6.1
An Exact Solution


Einstein had
been so preoccupied with other studies that he had not realized such
confirmation of his early theories had become an everyday affair in the
physical laboratory. He grinned like a small boy, and kept saying over and over
“Ist das wirklich so?”

A. E. Condon


The
special theory of relativity posits the existence of a unique class of global
coordinate systems  called inertial coordinates  with respect to which
inertia is homogeneous and isotropic. It was natural, then, to express
physical laws in terms of this preferred class of coordinate systems. The
special theory also strongly suggested the fundamental equivalence of mass
and energy, according to which light – and every other form of energy – must
be regarded as possessing inertia. It follows that the speed of light in
vacuum has an invariant value, c, in all directions in terms of any inertial
coordinate system. However, it soon became clear that the existence of global
inertial coordinate systems (with invariant light speed) together with the
idea that energy has inertia (as expressed in the famous relation E^{2}
= m^{2} + p^{2}) were incompatible with one of the most
firmly established empirical results of physics, namely, the exact
proportionality of inertial and gravitational mass. The latter implies that a
pulse of light must be deflected in a gravitational field, which clearly
requires the wavefront to have different speeds at different locations in
terms of suitable global coordinates. This incompatibility led Einstein, as
early as 1907, to the belief that the premise of global inertial coordinate
systems could not be maintained. We can establish inertial coordinates over
any sufficiently small region of space and time, but there do not exist any
global systems of inertial coordinates in regions where gravitating
massenergy is present.


Since
no preferred class of global coordinate systems exists, 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, in which
any “fictitious forces” that arise in accelerating coordinates affect the
values of the metric components g_{uv} for those coordinates. When
expressed in a rotating system of coordinates, the distant stars are indeed
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, for any given system of coordinates the velocity of light can always
be inferred from the components of the metric tensor, and typically looks
something like . To understand
this, recall that in special relativity we have global inertial coordinate
systems such that the metric tensor has the constant form




The
trajectory of a light ray follows a null path, i.e., a path with dt = 0, so dividing by (dt)^{2}
we see that the path of light everywhere satisfies the equation




Hence
the velocity of light is unambiguous in terms of these preferred systems of
coordinates. However, in the general theory we are no longer guaranteed the
existence of a global coordinate system with a constant metric of the simple
form (1). It is true that over a sufficiently small spatial and temporal
region surrounding any given event in spacetime there exists a coordinate
system of that simple Minkowskian form, but in the presence of a
nonvanishing gravitational field ("curvature") equation (1)
applies only with respect to "freefalling" inertial coordinates,
which are necessarily transient and don't extend globally.


So,
for an extended region of spacetime, instead of writing the metric in the xt
plane as (dt)^{2} =
(dt)^{2}  (dx)^{2}
, we must consider the more general form




where
the coefficients are functions of the coordinates. As always, the path of a
light ray is null, so dt = 0 and the
differentials dx and dt satisfy the equation




Solving
this gives




If
we diagonalize our metric we get g_{xt} = 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 in Euclidean space 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 c_{0} (1 + j/c_{0}^{2}),
where c_{0} is the nominal speed of light in the absence of gravity.
In geometrical units we define c_{0} = 1, so Einstein's 1911 formula
can be written simply as c = 1 + j.
However, this formula for the speed of light – indeed, this whole approach to
gravity – turned out to be incorrect. In the general theory of relativity,
completed in 1915, the speed of light in a gravitational field cannot
generally be represented by a simple scalar field of c values in Euclidean
space, due to the intrinsic curvature of spacetime. In terms of some quite
natural coordinate systems, the speed of light varies not only from place to
place, but also in different directions at any given place (even though the
speed of light always has the invariant value c in terms of local
freefalling inertial coordinates, consistent with the equivalence
principle). For example, near a spherically symmetrical and non rotating
mass, we can define stationary coordinates in which the speed of light is
isotropic, but in these coordinates the circumference of a circular orbit of
radius r is not equal to 2pr.
On the other hand, we can define stationary coordinates in which a circular
orbit of radius r does equal 2pr,
but in terms of these coordinates the circumferential speed of light differs
from the radial speed. The former is given by the same formula as in
Einstein’s 1911 paper, but the latter differs from the 1911 formula by a
factor of 2 on the “potential” term. To explain this in detail, we must first
consider how the Schwarzschild metric is derived 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
g_{qq} = r^{2}, g_{ff} = r^{2} sin(q)^{2}, and g_{tt}
and g_{rr} are functions of r and the gravitating mass m. (These
stipulations ensure that the circumference of a circular orbit of radius r is
2pr.) We expect
that if m = 0, and/or as r increases to infinity, we will have g_{tt}
= 1 and g_{rr} = 1
in order to give the flat Minkowski metric in the absence of gravity. We saw
in Section 5.5 that in this highly symmetrical context there is a fairly
plausible way to derive the metric coefficients g_{tt} and g_{rr}
simply from the requirement to satisfy Kepler's third law and the
inversesquare law, but with some ambiguity over the choice between proper
time and coordinate time. We can now determine unambiguously the values of
these metric coefficients consistent with Einstein's field equations.


