NationMaster - Encyclopedia: Mathematics of general relativity

The mathematics of general relativity refers to various mathematical structures and techniques that are used in Albert Einstein's theory of general relativity. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. An understanding of calculus and differential equations is necessary for the understanding of nonrelativistic physics. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... â€œEinsteinâ€ redirects here. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... â€œGravityâ€ redirects here. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ...

This article is a general description of the mathematics of general relativity. For a discussion of the minimal mathematics necessary to understand general relativity see Basic introduction to the mathematics of curved spacetime. This article is on the minimal body of mathematics necessary to understand general relativity. ...

Why tensors?

The principle of general covariance states that the laws of physics should take the same mathematical form in all reference frames and was one of the central principles in the development of general relativity. The term 'general covariance' was used in the early formulation of general relativity, but is now referred to by many as diffeomorphism covariance. Although diffeomorphism covariance is not the defining feature of general relativity^[1], and controversies remain regarding its present status in GR, the invariance property of physical laws implied in the principle coupled with the fact that the theory is essentially geometrical in character (making use of non-Riemannian geometry) suggested that general relativity be formulated using the language of tensors. This will be discussed further below. The principle of general covariance states that the laws of physics should take the same form in all coordinate systems. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ... In theoretical physics, general covariance (also known as diffeomorphism invariance) is the invariance of physical laws (for example, the equations of general relativity) under arbitrary coordinate transformations. ... 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. ...

Spacetime as a manifold

Most modern approaches to mathematical general relativity begin with the concept of a manifold. More precisely, the basic physical construct representing gravitation - a curved spacetime - is modelled by a four-dimensional, smooth, connected, Lorentzian manifold. Other physical descriptors are represented by various tensors, discussed below. In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... â€œGravityâ€ redirects here. ... Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...

The rationale for choosing a manifold as the fundamental mathematical structure is to reflect desirable physical properties. For example, in the theory of manifolds, each point is contained in a (by no means unique) coordinate chart and can be thought of as representing the 'local spacetime' around the observer (represented by the point). The principle of local Lorentz covariance, which states that the laws of special relativity hold locally about each point of spacetime, lends further support to the choice of a manifold structure for representing spacetime, as locally around a point on a general manifold, the region 'looks like', or approximates very closely Minkowski space (flat spacetime). In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... Observation is an activity of a sapient or sentient living being (e. ... â€¹The template below has been proposed for deletion. ... 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... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

The idea of coordinate charts as 'local observers who can perform measurements in their vicinity' also makes good physical sense, as this is how one actually collects physical data - locally. For cosmological problems, a coordinate chart may be quite large.

Local versus global structure

An important distinction in physics is the difference between local and global structures. Measurements in physics are performed in a relatively small region of spacetime and this is one reason for studying the local structure of spacetime in general relativity, whereas determining the global spacetime structure is important, especially in cosmological problems. This article or section does not cite its references or sources. ... This article or section does not cite its references or sources. ...

An important problem in general relativity is to tell when two spacetimes are 'the same', at least locally. This problem has its roots in manifold theory where determining if two Riemannian manifolds of the same dimension are locally isometric ('locally the same'). This latter problem has been solved and its adaptation for general relativity is called the Cartan-Karlhede algorithm. In the study of Riemannian geometry in mathematics, a local isometry from one (pseudo-)Riemannian manifold to another is a map which pulls back the metric tensor on the second manifold to the metric tensor on the first. ... One of the most fundamental problems of Riemannian geometry is this: given two Riemannian manifolds of the same dimension, how can one tell if they are locally isometric? This question was addressed by Elwin Christoffel, and completely solved by Ã‰lie Cartan using his exterior calculus with his method of moving...

Tensors in GR

For details on tensors, see the articles: Tensor, Tensor (intrinsic definition), Classical treatment of tensors, Intermediate treatment of tensors. 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. ... In mathematics, the modern component-free approach to the theory of tensors views tensors initially as abstract objects, expressing some definite type of multi-linear concept. ... The following is a component-based classical treatment of tensors. ... A tensor is the mathematical idealization of a geometric or physical quantity whose analytic description, relative to a fixed frame of reference, consists of an array of numbers. ...

