The

**Einstein field equations** (

**EFE**) or

**Einstein's equations** are a set of ten

equationAn equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

s in

Albert Einstein'sAlbert 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...

general theory of relativity which describe the

fundamental interactionIn particle physics, fundamental interactions are the ways that elementary particles interact with one another...

of

gravitationGravitation, 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...

as a result of

spacetimeIn 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...

being

curvedIn mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

by

matterMatter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

and

energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

. First published by Einstein in 1915 as a tensor equation, the EFE equate spacetime

curvatureIn mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

(expressed by the

Einstein tensorIn differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...

) with the energy and

momentumIn classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

within that spacetime (expressed by the

stress-energy tensorThe 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...

).

Similar to the way that

electromagnetic fieldAn 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 are determined using

chargeIn physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...

s and

currentsElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

via

Maxwell's equationsMaxwell'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...

, the EFE are used to determine the

spacetimeIn 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...

geometry resulting from the presence of mass-energy and linear momentum, that is, they determine the

metric tensorIn 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...

of spacetime for a given arrangement of stress-energy in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of non-linear

partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s when used in this way. The solutions of the EFE are the components of the metric tensor. The

inertiaInertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

l trajectories of particles and radiation (

geodesicsIn 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...

) in the resulting geometry are then calculated using the geodesic equation.

As well as obeying local energy-momentum conservation, the EFE reduce to Newton's law of gravitation where the gravitational field is weak and velocities are much less than the

speed of lightThe 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...

.

Solution techniques for the EFE include simplifying assumptions such as

symmetrySpacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems, spacetime symmetries finding ample application in the study of exact solutions of Einstein's field...

. Special classes of

exact solutionsIn 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....

are most often studied as they model many gravitational phenomena, such as

rotating black holeA rotating black hole is a black hole that possesses spin angular momentum.-Types of black holes:There are four known, exact, black hole solutions to Einstein's equations, which describe gravity in General Relativity. Two of these rotate...

s and the

expanding universeThe metric expansion of space is the increase of distance between distant parts of the universe with time. It is an intrinsic expansion—that is, it is defined by the relative separation of parts of the universe and not by motion "outward" into preexisting space...

. Further simplification is achieved in approximating the actual spacetime as

flat spacetimeIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

with a small deviation, leading to the linearised EFE. These equations are used to study phenomena such as gravitational waves.

## Mathematical form

The Einstein field equations (EFE) may be written in the form:

where

is the Ricci curvature tensor,

the

scalar curvatureIn Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...

,

the

metric tensorIn 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...

,

is the

cosmological constantIn 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

Newton's gravitational constantThe 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...

,

the

speed of lightThe 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 vacuum, and

the

stress-energy tensorThe 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...

.

The EFE is a

tensorTensors 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...

equation relating a set of

symmetric 4 x 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four

gauge fixingIn the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...

degrees of freedomDegrees of freedom can mean:* Degrees of freedom , independent displacements and/or rotations that specify the orientation of the body or system...

, which correspond to the freedom to choose a coordinate system.

Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in

*n* dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T is identically zero) define

Einstein manifoldIn differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor is proportional to the metric...

s.

Despite the simple appearance of the equations they are, in fact, quite complicated. Given a specified distribution of matter and energy in the form of a stress-energy tensor, the EFE are understood to be equations for the metric tensor

, as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. In fact, when fully written out, the EFE are a system of 10 coupled, nonlinear, hyperbolic-elliptic

partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s.

One can write the EFE in a more compact form by defining the

Einstein tensorIn differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...

which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as

Using geometrized units where

*G* =

*c* = 1, this can be rewritten as

The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how matter/energy determines the curvature of spacetime.

These equations, together with the

geodesic equationIn 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...

, which dictates how freely-falling matter moves through space-time, form the core of the

mathematical formulationThe 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...

of

general relativityGeneral 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...

.

### Sign convention

The above form of the EFE is the standard established by

Misner, Thorne, and WheelerIn physics, Gravitation is a very important reference book on Einstein's theory of gravity by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Often considered the "Bible" of General Relativity by researchers for its prominence. It is frequently called MTW after its authors' initials....

. The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):

The third sign above is related to the choice of convention for the Ricci tensor:

With these definitions

Misner, Thorne, and WheelerIn physics, Gravitation is a very important reference book on Einstein's theory of gravity by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Often considered the "Bible" of General Relativity by researchers for its prominence. It is frequently called MTW after its authors' initials....

classify themselves as

, whereas Weinberg (1972) is

, Peebles (1980) and Efstathiou (1990) are

while Peacock (1994), Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) are

.

Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative

The sign of the (very small) cosmological term would change in both these versions, if the +--- metric

sign conventionIn physics, a sign convention is a choice of the physical significance of signs for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of...

is used rather than the MTW −+++ metric sign convention adopted here.

### Equivalent formulations

Taking the

traceIn linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

of both sides of the EFE one gets

which simplifies to

If one adds

times this to the EFE, one gets the following equivalent "trace-reversed" form

Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace

in the expression on the right with the Minkowski metric without significant loss of accuracy).

