Introduction to the mathematics of general relativity: Difference between revisions

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{{General relativity sidebar}}

The '''mathematics of general relativity''' are complex. In [[Isaac Newton|Newton]]'s theories of motion, an object's length and the rate at which time passes remain constant while the object [[Acceleration|accelerates]], meaning that many problems in [[Classical mechanics|Newtonian mechanics]] may be solved by [[algebra]] alone. In [[Theory of relativity|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 [[Vector space|vectorvectors]]s, [[tensor]]s, [[pseudotensor]]s and [[curvilinear coordinates]].

For an introduction based on the example of particles following [[circular orbit]]s about a large mass, nonrelativistic and relativistic treatments are given in, respectively, [[Newtonian motivations for general relativity]] and [[Theoretical motivation for general relativity]].

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=== Coordinate transformation ===

<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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In physics, as well as mathematics, a vector is often identified with a [[tuple]], or list of numbers, which depend on some auxiliary coordinate system or [[frame of reference|reference frame]]. When the coordinates are transformed, for example by rotation or stretching of the coordinate system, then the components of the vector also transform. The vector itself has not changed, but the reference frame has, so the components of the vector (or measurements taken with respect to the reference frame) must change to compensate.

The vector is called [[Covariance and contravariance of vectors|''covariant'' or ''contravariant'']] depending on how the transformation of the vector's components is related to the transformation of coordinates.

* Contravariant vectors have units of distance (such as a displacement) or distance times some other unit (such as velocity or acceleration) and transform in the opposite way as the coordinate system. For example, in changing units from meters to millimeters the coordinate units get smaller, but the numbers in a vector become larger: 1&nbsp;m becomes 1000&nbsp;mm.

* Covariant vectors, on the other hand, have units of one-over-distance (such as a [[gradient]]) and transform in the same way as the coordinate system. For example, in changing from meters to millimeters, the coordinate units become smaller and the number measuring a gradient will also become smaller: 1&nbsp;[[Kelvin|K]]/m becomes 0.001&nbsp;K/mm.

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The Riemann tensor tells us, mathematically, how much curvature there is in any given region of space. Contracting the tensor produces 3 different mathematical objects:

# The [[Riemann curvature tensor]]: {{math|''R^ρ_σμν''}}, which gives the most information on the curvature of a space and is derived from derivatives of the [[metric tensor]]. In flat space this tensor is zero.

# The [[Ricci tensor]]: {{math|''R_σν''}}, comes from the need in Einstein's theory for a curvature tensor with only 2 indices. It is obtained by averaging certain portions of the Riemann curvature tensor.

# The [[scalar curvature]]: {{math|''R''}}, the simplest measure of curvature, assigns a single scalar value to each point in a space. It is obtained by averaging the Ricci tensor.

The Riemann curvature tensor can be expressed in terms of the covariant derivative.

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[[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 fieldfields]]s. 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]].

==Einstein equation==

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| last3=Wheeler |first3=John Archibald |authorlink3=John Archibald Wheeler

| year=1973

| title=[[Gravitation (book)|Gravitation]]

| url=

| publisher=[[W. H. Freeman]] |___location=San Francisco

| isbn=978-0-7167-0344-0

|title-link=Gravitation (book) }} Chapter 34, p 916</ref>

| postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}

}} Chapter 34, p 916</ref>

The Einstein Field Equations can be written as

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==See also==

* [[Differentiable manifold]]

* [[DifferentiableChristoffel manifoldsymbol]]

* [[ChristoffelRiemannian symbolgeometry]]

* [[RiemannianRicci geometrycalculus]]

* [[Differential geometry and topology]]

*[[Ricci calculus]]

* [[List of differential geometry topics]]

*[[Differential geometry and topology]]

* [[General relativity]]

*[[List of differential geometry topics]]

* [[Gauge gravitation theory]]

*[[General relativity]]

* [[Derivations of theGeneral Lorentzcovariant transformations]]

*[[Gauge gravitation theory]]

* [[GeneralDerivations of the covariantLorentz transformations]]

*[[Derivations of the Lorentz transformations]]

==Notes==

{{reflist}}

==References==

* {{cite book | author=P. A. M. Dirac | title=General Theory of Relativity | publisher=[[Princeton University Press]]| year=1996 | isbn=0-691-01146-X}}

* {{cite book |author1=Misner, Charles |author2=Thorne, Kip S. |author3=Wheeler, John Archibald | title=Gravitation | ___location=San Francisco | publisher=W. H. Freeman | year=1973 | isbn=0-7167-0344-0}}