The Plebanski tensor is an order 4 tensor in

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. ...

constructed from the

trace-free Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...

. It was first defined by

Jerzy Plebański Jerzy Franciszek Plebański (7 May 1928, Warsaw – 24 August 2005, Mexico) was a Polish theoretical physicist best known for his extensive research into general relativity and supergravity. Biography In 1954, Plebański received his Ph.D. u ...

in 1964. Let

S_

be the trace-free Ricci tensor: :

S_=R_-\fracRg_.

Then the Plebanski tensor is defined as :

P^_=S^_S^_+\delta^_S^S_-\frac\delta^_\delta^_S^S_.

The advantage of the Plebanski tensor is that it shares the same symmetries as 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 forc ...

. It therefore becomes possible to classify different

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why diffe ...

s based on additional algebraic symmetries of the Plebanski tensor in a manner analogous to the

Petrov classification In differential geometry and theoretical physics, the Petrov classification (also known as Petrov–Pirani–Penrose classification) describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold. It is ...

References

Tensors Tensors in general relativity {{math-physics-stub