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. ...

, a manifestly covariant equation is one in which all expressions are

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tens ...

s. 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 In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...

. 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 In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if ...

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 In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...

when evaluated instantaneously in a local inertial frame, 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 :

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, 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