This article is on the minimal body of mathematics necessary to understand general relativity. For a more complete overview see Mathematics of general relativity. General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915 [1][2]. It unifies special relativity and Isaac Newtons law of universal gravitation with the insight that gravitation is not due to a force but rather is a manifestation of curved space and time... Notational point: General relativity articles using tensors will use the abstract index notation . ...

An understanding of calculus and differential equations is necessary for the understanding of nonrelativistic physics. In order to understand special relativity one also needs an understanding of tensor calculus. To understand the general theory of relativity, one needs a basic introduction to the mathematics of curved spacetime that includes a treatment of curvilinear coordinates, nontensors, curved space, parallel transport, Christoffel symbols, geodesics, covariant differentiation, the curvature tensor, Bianchi identity, and the Ricci tensor. This article follows the basic treatment in the lecture series on the topic, intended for advanced undergraduates, given by Paul Dirac at Florida State University.[Ref. 1] Calculus is a central branch of mathematics, developed from algebra and geometry. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... For more technical Wiki articles on tensors, see the section later in this article. ... Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system. ... A pseudotensor is a generalization of the pseudovector concept, and changes its sign under inversion by some transformation matrix. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ... In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... In differential geometry, the curvature form describes curvature of principal bundle with connection. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... Paul Adrien Maurice Dirac Paul Adrien Maurice Dirac, OM FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Florida State University, is also commonly referred to as FSU or Florida State. ...

All the mathematics discussed in this article was known before Einstein's invention of the general theory of relativity. Albert Einstein, photographed in 1947 by Oren J. Turner. ...

For an introduction based on the specific physical example of particles orbiting a large mass in circular orbits, see Newtonian motivations for general relativity for a nonrelativistic treatment and Theoretical motivation for general relativity for a fully relativistic treatment. In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. ... Some of the basic concepts of General Relativity can be outlined outside the relativistic domain. ... A Theoretical motivation for general relativity, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. ...

Mathematics of special relativity

Main article: Special relativity

The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...

Vectors

Main article: Four-vector

In relativity, a four-vector is a vector in a four-dimensional real vector space, whose components transform like the space and time coordinates (ct, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). ...

Interval between two points

Spacetime physics requires four coordinates for the description of a point in spacetime: This article describes some of the common coordinate systems that appear in elementary mathematics. ...

where c is the speed of light and x, y, and z are spatial coordinates. The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. In metric units, c is exactly 299,792,458 metres per second (or 1,079,252,848. ...

A point very close to our original point is

.

The square of the distance, or interval, between the two points is

and is invariant under coordinate transformations. Here we are using the Minkowski metric. In mathematics, an invariant is something that does not change under a set of transformations. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

Coordinate transformations

Transformation of dx

If one defines a new coordinate system x^μ' such that

then

where repeated indices are summed according to the Einstein summation convention. For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...

The comma in the subscript of the last term indicates differentiation.

Transformation of a scalar

A scalar quantity transforms as In physics a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. ...

.

Contravariant vectors

Quantities A^μ that transform in the same way as dx^μ under a change of coordinates,

,

form a contravariant vector. The squared length of the vector is the invariant quantity This page does not deal with the statistical concept covariance of random variables, nor with the computer science concepts of covariance and contravariance. ...

.

The term on the left is the notation for the inner product of A with itself. In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

Covariant vectors

The covariant vector is defined as In category theory, see covariant functor. ...

.

It transforms as a scalar

.

Inner product

The inner product of two vectors is written

.

This quantity is also invariant under coordinate transformations.

Tensors

Main article: Tensor

In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

Definition

A rank 2 contravariant tensor can be constructed from the outer product of vectors as Note: This is a fairly abstract mathematical approach to tensors. ... Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ...

.

Contravariant tensor

The components of a rank 2 contravariant tensor transform in the same way as the quantities ,

.

Coavariant and mixed tensors

Higher rank tensors are constructed similarly as are covarariant and mixed tensors. For a rank 2 covariant tensor, the transformation is

.

