Einstein's Field Equations 117Step 4- Put u(9(s)) = l/r(s), using r{s) = -(u'(9)/u^2 (9))6(s) = -cm'(9),
to get
{u'(9))^2 + u^2 {9){\ - RoU{9)) - c^2 R 0 u{9)/a^2 = c^2 {(3^2 - I)/a^2.Step 5. Differentiate the last equality with respect to 6 to get«"(9) + «(9) = § + Y«^2 («)I
which is exactly equation (7.2.7) on page 415, with A$ = c^2 R 0 /(2a^2 ).
Another experimental verification of general relativity is the predicted
GRAVITATIONAL DEFLECTION OF LIGHT. For the flat Minkowski metric
(2.4.18), page 104, the trajectories of photons are lines satisfying x^2 + y^2 +
z^2 = c^2 t^2 or ds — 0. In a curved space, a photon geodesic is not necessarily
a line and can be curved (deflected) by the gravitational mass. Because of
the properties of g, this geodesic must still satisfy ds = 0.
EXERCISE 2.4.12.^3 Verify that the trajectory of a photon satisfies ds = 0 in
every curved space. Hint: change the metric on the flat space; the zero on the
right will stay zero.
Using the Schwarzschild metric (2.4.35), Einstein calculated the deflec-
tion of light rays passing near the Sun. The experimental validation of the
calculation came in 1919, when, during a total solar eclipse, the British
astronomer Sir ARTHUR STANLEY EDDINGTON (1882-1944) showed that a
star whose light passed close to the Sun appeared displaced by the amount
that corresponded to the deflection value calculated by Einstein.
EXERCISE 2.4.13? Consider a photon moving radially, that is, d6 = dip — 0.
Use (2.4-35) to conclude that ds^2 = 0 implies that, for r > R 0 , t — ±(r +
ro + R 0 In ((r/i?D) — l)- Therefore, the equation of trajectory (world line)
of the photon is different from the equation of the straight line in four-
dimensional space-time.
Yet another experimental verification of general relativity is GRAVITA-
TIONAL RED SHIFT, that is, the increase of wavelength of photons that
are mowing away from the source of the gravitational field Red light has
the longest wavelength in the optical spectrum, whence the term red shift.
On a deep level, the gravitational red shift illustrates two major results of
general relativity, the slow-down of time and shortening of length in a grav-
itational field; both these results can be deduced from the Schwarzschild