Mathematics of Physics and Engineering

Einstein's Field Equations 119

gravitational field of the Earth is much weaker in space. With communica-
tion signals travelling at the speed of light, even a small discrepancy in the
clocks can cause problems. For example, a 10-meter error in space can be
caused by a 3.3 • 10~^8 -second error in time (the time it take a radio-signal
to travel 10 meters). The relativity effects can be 10~^6 seconds per day or
more. The appropriate adjustment of the clock is necessary, and is indeed
implemented, on many navigation satellites.

EXERCISE 2.4.15.A+ Estimate the necessary time correction per day for a
satellite on a geostationary orbit (that is, a satellite moving in the equatorial
plane of the Earth in the direction of the rotation of the Earth and with
period of revolution around the Earth equal to the Earth period of rotation
about its axis). Take into account both special and general relativity effects.
The gravitational red shift is different from the cosmological red
shift, which is attributed to the expansion of the Universe, and which
was first observed in 1929 by the American astronomer EDWIN POWELL
HUBBLE (1889-1953). The cosmological red shift is consistent with general
relativity, and is connected with non-vacuum solutions of the field equa-
tions; it is a popular topic of research in modern cosmology. The idea is to
assume a particular form of the stress-energy tensor Ty on the right-hand
side of the Einstein field equations (2.4.34) so that the solution has a pre-
scribed form. A popular form of the solution produces the following line
element:

(ds)^2 = c^2 (dt)^2 - R(t) ({dr)^2 +r^2 (sm^2 p(de)^2 + (dip)^2 )) - c^2 {dt)^2 ,

where R(t) represents the radius of the Universe.

EXERCISE 2A.16.C (a) Verify that the radial motion of the photon, corre-

sponding to above metric, with ds = 0 and 6 = <p = 0, satisfies

/

4l Ht

Wr

(2A46)

(b) Assume that a pulse of electromagnetic radiation, lasting Aio seconds,
is emitted from a point in space at time moments to; the value of 1/Aio
can be interpreted as the frequency of the emitted radiation. The pulse is
received at a different point during the time interval [ti,ti + Aii], where
ti > h + Aio; the value 1/Aii can be interpreted as the frequency of the
received radiation. Assuming that both Ato and At\ are small, use (2.4-46)