General relativity

General relativity (GR) or general relativity theory (GRT) is a geometrical theory of gravitation and cosmology published by Albert Einstein in 1915. In this theory:

Overview

Fundamental principles

General relativity is based on a set of fundamental principles which guided its development. These are

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• The general principle of relativity: The laws of physics must be the same for all observers (accelerated or not).
• The principle of general covariance: The laws of physics must take the same form in all coordinate systems.
• The principle that inertial motion is geodesic motion: The world lines of particles unaffected by physical forces are timelike or null geodesics of spacetime.
• The principle of local Lorentz invariance: The laws of special relativity apply locally for all inertial observers.
• Spacetime is curved: This permits gravitational effects such as freefall to be described as a form of inertial motion. (See the discussion below of a person standing on Earth, under "Coordinate vs. physical acceleration.")
• Spacetime curvature is created by stress-energy within the spacetime: This is described in general relativity by the Einstein Field Equations.

(The equivalence principle, which was the starting point for the development of general relativity, ended up being a consequence of the general principle of relativity and the principle that inertial motion is geodesic motion.)

Related Topics:
Equivalence principle - Development of general relativity

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Spacetime as a curved Lorentzian manifold

In general relativity, the concept of spacetime (which was introduced by Hermann Minkowski for special relativity) is modified. In general relativity spacetime is

Related Topics:
Spacetime - Hermann Minkowski

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• curved: Spacetime has a non-Euclidean geometry. In special relativity, spacetime is flat.
• Lorentzian: The metrics of spacetime must have a mixed metric signature. This is inherited from special relativity.
• four dimensional: to cover the three spatial dimensions and time. This is also inherited from special relativity.

The curvature of spacetime (caused by the presence of stress-energy) can be viewed intuitively in the following way. Placing a heavy object such as a bowling ball or medicine ball on a trampoline will produce a 'dent' in the trampoline. This is analogous to a large mass such as the Earth causing the local spacetime geometry to curve. This is represented by the image at the beginning of this article. The larger the mass, the bigger the amount of curvature. A relatively light object such as a ping-pong ball placed in the vicinity of the 'dent' will accelerate towards the heavy object in an manner governed by the 'dent'. Firing the light object at just the right speed towards the 'dent' will result in the ping-pong ball 'orbiting' the heavy object. This is analogous to the Moon orbiting the Earth, for example.

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Similarly, in general relativity massive objects do not directly impart a force on other massive objects as hypothesized in Newton's action at a distance idea. Instead (in a manner analogous to how the light object responds to the dent caused by the heavy object as opposed to the heavy object itself), other massive objects respond to how the first massive object curves spacetime.

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The mathematics of general relativity

:Full article: Mathematics of general relativity

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Due to the expectation that spacetime is curved, Riemannian geometry (also known as non-Euclidean geometry) must be used. In essence, spacetime does not adhere to the "common sense" rules of Euclidean geometry, but instead objects that were initially traveling in parallel paths through spacetime (meaning that their velocities do not differ to first order in their separation) come to travel in a non-parallel fashion. This effect is called geodesic deviation, and it is used in general relativity as an alternative to gravity. For example, two people on the Earth heading due north from different positions on the equator are initially traveling on parallel paths, yet at the north pole those paths will cross. Similarly, two balls initially at rest with respect to and above the surface of the Earth (which are parallel paths by virtue of being at rest with respect to each other) come to have a converging component of relative velocity as both accelerate towards the center of the Earth due to their subsequent freefall. (Another way of looking at this is how a single ball moving in a purely timelike fashion parallel to the center of the Earth comes through geodesic motion to be moving towards the center of the Earth.)

Related Topics:
Riemannian geometry - Geodesic deviation - Equator - North pole

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The requirements of the mathematics of general relativity are further modified by the other principles. Local Lorentz Invariance requires that the manifolds described in GR be 4-dimensional and Lorentzian instead of Riemannian. In addition, the principle of general covariance forces that math to be expressed using tensor calculus. Tensor calculus permits a manifold as mapped with a coordinate system to be equipped with a metric tensor of spacetime which describes the incremental (spacetime) intervals between coordinates from which both the geodesic equations of motion and the curvature tensor of the spacetime can be ascertained.

Related Topics:
Riemannian - Tensor calculus - Map - Coordinate system - Metric tensor of spacetime - Geodesic equation - Curvature tensor

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The Einstein field equations

:Full article: Einstein field equations

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The Einstein field equations (EFE) are a tensor calculus expression which equates the curvature of spacetime with the presence of stress-energy within it. The field equations themselves can be written briefly in abstract index notation as

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: G_{ab} = kappa, T_{ab}

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where

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• $G\_\{ab\}$ is the Einstein tensor,
• $T\_\{ab\}$ is the stress-energy tensor,
• $kappa\; =\; 8\; pi\; G\; /\; c^4$ is the coupling constant of gravitation for general relativity,
• $G$ is the Newtonian gravitational constant and
• $c$ is the speed of light.

G_{ab} and T_{ab} are rank 2 symmetric tensors. This means that they are each composed of a 4×4 array of expressions which contains 10 independent terms.

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The EFE are non-linear, and often lack an exact solution. Even so, many exact solutions of the EFE are known.

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The solutions of the EFE are metrics of spacetime. These metrics describe the structure of spacetime given the stress-energy and coordinate mapping used to obtain that solution.

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The EFE reduce to Newton's law of gravity in the limiting cases of a weak gravitational field and slow speed relative to the speed of light. In fact, the value of kappa in the EFE is determined by making these two approximations.

Related Topics:
Newton's law of gravity - Weak gravitational field - Slow speed

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The EFE are the identifying feature of general relativity. Other theories built of the same premises include additional rules and/or constraints. The result almost invariably is a theory with different field equations (such as Brans-Dicke theory, Rosen bimetric theory, and Einstein-Cartan theory).

Related Topics:
Brans-Dicke theory - Rosen bimetric theory - Einstein-Cartan theory

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Coordinate vs. physical acceleration

One of the greatest sources of confusion about general relativity comes from the need to distinguish between coordinate and physical accelerations.

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In classical mechanics, space is preferentially mapped with a Cartesian coordinate system. Inertial motion then occurs as one moves through this space at a consistent coordinate rate with respect to time. Any change in this rate of progression must be due to a force, and therefore a physical and coordinate acceleration were in classical mechanics one and the same. It is important to note that in special relativity that same kind of Cartesian coordinate system was used, with time being added as a fourth dimension and defined for an observer using the Einstein synchronization procedure. As a result, physical and coordinate acceleration correspond in special relativity too, although their magnitudes may vary.

Related Topics:
Classical mechanics - Cartesian coordinate system - Special relativity - Einstein synchronization procedure

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In general relativity, the elegance of a flat spacetime and the ability to use a preferred coordinate system are lost (due to stress-energy curving spacetime and the principle of general covariance). Consequently, coordinate and physical accelerations become sundered. For example: Try using a radial coordinate system in classical mechanics. In this system, an inertially moving object which passes by (instead of through) the origin point is found to first be moving mostly inwards, then to be moving tangentially with respect to the origin, and finally to be moving outwards, yet is moving in a straight line. This is an example of an inertially moving object undergoing a coordinate acceleration, and the way this coordinate acceleration changes as the object travels are given by the geodesic equations for the manifold and coordinate system in use.

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Another more direct example is the case of someone standing on the Earth, where they are at rest with respect to the surface coordinates for the Earth (latitude, longitude, and elevation) but are undergoing a continuous physical acceleration because the mechanical resistance of the Earth's surface keeps them from free falling.

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