Derivations of Relativistic Light Deflection

 

One ought not to have to care

so much as you and I

care when the birds come round the house

to seem to say good-bye.

†††††††††††††††††††††††††††††††††† Robert Frost

 

In the note on Anomalous Precession we derived the relativistic equation of motion for a massive test particle in a spherically symmetrical gravitational field of a mass m. In terms of the usual Schwarzschild coordinates r,θ,ϕ,t (and using geometrical units so that c = G = 1) we considered the metric in the equatorial plane (θ = π/2)

 

 

and integrated two of the geodesic equations to find constants k and h (corresponding to the energy and angular momentum) given by

 

 

We then defined the auxiliary parameter u = 2m/r, and derived the equation of motion

 

 

where E = (k2 − 1)/2. We used this equation to determine the precession of bound orbits of material objects (such as the planet Mercury orbiting the Sun), but the same equation can be used to determine the deflection of light, provided we adapt it to account for the fact that dτ = 0 along the path of a light pulse. First, we note that k and h become infinite, so the coefficient of u in the equation of motion goes to zero. The constant coefficient has E (and hence k2) in the numerator and h2 in the denominator, so we need to evaluate that coefficient. In the limit as dτ goes to zero that coefficient goes to

 

 

Since this is a constant, we can evaluate it at any convenient point, such as the perigee of the light path, i.e., the point of nearest approach to the central mass. We will let r0 denote the radial coordinate of that point, and u0 the corresponding reciprocal coordinate. From the metric equation with dr = dτ = 0 (at the perigee) we have

 

 

Substituting this into the previous expression (evaluated at r = r0), we find that the constant coefficient of the equation of motion for a light-like path goes to

 

 

Therefore, our light-like pulse satisfies the equation of motion

 

 

The angular travel from the perigee (u = u0) out to infinity (u = 0) would be simply π/2 if the path was a straight line, but it will exceed this by a slight amount due to the effect of the gravitational field. To determine this amount, we re-arrange the equation of motion and perform the integration

 

 

Making a change of variables to ρ = u/u0 this can be written as

 

 

This is precisely the same integral we found by a completely different method in the note on Bending Light, leading to the overall deflection

 

 

Another (less rigorous) approach is to differentiate equation (1) again with respect to ϕ, and divide through by 2(du/dϕ) to give

 

 

In the note on Analysis of Relativistic Orbits we showed how, by expressing u(θ) as a power series in θ, substituting into this equation, and setting the coefficient of each power of θ to zero, we get conditions on the coefficients leading to the approximate solution

 

 

As a check, we can substitute this into the preceding differential equation and show that the left hand side differs from 0 only by terms on the order of u03 and u04. We also note that this solution gives u(0) = u0 (which is not true for the approximate solutions of most published derivations). The asymptotes of the path correspond to u = 0 (i.e., as r goes to infinity), so we need only solve the quadratic

 

 

for cos(θ) and then take the inverse cosine to give the angles of the asymptotes. The result is

 

 

To the lowest order, the difference between these two angles exceeds π by 2u0 = 4m/r0, which is the relativistic deflection to this level of approximation. Unlike the method of direct integration described above, this method canít be used to give higher orders, because the expansion is already based on an approximate solution of the differential equation.

 

Return to MathPages Main Menu