This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
In this section we will take a look at a few experiments which agree
with the predictions of GR.
For the first experiment, we use the effect mentioned in the previous
section whereby orbits which were supposed to be elliptical according to
Newtonian physics didn't actually close in on themselves according to GR
predictions. This effect can be seen as a rotation (or precession) of the
"long axis" of the elliptical orbit, whereas under Newtonian theory, this
axes doesn't move. Now, for the orbits of most planets, this effect is too
small to measure. However, for Mercury (which is closest to the sun and
would thus be the most affected) the effect is measurable. In fact,
measurements taken during the 1800s showed that Mercury's orbit precessed.
Now, much of this could be attributed to effects from the gravity of the
other planets, however, after all those effects were taken into account,
there was still a small amount of precession which wasn't accounted for. The
predictions of GR accounted for the left-over difference. It was Einstein
who first pointed this out, and this was the first evidence in favor of GR.
For the second experiment we want to consider, note that light, just
like anything else being acted on only by gravity, must follow a geodesic in
space-time. One can use the metric introduced in the previous section to
figure out how light would travel when passing near an approximately
spherically symmetric star. What one finds is that the light would be bent
by the presence of the star's gravitational field. Now, one might try to
make an argument using special relativity by which light with an energy E
would be said to have a "relativistic mass" defined by "m" = E/c^2. One
could then figure out how much the light with this "mass" would bend in the
presence of a Newtonian-type gravitational field. This, one might hope,
could allow the explanation of how light could be bent without considering
GR. However, one finds that the amount of bending predicted by this
SR-Newtonian method is exactly half as much as the bending predicted by GR.
Thus, if we could actually measure the bending of the light, we could figure
out which of the two predictions was correct.
Well, experiments to measure such bending can and have been performed
using the sun as the source of gravity and using light from particular
stars--light which passes near the sun on its way to us--as the light that
gets bent (it was Einstein who suggested this test, by the way). Normally,
of course, the sun would be too bright to see stars who's light passes near
the sun on its way to us. However, during a solar eclipse, the stars can be
seen. When one compares the positions of such stars which one sees during a
solar eclipse to the positions where the stars should actually be, one finds
that the difference can be attributed to the bending of the light as
predicted by GR, while the SR-Newtonian prediction was incorrect by a factor
of 2.
The third experiment we will look at involves using highly sensitive
atomic clocks taken aboard jets. When one compares the reading on such
clocks to clocks which remained on the ground, one finds that the difference
(though quite small) can only be accounted for completely if one includes
calculations for SR effects and acceleration along with the GR effects of
having the jet fly at high altitudes where the gravitational field is not as
strong as it is on the surface of the earth.
These are a few examples of experimental evidence that exists in favor
of GR. In many cases, more data and more precise measurements would be
needed to rule out all theories other than GR; however, all the evidence we
do have supports the theory.
 
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