semi-Riemannian geometry In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riem ...

, the Bel decomposition, taken with respect to a specific timelike congruence, is a way of breaking up the

Riemann tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...

of a

pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

into lower order tensors with properties similar to the electric field and magnetic field. Such a decomposition was partially described by Alphonse Matte in 1953 and by Lluis Bel in 1958. This decomposition is particularly important 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 ...

. This is the case of four-dimensional

Lorentzian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

s, for which there are only three pieces with simple properties and individual physical interpretations.

Decomposition of the Riemann tensor

In four dimensions the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field

\vec

, not necessarily geodesic or hypersurface orthogonal, consists of three pieces: # the ''electrogravitic tensor''

= R_ \, X^m \, X^n

#* Also known as the

tidal tensor Tidal is the adjectival form of tide. Tidal may also refer to: * ''Tidal'' (album), a 1996 album by Fiona Apple * Tidal (king), a king involved in the Battle of the Vale of Siddim * TidalCycles, a live coding environment for music * Tidal (servic ...

. It can be physically interpreted as giving the tidal stresses on small bits of a material object (which may also be acted upon by other physical forces), or the tidal accelerations of a small cloud of

test particle In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...

s in 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 no ...

electrovacuum solution In general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass–energy present is the field energy of an electromagnetic field, which must satisfy the (c ...

. # the ''magnetogravitic tensor''

= _ \, X^m \, X^n

#* Can be interpreted physically as a specifying possible spin-spin forces on spinning bits of matter, such as spinning

s. # the ''topogravitic tensor''

= _ \, X^m \, X^n

#* Can be interpreted as representing the sectional curvatures for the spatial part of a frame field. Because these are all ''transverse'' (i.e. projected to the spatial hyperplane elements orthogonal to our timelike unit vector field), they can be represented as linear operators on three-dimensional vectors, or as three-by-three real matrices. They are respectively symmetric,

traceless In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace o ...

, and symmetric (6,8,6 linearly independent components, for a total of 20). If we write these operators as E, B, L respectively, the principal invariants of the Riemann tensor are obtained as follows: *

K_1/4

is the trace of E² + L² - 2 B B^T, *

-K_2/8

is the trace of B ( E - L ), *

K_3/8

is the trace of E L - B².

References

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

Decomposition of the Riemann tensor

See also

References