The Goldberg–Sachs theorem is a result in Einstein's theory of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
about vacuum solutions of the
Einstein field equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Einstein in 1915 in the form ...
relating the existence of a certain type of
congruence with algebraic properties of the
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal f ...
.
More precisely, the theorem states that ''a
vacuum solution
In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non ...
of the Einstein field equations will admit a shear-free null geodesic congruence if and only if the Weyl tensor is
algebraically special.''
The theorem is often used when searching for algebraically special vacuum solutions.
Shear-Free Rays
A ray is a family of geodesic light-like curves. That is tangent vector field
is null and geodesic:
and
. At each point, there is a (nonunique) 2D spatial slice of the tangent space orthogonal to
. It is spanned by a complex null vector
and its complex conjugate
. If the metric is time positive, then the metric projected on the slice is
. Goldberg and Sachs considered the projection of the gradient on this slice.
A ray is shear-free if
. Intuitively, this means a small shadow cast by the ray will preserve its shape. The shadow may rotate and grow/shrink, but it will not be distorted.
The Theorem
''A vacuum metric,
, is algebraically special if and only if it contains a shear-free null geodesic congruence; the tangent vector obeys
.''
[; originally published in Acta Phys. Pol. 22, 13–23 (1962).]
This is the theorem originally stated by Goldberg and Sachs. While they stated it in terms of tangent vectors and the
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal f ...
, the proof is much simpler in terms of spinors. The
Newman-Penrose field equations give a natural framework for investigating Petrov classifications, since instead of proving
, one can just prove
. For these proofs, assume we have a spin frame with
having its flagpole aligned with the shear-free ray
.
Proof that a shear-free ray implies algebraic specialty: If a ray is geodesic and shear-free, then
. A complex rotation
does not affect
and can set
to simplify calculations. The first useful NP equation is
, which immediately gives
.
To show that
, apply the commutator
to it. The Bianchi identity gives the needed formulae:
and
. Working through the algebra of this commutator will show
, which completes this part of the proof.
Proof that algebraic specialty implies a shear-free ray: Suppose
is a degenerate factor of
. While this degeneracy could be n-fold (n=2..4) and the proof will be functionally the same, take it to be a 2-fold degeneracy. Then the projection
. The Bianchi identity in a vacuum spacetime is
, so applying a derivative to the projection will give
, which is equivalent to
The congruence is therefore shear-free and almost geodesic:
. A suitable rescaling of
exists which will make this congruence geodesic, and thus a shear-free ray. The shear of a vector field is invariant under rescaling, so it will remain shear-free.
Importance and Examples
In Petrov type D spacetimes, there are two algebraic degeneracies. By the Goldberg-Sachs theorem there are then two shear-free rays which point along these degenerate directions. Since the Newman-Penrose equations are written in a basis with two real null vectors, there is a natural basis which simplifies the field equations. Examples of such vacuum spacetimes are the
Schwarzschild metric
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an
exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumpti ...
and the
Kerr metric
The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of ge ...
, which describes a nonrotating and a rotating black hole, respectively. It is precisely this algebraic simplification which makes solving for the Kerr metric possible by hand.
In the Schwarzschild case with time-symmetric coordinates, the two shear-free rays are
Under the coordinate transformation
where
is the
tortoise coordinate, this simplifies to
.
Linearised gravity
It has been shown by Dain and Moreschi
that a corresponding theorem will not hold in
linearized gravity
In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects ...
, that is, given a solution of the
linearised Einstein field equations
In the theory of general relativity, linearized gravity is the application of perturbation theory to the metric tensor that describes the geometry of spacetime. As a consequence, linearized gravity is an effective method for modeling the effects ...
admitting a shear-free null congruence, then this solution need not be algebraically special.
See also
*
Geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
*
Optical scalars
References
{{DEFAULTSORT:Goldberg-Sachs theorem
Theorems in general relativity