One of the profound consequences of relativity theory was the abolishment of preferred reference frames. The description of physical phenomena should not depend upon who does the measuring - one reference frame should be as good as any other. Special relativity banished the singling out of inertial frames whereas general relativity eliminated any privileged reference frames for describing nature. In physics, an inertial frame of reference, or inertial frame for short (also descibed as absolute frame of reference), is a frame of reference in which the observers move without the influence of any accelerating or decelerating force. ... In theoretical physics, a preferred or privileged frame is usually a special hypothetical frame of reference in which the laws of physics might appear to be identifiably different from those in other frames. ...

Any observer can make measurements and the precise numerical quantities obtained only depend on the coordinate system used. This suggested a way of formulating relativity using 'invariant structures', those that are independent of the coordinate system (represented by the observer) used, yet still have an independent existence. The most suitable mathematical structure seemed to be a tensor. For example, when measuring the electric and magnetic fields produced by an accelerating charge, the values of the fields will depend on the coordinate system used, but the fields are regarded as having an independent existence, this independence represented by the electromagnetic field tensor . Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...

Mathematically, tensors are generalised linear operators - multilinear maps. As such, the ideas of linear algebra are employed to study tensors. In linear algebra, a multilinear map is a mathematical function of several vector variables that is linear in each variable. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...

At each point $, p$ of a manifold, the tangent and cotangent spaces to the manifold at that point may be constructed. Vectors (sometimes referred to as contravariant vectors) are defined as elements of the tangent space and covectors (sometimes termed covariant vectors, but more commonly dual vectors or one-forms) are elements of the cotangent space. On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... It has been suggested that this article or section be merged into Covariant transformation. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ... It has been suggested that this article or section be merged into Covariant transformation. ... In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

At $, p$ , these two vector spaces may be used to construct type $, (r,s)$ tensors, which are real-valued multilinear maps acting on the direct sum of $, r$ copies of the cotangent space with $, s$ copies of the tangent space. The set of all such multilinear maps forms a vector space, called the tensor product space of type $, (r,s)$ at $, p$ and denoted by $, (T_p)^r{}_sM$ . If the tangent space is n-dimensional, it can be shown that $dim (T_p)^r{}_sM = n^{r+s}$ . In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In abstract algebra, the direct sum is a construction which combines several modules into a new, bigger one. ...

In the general relativity literature, it is conventional to use the component syntax for tensors. An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

$T ;! = ;! {T^{a_1 ldots a_r}}_{{b_1} ldots {b_s}} frac {partial} {partial x^{a_1}} otimes ldots otimes frac {partial} {partial x^{a_r}} otimes dx^{b_1} otimes ldots otimes dx^{b_s}$

where $;!frac {partial} {partial x^{a_i}}$ is a basis for the i-th tangent space and a basis for the j-th cotangent space.

As spacetime is assumed to be four-dimensional, each index on a tensor can be one of four values. Hence, the total number of elements a tensor possesses equals 4^R, where R is the sum of the numbers of covariant and contravariant indices on the tensor (a number called the "rank" of the tensor). In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ...

Symmetric and antisymmetric tensors

Some physical quantities are represented by tensors not all of whose components are independent. Important examples of such tensors include symmetric and antisymmetric tensors. Antisymmetric tensors are commonly used to represent rotations (for example, the vorticity tensor).

Although a generic rank R tensor in 4 dimensions has 4^R components, constraints on the tensor such as symmetry or antisymmetry serve to reduce the number of distinct components. For example, a symmetric rank two tensor

T

satisfies T_ab = T_ba and possesses 10 independent components, whereas an antisymmetric (skew-symmetric) rank two tensor

P

satisfies P_ab = -P_ba and has 6 independent components. For ranks greater than two, the symmetric or antisymmetric index pairs must be explicitly identified. In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments. ... In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments. ...

Antisymmetric tensors of rank 2 play important roles in relativity theory. The set of all such tensors - often called bivectors - forms a vector space of dimension 6, sometimes called bivector space. THE TRIVECTOR: the sum of all coolness encapsulated in three rocking houses surrounding victoria park, located in Kingston Ontario Canada. ...

