Gravity and lightEinstein 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 gravityWithout 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 gravityNewtonian 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))1/2where 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,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 RelativityIn 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,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.