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6.1 An Exact Solution

The special theory of relativity assumes the existence of a unique
class of global coordinate systems - called inertial coordinates - with
respect to which the speed of light in vacuum is everywhere equal to the
constant c. It was natural, then, to express physical laws in terms of
this preferred class of coordinate systems, characterized by the global
invariance of the speed of light. In addition, the special theory also
strongly implied the fundamental equivalence of mass and energy, according
to which light (and every other form of energy) must be regarded as
possessing inertia. However, it soon became clear that the global
invariance of light speed together with the idea that energy has inertia
(as expressed in the famous relation E2 = m2 +
|p|2) were incompatible with one of the most firmly established
empirical results of physics, namely, the exact proportionality of
inertial and gravitational mass, which Einstein elevated to the status of
a Principle. This incompatibility led Einstein, as early as 1907, to the
belief that the global invariance of light speed, in the sense of the
special theory, could not be maintained. Indeed, he concluded that we
cannot assume, as do both Newtonian theory and special relativity, the
existence of any global inertial systems of coordinates (although we can
carry over the existence of a local system of inertial coordinates in a
vanishingly small region of spacetime around any event).

Since no preferred class of global coordinate systems is assumed, the
general theory essentially places all (smoothly related) systems of
coordinates on an equal footing, and expresses physical laws in a way that
is applicable to any of these systems. As a result, the laws of physics
will hold good even with respect to coordinate systems in which the speed
of light takes on values other than c. For example, the laws of general
relativity are applicable to a system of coordinates that is fixed rigidly
to the rotating Earth. According to these coordinates the distant galaxies
are "circumnavigating" nearly the entire universe in just 24 hours, so
their speed is obviously far greater than the constant c. The huge implied
velocities of the celestial spheres was always problematical for the
ancient conception of an immovable Earth, but it is beautifully
accommodated within general relativity by the effect which the
implied centrifugal acceleration field - whose strength increases in
direct proportion to the distance from the Earth - has on the values of
the metric components guv for this rotating system of
coordinates at those locations. It's true that, when expressed in this
rotating system of coordinates, those stars are moving with dx/dt values
that far exceed the usual numerical value of c, but they are not moving
faster than light, because the speed of light at those locations,
expressed in terms of those coordinates, is correspondingly
greater.

In general, the velocity of light can always be inferred from the
components of the metric tensor, and typically looks something like .
To understand why this is so, recall that in special relativity we
have

and the trajectory of a light ray follows a null path, i.e., a path
with dt = 0. Thus, dividing by (dt)2,
we see that the path of light through spacetime satisfies the equation

and so the velocity of light is unambiguous in the context of special
relativity, which is restricted to inertial coordinate systems with
respect to which equation (1) is invariant. However, in the general theory
we are no longer guaranteed the existence of a global coordinate system of
the simple form (1). It is true that over a sufficiently small spatial and
temporal region surrounding any given point in spacetime there exists a
coordinate system of that simple Minkowskian form, but in the presence of
a non-vanishing gravitational field ("curvature") equation (1) applies
only with respect to "free-falling" reference frames, which are
necessarily transient and don't extend globally.

So, for example, instead of writing the metric in the xt plane as
(dt)2 = (dt)2 - (dx)2 , we must consider the more general
form

As always, the path of a light ray is null, so we have dt = 0, and the differentials dx and dt must satisfy the
equation

Solving this gives

If we diagonalize our metric we get gxt = 0, in which case
the "velocity" of a null path in the xt plane with respect to this
coordinate system is simply dx/dt = . This
quantity can (and does) take on any value, depending on our choice of
coordinate systems.