The metric tensor

The metric tensor is a central object in general relativity that describes the local geometry of spacetime (as a result of solving the Einstein field equation). Using the weak-field approximation, the metric can also be thought of as representing the 'gravitational potential'. The metric tensor is often just called 'the metric'. In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... The weak-field approximation in general relativity is used to describe the gravitational field very far from the source of gravity. ...

The metric is a symmetric tensor and is an important mathematical tool. As well as being used to raise and lower tensor indices, it also generates the connections which are used to construct the geodesic equations of motion and the Riemann curvature tensor. In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ... 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. ...

A convenient means of expressing the metric tensor in combination with the incremental intervals of coordinate distance that it relates to is through the line element: The line element in mathematics can most generally be thought of as the square of the change in a position vector in an affine space equated to the square of the change of the arc length. ...

This way of expressing the metric was used by the pioneers of differential geometry. While some relativists consider the notation to be somewhat old-fashioned, many readily switch between this and the alternative notation: In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ...

The metric tensor is commonly written as a 4 by 4 matrix. Due to the symmetry of the metric, this matrix is symmetric and has 10 independent components.

Invariants

One of the central features of GR is the idea of invariance of physical laws. This invariance can be described in many ways, for example, in terms of local Lorentz covariance, the general principle of relativity, or diffeomorphism covariance. â€¹The template below has been proposed for deletion. ... The general principle of relativity as used in Einsteins general theory of relativity is that the laws of physics must take the same form in all reference frames. ... In theoretical physics, general covariance (also known as diffeomorphism invariance) is the invariance of physical laws (for example, the equations of general relativity) under arbitrary coordinate transformations. ...

A more explicit description can be given using tensors. The crucial feature of tensors used in this approach is the fact that (once a metric is given) the operation of contracting a tensor of rank R over all R indices gives a number - an invariant - that is independent of the coordinate chart one uses to perform the contraction. Physically, this means that if the invariant is calculated by any two observers, they will get the same number, thus suggesting that the invariant has some independent significance. Some important invariants in relativity include: In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

Other examples of invariants in relativity include the electromagnetic invariants, and various other curvature invariants, some of the latter finding application in the study of gravitational entropy and the Weyl curvature hypothesis. In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. ... In the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. ... In differential geometry and theoretical physics, the classification of electromagnetic fields is a pointwise classification of bivectors at each point of a Lorentzian manifold. ... It has been suggested that Kretschmann scalar be merged into this article or section. ... The Weyl curvature hypothesis, which arises in the application of Albert Einsteins general theory of relativity to cosmology, was introduced by the British mathematician and physicist Sir Roger Penrose in an article in 1979 [[ref label #pen1979_{{{1}}}]] in an attempt to provide explanations for two of the most...

Tensor classifications

The classification of tensors is a purely mathematical problem. In GR, however, certain tensors that have a physical interpretation can be classified with the different forms of the tensor usually corresponding to some physics. Examples of tensor classifications useful in general relativity include the Segre classification of the energy-momentum tensor and the Petrov classification of the Weyl tensor. There are various methods of classifying these tensors, some of which use tensor invariants. The Segre classification is an algebraic classification of rank two symmetric tensors. ... The stress tensor or energy-momentum tensor is the corresponding conserved Noether current of any theory which is invariant under spacetime translations. ... The Petrov classsification provides a means of algebraically classifying the Weyl tensor. ... In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is the traceless component of the Riemann curvature tensor. ...

Tensor fields in GR

Tensor fields on a manifold are maps which attach a tensor to each point of the manifold. This notion can be made more precise by introducing the idea of a fibre bundle, which in the present context means to collect together all the tensors at all points of the manifold, thus 'bundling' them all into one grand object called the tensor bundle. A tensor field is then defined as a map from the manifold to the tensor bundle, each point

p

being associated with a tensor at

p

. In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. ...

The notion of a tensor field is of major importance in GR. For example, the geometry around a star is described by a metric tensor at each point, so at each point of the spacetime the value of the metric should be given to solve for the paths of material particles. Another example is the values of the electric and magnetic fields (given by the electromagnetic field tensor) and the metric at each point around a charged black hole to determine the motion of a charged particle in such a field. STAR is an acronym for: Organizations Society of Ticket Agents and Retailers], the self-regulatory body for the entertainment ticket industry in the UK. Society for Telescopy, Astronomy, and Radio, a non-profit New Jersey astronomy club. ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... Simulated view of a black hole in front of the Milky Way. ...

