Notes on Relativity and Cosmology - Physics Department, UCSB

7.4. GOING BEYOND LOCALITY 181
ε
cε
This acceleration was
matched to g
δ T 1
T 2
∝ ǫ 2 , (7.4)
where δ denotes the error.
Let me pause here to say that the conceptual setup with which we have surrounded
equation (7.4) is much like what we find in calculus. In calculus, we
learned that locally any curve was essentially the same as a straight line. Of
course, over a region of finite size, curves are generally not straight lines. However,
the error we make by pretending a curve is straight over a small finite
region is small. Calculus is the art of carefully controlling this error to build up
curves out of lots of tiny pieces of straight lines 3 . Similarly, the main idea of
general relativity is to build up a gravitational field out of lots of tiny pieces of
inertial frames.
Suppose, for example, that we wish to compare clocks at the top and bottom of
a tall tower. We begin by breaking up this tower into a larger number of short
towers, each of size ∆l.
∆ l
g g
g g
g
0 1 2 3 4
If the tower is tall enough, the gravitational field may not be the same at the
top and bottom – the top might be enough higher up that the gravitational
field is measurably weaker. So, in general each little tower (0,1,2...) will have
a different value of the gravitational field g (g 0 , g 1 , g 2 ....). If l is the distance of
any given tower from the bottom, we might describe this by a function g(l).
3 In fact, in calculus we also have a result much like (7.4). When we match a straight line
to a curve at x = x 0 we get the slope right, but miss the curvature. Thus, the straight line
deviates from the curve by an amount proportional to (x − x 0 ) 2 , a quadratic expression.