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In
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, a manifestly covariant equation is one in which all expressions are
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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 vector space, dual. In components, it is expressed as a sum of products of scalar compo ...
, raising and lowering indices, and
covariant differentiation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
may appear in the equation. Forbidden terms include but are not restricted to
partial derivatives In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
of the metric. Writing an equation in manifestly covariant form is useful because it guarantees
general covarianceIn 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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
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 th ...
, then it is usually the correct generalization of the special relativistic equation in general relativity.


Example

An equation may be
Lorentz covariant In relativistic mechanics, relativistic physics, Lorentz symmetry, named after Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers t ...
even if it is not manifestly covariant. Consider the
electromagnetic field tensor In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ca ...
:F_ \, = \, \partial_a A_b \, - \, \partial_b A_a \, where A_a is the
electromagnetic four-potential An electromagnetic four-potential is a General relativity, 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.Gr ...
in the Lorenz gauge. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s 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 .


See also

*
Lorentz covariance In relativistic physics, Lorentz symmetry, named after Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers that are moving with re ...
*
Introduction to the mathematics of general relativity The mathematics of general relativity is complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved ...
* Introduction to special relativity


References

* * {{cite book, title=
Gravitation Gravity (), or gravitation, is a natural phenomenon Types of natural phenomena include: Weather, fog, thunder, tornadoes; biological processes, decomposition, germination seedlings, three days after germination. Germination is t ...
, 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