Einstein-Hilbert action
Encyclopedia
The Einstein–Hilbert action (also referred to as Hilbert action
) in general relativity
is the action
that yields the Einstein's field equations through the principle of least action
. With the (+ − − −) metric signature the action is given as
where
is the determinant of the metric tensor
,
is the Ricci scalar, and 
, where 
is the Newton's gravitational constant
and
is the speed of light
in vacuum. The integral is taken over the whole spacetime
if it converges. If it does not converge,
is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the Euler-Lagrange equation
of the Einstein–Hilbert action.
The action was first proposed by David Hilbert
in 1915.
by studying symmetries of the action.
In general relativity, the action is usually assumed to be a functional
of the metric (and matter fields), and the connection
is given by the Levi-Civita connection
. The Palatini formulation of general relativity assumes the metric and connection to be independent, and varies with respect to both independently, which makes it possible to include fermionic matter fields with non-integral spin.
The Einstein equations in the presence of matter are given by adding the matter action to the Hilbert-Einstein action.
describing any matter fields appearing in the theory.
The action principle then tells us that the variation of this action with respect to the inverse metric is zero, yielding
Since this equation should hold for any variation
, it implies that
is the equation of motion
for the metric field. The right hand side of this equation is (by definition) proportional to the stress-energy tensor
,
To calculate the left hand side of the equation we need the variations of the Ricci scalar R and the determinant of the metric. These can be obtained by standard text book calculations such as the one given below, which is strongly based on the one given in .
, and then the variation of the Ricci tensor. So, the Riemann curvature tensor is defined as,
Since the Riemann curvature depends only on the Levi-Civita connection
, the variation of the Riemann tensor can be calculated as,
Now, since
is the difference of two connections, it is a tensor and we can thus calculate its covariant derivative
,
We can now cleverly observe that the expression for the variation of Riemann curvature tensor above is equal to the difference of two such terms,
We may now obtain the variation of the Ricci curvature tensor simply by contracting two indices of the variation of the Riemann tensor,
The Ricci scalar is defined as
Therefore, its variation with respect to the inverse metric
is given by
In the second line we used the previously obtained result for the variation of the Ricci curvature and the metric compatibility of the covariant derivative,
.
The last term,
,
multiplied by
becomes a total derivative
, since
and thus by Stokes' theorem
only yields a boundary term when integrated. Hence when the variation of the metric
vanishes at infinity, this term does not contribute to the variation of the action. And we thus obtain,
, the rule for differentiating a determinant, gives:
or one could transform to a coordinate system where
is diagonal and then apply the product rule to differentiate the product of factors on the main diagonal.
Using this we get
In the last equality we used the fact that:
Which follows from the requirement that the inverse of the variationed metric matrix
is 
.
Thus we conclude that
which is Einstein's field equation and
has been chosen such that the non-relativistic limit yields the usual form of Newton's gravity law
, where G is the gravitational constant
(see here for details).
Λ is included in the Lagrangian
so that the new action
yields the field equations:
Relativity priority dispute
Albert Einstein presented the theories of Special Relativity and General Relativity in groundbreaking publications that either contained no formal references to previous literature, or referred only to a small number of his predecessors for fundamental results on which he based his theories, most...
) 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 action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
that yields the Einstein's field equations through the principle of least action
Principle of least action
In physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
. With the (+ − − −) metric signature the action is given as
where
is the determinant of the metric tensorMetric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
,
is the Ricci scalar, and , where is the Newton's 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...
and
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 vacuum. The integral is taken over the whole 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...
if it converges. If it does not converge,
is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the Euler-Lagrange equationEuler-Lagrange equation
In calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation, is a differential equation whose solutions are the functions for which a given functional is stationary...
of the Einstein–Hilbert action.
The action was first proposed by David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
in 1915.
Discussion
The derivation of equations from an action has several advantages. First of all, it allows for easy unification of general relativity with other classical fields theories (such as Maxwell theory), which are also formulated in terms of an action. In the process the derivation from an action identifies a natural candidate for the source term coupling the metric to matter fields. Moreover, the action allows for the easy identification of conserved quantities through Noether's theoremNoether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
by studying symmetries of the action.
In general relativity, the action is usually assumed to be a functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
of the metric (and matter fields), and the connection
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...
is given by 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...
. The Palatini formulation of general relativity assumes the metric and connection to be independent, and varies with respect to both independently, which makes it possible to include fermionic matter fields with non-integral spin.
