Mathematics of general relativity
Encyclopedia
The mathematics of general relativity refers to various mathematical
structures and techniques that are used in studying and formulating 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
. This article is a general description of the mathematics of general relativity.
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 geometries
which are not Euclidean) suggested that general relativity be formulated using the language of tensor
s. This will be discussed further below.
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.
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 this chart 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).
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.
in general relativity, whereas determining the global spacetime structure is important, especially in cosmological problems.
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
.
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 .
Mathematically, tensors are generalised linear operators - multilinear map
s. As such, the ideas of linear algebra
are employed to study tensors.
At each point
of a manifold
, the tangent
and cotangent space
s 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.
At
, these two vector space
s may be used to construct type
tensors, which are real-valued multilinear maps acting on the direct sum of 
copies of the cotangent space with 
copies of the tangent space. The set of all such multilinear maps forms a vector space, called the tensor product space of type 
at 
and denoted by 
. If the tangent space is n-dimensional, it can be shown that 
.
In the general relativity
literature, it is conventional to use the component syntax for tensors.
A type (r,s) tensor may be written as
where
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 4R, where R is the sum of the numbers of covariant and contravariant indices on the tensor (a number called the rank of the tensor).
Although a generic rank R tensor in 4 dimensions has 4R 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
satisfies Tab = Tba and possesses 10 independent components, whereas an antisymmetric (skew-symmetric) rank two tensor 
satisfies Pab = -Pba and has 6 independent components. For ranks greater than two, the symmetric or antisymmetric index pairs must be explicitly identified.
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 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
.
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
:
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:
The metric tensor is commonly written as a 4 by 4 matrix. This matrix is symmetric and thus has 10 independent components.
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:
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
.
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.
. 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
being associated with a tensor at 
.
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.
Vector fields are contravariant rank one tensor fields. Important vector fields in relativity
include the four-velocity
,
, which is the coordinate distance travelled per unit of proper time, the four-acceleration
and the four-current 
describing the charge and current densities. Other physically important tensor fields in relativity include the following:
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'.
At each point of a spacetime
on which a metric is defined, the metric can be reduced to the Minkowski form using Sylvester's Law of Inertia
.
s, for example, in describing changes in electromagnetic field
s (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 curve
s of vector fields.
The problem in defining derivatives on manifold
s 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.
can be characterised by taking a vector at some point and parallel transport
ing 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.
By definition, an affine connection is a bilinear map
, where 
is a space of all vector fields on the spacetime. This bilinear map can be described in terms of a set of connection coefficients (also known as Christoffel symbols
) specifying what happens to components of basis vectors under infinitesimal parallel transport:
Despite their 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 or torsion-free, if
. A symmetric connection has at most D2(D+1)/2 unique coefficients.
For any curve
and two points 
and 
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:
,
and
can be computed component-wise by solving the differential equation
being the vector tangent to the curve at the point 
.
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 (Christoffel symbols
) can be calculated directly from the metric. For this reason, this type of connection is often called a metric connection.
be a point, 
a vector located at 
, and 
a vector field.
The idea of differentiating
at 
along the direction of 
in a physically meaningful way can be made sense of by choosing an affine connection and a parameterized smooth curve 
such that 
and 
. The formula
for a covariant derivative of
along 
associated with connection 
turns out to give curve-independent results and can be used as a "physical definition" of a covariant derivative.
It can be expressed using connection coefficients:
The expression in brackets, called a covariant derivative of
(with respect to the connection) and denoted by 
, is more often used in calculations:
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 the literature, there are three common methods of denoting covariant differentiation:
Many standard properties of regular partial derivatives also apply to covariant derivatives:
, if c is a constant
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:
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.
The Lie derivative is usually denoted by
, where 
is the vector field along whose congruence
the Lie derivative is taken.
The Lie derivative of any tensor along a vector field can be expressed through the covariant derivatives of that tensor and vector field. 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
More generally,
In fact in the above expression, one can replace the covariant derivative
with any torsion free connection 
or locally, with the coordinate dependent derivative 
, showing that the Lie derivative is independent of the metric. The covariant derivative is convenient however because it commutes with raising and lowering indices.
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:
which is equivalent to 
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.
