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{{Introductory article|Mathematics of general relativity}}
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The '''mathematics of general relativity''' is complex. In [[Biography:Isaac Newton|Newton]]'s theories of motion, an object's length and the rate at which time passes remain constant while the object [[Physics:Acceleration|accelerates]], meaning that many problems in [[Physics:Classical mechanics|Newtonian mechanics]] may be solved by [[Algebra|algebra]] alone. In [[Physics: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 [[Physics:Speed of light|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|vectors]], [[Tensor|tensor]]s, [[Pseudotensor|pseudotensor]]s and [[Curvilinear coordinates|curvilinear coordinates]].

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

==Vectors and tensors==
{{main|Euclidean vector|Tensor}}

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

In [[Mathematics|mathematics]], [[Physics:Physics|physics]], and [[Engineering:Engineering|engineering]], a '''Euclidean vector''' (sometimes called a '''geometric'''<ref>{{harvnb|Ivanov|2001}}</ref> or '''spatial vector''',<ref>{{harvnb|Heinbockel|2001}}</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|algebraic operation]]s on [[Real number|real number]]s such as [[Addition|addition]], [[Subtraction|subtraction]], [[Multiplication|multiplication]], and [[Negation|negation]] have close analogues for vectors, operations which obey the familiar algebraic laws of [[Commutative property|commutativity]], [[Associative property|associativity]], and [[Distributive property|distributivity]].

===Tensors===
[[Image:Components stress tensor cartesian.svg|300px|right|thumb|Stress is a second-order tensor that represents the response of a material to force applied at an angle. The two directions of the tensor represent the "normal" (at right angles to the surface) force, and "shear" (parallel to the surface) force.]]

A tensor extends the concept of a vector to additional directions. A [[Scalar (mathematics)|scalar]], that is, a simple number without a direction, would be shown on a graph as a point, a zero-dimensional object. A vector, which has a magnitude and direction, would appear on a graph as a line, which is a one-dimensional object. A vector is a first-order tensor, since it holds one direction.
A second-order tensor has two magnitudes and two directions, and would appear on a graph as two lines similar to the hands of a clock. The "order" of a tensor is the number of directions contained within, which is separate from the dimensions of the individual directions. A second-order tensor in two dimensions might be represented mathematically by a 2-by-2 matrix, and in three dimensions by a 3-by-3 matrix, but in both cases the matrix is "square" for a second-order tensor. A third-order tensor has three magnitudes and directions, and would be represented by a cube of numbers, 3-by-3-by-3 for directions in three dimensions, and so on.

=== Applications ===

Vectors are fundamental in the physical sciences.  They can be used to represent any quantity that has both a magnitude and direction, such as [[Physics:Velocity|velocity]], the magnitude of which is [[Physics:Speed|speed]]. For example, the velocity ''5 meters per second upward'' could be represented by the vector {{nowrap|(0, 5)}} (in 2 dimensions with the positive {{math|''y''}} axis as 'up'). Another quantity represented by a vector is [[Physics:Force|force]], since it has a magnitude and direction. Vectors also describe many other physical quantities, such as [[Displacement (vector)|displacement]], [[Physics:Acceleration|acceleration]], [[Physics:Momentum|momentum]], and [[Physics:Angular momentum|angular momentum]].  Other physical vectors, such as the [[Physics:Electric field|electric]] and [[Physics:Magnetic field|magnetic field]], are represented as a system of vectors at each point of a physical space; that is, a [[Vector field|vector field]].

Tensors also have extensive applications in physics:
* [[Physics:Electromagnetic tensor|Electromagnetic tensor]] (or Faraday's tensor) in [[Physics:Electromagnetism|electromagnetism]]
* Finite deformation tensors for describing deformations and strain tensor for [[Physics:Strain (materials science)|strain]] in [[Physics:Continuum mechanics|continuum mechanics]]
* [[Physics:Permittivity|Permittivity]] and [[Physics:Electric susceptibility|electric susceptibility]] are tensors in anisotropic media
* [[Stress–energy tensor]] in [[Physics:General relativity|general relativity]], used to represent [[Physics:Momentum|momentum]] [[Physics:Flux|flux]]es
* Spherical tensor operators are the eigenfunctions of the quantum [[Physics:Angular momentum|angular momentum]] operator in spherical coordinates
* Diffusion tensors, the basis of diffusion tensor imaging, represent rates of diffusion in biologic environments

