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 ...
, 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 tensor ...
s. The operations of addition,
tensor multiplication,
tensor contraction
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tens ...
,
raising and lowering indices In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.
Vectors, covectors and the metric
Mat ...
, 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). Pa ...
.
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 a ...
of the metric.
Writing an equation in manifestly covariant form is useful because it guarantees
general covariance
In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the ''form'' of physical laws under arbitrary differentiable coordinate transformations. The essential idea ...
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
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 ...
:
where
is the
electromagnetic four-potential
An electromagnetic four-potential is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.Gravitation, J.A. W ...
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 symbol
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing dist ...
s as
:
:
For a
torsion-free metric assumed in general relativity, we may appeal to the symmetry of the Christoffel symbols
:
which allows the field tensor to be written in manifestly covariant form
:
See also
*
Lorentz covariance
*
Introduction to the mathematics of general relativity
*
Introduction to 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 laws o ...
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