Introduction to the mathematics of general relativity: Difference between revisions

Revision as of 20:39, 19 April 2022

Revision as of 18:37, 15 June 2022

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===Vectors===

[[Image:Vector by Zureks.svg|right|thumb|Illustration of a typical vector.]]

In [[mathematics]], [[physics]], and [[engineering]], a '''Euclidean vector''' (sometimes called a '''geometric'''<ref>{{harvnb|Ivanov|2001}}{{Citation not found}}</ref> or '''spatial vector''',<ref>{{harvnb|Heinbockel|2001}}{{Citation not found}}</ref> or – as here – simply a vector) is a geometric object that has both a [[Magnitude (mathematics)|magnitude]] (or [[Norm (mathematics)#Euclidean norm|length]]) and direction. A vector is what is needed to "carry" the point {{math|''A''}} to the point {{math|''B''}}; the Latin word ''vector'' means "one who carries".<ref>From Latin ''vectus'', [[perfect participle]] of ''vehere'', "to carry". For historical development of the word ''vector'', see {{OED|vector ''n.''}} and {{cite web|author = Jeff Miller| url = http://jeff560.tripod.com/v.html | title = Earliest Known Uses of Some of the Words of Mathematics | access-date = 2007-05-25}}</ref> The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from {{math|''A''}} to {{math|''B''}}. Many [[algebraic operation]]s on [[real number]]s such as [[addition]], [[subtraction]], [[multiplication]], and [[negation]] have close analogues for vectors, operations which obey the familiar algebraic laws of [[Commutative property|commutativity]], [[Associative property|associativity]], and [[Distributive property|distributivity]].

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<div style="float:right; border:1px solid black; padding:3px; margin-right: 1em;

text-align:left"><gallery widths="200px" heights="200px">

Image:Transformation2polar_basis_vectors.svg|A vector {{math|'''v'''}}, is shown with two coordinate grids, {{math|''e_x''}} and {{math|''e_r''}}. In space, there is no clear coordinate grid to use. This means that the coordinate system changes based on the ___location and orientation of the observer. Observer {{math|''e_x''}} and {{math|''e_r''}} in this image are facing different directions.

Image:Transformation2polar contravariant vector.svg|Here we see that {{math|''e_x''}} and {{math|''e_r''}} see the vector differently. The direction of the vector is the same. But to {{math|''e_x''}}, the vector is moving to its left. To {{math|''e_r''}}, the vector is moving to its right.

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{{see also|Pseudotensor}}

A nontensor is a tensor-like quantity that behaves like a tensor in the raising and lowering of indices, but that does not transform like a tensor under a coordinate transformation. For example, [[Christoffel symbols]] cannot be tensors themselves if the coordinates don'tdo not change in a linear way.

In general relativity, one cannot describe the energy and momentum of the gravitational field by an energy–momentum tensor. Instead, one introduces objects that behave as tensors only with respect to restricted coordinate transformations. Strictly speaking, such objects are not tensors at all. A famous example of such a pseudotensor is the [[Landau–Lifshitz pseudotensor]].

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==Stress–energy tensor==

{{main|Stress–energy tensor}}

[[File:StressEnergyTensor contravariant.svg|right|250px|thumb|Contravariant components of the stress–energy tensor.]]

The '''stress–energy tensor''' (sometimes '''stress–energy–momentum tensor''' or '''energy–momentum tensor''') is a [[tensor]] quantity in [[physics]] that describes the [[density]] and [[flux]] of [[energy]] and [[momentum]] in [[spacetime]], generalizing the [[stress (physics)|stress tensor]] of Newtonian physics. It is an attribute of [[matter]], [[radiation]], and non-gravitational [[force field (physics)|force fields]]. The stress–energy tensor is the source of the [[gravitational field]] in the [[Einstein field equations]] of [[general relativity]], just as mass density is the source of such a field in [[Newtonian gravity]]. Because this tensor has 2 indices (see next section) the Riemann curvature tensor has to be contracted into the Ricci tensor, also with 2 indices.

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==Schwarzschild solution and black holes==

{{main|Schwarzschild metric}}

In [[Albert Einstein|Einstein]]'s theory of [[general relativity]], the '''Schwarzschild metric''' (also '''Schwarzschild vacuum''' or '''Schwarzschild solution'''), is a solution to the [[Einstein field equations]] which describes the [[gravitational field]] outside a spherical mass, on the assumption that the [[electric charge]] of the mass, the [[angular momentum]] of the mass, and the universal [[cosmological constant]] are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many [[star]]s and [[planet]]s, including Earth and the Sun. The solution is named after [[Karl Schwarzschild]], who first published the solution in 1916, just before his death.

According to [[Birkhoff's theorem (relativity)|Birkhoff's theorem]], the Schwarzschild metric is the most general [[rotational symmetry|spherically symmetric]], [[Vacuum solution (general relativity)|vacuum solution]] of the [[Einstein field equations]]. A '''Schwarzschild black hole''' or '''static black hole''' is a [[black hole]] that has no [[Charge (physics)|charge]] or [[angular momentum]]. A Schwarzschild black hole is described by the Schwarzschild metric, and cannot be distinguished from any other Schwarzschild black hole except by its mass.