Oblique axes

Main article: Metric tensor

In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

The interval and the metric tensor

An oblique coordinate system is one in which the axis are not necessarily orthogonal to each other. For oblique axes, the interval is In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

where the coefficients g_μν, called the metric tensor depend on the system of oblique axes. In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

Determinant of the metric tensor

The determinant of g_μν is denoted g and is always negative for any real coordinate axes.

Inner product

The inner product of any two vectors

is invariant.

Relation between covariant and contravariant tensors

Covariant tensors are related to contravariant tensors by

and

where g^μν is the cofactor of the corresponding g_μν

and

.

Nontensors

Main article: Pseudotensor

A nontensor is a tensor-like quantity N_μ that behaves like a tensor in the raising and lowering of indices, A pseudotensor is a generalization of the pseudovector concept, and changes its sign under inversion by some transformation matrix. ...

and

,

but that does not transform like a tensor under a coordinate transformation.

Mathematics of general relativity

Curvilinear coordinates and curved spacetime

Curvilinear coordinates are coordinates in which the angles between axes can change from point to point. In other words, the metric tensor g_μν in curvilinear coordinates is no longer a constant, but depends on the spacetime location of the metric tensor. It is therefore a field quantity. Curvilinear coordinates are a coordinate system based on some transformation of the standard coordinate system. ...

Like the surface of a ball embedded in three-dimensional space, we can imagine four dimensional spacetime as embedded in a flat space of a higher dimension. The coordinates on the surface of the ball are curvilinear, while the coordinates in three dimensional space can be rectilinear. The coordinates of four dimensional curved spacetime are curvilinear, while the four space is embedded in a larger dimensional space of rectilinear coordinates. Rectilinear: Characterized by straight lines, as opposed to curvilinear which is characterized by curved lines. ...

Parallel transport

Main article: Parallel transport

Example: Parallel displacement along a circle embedded in two dimensions. The circle of radius r is embedded in a two dimensional space characterized by the coordinates z¹ and z². The circle itself is characterized by coordinates y¹ and y² in the two dimensional space. The circle itself is one dimensional and can be characterized by its arc length x. The coordinates y are related to the coordinate x through the relation y¹ = rsin(x / r) and y² = rcos(x / r). This gives and . In this case the metric is a scalar and is given by g = cos²(x / r) + sin²(x / r) = 1. The interval is then ds² = gdx² = dx². The interval is just equal to the arc length as expected.

In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ... Image File history File links Download high-resolution version (939x788, 22 KB) Summary author = complexica Licensing I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high-resolution version (939x788, 22 KB) Summary author = complexica Licensing I, the creator of this work, hereby release it into the public domain. ...

The interval in a high dimensional space

Imagine our four dimensional curved spacetime is embeded in a larger N dimensional flat space. Any true physical vector lies entirely in the curved physical space. In other words, the vector is tangent to the curved physical spacetime. It has no component normal to the four dimensional curved spacetime. In mathematics, the word tangent has two distinct, but etymologically-related meanings: one in geometry, and one in trigonometry. ...

In the N dimensional flat space with coordinates the interval between neighboring points is

where h_nm is the metric for the flat space. We do not assume the coordinates are orthogonal, only rectilinear.

The interval between two point in physical spacetime

To quote Dirac:

Physical spacetime forms a four dimensional "surface" in the flat N-dimensional space. Each point x^μ determines a definite point yⁿ in the N-dimensional space. Each coordinate yⁿ is a function of the four x's; say yⁿ(x). There are N-4 such equations.

The relation between neighboring contravariant vectors: Christoffel symbols

Main article: Christoffel symbol

The difference in y for two neighboring points in the surface differing by δx^μ is In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

where

.

The interval between two neighboring points in physical spacetime becomes

where

.

A contravariant vector at a point x in physical spacetime is related to the same contravariant vector at the same point y(x) in N-dimensional space by the relation

.

