Notes on Relativity and Cosmology - Physics Department, UCSB

8.3. GRAVITY AND THE METRIC 209
Now, why is there a 2 with the dxdy term? Note that since dxdy = dydx, there
is no need to have a separate g yx term. The metric is always symmetric, with
g yx = g xy . So, g xy dxdy + g yx dydx = 2g xy dxdy.
If you are familiar with vectors, then I can tell you a bit more about how lengths
and angles are encoded. Consider the ‘unit’ vectors ˆx and ŷ. By ‘unit’ vectors,
I mean the vectors that go from x = 0 to x = 1 and from y = 0 to y = 1. As
a result, their length is one in terms of the coordinates. This may or may not
be the physical length of the vectors. For example, I might have decided to use
coordinates with a tiny spacing (so that ˆx is very short) or coordinates with a
huge spacing (so that ˆx is large). What the metric tells us directly are the dot
products of these vectors:
ˆx · ˆx = g xx ,
ˆx · ŷ = g xy ,
ŷ · ŷ = g yy . (8.7)
Anyway, this object (g αβ ) is called the metric (or, the metric tensor) for the
space. It tells us how to measure all lengths and angles. The corresponding
object for a spacetime will tell us how to measure all proper lengths, proper
times, angles, etc. It will be much the same except that it will have a time part
with g tt negative 4 instead of positive, as did the flat Minkowski space.
Rather than write out the entire expression (8.6) all of the time (especially when
working in, say, four dimensions rather than just two) physicists use a condensed
notation called the ‘Einstein summation convention’. To see how this works, let
us first relabel our coordinates. Instead of using x and y, let’s use x 1 , x 2 with
x 1 = x and x 2 = y. Then we have:
ds 2 =
2∑
α=1 β=1
2∑
g αβ dx α dx β = g αβ dx α dx β . (8.8)
It is in the last equality that we have used the Einstein summation convention –
instead of writing out the summation signs, the convention is that we implicitly
sum over any repeated index.
8.3.4 A first example
To get a better feel for how the metric works, let’s look at the metric for a flat
plane in polar coordinates (r, θ). It is useful to think about this in terms of the
unit vectors ˆr, ˆθ.
4 More technically, g should have one negative and three positive eigenvalues at each point
in spacetime.