Natural Science Forum / Physics / Relativity / April 2007

[...]

Here is the relevant mathematics. You'll see precisely how in the
Schwarzschild case the independence of metric coefficients of t
follows from the EFE, not staticity everywhere.

We seek a spherically symmetric metric of signature 2 satisfying the
Einstein equation in vacuum:

   R_ab = 0

The assumption of symmetry allows us to assume WLOG that the metric
has the general form like so:

   ds^2 = f(r,t) dt^2 + g(r,t) dr^2 + h(r,t) dr dt + k(r,t) dO^2,

where  r  is a coordinate labelling the spheres of symmetry and:

   dO^2 = dtheta^2 + sin^2(theta) dphi^2,  as usual.

It turns out that even this reasonably nice general form results in an
unmanageable computational mess so one simplifies the general form
further by a local change of coordinates which cleverly removes the
"dr dt" term and also relabels the spheres of symmetry by their areas
(divided by 4pi). (I can supply more details on this final coordinate
change if you wish.) This results in the following two possible
simplified, diagonal, forms consistent with the given signature
constraint:

either:

   ds^2 = -A^2 dt^2 + B^2 dr^2 + r^2 dO^2     [1]

or:

   ds^2 = +A^2 dt^2 - B^2 dr^2 + r^2 dO^2     [2]

where A(r,t) and B(r,t) are some functions which are never zero (or
else the metric would not be rank-4).

NB: this local coordinate change happens to rely on dividing by
certain function, so any solutions for metric coefficients we may
obtain by proceeding from these two general forms may not be defined
over the spacetime events for which the function we divided by is
zero.

[In fact (getting slightly ahead of myself here) this is precisely
what happens: the function we divide by is zero for r=2M, hence the
(in)famous exclusion of the locus r=2M from the domain of the metric
coefficients in the Schwarzschild form. There is of course no reason
to presume this is an actual physical exclusion as it potentially
results from the man-made work-saving transformation. So whether it's
a true singularity must be verified separately. It turns out - as is
well known - that the metric is well-defined for r=2M, it is merely
non-diagonalizable there hence the forms [1] and [2] break down there
- it's just a case of being too greedy/lazy.]

OK, so we have those two forms potentially satisfying the EFE. The
maximal domain possible (unless the EFE themselves somehow end up
restricting it) is:

   -infty < t < +infty,
        0 < r < +infty,
        0 < theta < 2pi,
        0 < phi < pi.

I'm going to skip the exact details of deriving the equations below
(we can go into it later if necessary). I'm using Cartan's method of
moving frames, and after computing the connection and curvature forms
the Einstein equation R_ab=0 reduces to the following four PDEs:

   R_tt = 0
    R_rr = 0
   R_theta theta = 0
   R_rt = 0

...with the remaining equations being either identities or equivalent
to those four ones. (For example, R_phi phi = R_theta theta.)

The exact forms of the four left-hand sides are, few pages of
calculations later, respectively (excuse the mess, it's the fourth,
simplest, equation that's interesting to us):

(1/AB)(-d/dt((1/A)*dB/dt) + d/dr((1/B)*dA/dr) + (2/rB)*dA/dr)     = 0
(1/AB)( d/dt((1/A)*dB/dt) - d/dr((1/B)*dA/dr)) - (2/rB)*d/dr(1/B) = 0
-+(1/rAB^2)*dA/dr -+ (1/rB)*d/dr(1/B) + (1/r^2)*(1 -+ 1/B^2)      = 0
(2/rAB^2)*dB/dt                                                   = 0

...where "-+" means use the "-" sign for the first general form of the
diagonal metric [1], use "+" for the second [2]. Also "d" obviously
denotes partial derivatives.

Note the fourth equation R_rt = 0 says then (cancelling the factor in
front):

   dB/dt = 0

...in other words, B(r,t) is independent of t, both in case [1] and
[2]. This means that B is only a function of r which means that the
first two PDEs simplify quite a bit:

 (1/AB)( d/dr((1/B)*dA/dr) + (2/rB)*dA/dr)     = 0
 (1/AB)(-d/dr((1/B)*dA/dr)) - (2/rB)*d/dr(1/B) = 0

This is actually a pair of _ODEs_ (r is the only variable in them)
which remarkably simplify to the following (denoting the r-derivatives
by primes now):

   A'/A + B'/B = 0

i.e.,

   (ln |A|)' + (ln |B|)' = 0

hence:

   |AB| = C   (a positive constant of integration)

i.e.,

   AB = C    (an arbitary nonzero constant of integration).

The important thing as far as Birkhoff is concerned is that this means
that:

   B = constant/A

...in other words B(r,t) is independent of t as well, in both [1] and
[2].

Plugging A(r) and B(r)=1/A(r) into the remaining (third) PDE yields
another _ordinary_ differential equation whose exact form and details
of the solution are less interesting to us at this point. Suffice to
say it yields the well-known Schwarzschild form of the metric
components:

 ds^2 = -(1 - 2M/r) dt^2 + 1/(1 - 2M/r) dr^2 + r^2 dO^2       (case
[1])

...with the domain restricted to r > 2M, and:

 ds^2 =  (2M/r - 1) dt^2 - 1/(2M/r - 1) dr^2 + r^2 dO^2       (case
[2])

...with the domain restricted to 0 < r < 2M.

[As I mentioned earlier, the case of r=2M must be verified separately,
as logically at this point we don't know if the exclusion is real or
merely due to our coordinate change in the beginning.]

A quick examination of these two formulas shows that they can be
written as a single equation by a simple sign switch:

 ds^2 = -(1 - 2M/r) dt^2 + 1/(1 - 2M/r) dr^2 + r^2 dO^2

...with the domain restricted to r>0 AND r=/=2M.

We note that this solution is static for r>2M and spatially
homogeneous for r<2M. It is also unique over this domain as it's a
solution to an ODE system.

Incidentally, since the coordinate denoted by "r" inside the horizon
is timelike, it's instructive to switch around the letters "r" and "t"
in case [2] because of our strong mental habit of thinking of "t" as
denoting time:

 ds^2 = -1/(2M/t - 1) dt^2 + (2M/t - 1) dr^2 + t^2 dO^2,

...with r arbitrary and t > 0 only (we "fall off" spacetime at t=0).

This is what the metric inside "really" is.

--
Jan Bielawski