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Deriving the Schwarzschild solution

Index Deriving the Schwarzschild solution

The Schwarzschild solution describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object. [1]

32 relations: Arthur Eddington, Asymptotically flat spacetime, Atlas (topology), Birkhoff's theorem (relativity), Christoffel symbols, Coordinate system, Cosmological constant, Einstein field equations, Euler–Lagrange equation, Event horizon, Gravitational constant, Gravitational wave, Gravity, Hypersurface, Isotropic coordinates, Karl Schwarzschild, Kepler's laws of planetary motion, Kerr metric, Kruskal–Szekeres coordinates, Lagrangian mechanics, Linearized gravity, Metric signature, Minkowski space, Reissner–Nordström metric, Schwarzschild metric, Schwarzschild radius, Singularity (mathematics), Spacetime, Spherically symmetric spacetime, Static spacetime, Stationary spacetime, Tensor contraction.

Arthur Eddington

Sir Arthur Stanley Eddington (28 December 1882 – 22 November 1944) was an English astronomer, physicist, and mathematician of the early 20th century who did his greatest work in astrophysics.

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Asymptotically flat spacetime

An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski spacetime.

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Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an atlas.

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Birkhoff's theorem (relativity)

In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat.

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Christoffel symbols

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.

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Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

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Cosmological constant

In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ) is the value of the energy density of the vacuum of space.

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Einstein field equations

The Einstein field equations (EFE; also known as Einstein's equations) comprise the set of 10 equations in Albert Einstein's general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.

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Euler–Lagrange equation

In the calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.

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Event horizon

In general relativity, an event horizon is a region in spacetime beyond which events cannot affect an outside observer.

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Gravitational constant

The gravitational constant (also known as the "universal gravitational constant", the "Newtonian constant of gravitation", or the "Cavendish gravitational constant"), denoted by the letter, is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general theory of relativity.

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Gravitational wave

Gravitational waves are the disturbance in the fabric ("curvature") of spacetime generated by accelerated masses and propagate as waves outward from their source at the speed of light.

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Gravity

Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or gravitate toward) one another.

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Hypersurface

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.

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Isotropic coordinates

In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres.

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Karl Schwarzschild

Karl Schwarzschild (October 9, 1873 – May 11, 1916) was a German physicist and astronomer.

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Kepler's laws of planetary motion

In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun.

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Kerr metric

The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially-symmetric black hole with a spherical event horizon.

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Kruskal–Szekeres coordinates

In general relativity Kruskal–Szekeres coordinates, named after Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole.

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Lagrangian mechanics

Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.

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Linearized gravity

Linearized gravity is an approximation scheme in general relativity in which the nonlinear contributions from the spacetime metric are ignored, simplifying the study of many problems while still producing useful approximate results.

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Metric signature

The signature of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive and zero eigenvalues of the real symmetric matrix of the metric tensor with respect to a basis.

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Minkowski space

In mathematical physics, Minkowski space (or Minkowski spacetime) is a combining of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.

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Reissner–Nordström metric

In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M. The metric was discovered by Hans Reissner, Hermann Weyl, Gunnar Nordström and G. B. Jeffery.

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Schwarzschild metric

In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild vacuum or Schwarzschild solution) is the solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero.

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Schwarzschild radius

The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a physical parameter that shows up in the Schwarzschild solution to Einstein's field equations, corresponding to the radius defining the event horizon of a Schwarzschild black hole.

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Singularity (mathematics)

In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

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Spacetime

In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

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Spherically symmetric spacetime

A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the rotation group SO(3) and the orbits of this group are 2-spheres (ordinary 2-dimensional spheres in 3-dimensional Euclidean space).

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Static spacetime

In general relativity, a spacetime is said to be static if it does not change over time and is also irrotational.

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Stationary spacetime

In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike.

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Tensor contraction

In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual.

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References

[1] https://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution

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