# Einstein–Hilbert action

Einstein-Hilbert actionEinstein–Hilbert LagrangianEinstein-Hilbert LagrangianEinstein–HilbertEinstein–Hilbert functionalEinstein–Hilbert termgravityHilbert's actionHilbert's action principleHilbert-Einstein Lagrangian
The Einstein–Hilbert action (also referred to as [[Relativity priority dispute#Did Hilbert claim priority for parts of General Relativity?|Hilbert action]] ) in general relativity is the action that yields the Einstein field equations through the principle of least action.wikipedia
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### Principle of least action

principle of stationary actionleast action principleleast action
The Einstein–Hilbert action (also referred to as [[Relativity priority dispute#Did Hilbert claim priority for parts of General Relativity?|Hilbert action]] ) in general relativity is the action that yields the Einstein field equations through the principle of least action.
The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity (see Einstein–Hilbert action).

### Scalar curvature

Ricci scalarcurvaturecurvature scalar
where is the determinant of the metric tensor matrix, R is the Ricci scalar, and is Einstein's constant (G is the gravitational constant and c is the speed of light in vacuum).
In general relativity, the scalar curvature is the Lagrangian density for the Einstein–Hilbert action.

### Relativity priority dispute

Hilbert actionprioritythe article on this dispute
The Einstein–Hilbert action (also referred to as [[Relativity priority dispute#Did Hilbert claim priority for parts of General Relativity?|Hilbert action]] ) in general relativity is the action that yields the Einstein field equations through the principle of least action.

### David Hilbert

HilbertHilbert, DavidD. Hilbert
The action was first proposed by David Hilbert in 1915.
Nearly simultaneously David Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations (see Einstein–Hilbert action).

### Gibbons–Hawking–York boundary term

GHY boundary termsGibbons-Hawking-York boundary term
The boundary term is in general non-zero, because the integrand depends not only on but also on its partial derivatives ; see the article Gibbons–Hawking–York boundary term for details.
In 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.

### General relativity

general theory of relativitygeneral relativity theoryrelativity
The Einstein–Hilbert action (also referred to as [[Relativity priority dispute#Did Hilbert claim priority for parts of General Relativity?|Hilbert action]] ) in general relativity is the action that yields the Einstein field equations through the principle of least action.

### Palatini identity

We may now obtain the variation of the Ricci curvature tensor simply by contracting two indices of the variation of the Riemann tensor, and get the Palatini identity:
A proof can be found in the entry Einstein–Hilbert action.

### Action (physics)

actionaction principleaction integral
The Einstein–Hilbert action (also referred to as [[Relativity priority dispute#Did Hilbert claim priority for parts of General Relativity?|Hilbert action]] ) in general relativity is the action that yields the Einstein field equations through the principle of least action.
The Einstein equation utilizes the Einstein–Hilbert action as constrained by a variational principle.

### Einstein–Cartan theory

Einstein-CartanEinstein-Cartan gravityEinstein–Cartan
The theory of general relativity was originally formulated in the setting of Riemannian geometry by the Einstein-Hilbert action, out of which arise the Einstein field equations.

### Tetradic Palatini action

The Einstein–Hilbert action for general relativity was first formulated purely in terms of the space-time metric.

### Komar superpotential

In general relativity, the Komar superpotential, corresponding to the invariance of the Hilbert-Einstein Lagrangian, is the tensor density:

### Stress–energy tensor

energy–momentum tensorenergy-momentum tensorstress-energy tensor
The right hand side of this equation is (by definition) proportional to the stress-energy tensor,
See Einstein–Hilbert action for more information.

### F(R) gravity

f(R) theoryf''(''R'') modified gravityf''(''R'') gravity
In f(R) gravity, one seeks to generalize the Lagrangian of the Einstein–Hilbert action:

### Lagrangian (field theory)

LagrangianLagrangian densityLagrangian field theory
When a cosmological constant Λ is included in the Lagrangian, the action:
The integral of is known as the Einstein-Hilbert action.

### Kaluza–Klein theory

Kaluza–KleinKaluza–Klein theoriesKaluza-Klein
Next, one takes this scalar curvature as the Lagrangian density, and, from this, constructs the Einstein–Hilbert action for the bundle, as a whole.

### Einstein field equations

Einstein field equationEinstein's field equationsEinstein's field equation
The Einstein–Hilbert action (also referred to as [[Relativity priority dispute#Did Hilbert claim priority for parts of General Relativity?|Hilbert action]] ) in general relativity is the action that yields the Einstein field equations through the principle of least action.

### Variational methods in general relativity

variational approach to general relativityvariational-principle formulation
This used the functional now called the Einstein-Hilbert action.

### Metric tensor

metricmetricsround metric
where is the determinant of the metric tensor matrix, R is the Ricci scalar, and is Einstein's constant (G is the gravitational constant and c is the speed of light in vacuum).

### Einstein's constant

Einstein constantEinstein's constant of gravitation
where is the determinant of the metric tensor matrix, R is the Ricci scalar, and is Einstein's constant (G is the gravitational constant and c is the speed of light in vacuum).

### Gravitational constant

Newton's constantGuniversal gravitational constant
where is the determinant of the metric tensor matrix, R is the Ricci scalar, and is Einstein's constant (G is the gravitational constant and c is the speed of light in vacuum).

### Speed of light

clight speedspeed of light in vacuum
where is the determinant of the metric tensor matrix, R is the Ricci scalar, and is Einstein's constant (G is the gravitational constant and c is the speed of light in vacuum).

### Spacetime

space-timespace-time continuumspace and time
If it converges, the integral is taken over the whole spacetime.

### Euler–Lagrange equation

Euler–Lagrange equationsEuler-Lagrange equationLagrange's equation
If it does not converge, S 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.

### Maxwell's equations

Maxwell equationsMaxwell equationMaxwell’s equations
First of all, it allows for easy unification of general relativity with other classical field theories (such as Maxwell theory), which are also formulated in terms of an action.