This tensor measures curvature by use of an affine connection
by considering the effect of parallel transport
ing 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.
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 force
s experienced in a gravitational field are a result of the curvature of spacetime
.
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 geodesic
s in that region of spacetime will stay parallel.
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.
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.
The corresponding statement of local energy conservation in special relativity
is:
This illustrates the rule of thumb that 'partial derivatives go to covariant derivatives'.
) are related to the curvature of space-time (as represented in the Einstein tensor
). In abstract index notation
, the EFE reads as follows:
where
is the Einstein tensor
,
is the cosmological constant
,
is the speed of light
in a vacuum and
is the gravitational constant
, which comes from Newton's law of universal gravitation
.
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.
The EIH approximation plus other references (e.g. Geroch and Jang, 1975 - 'Motion of a body in general relativity', JMP, Vol. 16 Issue 1).
. Geodesics are curves that parallel transport
their own tangent vector
, i.e. 
. This condition - the geodesic equation - can be written in terms of a coordinate system 
with the tangent vector 
:
where
, τ parametrises proper time
along the curve and the presence of the Christoffel symbols
is made manifest.
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
.
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.
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.
s.
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).
set of 4 vector field
s (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.
and the set of all such vector fields usually forms a finite-dimensional Lie algebra
.
(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 equation
s - 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. Typically, solving this initial value problem requires selection of particular coordinate conditions
.
s find several important applications in relativity. Their use as a method of analysing spacetimes using tetrad
s, in particular, in the Newman-Penrose formalism
is important.
Another appealing feature of spinors in general relativity
is the condensed 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.
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.
. A singularity is a point where the solutions to the equations become infinite, indicating that the theory has been probed at inappropriate ranges.
, finite element and pseudo-spectral
methods are used to approximate the solution to the partial differential equations which arise. Novel techniques developed by numerical relativity include the excision method and the puncture method for dealing with the singularities arising in black hole spacetimes. Common research topics include black holes and neutron stars.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
structures and techniques that are used in studying and formulating Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
's theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. The main tools used in this geometrical
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
theory
Theory
The English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...
of gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
are tensor fields defined on a Lorentzian manifold representing spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. This article is a general description of the mathematics of general relativity.
- Note: General relativity articles using tensors will use the abstract index notationAbstract index notationAbstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical...
.
Why tensors?
The principle of general covariance states that the laws of physicsPhysics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
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 geometries
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
which are not Euclidean) suggested that general relativity be formulated using the language of tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s. This will be discussed further below.
Spacetime as a manifold
Most modern approaches to mathematical general relativityGeneral relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
begin with the concept of a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. More precisely, the basic physical construct representing gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
- a curved spacetime - is modelled by a four-dimensional, smooth, connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
, Lorentzian manifold. Other physical descriptors are represented by various tensors, discussed below.
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 this chart can be thought of as representing the 'local spacetime' around the observer
Observation
Observation is either an activity of a living being, such as a human, consisting of receiving knowledge of the outside world through the senses, or the recording of data using scientific instruments. The term may also refer to any data collected during this activity...
(represented by the point). The principle of local Lorentz covariance, which states that the laws of special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
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
Minkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
(flat spacetime).
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 spacetimeLocal spacetime structure
Local spacetime structure refers to the structure of spacetime on a local level, i.e. only considering those points in an open region of a point. This notion is useful in many areas of physics, most notably in Einstein's theory of general relativity....
in general relativity, whereas determining the global spacetime structure is important, especially in cosmological problems.
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
Cartan-Karlhede algorithm
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 Élie Cartan using his exterior calculus with his...
.
Tensors in General Relativity
One of the profound consequences of relativity theory was the abolition of privileged 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 demonstrated that no inertial reference frame was preferential to any other inertial reference frame, but preferred inertial reference frames over noninertial reference frames. General relativity eliminated preference for inertial reference frames by showing that there is no preferred reference frame (inertial or not) for describing nature.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 .
Mathematically, tensors are generalised linear operators - multilinear map
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
s. As such, the ideas of linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
are employed to study tensors.
At each point
of a manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, the tangent
Tangent space
In mathematics, the tangent space of a manifold 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....
and cotangent space
Cotangent space
In differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...
s to the manifold at that point may be constructed. Vectors
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
(sometimes referred to as contravariant vectors
Covariance and contravariance
In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. When one coordinate system is just a rotation of the other, this...