Around 1911 Einstein proposed to incorporate gravitation into a
modified version of special relativity by allowing the speed of light to
vary as a scalar from place to place as a function of the gravitational
potential. This "scalar c field" is remarkably similar to a simple
refractive medium, in which the speed of light varies as a function of the
density. Fermat's principle of least time can then be applied to define
the paths of light rays as geodesics in the spacetime manifold (as
discussed in Section 8.4). Specifically, Einstein wrote in 1911 that the
speed of light at a place with the gravitational potential j would be c0 (1 + j/c02), where c0 is
the nominal speed of light in the absence of gravity. In geometrical units
we define c0 = 1, so Einstein's 1911 formula can be written
simply as c = 1 + j. However, this formula for
the speed of light (not to mention this whole approach to gravity) turned
out to be incorrect, as Einstein realized during the years leading up to
1915 and the completion of the general theory. In fact, the general theory
of relativity doesn't give any equation for the speed of light at a
particular location, because the effect of gravity
cannot be represented by a simple scalar field of c values. Instead, the
"speed of light" at a each point depends on the direction of the
light ray through that point, as well as on the choice of coordinate
systems, so we can't generally talk about the value of c at a given
point in a non-vanishing gravitational field. However, if we consider just
radial light rays near a spherically symmetrical (and non-
rotating) mass, and if we agree to use a specific set of coordinates,
namely those in which the metric coefficients are independent of t, then
we can read a formula analogous to Einstein's 1911 formula directly from
the Schwarzschild metric. But how does the Schwarzschild metric follow
from the field equations of general relativity?

To deduce the implications of the field equations for observable
phenomena Einstein originally made use of approximate methods, since no
exact solutions were known. These approximate methods were adequate to
demonstrate that the field equations lead in the first approximation to
Newton's laws, and in the second approximation to a natural explanation
for the anomalous precession of Mercury (see Section 6.2). However, these
results can now be directly computed from the exact solution for a
spherically symmetric field, found by Karl Schwarzschild in 1916. As
Schwarzschild wrote, it's always pleasant to find exact solutions, and the
simple spherically symmetrical line element "let's Mr. Einstein's result
shine with increased clarity". To this day, most of the empirically
observable predictions of general relativity are consequences of this
simple solution.

We will discuss Schwarzschild's original derivation in Section 8.7, but
for our present purposes we will take a slightly different approach.
Recall from Section 5.5 that the most general form of the metrical
spacetime line element for a spherically symmetrical static field
(although it is not strictly necessary to assume the field is static) can
be written in polar coordinates as

where gqq = -r2, gff
= -r2 sin(q)2, and gtt and grr
are functions of r and the gravitating mass m. We expect that if m = 0,
and/or as r increases to infinity, we will have gtt = 1 and
grr = -1 in order to give the flat
Minkowski metric in the absence of gravity. We've seen that in this highly
symmetrical context there is a very natural way to derive the metric
coefficients gtt and grr simply from the requirement
to satisfy Kepler's third law and the principle of symmetry between space
and time. However, we now wish to know what values for these metric
coefficients are implied by Einstein's field equations.

In any region that is free of (non-gravitational) mass-energy the
vacuum field equations must apply, which means the Ricci tensor

must vanish, i.e., all the components are zero. Since our metric is in
diagonal form, it's easy to see that the Christoffel symbols for any three
distinct indices a,b,c reduce to

with no summations implied. In two of the non-vanishing cases the
Christoffel symbols are of the form qa/(2q), where q is a
particular metric component and subscripts denote partial differentiation
with respect to xa. By an elementary identity these can also be
written as . Hence if
we define the new variable we can
write the Christoffel symbol in the form Qa with q =
e2Q. Accordingly if we define the variables (functions of
r)

then we have

and the non-vanishing Christoffel symbols (as given in Section 5.5) can
be written as

We can now write down the components of the Ricci tensor, each of which
must vanish in order for the field equations to be satisfied. Writing them
out explicitly and expanding all the implied summations for our line
element, we find that all the non-diagonal components are identically zero
(which we might have expected from symmetry arguments), so the only
components of interest in our case are the diagonal elements

Inserting the expressions for the Christoffel symbols gives the
equations for the four diagonal components of the Ricci tensor as
functions of u and v:

The necessary and sufficient condition for the field equations to be
satisfied by a line element of the form (2) is that these four quantities
each vanish. Combining the expressions for Rtt and
Rrr we immediately have ur = -vr , which
implies u = -v + k for some arbitrary constant k. Making these
substitutions into the equation for Rqq and setting the constant of integration to k =
pi/2 gives the condition

