Gravity and light
Einstein was very lucky because data that fit
predictions of his new mathematical model was observed
in light from stars a solar eclipse in 1919, very soon
after his model became public. The deflection
of light around the Sun was one such predictive success of
General Relativity.
The figure to the left shows three different possible (mathematical)
paths for a pulse of light travelling around the Sun: the path
with no gravity, the path as predicted by
Newtonian gravity, and the
path as predicted by Einstein's General Theory of Relativity.
The deflection angle df tells us how
far away from a straight line the path of the light pulse in question
was deflected by the Sun. The deflection angle is by definition
zero when there is no gravity. We need to compare the deflection
angles calculated using the Newtonian and relativistic models
for gravity and spacetime.
The turning point R0 is
the closest distance that the light pulse gets to the Sun.
We'll standardize our coordinate system
so that f=0 corresponds to
R = R0, and calculate df that way.
No gravity
Without gravity, both Newtonian and relativistic models say
the path is a straight line. The path of a straight line in
polar coordinates centered at the center of the Sun would be:
1/r = (1/R0) cos(f)
where R0 is the turning point
mentioned above.
First we want to find Df,
which is the total angle swept out by the light pulse
from the start to the end of its journey across spacetime.
Look at the figure to the left and imagine the straight line path
extending infinitely far to the right and left of your screen.
When r = infinity, by symmetry of our coordinate
system we have
0 = (1/R0) cos(Df/2).
Therefore Df = p is the
total difference in angle swept out by the light pulse as it
comes in from infinitely far away and travels back out
infinitely far away.
The deflection angle here is
df = Df - p = 0,
as it should be for a straight line.
Newtonian gravity
Newtonian gravity doesn't work well for describing
the properties of light, which can be modeled like
the propagation of a massless particle.
But it is possible to fake it by using the equation for a
Newtonian hyperbolic orbit:
1/r = (G M(m/L)2)(1 + e cos(f)),
e = (1 + (2E/m)(L/GMm)2))1/2
where the eccentricity e is a function of the
incoming particle's energy E, mass m and
angular momentum L. The turning point
R0 = (L/m)2/(G M (1 + e)).
If we want to fake the propagation of light in Newtonian gravity,
we can set the energy E = m v2/2 = m c2/2
so that (2 E/m) = c2. The angular momentum
per unit incoming mass (L/m) becomes L/m = R0 c.
The total angular sweep
Df = p + df is given by
0 = (1/R0) cos(Df/2)
+ (G M/c2)/R02,
- cos(p/2 + df/2) =
sin(df/2) ~
df/2 = (G M/c2)/R0
so finally dfN =
2 (G M/c2)/R0 is the deflection angle
for light found by naively using the Newtonian model for
a particle with velocity c.
Einstein's General Theory of Relativity
In General Relativity, the path of a light pulse is described
as a null geodesic satisfying the geodesic equation
for the Schwarzschild metric, the distance function that
solves the Einstein equations around a massive object in outer space such
as the Sun. An approximate equation for the trajectory is
1/r = (1/R0) cos(f) +
((G M/c2)/R02)
(2 - cos2(f)).
The term cos2(f)
can be neglected if the deflection angle
df is very small and
Df/2 is close to
p/2. Therefore, to lowest order
in df we get
0 = (1/R0) cos(Df/2) +
2 (G M/c2)/R02,
- cos(p/2 + df/2) =
sin(df/2) ~
df/2 = 2 (G M/c2)/R0.
Therefore dfE =
4 (G M/c2)/R0 =
2 dfN
is the deflection angle
for light found by using null geodesics in the Schwarzschild
metric according to General Relativity.
Observations of starlight deflected around the Sun were made
during solar eclipses beginning in 1919, and the measurements
supported Einstein's model, not Newton's which predicts an
angular deflection of half the size that was observed.
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