## The cosmological constant

Einstein modified his original field equations to include a cosmological term proportional to the

metricIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

The constant

is the

cosmological constantIn 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...

. Since

is constant, the energy conservation law is unaffected.

The cosmological constant term was originally introduced by Einstein to allow for a static universe (i.e., one that is not expanding or contracting). This effort was unsuccessful for two reasons: the static universe described by this theory was unstable, and observations of distant galaxies by

HubbleEdwin Powell Hubble was an American astronomer who profoundly changed the understanding of the universe by confirming the existence of galaxies other than the Milky Way - our own galaxy...

a decade later confirmed that our universe is, in fact, not static but expanding. So

was abandoned, with Einstein calling it the "biggest blunder [he] ever made". For many years the cosmological constant was almost universally considered to be 0.

Despite Einstein's misguided motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations. Indeed, recent improved

astronomicalAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

techniques have found that a positive value of

is needed to explain the

accelerating universeThe accelerating universe is the observation that the universe appears to be expanding at an increasing rate, which in formal terms means that the cosmic scale factor a has a positive second derivative, implying that the velocity at which a given galaxy is receding from us should be continually...

.

Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side, written as part of the stress-energy tensor:

The resulting

vacuum energyVacuum energy is an underlying background energy that exists in space even when the space is devoid of matter . The concept of vacuum energy has been deduced from the concept of virtual particles, which is itself derived from the energy-time uncertainty principle...

is constant and given by

The existence of a cosmological constant is thus equivalent to the existence of a non-zero vacuum energy. The terms are now used interchangeably in general relativity.

### Conservation of energy and momentum

General relativity is consistent with the local conservation of energy and momentum expressed as

.

which expresses the local conservation of stress-energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition.

### Nonlinearity

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example,

Maxwell's equationsMaxwell'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...

of

electromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

are linear in the

electricIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

and

magnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

s, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation of

quantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

which is linear in the

wavefunctionNot to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

.

### The correspondence principle

The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation. In fact, the constant appearing in the EFE is determined by making these two approximations.

## Vacuum field equations

If the energy-momentum tensor

is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting

in the trace -reversed field equations, the vacuum equations can be written as

In the case of nonzero cosmological constant, the equations are

The solutions to the vacuum field equations are called

vacuum solutionsIn general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress-energy tensor also vanishes identically, so that no matter or non-gravitational fields are present.More generally, a...

. Flat

Minkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.

ManifoldIn 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 with a vanishing Ricci tensor,

, are referred to as

Ricci-flat manifoldIn mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes. In physics, they represent vacuum solutions to the analogues of Einstein's equations for Riemannian manifolds of any dimension, with vanishing cosmological constant...

s and manifolds with a Ricci tensor proportional to the metric as

Einstein manifoldIn differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor is proportional to the metric...

s.

## Einstein–Maxwell equations

If the energy-momentum tensor

is that of an

electromagnetic fieldAn 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...

in free space, i.e. if the electromagnetic stress-energy tensor

is used, then the Einstein field equations are called the

*Einstein–Maxwell equations* (with

cosmological constantIn 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...

Λ, taken to be zero in conventional relativity theory):

Additionally, the covariant Maxwell Equations are also applicable in free space:

where the semicolon represents a

covariant derivativeIn mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...

, and the brackets denote anti-symmetrization. The first equation asserts that the 4-

divergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

of the

two-formIn linear algebra, a two-form is another term for a bilinear form, typically used in informal discussions, or sometimes to indicate that the bilinear form is skew-symmetric....

*F* is zero, and the second that its

exterior derivativeIn differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....

is zero. From the latter, it follows by the Poincaré lemma that in a coordinate chart it is possible to introduce an electromagnetic field potential

*A*_{α} such that

in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived. However, there are global solutions of the equation which may lack a globally defined potential.

## Solutions

The solutions of the Einstein field equations are

metricsIn 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...

of

spacetimeIn 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...

. The solutions are hence often called 'metrics'. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as post-Newtonian approximations. Even so, there are numerous cases where the field equations have been solved completely, and those are called exact solutions.

The study of exact solutions of Einstein's field equations is one of the activities of

cosmologyPhysical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

. It leads to the prediction of

black holeA 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...

s and to different models of evolution of the

universeThe Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...

.

## The linearised EFE

The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the

gravitational fieldThe 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...

is very weak and the

spacetimeIn 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...

approximates that of

Minkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the Minkowski metric. This linearisation procedure can be used to discuss the phenomena of gravitational radiation.

## See also

- Einstein–Hilbert action
- Exact solutions of Einstein's field equations
- Equivalence principle
In the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...

- 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...

- General relativity resources
- History of general relativity
-Overview:General relativity is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915, with contributions by many others after 1915...

- Mathematics of general relativity
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...

- Solutions of the Einstein field equations
Where appropriate, this article will use the abstract index notation.Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations of general relativity. Solving the field equations actually gives Lorentz metrics...

## External links