion for this relative
acceleration a as a function of the separation δx. In general, we have
a(δx) = a 0 + a 1 δx + O(δx 2 ). (7.3)
But, we know that this acceleration vanishes in the limit δx → 0 where the two
observers have zero separation. As a result, a 0 = 0 and for small δx we have
the approximation a ≈ a 1 δx.