) are defined as elements of the tangent space and covectors
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
(sometimes termed covariant vectors
Covariance and contravariance
In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis from one coordinate system to another. When one coordinate system is just a rotation of the other, this...
, but more commonly dual vectors
Dual space
In mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
or one-forms) are elements of the cotangent space.
At
, these two vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s may be used to construct type
tensors, which are real-valued multilinear maps acting on the direct sum of copies of the cotangent space with copies of the tangent space. The set of all such multilinear maps forms a vector space, called the tensor product space of type at and denoted by . If the tangent space is n-dimensional, it can be shown that .In the general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
literature, it is conventional to use the component syntax for tensors.
A type (r,s) tensor may be written as
where
is a basis for the i-th tangent space and a basis for the j-th cotangent space.As spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
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 4R, where R is the sum of the numbers of covariant and contravariant indices on the tensor (a number called the rank of the tensor).
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 4R 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
satisfies Tab = Tba and possesses 10 independent components, whereas an antisymmetric (skew-symmetric) rank two tensor satisfies Pab = -Pba and has 6 independent components. For ranks greater than two, the symmetric or antisymmetric index pairs must be explicitly identified.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 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'.The metric is a symmetric tensor and is an important mathematical tool. As well as being used to raise and lower tensor indices
Metric tensor (general relativity)
In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...
, it also generates the connections
Connection (mathematics)
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. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
which are used to construct the geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
equations of motion and the Riemann curvature tensor
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...
.
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
Line element
A line element ds in mathematics can most generally be thought of as the change in a position vector in an affine space expressing the change of the arc length. An easy way of visualizing this relationship is by parametrizing the given curve by Frenet–Serret formulas...
:
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:
The metric tensor is commonly written as a 4 by 4 matrix. This matrix is symmetric and thus 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.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:
- The Ricci scalar:
- The Kretschmann scalarKretschmann scalarIn the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. It was introduced by Erich Kretschmann.-Definition:...
:
Other examples of invariants in relativity include the electromagnetic invariants, and various other curvature invariants
Curvature invariant (general relativity)
Curvature invariants in general relativity are a set of scalars called curvature invariants that arise in general relativity. They are formed from the Riemann, Weyl and Ricci tensors - which represent curvature - and possibly operations on them such as contraction, covariant differentiation and...
, some of the latter finding application in the study of gravitational entropy and the Weyl curvature hypothesis
Weyl curvature hypothesis
The Weyl curvature hypothesis, which arises in the application of Albert Einstein's general theory of relativity to physical cosmology, was introduced by the British mathematician and theoretical physicist Sir Roger Penrose in an article in 1979 in an attempt to provide explanations for two of the...
.
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 classificationSegre classification
The Segre classification is an algebraic classification of rank two symmetric tensors. The resulting types are then known as Segre types. It is most commonly applied to the energy-momentum tensor and primarily finds application in the classification of exact solutions in general relativity....
of the energy-momentum tensor and the Petrov classification
Petrov classification
In differential geometry and theoretical physics, the Petrov classification describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold....
of the Weyl tensor
Weyl tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic...
. There are various methods of classifying these tensors, some of which use tensor invariants.
Tensor fields in General Relativity
Tensor fields on a manifold are maps which attach a tensor to each point of the manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension 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
Tensor bundle
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. To do calculus on the tensor bundle a connection is needed....
. A tensor field is then defined as a map from the manifold to the tensor bundle, each point
being associated with a tensor at .The notion of a tensor field is of major importance in GR. For example, the geometry around a star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
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
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
tensor) and the metric at each point around a charged black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
to determine the motion of a charged particle in such a field.
Vector fields are contravariant rank one tensor fields. Important vector fields in relativity
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
include the four-velocity
Four-velocity
In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector that replaces classicalvelocity...
,
, which is the coordinate distance travelled per unit of proper time, the four-accelerationFour-acceleration
In special relativity, four-acceleration is a four-vector and is defined as the change in four-velocity over the particle's proper time:whereandand \gamma_u is the Lorentz factor for the speed u...
and the four-current describing the charge and current densities. Other physically important tensor fields in relativity include the following:- The stress-energy tensor
, a symmetric rank-two tensor. - The electromagnetic field tensor
, a rank-two antisymmetric tensor.
