The Goldberg–Sachs theorem is a result in Einstein's theory of general relativity about vacuum solutions of the Einstein field equations relating the existence of a certain type of congruence with algebraic properties of the Weyl tensor.
More precisely, the theorem states that a vacuum solution 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.
A ray is a family of geodesic light-like curves.
At each point, there is a (nonunique) 2D spatial slice of the tangent space orthogonal to
It is spanned by a complex null vector
Goldberg and Sachs considered the projection of the gradient on this slice.
{\displaystyle A^{ab}={\tilde {g}}^{ap}{\tilde {g}}^{bq}\nabla _{p}l_{q}=z{\bar {m}}^{a}m^{b}+{\bar {z}}m^{a}{\bar {m}}^{b}+{\bar {\sigma }}m^{a}m^{b}+\sigma {\bar {m}}^{a}{\bar {m}}^{b}.}
Intuitively, this means a small shadow cast by the ray will preserve its shape.
, is algebraically special if and only if it contains a shear-free null geodesic congruence; the tangent vector obeys
[1] This is the theorem originally stated by Goldberg and Sachs.
While they stated it in terms of tangent vectors and the Weyl tensor, the proof is much simpler in terms of spinors.
The Newman-Penrose field equations[2] give a natural framework for investigating Petrov classifications, since instead of proving
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
The Bianchi identity gives the needed formulae:
[3] Working through the algebra of this commutator will show
Proof that algebraic specialty implies a shear-free ray: Suppose
The Bianchi identity in a vacuum spacetime is
, so applying a derivative to the projection will give
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.
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 and the Kerr metric, 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
It has been shown by Dain and Moreschi[4] that a corresponding theorem will not hold in linearized gravity, that is, given a solution of the linearised Einstein field equations admitting a shear-free null congruence, then this solution need not be algebraically special.