=== Dimensions ===

In general relativity, four-dimensional vectors, or [[Physics:Four-vector|four-vector]]s, are required. These four dimensions are length, height, width and time. A "point" in this context would be an event, as it has both a location and a time. Similar to vectors, tensors in relativity require four dimensions. One example is the [[Riemann curvature tensor]].
{{clear}}
=== 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<sub>x</sub>''}} and {{math|''e<sub>r</sub>''}}. 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<sub>x</sub>''}} and {{math|''e<sub>r</sub>''}} in this image are facing different directions.
Image:Transformation2polar contravariant vector.svg|Here we see that {{math|''e<sub>x</sub>''}} and {{math|''e<sub>r</sub>''}} see the vector differently. The direction of the vector is the same. But to {{math|''e<sub>x</sub>''}}, the vector is moving to its left. To {{math|''e<sub>r</sub>''}}, the vector is moving to its right.
</gallery></div>

In physics, as well as mathematics, a vector is often identified with a [[Tuple|tuple]], or list of numbers, which depend on a coordinate system or [[Astronomy:Frame of reference|reference frame]].  If the coordinates are transformed, such as by rotation or stretching the coordinate system, the components of the vector also transform. The vector itself does not change, but the reference frame does. This means that the components of the vector have to 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 (as in a [[Gradient|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 per m becomes 0.001&nbsp;Kelvin per mm.

In [[Einstein notation]], contravariant vectors and components of tensors are shown with superscripts, e.g. {{math|''x<sup>i</sup>''}}, and covariant vectors and components of tensors with subscripts, e.g. {{math|''x<sub>i</sub>''}}.  Indices are "raised" or "lowered" by multiplication by an appropriate matrix, often the identity matrix.

Coordinate transformation is important because relativity states that there is not one reference point (or perspective) in the universe that is more favored than another. On earth, we use dimensions like north, east, and elevation, which are used throughout the entire planet. There is no such system for space. Without a clear reference grid, it becomes more accurate to describe the four dimensions as towards/away, left/right, up/down and past/future. As an example event, assume that Earth is a motionless object, and consider the signing of the Declaration of Independence. To a modern observer on [[Earth:Mount Rainier|Mount Rainier]] looking east, the event is ahead, to the right, below, and in the past. However, to an observer in medieval England looking north, the event is behind, to the left, neither up nor down, and in the future. The event itself has not changed: the location of the observer has.

==Oblique axes==

An oblique coordinate system is one in which the axes are not necessarily [[Orthogonality|orthogonal]] to each other; that is, they meet at angles other than [[Right angle|right angle]]s. When using coordinate transformations as described above, the new coordinate system will often appear to have oblique axes compared to the old system.

==Nontensors==

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 do 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.

==Curvilinear coordinates and curved spacetime==

[[Image:Cassini-science-br.jpg|thumb|right|250px|High-precision test of general relativity by the [[Astronomy:Cassini–Huygens|Cassini]] space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) are delayed by the warping of space and time (blue lines) due to the Sun's mass. That is, the Sun's mass causes the regular grid coordinate system (in blue) to distort and have curvature. The radio wave then follows this curvature and moves toward the Sun.]]

[[Curvilinear coordinates]] are coordinates in which the angles between axes can change from point to point. This means that rather than having a grid of straight lines, the grid instead has curvature.

A good example of this is the surface of the Earth. While maps frequently portray north, south, east and west as a simple square grid, that is not in fact the case. Instead, the longitude lines running north and south are curved and meet at the north pole. This is because the Earth is not flat, but instead round.

In general relativity, energy and mass have curvature effects on the four dimensions of the universe (= spacetime). This curvature gives rise to the gravitational force. A common analogy is placing a heavy object on a stretched out rubber sheet, causing the sheet to bend downward. This curves the coordinate system around the object, much like an object in the universe curves the coordinate system it sits in. The mathematics here are conceptually more complex than on Earth, as it results in [[Four-dimensional space|four dimensions]] of curved coordinates instead of three as used to describe a curved 2D surface.