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'.
At each point of a spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
on which a metric is defined, the metric can be reduced to the Minkowski form using Sylvester's Law of Inertia
Sylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates...
.
Tensorial derivatives
Before the advent of general relativity, changes in physical processes were generally described by partial derivativePartial derivative
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...
s, for example, in describing changes in electromagnetic field
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
s (see Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
). Even in special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, 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 curve
Integral curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...
s of vector fields.
The problem in defining derivatives on manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s 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.
Affine connections
The curvature of a spacetimeSpacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
can be characterised by taking a vector at some point and parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
ing it along a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
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.
By definition, an affine connection is a bilinear map
, where is a space of all vector fields on the spacetime. This bilinear map can be described in terms of a set of connection coefficients (also known as Christoffel symbolsChristoffel symbols
In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
) specifying what happens to components of basis vectors under infinitesimal parallel transport:
Despite their 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 or torsion-free, if
. A symmetric connection has at most D2(D+1)/2 unique coefficients.For any curve
and two points and 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:,and
can be computed component-wise by solving the differential equation being the vector tangent to the curve at the point .An important affine connection in general relativity is the Levi-Civita connection
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
, 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 (Christoffel symbols
Christoffel symbols
In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
) can be calculated directly from the metric. For this reason, this type of connection is often called a metric connection.
The covariant derivative
Let be a point, a vector located at , and a vector field.The idea of differentiating
at along the direction of in a physically meaningful way can be made sense of by choosing an affine connection and a parameterized smooth curve such that and . The formulafor a covariant derivative of
along associated with connection turns out to give curve-independent results and can be used as a "physical definition" of a covariant derivative.It can be expressed using connection coefficients:
The expression in brackets, called a covariant derivative of
(with respect to the connection) and denoted by , is more often used in calculations:A covariant derivative of X can thus be viewed as a differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
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 the literature, there are three common methods of denoting covariant differentiation:
Many standard properties of regular partial derivatives also apply to covariant derivatives:
, if c is a constantIn 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. Unlike the covariant derivative, the Lie derivative is independent of the metric, although in general relativity one usually uses an expression that seemingly depends on the metric through the affine connection. 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 draggingLie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
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.
The Lie derivative is usually denoted by
, where is the vector field along whose congruenceCongruence (general relativity)
In general relativity, a congruence is the set of integral curves of a vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime...
the Lie derivative is taken.
The Lie derivative of any tensor along a vector field can be expressed through the covariant derivatives of that tensor and vector field. 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
More generally,
In fact in the above expression, one can replace the covariant derivative
with any torsion free connection or locally, with the coordinate dependent derivative , showing that the Lie derivative is independent of the metric. The covariant derivative is convenient however because it commutes with raising and lowering indices.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:
which is equivalent to The Riemann curvature tensor
A crucial feature of general relativityGeneral relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
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.
This tensor measures curvature by use of an affine connection
Affine connection
In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
by considering the effect of parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
ing 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.
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 force
Tidal force
The tidal force is a secondary effect of the force of gravity and is responsible for the tides. It arises because the gravitational force per unit mass exerted on one body by a second body is not constant across its diameter, the side nearest to the second being more attracted by it than the side...
s experienced in a gravitational field are a result of the curvature of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
.
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
Christoffel symbols
In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
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
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
in that region. From the viewpoint of geodesic deviation, this means that initially parallel geodesic
Geodesic (general relativity)
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational, force is a particular type of geodesic...
s in that region of spacetime will stay parallel.
The Riemann tensor has a number of properties sometimes referred to as the symmetries of the Riemann tensor. Of particular relevance to general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
are the algebraic and differential Bianchi identities.
The connection and curvature of any Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
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.
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.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
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
is:
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 tensorStress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
) are related to the curvature of space-time (as represented in the Einstein tensor
Einstein tensor
In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
). In abstract index notation
Abstract index notation
Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical...