Vector fields are contravariant rank one tensor fields. Important vector fields in relativity include the four-velocity, $U^a = dot{x}^a$ , which is the coordinate distance travelled per unit of proper time, the four-acceleration $A^a = ddot{x}^a$ and the four-current $, J^a$ describing the charge and current densities. Other physically important tensor fields in relativity include the following: Two-dimensional analogy of space-time curvature described in General Relativity. ... In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector (vector in four-dimensional spacetime) that replaces classical velocity (a three-dimensional vector). ... In special relativity, four-acceleration is a four-vector and is defined as the change in four-velocity over the particles proper time: where and and is the Lorentz factor for the speed . ...

Although the word 'tensor' refers to an object at a point, it is common practice to refer to tensor fields on a spacetime (or a region of it) as just 'tensors'. The stress tensor or energy-momentum tensor is the corresponding conserved Noether current of any theory which is invariant under spacetime translations. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...

At each point of a spacetime on which a metric is defined, the metric can be reduced to the Minkowski form (by Sylvester's Law of Inertia). In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... In linear algebra, Sylvesters law of inertia states that the inertia of a symmetric matrix A is invariant under congruence transformations. ...

Tensorial derivatives

Before the advent of general relativity, changes in physical processes were generally described by partial derivatives, for example, in describing changes in electromagnetic fields (see Maxwell's equations). Even in special relativity, the partial derivative is still sufficient to describe such changes. However, in general relativity, it is found that derivatives which are also tensors must be used. The derivatives have some common features including that they are derivatives along integral curves of vector fields. 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). ... The electromagnetic field is a physical field that is produced by electrically charged objects and which affects the behaviour of charged objects in the vicinity of the field. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... 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... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ...

The problem in defining derivatives on manifolds that are not flat is that there is no natural way to compare vectors at different points. An extra structure on a general manifold is required to define derivatives. Below are described two important derivatives that can be defined by imposing an additional structure on the manifold in each case. On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

Affine connections

The curvature of a spacetime can be characterised by taking a vector at some point and parallel transporting it along a curve on the spacetime. An affine connection is a rule which describes how to legitimately move a vector along a curve on the manifold without changing its direction. An affine connection is a connection on the tangent bundle of a differentiable manifold. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... 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, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

By definition, an affine connection is a bilinear map $Gamma(TM)timesGamma(TM) rightarrow Gamma(TM)$ , where $,Gamma(TM)$ is a space of all vector fields on the spacetime. This bilinear map can be described in terms of a set of connection coefficients specifying what happens to components of basis vectors under infinitesimal parallel transport:

Despite their tempting appearance, the connection coefficients are not the components of a tensor.

Generally speaking, there are D³ independent connection coefficients at each point of spacetime. The connection is called symmetric if $Gamma^k_{ji} = Gamma^k_{ij}$ . A symmetric connection has D²(D+1)/2 coefficients.

For any curve

γ

and two points

A = γ(0)

and

B = γ(t)

on this curve, an affine connection gives rise to a map of vectors in the tangent space at A into vectors in the tangent space at B:

An important affine connection in general relativity is the Levi-Civita connection, which is a symmetric connection obtained from parallel transporting a tangent vector along a curve whilst keeping the inner product of that vector constant along the curve. The resulting connection coefficients are called Christoffel symbols and can be calculated directly from the metric. For this reason, this type of connection is often called a metric connection. In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829â€“1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ...

The covariant derivative

Let

X

be a point, $vec A$ a vector located at

X

, and $vec B$ a vector field. The idea of differentiating $vec B$ at

X

along the direction of $vec A$ in a physically meaningful way can be made sense of by choosing an affine connection and a parameterized smooth curve $gamma,(t)$ such that and $vec A = {d over dt}gamma(0)$ . The formula In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ...

$nabla _{vec A} vec B(X) = lim_{epsilon rightarrow 0} frac{Pi_{(epsilon,0,gamma)} vec B(gamma(epsilon)) - vec B(X)}{epsilon}$

for a covariant derivative of $vec B$ along $vec A$ associated with connection $,Pi$ turns out to give curve-independent results and can be used as a "physical definition" of a covariant derivative.