==Parallel transport==
{{main|Parallel transport}}

[[Image:parallel displacement.svg|thumb|450px|right|Example: Parallel displacement along a circle of a three-dimensional ball embedded in two dimensions. The circle of radius {{math|''r''}} is embedded in a two-dimensional space characterized by the coordinates {{math|''z''<sup>1</sup>}} and {{math|''z''<sup>2</sup>}}. The circle itself is characterized by coordinates {{math|''y''<sup>1</sup>}} and {{math|''y''<sup>2</sup>}} in the two-dimensional space. The circle itself is one-dimensional and can be characterized by its arc length {{math|''x''}}. The coordinate {{math|''y''}} is related to the coordinate {{math|''x''}} through the relation {{math|''y''<sup>1</sup> {{=}} ''r'' cos {{sfrac|''x''|''r''}}}} and {{math|''y''<sup>2</sup> {{=}} ''r'' sin {{sfrac|''x''|''r''}}}}. This gives {{math|{{sfrac|∂''y''<sup>1</sup>|∂''x''}} {{=}} −sin {{sfrac|''x''|''r''}}}} and {{math|{{sfrac|∂''y''<sup>2</sup>|∂''x''}} {{=}} cos {{sfrac|''x''|''r''}}}} In this case the metric is a scalar and is given by {{math|''g'' {{=}} cos<sup>2</sup> {{sfrac|''x''|''r''}} + sin<sup>2</sup> {{sfrac|''x''|''r''}} {{=}} 1}}. The interval is then {{math|''ds''<sup>2</sup> {{=}} ''g dx''<sup>2</sup> {{=}} ''dx''<sup>2</sup>}}. The interval is just equal to the arc length as expected.]]

===The interval in a high-dimensional space===
In a [[Euclidean space]], the separation between two points is measured by the distance between the two points. The distance is purely spatial, and is always positive. In spacetime, the separation between two events is measured by the ''invariant interval'' between the two events, which takes into account not only the spatial separation between the events, but also their separation in time. The interval, {{math|''s''<sup>2</sup>}}, between two events is defined as:

:<math>s^2 = \Delta r^2 - c^2\Delta t^2 \,</math>{{spaces|5}}(spacetime&nbsp;interval),

where {{math|''c''}} is the speed of light, and {{math|Δ''r''}} and {{math|Δ''t''}} denote differences of the space and time coordinates, respectively, between the events. The choice of signs for {{math|''s''<sup>2</sup>}} above follows the [[Sign convention#Relativity|space-like convention (−+++)]]. A notation like {{math|Δ''r''<sup>2</sup>}} means {{math|(Δ''r'')<sup>2</sup>}}. The reason {{math|''s''<sup>2</sup>}} is called the interval and not {{math|''s''}} is that {{math|''s''<sup>2</sup>}} can be positive, zero or negative.

Spacetime intervals may be classified into three distinct types, based on whether the temporal separation ({{math|''c''<sup>2</sup>Δ''t''<sup>2</sup>}}) or the spatial separation ({{math|Δ''r''<sup>2</sup>}}) of the two events is greater: time-like, light-like or space-like.

Certain types of [[World line|world line]]s are called [[Geodesic|geodesic]]s of the spacetime – straight lines in the case of flat Minkowski spacetime and their closest equivalent in the curved spacetime of general relativity. In the case of purely time-like paths, geodesics are (locally) the paths of greatest separation (spacetime interval) as measured along the path between two events, whereas in Euclidean space and Riemannian manifolds, geodesics are paths of shortest distance between two points.<ref>This characterization is not universal: both the arcs between two points of a [[Great circle|great circle]] on a sphere are geodesics.</ref><ref>{{cite book |title=Principles of Cosmology and Gravitation |first1=Michael V. |last1=Berry |publisher=CRC Press |year=1989 |isbn=0-85274-037-9 |page=58 |url=https://books.google.com/books?id=wTvFxXguod0C&pg=PA58}}</ref> The concept of geodesics becomes central in [[Physics:General relativity|general relativity]], since geodesic motion may be thought of as "pure motion" ([[Physics:Fictitious force|inertial motion]]) in spacetime, that is, free from any external influences.

===The covariant derivative===

{{main|Covariant derivative}}

The covariant derivative is a generalization of the directional derivative from vector calculus. As with the directional derivative, the covariant derivative is a rule, which takes as its inputs: (1) a vector, {{math|'''u'''}}, (along which the derivative is taken) defined at a point {{math|''P''}}, and (2) a vector field, {{math|'''v'''}}, defined in a neighborhood of {{math|''P''}}. The output is a vector, also at the point {{math|''P''}}. The primary difference from the usual directional derivative is that the covariant derivative must, in a certain precise sense, be independent of the manner in which it is expressed in a coordinate system.

