Deriving the Schwarzschild solution
Encyclopedia
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 non-rotating massive spherically-symmetric 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.
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 rotation-reversal
or 
.
(2) A static spacetime
is one in which all metric components are independent of the time coordinate
(so that 
) and the geometry of the spacetime is unchanged under a time-reversal 
.
(3) A vacuum solution is one which satisfies the equation
. From the Einstein field equations
(with zero cosmological constant
), this implies that
(after contracting 
and putting 
).
, all metric components should remain the same. The metric components 
(
) change under this transformation as:
(
)
But, as we expect
(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
(by the static assumption).
of constant
, constant 
and constant 
(i.e., on each radial line), 
should only depend on 
(by spherical symmetry). Hence 
is a function of a single variable:
A similar argument applied to
shows that:
On the hypersurfaces of constant
and constant 
, it is required that the metric be that of a 2-sphere:
Choosing one of these hypersurfaces (the one with radius
, say), the metric components restricted to this hypersurface (which we denote by 
and 
) should be unchanged under rotations through 
and 
(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
and 
as yet undetermined functions of 
. Note that if 
or 
is equal to zero at some point, the metric would be singular
at that point.
Calculating the Christoffel symbols
Another well-known notation for the metric tensor is
Einstein field equations
Einstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
(see general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
). It describes spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
in the vicinity of a non-rotating massive spherically-symmetric 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
Spherically symmetric spacetime
A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the group SO and the orbits of this group are 2-dimensional spheres . The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one...
is one in which all metric components are unchanged under any rotation-reversal
or .(2) A static spacetime
Static spacetime
In general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field K which is irrotational, i.e., whose orthogonal distribution is involutive...
is one in which all metric components are independent of the time coordinate
(so that ) and the geometry of the spacetime is unchanged under a time-reversal .(3) A vacuum solution is one which satisfies the equation
. From the Einstein field equationsEinstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
(with zero cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
), this implies that
(after contracting and putting ).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 () change under this transformation as: ()But, as we expect
(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
(by the static assumption).Simplifying the components
On each hypersurfaceHypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
of constant
, constant and constant (i.e., on each radial line), should only depend on (by spherical symmetry). Hence is a function of a single variable:A similar argument applied to
shows that:On the hypersurfaces of constant
and constant , it is required that the metric be that of a 2-sphere:Choosing one of these hypersurfaces (the one with radius
, say), the metric components restricted to this hypersurface (which we denote by and ) should be unchanged under rotations through and (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
and as yet undetermined functions of . Note that if or is equal to zero at some point, the metric would be singularMathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
at that point.
Calculating the Christoffel symbolsChristoffel symbolsIn mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
Another well-known notation for the metric tensor is-
From this form of the metric tensor one can calculate the Christoffel symbolsChristoffel symbolsIn mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
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Here comma means the r derivative of the functions.
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Using the field equations to find
To determine
and 
and 
, the vacuum field equations are employed:
Only four of these equations are nontrivial and upon simplification become:
(The fourth equation is just
times the second equation)
Here dot means the r derivative of the functions. Subtracting the first and third equations produces:
where
is a non-zero real constant. Substituting 
into the second equation and tidying up gives:
which has general solution:
for some non-zero real constant
. 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 spaceMinkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
.
Using the Weak-Field Approximation to find
The geodesics of the metric (obtained where
and 
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 spaceMinkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
when the mass it represents vanishes.)
(where E and Eg are _____?) The constants
and 
are fully determined by some variant of this approach; from the weak-field approximation one arrives at the result:
where
is the gravitational constantGravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
,
is the mass of the gravitational source and 
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 gave alternative forms in isotropic coordinatesIsotropic coordinatesIn the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres. There are several different types of coordinate chart which are adapted to this family of nested spheres; the best known is the Schwarzschild chart, but the isotropic chart is also often...
. For isotropic spherical coordinates
, 
, 
, coordinates 
and 
are unchanged, and then (provided r >= 2Gm/c2 )
. . ., 
. . ., and
. . .
Then for isotropic rectangular coordinates
, 
, 
,
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 staticStatic spacetimeIn general relativity, a spacetime is said to be static if it admits a global, non-vanishing, timelike Killing vector field K which is irrotational, i.e., whose orthogonal distribution is involutive...
. In fact, the static assumption is stronger than required, as Birkhoff's theoremBirkhoff's theorem (relativity)In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution must be given by the Schwarzschild metric....
states that any spherically symmetric vacuum solution of Einstein's field equations is stationaryStationary spacetimeIn general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike....
; then one obtains the Schwarzschild solution. Birkhoff's theorem has the consequence that any pulsating star which remains spherically symmetric cannot generate gravitational waveGravitational waveIn physics, gravitational waves are theoretical ripples in the curvature of spacetime which propagates as a wave, traveling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as...
s (as the region exterior to the star must remain static).
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