Variational methods in general relativity
Encyclopedia
Variational methods in general relativity refers to various mathematical techniques that employ the use of variational calculus in Einstein
's theory of general relativity
. The most commonly used tools are Lagrangian
s and Hamiltonian
s and are used to derive the Einstein field equations
.
. In classical mechanics
, this object is usually of the form, 'kinetic energy − potential energy'. For more general theories, the Lagrangian is some functional, such that the formation of the Euler–Lagrange equations from it recovers the required equations.
David Hilbert
gave an early and classic formulation of the equations in Einstein's general relativity. This used the functional now called the Einstein-Hilbert action.
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 most commonly used tools are 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...
s and Hamiltonian
Hamiltonian
Hamiltonian may refer toIn mathematics :* Hamiltonian system* Hamiltonian path, in graph theory** Hamiltonian cycle, a special case of a Hamiltonian path* Hamiltonian group, in group theory* Hamiltonian...
s and are used to derive the Einstein field equations
Einstein 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...
.
Lagrangian methods
The equations of motion in physical theories can often be derived from an object called the 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...
. In classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, this object is usually of the form, 'kinetic energy − potential energy'. For more general theories, the Lagrangian is some functional, such that the formation of the Euler–Lagrange equations from it recovers the required equations.
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...
gave an early and classic formulation of the equations in Einstein's general relativity. This used the functional now called the Einstein-Hilbert action.
See also
- Palatini action
- Plebanski actionPlebanski actionGeneral relativity and supergravity in all dimensions meet each other at a common assumption:Using these assumptions one can construct an effective field theory in low energies for both...
- MacDowell-Mansouri actionMacDowell-Mansouri actionThe MacDowell–Mansouri action is a mathematical object that is used to derive Einstein's field equations of general relativity....
- Freidel-Starodubtsev action
- Mathematics of general relativityMathematics of general relativityThe 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...