$nabla _{vec Y} vec X = X^a{}_{;b}Y^b frac {partial} {partial x^a} = (X^a{}_{,b}+Gamma ^a _{bc}X^c)Y^b frac {partial} {partial x^a}$

The expression in brackets, called a covariant derivative of $X$ (with respect to the connection) and denoted by $nabla vec X$ , is more often used in calculations:

$nabla vec X = X^a{}_{;b} frac {partial} {partial x^a} otimes dx^b = (X^a{}_{,b}+Gamma ^a _{bc}X^c) frac {partial} {partial x^a} otimes dx^b$

A covariant derivative of X can thus be viewed as a differential operator acting on a vector field sending it to a type (1,1) tensor ('increasing the covariant index by 1') and can be generalised to act on type (r,s) tensor fields sending them to type (r, s+1) tensor fields. Notions of parallel transport can then be defined similarly as for the case of vector fields. By definition, a covariant derivative of a scalar field is equal to the regular derivative of the field. In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ...

In the literature, there are three common methods of denoting covariant differentiation:

$D_a T^{bdots c}_{ddots e} = nabla_a T^{bdots c}_{ddots e} = T^{bdots c}_{ddots e;a}$

Many standard properties of regular partial derivatives also apply to covariant derivatives:

In General Relativity, one usually refers to "the" covariant derivative, which is the one associated with Levi-Civita affine connection. By definition, Levi-Civita connection preserves the metric under parallel transport, therefore, the covariant derivative gives zero when acting on a metric tensor (as well as its inverse). It means that we can take the (inverse) metric tensor in and out of the derivative and use it to raise and lower indices:

The Lie derivative

Another important tensorial derivative is the Lie derivative. Whereas the covariant derivative required an affine connection to allow comparison between vectors at different points, the Lie derivative uses a congruence from a vector field to achieve the same purpose. The idea of Lie dragging a function along a congruence leads to a definition of the Lie derivative, where the dragged function is compared with the value of the original function at a given point. The Lie derivative can be defined for type (r,s) tensor fields and in this respect can be viewed as a map that sends a type (r,s) to a type (r,s) tensor. In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... The role of symmetry in physics is important, for example, in simplifying solutions to many problems. ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...

The Lie derivative is usually denoted by $mathcal L_X$ , where

X

is the vector field along whose congruence the Lie derivative is taken. In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. ...

The Lie derivative of any tensor along a vector field can be expressed through the covariant derivatives of that tensor and vector field. (In fact, any derivative will work, but covariant derivative is convenient because it commutes with raising and lowering indices.) The Lie derivative of a scalar is just the directional derivative:

Higher rank objects pick up additional terms when the Lie derivative is taken. For example, the Lie derivative of a type (0,2) tensor is

One of the main uses of the Lie derivative in general relativity is in the study of spacetime symmetries where tensors or other geometrical objects are preserved. In particular, Killing symmetry (symmetry of the metric tensor under Lie dragging) occurs very often in the study of spacetimes. Using the formula above, we can write down the condition that must be satisfied for a vector field to generate a Killing symmetry:

The Riemann curvature tensor

A crucial feature of general relativity is the concept of a curved manifold. A useful way of measuring the curvature of a manifold is with an object called the Riemann (curvature) tensor. To meet Wikipedias quality standards, this article or section may require cleanup. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

This tensor measures curvature by use of an affine connection by considering the effect of parallel transporting a vector between two points along two curves. The discrepancy between the results of these two parallel transport routes is essentially quantified by the Riemann tensor. An affine connection is a connection on the tangent bundle of a differentiable manifold. ... 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 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. ...

This property of the Riemann tensor can be used to describe how initially parallel geodesics diverge. This is expressed by the equation of geodesic deviation and means that the tidal forces experienced in a gravitational field are a result of the curvature of spacetime. In differential geometry, the geodesic deviation equation is an equation involving the Riemann curvature tensor, which measures the change in separation of neighbouring geodesics. ... Comet Shoemaker-Levy 9 after breaking up under the influence of Jupiters tidal forces. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ...