===Parallel transport===

Given the covariant derivative, one can define the [[Parallel transport|parallel transport]] of a vector {{math|'''v'''}} at a point {{math|''P''}} along a curve {{math|''γ''}} starting at {{math|''P''}}. For each point {{math|''x''}} of {{math|''γ''}}, the parallel transport of {{math|'''v'''}} at {{math|''x''}} will be a function of {{math|''x''}}, and can be written as {{math|'''v'''(''x'')}}, where {{math|'''v'''(0) {{=}} '''v'''}}. The function {{math|'''v'''}} is determined by the requirement that the covariant derivative of {{math|'''v'''(''x'')}} along {{math|''γ''}} is 0. This is similar to the fact that a constant function is one whose derivative is constantly 0.

===Christoffel symbols===
{{main|Christoffel symbols}}

The equation for the covariant derivative can be written in terms of Christoffel symbols. The Christoffel symbols find frequent use in Einstein's theory of [[Physics:General relativity|general relativity]], where [[Physics:Spacetime|spacetime]] is represented by a curved 4-dimensional Lorentz manifold with a [[Levi-Civita connection]]. The [[Physics:Einstein field equations|Einstein field equations]] – which determine the geometry of spacetime in the presence of matter – contain the Ricci tensor. Since the Ricci tensor is derived from the Riemann curvature tensor, which can be written in terms of Christoffel symbols, a calculation of the Christoffel symbols is essential. Once the geometry is determined, the paths of particles and light beams are calculated by [[Solving the geodesic equations|solving the geodesic equations]] in which the Christoffel symbols explicitly appear.

==Geodesics==
{{main|Geodesics in general relativity}}

In [[Physics:General relativity|general relativity]], a '''geodesic''' generalizes the notion of a "straight line" to curved [[Physics:Spacetime|spacetime]]. Importantly, the [[World line|world line]] of a particle free from all external, non-gravitational force, is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic.

In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the [[Stress–energy tensor|stress–energy tensor]] (representing matter, for instance).  Thus, for example, the path of a planet orbiting around a star is the projection of a geodesic of the curved 4-dimensional spacetime geometry around the star onto 3-dimensional space.

A curve is a geodesic if the [[Tangent vector|tangent vector]] of the curve at any point is equal to the [[Parallel transport|parallel transport]] of the [[Tangent vector|tangent vector]] of the base point.

==Curvature tensor==
{{main|Riemann curvature tensor}}

The [[Riemann curvature tensor]] {{math|''R<sup>ρ</sup><sub>σμν</sub>''}} tells us, mathematically, how much curvature there is in any given region of space. In flat space this tensor is zero.

Contracting the tensor produces 2 more mathematical objects:

# The Ricci tensor: {{math|''R<sub>σν</sub>''}}, 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|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.

The Einstein tensor {{math|'''G'''}} is a rank-2 [[Tensor|tensor]] defined over [[Pseudo-Riemannian manifold|pseudo-Riemannian manifold]]s.  In index-free notation it is defined as

:<math>\mathbf{G}=\mathbf{R}-\tfrac12\mathbf{g}R,</math>

where {{math|'''R'''}} is the Ricci tensor, {{math|'''g'''}} is the [[Physics:Metric tensor|metric tensor]] and {{math|''R''}} is the [[Scalar curvature|scalar curvature]]. It is used in the [[Physics:Einstein field equations|Einstein field equations]].

==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|tensor]] quantity in [[Physics:Physics|physics]] that describes the [[Physics:Density|density]] and [[Physics:Flux|flux]] of [[Physics:Energy|energy]] and [[Physics:Momentum|momentum]] in [[Physics:Spacetime|spacetime]], generalizing the [[Physics:Stress|stress tensor]] of Newtonian physics. It is an attribute of [[Physics:Matter|matter]], [[Physics:Radiation|radiation]], and non-gravitational [[Physics:Force field|force fields]]. The stress–energy tensor is the source of the [[Physics:Gravitational field|gravitational field]] in the [[Physics:Einstein field equations|Einstein field equations]] of [[Physics:General relativity|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.