Remembering that e2u = gtt, and that the
derivative of e2u is 2ur e2u, this
condition expresses the requirement

The left side is just the chain rule for the derivative of the product
r gtt, and since this derivative equals 1 we immediately have
rgtt = r + a for some constant a . Also, since grr = e2v where v
= -u + pi/2, it follows
that grr = -1/gtt, and so
we have the results

To match the Newtonian limit we set a = -2m where m is classically identified with the mass of
the gravitating body. These metric coefficients were derived by combining
the expressions for Rtt and Rrr, but it's easy to
verify that they also satisfy each of those equations separately, so this
is indeed the unique spherically symmetrical static solution of Einstein's
field equations.

Now that we have derived the Schwarzschild metric, we can easily
correct the "speed of light" formula that Einstein gave in 1911. A ray of
light always travels along a null trajectory, i.e., with dt = 0, and for a radial ray we have dq and df both equal to zero,
so the equation for the light ray trajectory through spacetime, in
Schwarzschild coordinates (which are the only spherically symmetrical ones
in which the metric is independent of t) is simply

from which we get

where the ± sign just indicates that the light
can be going radially inward or outward. (Note that we're using geometric
units, so c = 1.) In the Newtonian limit the classical gravitational
potential at a distance r from mass m is j =
- m/r, so if we let cr = dr/dt denote
the radial speed of light in Schwarzschild coordinates, we have

cr = 1 + 2j

which corresponds to Einstein's 1911 equation, except that we have a
factor of 2 instead of 1 on the potential term. Thus, as j becomes increasingly negative (i.e., as the magnitude
of the potential increases), the radial "speed of light" cr
defined in terms of the Schwarzschild parameters t and r is reduced to
less than the nominal value of c.

On the other hand, if we define the tangential speed of light at a
distance r from a gravitating mass center in the equatorial plane (q = p/2) in terms of the
Schwarzschild coordinates as ct = r(df/dt), then the metric divided by (dt)2
immediately gives

Thus, we again find that the "velocity of light" is reduced a region
with a strong gravitational field, but this speed is the square root of
the radial speed at the same point, and to the first order in m/r this is
the same as Einstein's 1911 formula, although it is understood now to
signify just the tangential speed. This illustrates the fact that the
general theory doesn't lead to a simple scalar field of c values. The
effects of gravitation can only
be accurately represented by a tensor field.

One of the observable implications of general relativity (as well as
any other theory that respects the equivalence principle) is that the rate
of proper time at a fixed radial position in a gravitational field
relative to the coordinate time (which corresponds to proper time
sufficiently far from the gravitating mass) is given by

It follows that the characteristic frequency n
1 of light emitted by some known physical process at a radial
location r1 will represent a different frequency n 1 with respect to the proper time at some
other radial location r2 according to the formula

From the Schwarzschild metric we have gtt(rj) =
1+2jj where
jj = -m/rj is the
gravitational potential at rj, so

Neglecting the higher-order terms and rearranging, this can also be
written as

Observations of the light emitted from the surface of the Sun, and from
other stars, is consistent with this predicted amount of gravitational
redshift (up to first order), although measurements of this slight effect are difficult.
A terrestrial experiment performed by Rebka and Pound in 1960 exploited
the Mossbauer effect to precisely
determine the redshift between the top and bottom of a tower. The results
were in good agreement with the above formula, and subsequent experiments
of the same kind have improved the accuracy to within about 1 percent.
(Note that if r1 and r2 are nearly equal, as, for
example, at two heights near the Earth's surface, then the leading factor
of the right-most expression is essentially just the acceleration of
gravity a = -m/r2, and the factor in parentheses is the
difference in heights Dh, so we have Dn/n = a Dh.)

However, it's worth noting that this amount of gravitational redshift
is a feature of just about any viable theory of gravity that includes the
equivalence principle, so these experimental results, although useful for
validating that principle, are not very robust for distinguishing between
competing theories of gravity. For this we need to consider other
observations, such as the paths of light near a gravitating body, and the
precise orbits of planets. These phenomena are discussed in the subsequent
sections.

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