Using the above procedure, the Riemann tensor is defined as a type (1,3) tensor and when fully written out explicitly contains the Christoffel symbols and its first partial derivatives. The Riemann tensor has 20 independent components. The vanishing of all these components over a region indicates that the spacetime is flat in that region. From the viewpoint of geodesic deviation, this means that initially parallel geodesics in that region of spacetime will stay parallel. In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829â€“1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... In physics, and specifically general relativity, geodesics are the world lines of a particle free from all external force. ...

The Riemann tensor has a number of properties sometimes referred to as the symmetries of the Riemann tensor. Of particular relevance to general relativity are the algebraic and differential Bianchi identities. 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. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

The connection and curvature of any Riemannian manifold are closely related, the theory of holonomy groups, which are formed by taking linear maps defined by parallel transport around curves on the manifold, providing a description of this relationship. In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...

The energy-momentum tensor

The sources of any gravitational field (matter and energy) are represented in relativity by a type (0,2) symmetric tensor called the energy-momentum tensor. It is closely related to the Ricci tensor. Being a second rank tensor in four dimensions, the energy-momentum tensor may be viewed as a 4 by 4 matrix. The various admissible matrix types, called Jordan forms cannot all occur, as the energy conditions that the energy-momentum tensor is forced to satisfy rule out certain forms. To meet Wikipedias quality standards, this article or section may require cleanup. ... The stress tensor or energy-momentum tensor is the corresponding conserved Noether current of any theory which is invariant under spacetime translations. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K, to what extent M can be simplified into a standard... The energy conditions refer to various constraints which can be imposed on a spacetime that any physically reasonable matter distributions in physics are expected to satisfy. ...

Energy conservation

In GR, there is a local law for the conservation of energy-momentum. It can be succinctly expressed by the tensor equation:

The corresponding statement of local energy conservation in special relativity is: 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...

This illustrates the rule of thumb that 'partial derivatives go to covariant derivatives'.

The Einstein field equations

The Einstein field equations (EFE) are the core of general relativity theory. The EFE describe how mass and energy (as represented in the stress-energy tensor) are related to the curvature of space-time (as represented in the Einstein tensor). In abstract index notation, the EFE reads as follows: This article or section is in need of attention from an expert on the subject. ... This article or section is in need of attention from an expert on the subject. ... This article is in need of attention from an expert on the subject. ... Definition In differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds. ... Abstract index notation - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...

where

G a b

is the Einstein tensor,

Λ

is the cosmological constant,

c

is the speed of light in a vacuum and

G

is the gravitational constant, which comes from Newton's law of universal gravitation. Definition In differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds. ... The cosmological constant (usually denoted by the Greek capital letter lambda: Î›) was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe. ... A line showing the speed of light on a scale model of Earth and the Moon 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.[1] It is the speed of all electromagnetic... According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. ... Isaac Newtons theory of universal gravitation (part of classical mechanics) states the following: Every single point mass attracts every other point mass by a force pointing along the line combining the two. ...

The solutions of the EFE are metric tensors. The EFE, being non-linear differential equations for the metric, are often difficult to solve. There are a number of strategies used to solve them. For example, one strategy is to start with an ansatz (or an educated guess) of the final metric, and refine it until it is specific enough to support a coordinate system but still general enough to yield a set of simultaneous differential equations with unknowns that can be solved for. Metric tensors resulting from cases where the resultant differential equations can be solved exactly for a physically reasonable distribution of energy-momentum are called exact solutions. Examples of important exact solutions include the Schwarzschild solution and the Friedman-Lemaître-Robertson-Walker solution. Ansatz (Ger. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Strictly speaking, any Lorentz metric is a solution of the Einstein field equation, as this amounts to nothing more than a mathematical definition of the energy-momentum tensor (by the field equations). ... Introduction In Einsteins theory of general relativity, the Schwarzschild metric is the most general static, spherically symmetric solution of the vacuum field equations. ... The Friedmann-LemaÃ®tre-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. ...

The EIH approximation plus other references (e.g. Geroch and Jang, 1975 - 'Motion of a body in general relativity', JMP, Vol. 16 Issue 1).