, the EFE reads as follows:
where
is the Einstein tensorEinstein tensor
In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
,
is the cosmological constantCosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
,
is the speed of lightSpeed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
in a vacuum and
is the gravitational constantGravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
, which comes from Newton's law of universal gravitation
Newton's law of universal gravitation
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...
.
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
Ansatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...
(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.
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 timeProper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
. Geodesics are curves that parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
their own tangent vector
, i.e. . This condition - the geodesic equation - can be written in terms of a coordinate system with the tangent vector :where
, τ parametrises proper timeProper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
along the curve and the presence of the Christoffel symbols
Christoffel symbols
In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
is made manifest.
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
Solving the geodesic equations
Solving the geodesic equations is a procedure used in mathematics, particularly Riemannian geometry, and in physics, particularly in general relativity, that results in obtaining geodesics. Physically, these represent the paths of particles with no proper acceleration, their motion satisfying the...
.
The EFE relate the total matter (energy) distribution to the curvature of spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. 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
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
, the motion of the planet is determined by solving the field equations with the energy-momentum tensor the sum of that for the planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
and the star. The gravitational field
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
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.
When the energy-momentum tensor for a system is that of dust
Perfect fluid
In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame energy density ρ and isotropic pressure p....
, it may be shown by using the local conservation law for the energy-momentum tensor that the geodesic equations are satisfied exactly.
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 LagrangianLagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
s.
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 orthonormalOrthonormality
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and both of unit length. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length...
set of 4 vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s (1 timelike, 3 spacelike) defined on a spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. 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
Metric tensor (general relativity)
In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...
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.
Symmetry vector fields
Some modern techniques in analysing spacetimes rely heavily on using spacetime symmetries, which are infinitesimally generated by 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 relativityExact solutions in general relativity
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....
and the set of all such vector fields usually forms a finite-dimensional Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
.
The Cauchy problem
The Cauchy problemCauchy problem
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain. Cauchy problems are an extension of initial value problems and are to be contrasted with boundary value problems...
(sometimes called the initial value problem) is the attempt at finding a solution to a differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
given initial conditions. In the context of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
, it means the problem of finding solutions to Einstein's field equations - a system of hyperbolic partial differential equation
Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...
s - 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. Typically, solving this initial value problem requires selection of particular coordinate conditions
Coordinate conditions
In general relativity, the laws of physics can be expressed in a generally covariant form. In other words, the real world does not care about our coordinate systems. However, it is often useful to fix upon a particular coordinate system, in order to solve actual problems or make actual predictions...
.
Spinor formalism
SpinorSpinor
In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
s find several important applications in relativity. Their use as a method of analysing spacetimes using tetrad
Cartan connection applications
The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. This section is an approach to tetrads, but written in general terms...
s, in particular, in the Newman-Penrose formalism
Newman-Penrose Formalism
The Newman-Penrose Formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for General Relativity. Their notation is an effort to treat General Relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR.The NP formalism is itself a...
is important.
Another appealing feature of spinors in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
is the condensed 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.
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 actionEinstein-Hilbert action
The Einstein–Hilbert action in general relativity is the action that yields the Einstein's field equations through the principle of least 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
Numerical relativity
Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars and many other phenomena governed by Einstein's Theory...
and quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
, the latter using a generalisation of Regge calculus.
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 singularityGravitational singularity
A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...
. A singularity is a point where the solutions to the equations become infinite, indicating that the theory has been probed at inappropriate ranges.
Numerical relativity
Numerical relativity is the sub-field of general relativity which seeks to solve Einstein's equations through the use of numerical methods. Finite differenceFinite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
, finite element and pseudo-spectral
Pseudo-spectral method
Pseudo-spectral methods are a class of numerical methods used in applied mathematics and scientific computing for the solution of PDEs, such as the direct simulation of a particle with an arbitrary wavefunction interacting with an arbitrary potential...
methods are used to approximate the solution to the partial differential equations which arise. Novel techniques developed by numerical relativity include the excision method and the puncture method for dealing with the singularities arising in black hole spacetimes. Common research topics include black holes and neutron stars.
Perturbation methods
The nonlinearity of the Einstein field equationsEinstein field equations
The Einstein field equations or Einstein's equations are a set of ten equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy...
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
Perturbation theory
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...
find ample application in such areas.