Notes on Relativity and Cosmology - Physics Department, UCSB

9.4. STRETCHING AND SQUISHING 253
that it stretches the ocean in the direction pointing toward (and away from)
the moon, while it squishes (or compresses) the ocean in the perpendicular
directions. This is because different parts of the ocean would like to separate
from each other along the direction toward the moon, while they would like to
come closer together in the other directions:
Ocean
Wants to come
together in this
direction
Earth
Moon
Wants to separate this way.
As stated in the homework solutions, this effect is responsible for the tides
in the earth’s oceans. (You know: if you stand at the beach for 24 hours, the
ocean level rises, falls, then rises and falls again.) Whenever gravity causes
freely falling observers (who start with no relative velocity) to come together
or to separate, we call this a tidal effect. As we have seen, tidal effects are the
fundamental signature of spacetime curvature, and in fact tidal effects are a
direct measure of spacetime curvature.
Of course any other object (a person, rocket ship, star, etc.) would feel a
similar stretching or squishing in a gravitational field. Depending on how you
are lined up, your head might be trying to follow a geodesic which would cause
it to separate from your feet, or perhaps to move closer to your feet. If this
effect were large, it would be quite uncomfortable, and could even rip you into
shreds (or squash you flat).
However, as we have seen, this effect is generally small for geodesics that are
close together: if the stones in problem 1 were only as far apart as your head
and feet are, the effect would have been completely negligible! So, unless the
spacetime curvature is really big, it usually does not harm small objects like
people – only large objects like planets and stars.
On the other hand, we argued that this tidal effect will become infinitely
large at the singularity of a black hole. There the effect certainly will be strong
enough to rip apart even tiny objects like humans, or cells, or atoms, or even
subatomic particles. It is therefore of interest to learn how to compute how
strong this effect actually is. We know that it is small far away from a black
hole and that it is large at the singularity, but how big is it at the horizon?
This last question is the key to understanding what you would feel as you fell
through the horizon of a Schwarzschild black hole.
These are exactly the questions we’re going to ask (and answer) in the current
section. We will investigate tidal effects in more detail and discover how to
compute them directly from the metric. Before we begin, I should warn you that
this will be one of the more technical sections of these notes as the derivations
are quite involved. It is sufficiently technical that I will almost certainly not go