The geodesic equations

Once the EFE are solved to obtain a metric, it remains to determine the motion of inertial objects in the spacetime. In general relativity, it is assumed that inertial motion occurs along timelike and null geodesics of spacetime as parameterized by proper time. Geodesics are curves that parallel transport their own tangent vector $vec U$ , i.e. . This condition - the geodesic equation - can be written in terms of a coordinate system

x a

with the tangent vector : In physics, and specifically general relativity, geodesics are the world lines of a particle free from all external force. ... In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... 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, a geodesic is a generalization of the notion of a straight line to curved spaces. ...

where $dot{} = d/dtau$ , τ parametrises proper time along the curve and the presence of the Christoffel symbols is made manifest. In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. ... In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829â€“1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

A principal feature of general relativity is to determine the paths of particles and radiation in gravitational fields. This is accomplished by solving the geodesic equations. This article is in need of attention from an expert on the subject. ...

The EFE relate the total matter (energy) distribution to the curvature of spacetime. Their nonlinearity leads to a problem in determining the precise motion of matter in the resultant spacetime. For example, in a system composed of one planet orbiting a star, the motion of the planet is determined by solving the field equations with the energy-momentum tensor the sum of that for the planet and the star. The gravitational field of the planet affects the total spacetime geometry and hence the motion of objects. It is therefore reasonable to suppose that the field equations can be used to derive the geodesic equations. In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... STAR is an acronym for: Organizations Society of Ticket Agents and Retailers], the self-regulatory body for the entertainment ticket industry in the UK. Society for Telescopy, Astronomy, and Radio, a non-profit New Jersey astronomy club. ... The eight planets and three dwarf planets of the Solar System. ... The gravitational field is a field (physics), generated by massive objects, that determines the magnitude and direction of gravitation experienced by other massive objects. ...

When the energy-momentum tensor for a system is that of dust, it may be shown by using the local conservation law for the energy-momentum tensor that the geodesic equations are satisfied exactly. In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame energy density Ï and isotropic pressure p. ...

Lagrangian formulation

The issue of deriving the equations of motion or the field equations in any physical theory is considered by many researchers to be appealing. A fairly universal way of performing these derivations is by using the techniques of variational calculus, the main objects used in this being Lagrangians. To meet Wikipedias quality standards, this article or section may require cleanup. ... Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

Many consider this approach to be an elegant way of constructing a theory, others as merely a formal way of expressing a theory (usually, the Lagrangian construction is performed after the theory has been developed).

Mathematical techniques for analysing spacetimes

Having outlined the basic mathematical structures used in formulating the theory, some important mathematical techniques that are employed in investigating spacetimes will now be discussed.

Frame fields

A frame field is an orthonormal set of 4 vector fields (1 timelike, 3 spacelike) defined on a spacetime. Each frame field can be thought of as representing an observer in the spacetime moving along the integral curves of the timelike vector field. Every tensor quantity can be expressed in terms of a frame field, in particular, the metric tensor takes on a particularly convenient form. When allied with coframe fields, frame fields provide a powerful tool for analysing spacetimes and physically interpreting the mathematical results. In general relativity, a frame field (also called a tetrad or vierbein) is an orthonormal set of four vector fields, one timelike and three spacelike, defined on a Lorentzian manifold which is physically interpreted as a model of spacetime. ... In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... In Riemannian geometry and pseudo-Riemannian geometry, a coframe field is a set of pairwise orthogonal unit covector fields. ...

Symmetry vector fields

Some modern techniques in analysing spacetimes rely heavily on using spacetime symmetries, which are vector fields (usually defined locally) on a spacetime that preserve some feature of the spacetime. The most common type of such symmetry vector fields include Killing vector fields (which preserve the metric structure) and their generalisations called generalised Killing vector fields. Symmetry vector fields find extensive application in the study of exact solutions in general relativity and the set of all such vector fields usually forms a finite-dimensional Lie algebra. The role of symmetry in physics is important, for example, in simplifying solutions to many problems. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ... In mathematics, a Killing vector field is a vector field on a Riemannian manifold that preserves the metric. ... In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

The Cauchy problem

The Cauchy problem (sometimes called the initial value problem) is the attempt at finding a solution to a differential equation given initial conditions. In the context of general relativity, it means the problem of finding solutions to Einstein's field equations - a system of hyperbolic partial differential equations - given some initial data on a hypersurface. Studying the Cauchy problem allows one to formulate the concept of causality in general relativity, as well as 'parametrising' solutions of the field equations. Ideally, one desires global solutions, but usually local solutions are the best that can be hoped for. Consider a smooth hypersurface having a continuous, non-tangential direction field described by unitary vectors , i. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... A hyperbolic partial differential equation is usually a second-order partial differential equation of the form with . ...

