In general relativity, a manifestly covariant equation is one in which all expressions are tensors. The operations of addition,

tensor multiplication In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...

, tensor contraction, raising and lowering indices, and covariant differentiation may appear in the equation. Forbidden terms include but are not restricted to partial derivatives. Tensor densities, especially integrands and variables of integration, may be allowed in manifestly covariant equations if they are clearly weighted by the appropriate power of the determinant of the metric. Writing an equation in manifestly covariant form is useful because it guarantees general covariance upon quick inspection. If an equation is manifestly covariant, and if it reduces to a correct, corresponding equation in special relativity when evaluated instantaneously in a

local inertial frame In theoretical physics, a local reference frame (local frame) refers to a coordinate system or frame of reference that is only expected to function over a small region or a restricted region of space or spacetime. The term is most often used in t ...

, then it is usually the correct generalization of the special relativistic equation in general relativity.

Example

An equation may be Lorentz covariant even if it is not manifestly covariant. Consider the

electromagnetic field tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...

F_ \, = \, \partial_a A_b \, - \, \partial_b A_a \,

where

A_a

is the electromagnetic four-potential in the

Lorenz gauge In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ha ...

. The equation above contains partial derivatives and is therefore not manifestly covariant. Note that the partial derivatives may be written in terms of covariant derivatives and Christoffel symbols as :

\partial_a A_b = \nabla_a A_b + \Gamma^c_ A_c

\partial_b A_a = \nabla_b A_a + \Gamma^c_ A_c

For a torsion-free metric assumed in general relativity, we may appeal to the symmetry of the Christoffel symbols :

\Gamma^c_ - \Gamma^c_ = 0,

which allows the field tensor to be written in manifestly covariant form :

F_ \, = \, \nabla_a A_b \, - \, \nabla_b A_a .

References

* * {{cite book, title=

Gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...

, author1=John Archibald Wheeler, author2=C. Misner, author3=K. S. Thorne, author-link1=John Archibald Wheeler, author-link2=Charles W. Misner, author-link3=Kip Thorne, publisher=W.H. Freeman & Co, year=1973, isbn=0-7167-0344-0 General relativity Tensors

Example

See also

References