 Deriving the Schwarzschild solution

The Schwarzschild solution is one of the simplest and most useful solutions of the Einstein field equations (see general relativity). It describes spacetime in the vicinity of a nonrotating massive sphericallysymmetric object. It is worthwhile deriving this metric in some detail; the following is a reasonably rigorous derivation that is not always seen in the textbooks.
Assumptions and notation
Working in a coordinate chart with coordinates labelled 1 to 4 respectively, we begin with the metric in its most general form (10 independent components, each of which is an arbitrary function of 4 variables). The solution is assumed to be spherically symmetric, static and vacuum. For the purposes of this article, these assumptions may be stated as follows (see the relevant links for precise definitions):
(1) A spherically symmetric spacetime is one in which all metric components are unchanged under any rotationreversal or .
(2) A static spacetime is one in which all metric components are independent of the time coordinate t (so that ) and the geometry of the spacetime is unchanged under a timereversal .
(3) A vacuum solution is one which satisfies the equation T_{ab} = 0. From the Einstein field equations (with zero cosmological constant), this implies that R_{ab} = 0 (after contracting and putting R = 0).
Diagonalising the metric
The first simplification to be made is to diagonalise the metric. Under the coordinate transformation, , all metric components should remain the same. The metric components g_{μ4} () change under this transformation as:
 ()
But, as we expect g'_{μ4} = g_{μ4} (metric components remain the same), this means that:
 ()
Similarly, the coordinate transformations and respectively give:
 ()
 ()
Putting all these together gives:
 ()
and hence the metric (line element) must be of the form:
where the four metric components are independent of the time coordinate t (by the static assumption).
Simplifying the components
On each hypersurface of constant t, constant θ and constant ϕ (i.e., on each radial line), g_{11} should only depend on r (by spherical symmetry). Hence g_{11} is a function of a single variable:
A similar argument applied to g_{44} shows that:
On the hypersurfaces of constant t and constant r, it is required that the metric be that of a 2sphere:
Choosing one of these hypersurfaces (the one with radius r_{0}, say), the metric components restricted to this hypersurface (which we denote by and ) should be unchanged under rotations through θ and ϕ (again, by spherical symmetry). Comparing the forms of the metric on this hypersurface gives:
which immediately yields:
 and
But this is required to hold on each hypersurface; hence,
 and
Thus, the metric can be put in the form:
with A and B as yet undetermined functions of r. Note that if A or B is equal to zero at some point, the metric would be singular at that point.
Calculating the Christoffel symbols
Another wellknown notation for the metric tensor is
From this form of the metric tensor one can calculate the Christoffel symbols
Here comma means the r derivative of the functions.
Using the field equations to find A(r) and B(r)
To determine A and B, the vacuum field equations are employed:
Only four of these equations are nontrivial and upon simplification become:
(The fourth equation is just sin ^{2}θ times the second equation)
Here dot means the r derivative of the functions. Subtracting the first and third equations produces:
where K is a nonzero real constant. Substituting into the second equation and tidying up gives:
which has general solution:
for some nonzero real constant S. Hence, the metric for a static, spherically symmetric vacuum solution is now of the form:
Note that the spacetime represented by the above metric is asymptotically flat, i.e. as , the metric approaches that of the Minkowski metric and the spacetime manifold resembles that of Minkowski space.
Using the WeakField Approximation to find K and S
The geodesics of the metric (obtained where ds is extremised) must, in some limit (e.g., toward infinite speed of light), agree with the solutions of Newtonian motion (e.g., obtained by Lagrange equations). (The metric must also limit to Minkowski space when the mass it represents vanishes.)
(where E and Eg are _____?) The constants K and S are fully determined by some variant of this approach; from the weakfield approximation one arrives at the result:
where G is the gravitational constant, m is the mass of the gravitational source and c is the speed of light. It is found that:
and
Hence:
and
So, the Schwarzschild metric may finally be written in the form:
Alternative form in isotropic coordinates
The original formulation of the metric uses anisotropic coordinates in which the velocity of light is not the same in the radial and transverse directions. A S Eddington^{[1]} gave alternative forms in isotropic coordinates. For isotropic spherical coordinates r_{1}, θ, ϕ , coordinates θ and ϕ are unchanged, and then (provided r >= 2Gm/c^{2} ^{[2]})
. . ., . . ., and
. . .
Then for isotropic rectangular coordinates x, y, z,
The metric then becomes, in isotropic rectangular coordinates:
. . .
Dispensing with the static assumption  Birkhoff's theorem
In deriving the Schwarzschild metric, it was assumed that the metric was vacuum, spherically symmetric and static. In fact, the static assumption is stronger than required, as Birkhoff's theorem states that any spherically symmetric vacuum solution of Einstein's field equations is stationary; then one obtains the Schwarzschild solution. Birkhoff's theorem has the consequence that any pulsating star which remains spherically symmetric cannot generate gravitational waves (as the region exterior to the star must remain static).
References
 ^ A S Eddington, "Mathematical Theory of Relativity", Cambridge UP 1922 (2nd ed.1924, repr.1960), at page 85 and page 93. Symbol usage in the Eddington source for interval s and timelike coordinate t has been converted for compatibility with the usage in the derivation above.
 ^ H. A. Buchdahl, "Isotropic coordinates and Schwarzschild metric", International Journal of Theoretical Physics, Vol.24 (1985) pp.731739.
See also
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