Spinor formalism

Spinors find several important applications in relativity. Their use as a method of analysing spacetimes using tetrads, in particular, in the Newman-Penrose formalism is important. To meet Wikipedias quality standards, this article or section may require cleanup. ... This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. ... The Newman-Penrose Formalism is a set of notation developed by Ezra T. Newman and Roger Penrose[1] for General Relativity. ...

Another appealing feature of spinors in general relativity is the condense way in which some tensor equations may be written using the spinor formalism. For example, in classifying the Weyl tensor, determining the various Petrov types becomes much easier when compared with the tensorial counterpart. An illustration of a rotating black hole at the center of a galaxy General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... The Petrov classsification provides a means of algebraically classifying the Weyl tensor. ...

Regge calculus

Regge calculus is a formalism which chops up a Lorentzian manifold into discrete 'chunks' (four-dimensional simplicial blocks) and the block edge lengths are taken as the basic variables. A discrete version of the Einstein-Hilbert action is obtained by considering so called 'deficit angles' of these blocks, a zero deficit angle corresponding to no curvature. This novel idea finds application in approximation methods in numerical relativity and quantum gravity, the latter using a generalisation of Regge calculus. In theoretical physics, Regge calculus is a simplified form of general relativity, introduced by the Italian theoretician Tullio Regge in the early 1960s. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen... This article is in need of attention from an expert on the subject. ... This article does not cite any references or sources. ...

Singularity theorems

In general relativity, a new idea burst forth in physics with the realisation that under fairly generic conditions, gravitational collapse will inevitably result in a so-called singularity. This article or section is in need of attention from an expert on the subject. ... A gravitational singularity (sometimes spacetime singularity) is, approximately, a place where quantities which are used to measure the gravitational field become infinite. ...

Numerical relativity

Perturbation methods

The nonlinearity of the Einstein field equations often leads one to consider approximation methods in solving them. For example, an important approach is to linearise the field equations. Techniques from perturbation theory find ample application in such areas. This article or section is in need of attention from an expert on the subject. ... This article is in need of attention from an expert on the subject. ... Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ...

Notes

^[1] The defining feature (central physical idea) of general relativity is that matter and energy cause the surrounding spacetime geometry to be curved.

External links

Video lecture on the Mathematics of General Relativity (Part 1) by Kip Thorne, the Feynman Physics Professor of Caltech
Video lecture on the Mathematics of General Relativity (Part 2) by Kip Thorne, the Feynman Physics Professor of Caltech

References

[1] Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0-517-02961-8.

[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.

v • d • e General subfields within physics
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Categories: Articles with sections needing expansion | Mathematical methods in general relativity | Articles with separate introductions Kip S. Thorne Professor Kip Stephen Thorne, Ph. ... Richard Feynman Richard Phillips Feynman (May 11, 1918–February 15, 1988) (surname pronounced FINE-man) was one of the most influential American physicists of the 20th century, expanding greatly the theory of quantum electrodynamics. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... California Institute of Technology The California Institute of Technology (commonly known as Caltech) is a private, coeducational university located in Pasadena, California, in the United States. ... Kip S. Thorne Professor Kip Stephen Thorne, Ph. ... Richard Feynman Richard Phillips Feynman (May 11, 1918–February 15, 1988) (surname pronounced FINE-man) was one of the most influential American physicists of the 20th century, expanding greatly the theory of quantum electrodynamics. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... California Institute of Technology The California Institute of Technology (commonly known as Caltech) is a private, coeducational university located in Pasadena, California, in the United States. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dunamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Fig. ... Two-dimensional analogy of space-time curvature described in General Relativity. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per ion) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Condensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. ... Atomic, molecular, and optical physics is the study of matter-matter and light-matter interactions on the scale of single atoms or structures containing a few atoms. ...

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