The vector lies in the surface of physical spacetime.

Now shift the vector Aⁿ to the point yⁿ(x + dx) keeping it parallel to itself. In other words, we hold the comonents of the vector constant during the shift. The vector no longer lies in the surface because of curvature of the surface.

The shifted vector can be split into two parts, one tangent to the surface and one normal to surface, as

.

The vector as a function of y tangent to the surface can be written in terms of the vector K in terms of x as

.

The normal vector is normal to every vector in the surface including the unit vectors that define the comonents of x^μ. Therefore

.

This allows us to write

or

where

is a nontensor called the Christoffel symbol of the first kind. It can be shown to be related to the metric tensor through the relation In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

.

Since the Christoffel symbol can be written entirely in terms of the metric in physical spacetime, all reference to ther N-dimensional space has disappeared.

Christoffel symbol of the second kind

The Christoffel symbol of the second kind is defined as

.

This operation is allowed for nontensors.

This allows us to write

and

.

The minus sign in the second expression can be seen from the invariance of an inner product of two vectors

.

The constancy of the length of the parallel displaced vector

From Dirac:

The constancy of the length of the vector follows from geometrical arguments. When we split up the vector into tangential and normal parts ... the normal part is infinitesimal and is orthogonal to the tangential part. It follows that, to the first order, the length of the whole vector equals that of its tangential part.

The covariant derivative

Main article: Covariant derivative

The partial derivative of a vector with respect to a spacetime coordinate is composed of two parts, the normal partial derivative minus the the change in the vector due to parallel transport In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...

.

The covariant derivative of a product is

.

Geodesics

Main article: Geodesic

Suppose we have a point z^μ that moves along a track in physical spacetime. Suppose the track is parameterized with the quantity τ. The a "velocity" vector that points in the direction of motion in spacetime is In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ...

.

The variation of the velocity upon parallel displacement along the track is then

.

If there are no "forces" acting on the point, then the velocity is unchanged along the track and we have

,

which is called the geodesic equation.

Curvature tensor

Main article: Riemann curvature tensor

In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

Definition

The curvature K of a surface is simply the angle through which a vector is turned as we take it around an infinitesimal closed path. For a two dimensional Euclidean surface we have

.

For a triangle on a sperical surface the angle is the excess (over 180 degrees) of the sum of the angles of the triangle. For a spherical surface of radius r, the curvature is

.

The definition of curvature generalizes to

where A^μ is an arbitrary vector tgransported around a closed loop of area Δ^ρσ along the x^ρ and x^σ directions. Here,

.

This expression can be reduced to the commutation relation In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ...

where

.

In flat spacetime, the derivatives commute and the curvature is zero.

Symmetries of the curvature tensor

The curvature tensor is antisymmetric in the last two indices

.

Also

and

.

A consequence of the symmetries is that the curvature tensor has only 20 independent components.

Bianchi identity

The following differential relation, known as the Bianchi identity is true. In differential geometry, the curvature form describes curvature of principal bundle with connection. ...

Ricci tensor and scalar curvature

The Ricci tensor is defined as the contraction In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In multilinear algebra, a tensor contraction is a sum of products of scalar components of one or more tensors caused by applying the summation convention to a pair of dummy indices which are bound to each other in an expression. ...

.

A second contraction yields the scalar curvature In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. ...

.

It can be shown that consequence of the Bianchi identity is

.

See also

• Riemannian geometry
• Differential geometry and topology
• List of differential geometry topics

In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... This is a list of differential geometry topics, by Wikipedia page. ...

References

[1] P. A. M. Dirac (1996). General Theory of Relativity. Princeton University Press. ISBN 0-691-01146-X.

[2] Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.

[3] Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0-08-018176-7.

[4] R. P. Feynman, F. B. Moringo, and W. G. Wagner (1995). Feynman Lectures on Gravitation. Addison-Wesley. ISBN 0-201-62734-5.

[5] Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0-517-029618.