In
any region that is free of (nongravitational) massenergy 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 nonvanishing cases the Christoffel
symbols are of the form q_{a}/(2q), where q is a particular metric
component and subscripts denote partial differentiation with respect to x^{a}.
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 Q_{a} with q = e^{2Q}. Accordingly if we define the
variables (functions of r)




then
we have



and
the nonvanishing 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 nondiagonal 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 R_{tt} and R_{rr} we
immediately have u_{r} = v_{r}
, which implies u = v + k for some
arbitrary constant k. Making these substitutions into the equation for R_{qq} we get the
condition




Remembering
that e^{2u} = g_{tt}, and that the derivative of e^{2u}
is 2u_{r }e^{2u}, this condition expresses the requirement




The
left side is just the chain rule for the derivative of the product rg_{tt},
and since this derivative equals the constant –e^{2k} we immediately
have rg_{tt} = e^{2k}r
+ a for some
constant a, and hence g_{tt}
= e^{2k} +
a/r. As r
increases to infinity the metric must go over to the Minkowski metric, which
has g_{tt} = 1, so we must have –e^{2k} = 1, which implies
that k = pi/2. Also, since
g_{rr} = e^{2v} where v = u
+ pi/2, it follows
that g_{rr} = 1/g_{tt},
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 R_{tt} and
R_{rr}, but it's easy to verify that they also satisfy each of those
equations separately. Thus, substituting these expressions into the line
element (2), we arrive at the essentially unique (up to changes in coordinate
systems) 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 essentially the only spherically symmetrical ones in
which the metric is independent of t and the circumference of a circle of
radius r is 2pr) is simply




from
which we get




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




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"
c_{r} defined in terms of the Schwarzschild parameters t and r is
reduced to less than the nominal value of c. The factor of 2 relative to the
equation of 1911 arises because in the full theory there is gravitational
length contraction as well as time dilation. The length contraction doesn’t
affect the gravitational redshift, which is purely a function of the time
dilation, so the redshift prediction of 1911 remains valid. Only the radial
speed of light (in terms of Schwarzschild coordinates) is changed.


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 c_{t}
= 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 in Euclidean space. The effects of
gravitation can only be accurately represented by a tensor field. (It’s
possible to define socalled isotropic coordinates, as discussed in Section
8.4, in terms of which the speed of light is the same in all directions, but
only by using a radial coordinate in terms of which the circumference of a
circular orbit of radius r is not 2pr,
which shows the nonEuclidean character of the space.)


As
mentioned, one of the observable implications of general relativity (as well
as any other metrical theory that respects the equivalence principle) is
gravitational redshift, which is a consequence of the fact that, for any
stationary metric, the rate of proper time at a fixed radial position in a
gravitational field relative to the coordinate time is given by




Since
the coordinate time for successive wavecrests to traverse a fixed interval is
the same, the characteristic frequency n_{1} of light emitted by some known physical
process at a radial location r_{1} will represent a different
frequency n_{2 }with respect to
the proper time at some other radial location r_{2} according to the
formula




From
the Schwarzschild metric we have g_{tt}(r_{j}) = 1+2j_{j} where j_{j} = m/r_{j}
is the gravitational potential at r_{j}, so




Neglecting
the higherorder 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 r_{1} and r_{2} are nearly
equal, as, for example, at two heights near the Earth's surface, then the
leading factor of the rightmost expression is essentially just the
acceleration of gravity a = m/r^{2},
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 metrical theory of gravity that includes the
equivalence principle (e.g., Nordstrom’s scalar theory), 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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