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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 Lagrangians and Hamiltonians and are used to derive the Einstein field equations.
The equations of motion in physical theories can often be derived from an object called the Lagrangian. In classical mechanics, this object is usually of the form, 'kinetic energy − potential energy'. In general, the Lagrangian is that function which when integrated over produces the Action functional.
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.
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference.
Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the previous theories, or new theories based on the older paradigm, will often be referred to as belonging to the area of "classical physics".
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics.
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the action of a mechanical system, yields the equations of motion for that system. The principle states that the trajectories are stationary points of the system's action functional. The term "least action" is a historical misnomer since the principle has no minimality requirement: the value of the action functional need not be minimal on the trajectories. Least action refers to the absolute value of the action functional being minimized.
In physics, action is a numerical value describing how a physical system has changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action.
In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation similar to general relativity. The theory was first proposed by Élie Cartan in 1922. Einstein–Cartan theory is the simplest Poincaré gauge theory.
In general relativity and gravitation the Palatini variation is nowadays thought of as a variation of a Lagrangian with respect to the connection.
Since the 19th century, some physicists, notably Albert Einstein, have attempted to develop a single theoretical framework that can account for all the fundamental forces of nature – a unified field theory. Classical unified field theories are attempts to create a unified field theory based on classical physics. In particular, unification of gravitation and electromagnetism was actively pursued by several physicists and mathematicians in the years between the two World Wars. This work spurred the purely mathematical development of differential geometry.
In theoretical physics, the Brans–Dicke theory of gravitation is a theoretical framework to explain gravitation. It is a competitor to Einstein's theory of general relativity. It is an example of a scalar–tensor theory, a gravitational theory in which the gravitational interaction is mediated by a scalar field as well as the tensor field of general relativity. The gravitational constant G is not presumed to be constant but instead 1/G is replaced by a scalar field which can vary from place to place and with time.
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically meant to describe electromagnetism and gravitation, two of the fundamental forces of nature.
When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are used. 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.
General relativity is a theory of gravitation that was developed by Albert Einstein between 1907 and 1915, with contributions by many others after 1915. According to general relativity, the observed gravitational attraction between masses results from the warping of space and time by those masses.
In theoretical physics, a scalar–tensor theory is a field theory that includes both a scalar field and a tensor field to represent a certain interaction. For example, the Brans–Dicke theory of gravitation uses both a scalar field and a tensor field to mediate the gravitational interaction.
The ADM formalism is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was first published in 1959.
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, the Lagrangian, which may contain all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories.
The mathematics of general relativity is complex. In Newton's theories of motion, an object's length and the rate at which time passes remain constant while the object accelerates, meaning that many problems in Newtonian mechanics may be solved by algebra alone. In relativity, however, an object's length and the rate at which time passes both change appreciably as the object's speed approaches the speed of light, meaning that more variables and more complicated mathematics are required to calculate the object's motion. As a result, relativity requires the use of concepts such as vectors, tensors, pseudotensors and curvilinear coordinates.
Gauge theory gravity (GTG) is a theory of gravitation cast in the mathematical language of geometric algebra. To those familiar with general relativity, it is highly reminiscent of the tetrad formalism although there are significant conceptual differences. Most notably, the background in GTG is flat, Minkowski spacetime. The equivalence principle is not assumed, but instead follows from the fact that the gauge covariant derivative is minimally coupled. As in general relativity, equations structurally identical to the Einstein field equations are derivable from a variational principle. A spin tensor can also be supported in a manner similar to Einstein–Cartan–Sciama–Kibble theory. GTG was first proposed by Lasenby, Doran, and Gull in 1998 as a fulfillment of partial results presented in 1993. The theory has not been widely adopted by the rest of the physics community, who have mostly opted for differential geometry approaches like that of the related gauge gravitation theory.
In general relativity, the Hamilton–Jacobi–Einstein equation (HJEE) or Einstein–Hamilton–Jacobi equation (EHJE) is an equation in the Hamiltonian formulation of geometrodynamics in superspace, cast in the "geometrodynamics era" around the 1960s, by Asher Peres in 1962 and others. It is an attempt to reformulate general relativity in such a way that it resembles quantum theory within a semiclassical approximation, much like the correspondence between quantum mechanics and classical mechanics.
Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
Albert Einstein presented the theories of special relativity and general relativity in 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 notably to the work of Henri Poincaré and Hendrik Lorentz for special relativity, and to the work of David Hilbert, Carl F. Gauss, Bernhard Riemann, and Ernst Mach for general relativity. Subsequently, claims have been put forward about both theories, asserting that they were formulated, either wholly or in part, by others before Einstein. At issue is the extent to which Einstein and various other individuals should be credited for the formulation of these theories, based on priority considerations.
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