The Einstein equations in the presence of matter are given by adding the matter action to the Hilbert-Einstein action.
Derivation of Einstein's field equations
Suppose that the full action of the theory is given by the Einstein-Hilbert term plus a term describing any matter fields appearing in the theory.The action principle then tells us that the variation of this action with respect to the inverse metric is zero, yielding
Since this equation should hold for any variation
, it implies thatis the equation of motion
Equation of motion
Equations of motion are equations that describe the behavior of a system in terms of its motion as a function of time...
for the metric field. The right hand side of this equation is (by definition) proportional to the stress-energy tensor
Stress-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...
,
To calculate the left hand side of the equation we need the variations of the Ricci scalar R and the determinant of the metric. These can be obtained by standard text book calculations such as the one given below, which is strongly based on the one given in .
Variation of the Riemann tensor, the Ricci tensor, and the Ricci scalar
To calculate the variation of the Ricci scalar we calculate first the variation of the Riemann curvature tensorRiemann 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...
, and then the variation of the Ricci tensor. So, the Riemann curvature tensor is defined as,
Since the Riemann curvature depends only on 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...
, the variation of the Riemann tensor can be calculated as,Now, since
is the difference of two connections, it is a tensor and we can thus calculate its covariant derivativeCovariant derivative
In 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...
,
We can now cleverly observe that the expression for the variation of Riemann curvature tensor above is equal to the difference of two such terms,
We may now obtain the variation of the Ricci curvature tensor simply by contracting two indices of the variation of the Riemann tensor,
The Ricci scalar is defined as
Therefore, its variation with respect to the inverse metric
is given byIn the second line we used the previously obtained result for the variation of the Ricci curvature and the metric compatibility of the covariant derivative,
.The last term,
,multiplied by
becomes a total derivativeTotal derivative
In the mathematical field of differential calculus, the term total derivative has a number of closely related meanings.The total derivative of a function f, of several variables, e.g., t, x, y, etc., with respect to one of its input variables, e.g., t, is different from the partial derivative...
, since
and thus by Stokes' theorem
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
only yields a boundary term when integrated. Hence when the variation of the metric
vanishes at infinity, this term does not contribute to the variation of the action. And we thus obtain,Variation of the determinant
Jacobi's formulaJacobi's formula
In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A...
, the rule for differentiating a determinant, gives:
or one could transform to a coordinate system where
is diagonal and then apply the product rule to differentiate the product of factors on the main diagonal.Using this we get
In the last equality we used the fact that:
Which follows from the requirement that the inverse of the variationed metric matrix
is .Thus we conclude that
Equation of motion
Now that we have all the necessary variations at our disposal, we can insert them into the equation of motion for the metric field to obtain,which is Einstein's field equation and
has been chosen such that the non-relativistic limit yields the usual form of Newton's gravity law
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...
, where G is the gravitational constant
Gravitational 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...
(see here for details).
Cosmological constant
Sometimes, a 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...
Λ is included in the Lagrangian
Lagrangian
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...
so that the new action
yields the field equations:
See also
- Belinfante-Rosenfeld tensor
- Brans-Dicke theoryBrans-Dicke theoryIn theoretical physics, the Brans–Dicke theory of gravitation is a theoretical framework to explain gravitation. It is a well-known competitor of Einstein's more popular theory of general relativity...
(in which the constant k is replaced by a scalar field). - Einstein-Cartan theory
- f(R) gravityF(R) gravityf gravity is a type of modified gravity theory first proposed in 1970by Buchdahl as a generalisation of Einstein's General Relativity. Although it is an active field of research, there are known problems with the theory...
- Gibbons-Hawking-York boundary termGibbons-Hawking-York boundary termIn general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein-Hilbert action when the underlying spacetime manifold has a boundary....
- Palatini action
- TeleparallelismTeleparallelismTeleparallelism , was an attempt by Einstein to unify electromagnetism and gravity...
- Variational methods in general relativityVariational methods in general relativityVariational methods in general relativity refers to various mathematical techniques that employ the use of variational calculus in Einstein's theory of general relativity...
- Einstein-Maxwell-Dirac equationsEinstein-Maxwell-Dirac equationsEinstein-Maxwell-Dirac equations are related to quantum field theory. The current Big Bang Model is a quantum field theory in a curved spacetime. Unfortunately, no such theory is mathematically well-defined; in spite of this, theoreticians claim to extract information from this hypothetical theory...