==Einstein equation==

{{main|Physics:Einstein field equations}}
The '''Einstein field equations''' ('''EFE''') or '''Einstein's equations''' are a set of 10 [[Equation|equation]]s in [[Biography:Albert Einstein|Albert Einstein's]] [[Physics:General relativity|general theory of relativity]] which describe the [[Physics:Fundamental interaction|fundamental interaction]] of gravitation as a result of [[Physics:Spacetime|spacetime]] being [[Curvature|curved]] by [[Physics:Matter|matter]] and [[Physics:Energy|energy]].<ref name=ein>{{cite journal|last=Einstein |first=Albert |title=The Foundation of the General Theory of Relativity |journal=[[Physics:Annalen der Physik|Annalen der Physik]] |volume=354 |issue=7 |pages=769 |year=1916 |url=http://www.alberteinstein.info/gallery/gtext3.html |doi=10.1002/andp.19163540702 |format=[[PDF]] |bibcode=1916AnP...354..769E |url-status=dead |archive-url=https://web.archive.org/web/20060829045130/http://www.alberteinstein.info/gallery/gtext3.html |archive-date=2006-08-29 }}</ref> First published by Einstein in 1915<ref name=Ein1915>{{cite journal|last=Einstein| first=Albert|  date=November 25, 1915| title=Die Feldgleichungen der Gravitation| journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin| pages=844–847 | url=http://nausikaa2.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.x.cgi?dir=6E3MAXK4&step=thumb | access-date=2006-09-12}}</ref> as a tensor equation, the EFE equate local spacetime [[Curvature|curvature]] (expressed by the [[Physics:Einstein tensor|Einstein tensor]]) with the local energy and [[Physics:Momentum|momentum]] within that spacetime (expressed by the [[Stress–energy tensor|stress–energy tensor]]).<ref>{{Cite book
 | year=1973
 | title=Gravitation
 | publisher=W. H. Freeman |location=San Francisco
 | isbn=978-0-7167-0344-0
 |title-link=Gravitation (book) }} Chapter 34, p 916</ref>

The Einstein field equations can be written as
:<math>G_{\mu \nu}= {8 \pi G \over c^4} T_{\mu \nu} ,</math>

where {{math|''G<sub>μν</sub>''}} is the [[Physics:Einstein tensor|Einstein tensor]] and {{math|''T<sub>μν</sub>''}} is the [[Stress–energy tensor|stress–energy tensor]].

This implies that the curvature of space (represented by the Einstein tensor) is directly connected to the presence of matter and energy (represented by the stress–energy tensor).

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

According to [[Physics:Birkhoff's theorem (relativity)|Birkhoff's theorem]], the Schwarzschild metric is the most general [[Rotational symmetry|spherically symmetric]], [[Physics:Vacuum solution (general relativity)|vacuum solution]] of the [[Physics:Einstein field equations|Einstein field equations]]. A '''Schwarzschild black hole''' or '''static black hole''' is a [[Astronomy:Black hole|black hole]] that has no [[Physics:Charge|charge]] or [[Physics:Angular momentum|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.

==See also==
* [[Differentiable manifold]]
* Christoffel symbol
* [[Riemannian geometry]]
* [[Ricci calculus]]
* Differential geometry and topology
* [[List of differential geometry topics]]
* [[Physics:General relativity|General relativity]]
* [[Physics:Gauge gravitation theory|Gauge gravitation theory]]
* [[General covariant transformations]]
* [[Physics:Derivations of the Lorentz 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}}
*{{Citation
 | title = Introduction to Tensor Calculus and Continuum Mechanics
 | first = J. H.
 | last = Heinbockel
 | publisher = Trafford Publishing
 | year = 2001
 | isbn = 1-55369-133-4
 | url = http://www.math.odu.edu/~jhh/counter2.html
}}.
* {{springer|id=V/v096340|title=Vector|first=A.B.|last=Ivanov}}.
* {{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}}
* {{cite book |author1=Landau, L. D.  |author2=Lifshitz, E. M.| title=Classical Theory of Fields | location=Oxford | publisher=Pergamon | year=1975 | isbn=0-08-018176-7|edition=Fourth Revised English}}
* {{cite book |author1=R. P. Feynman |author2=F. B. Moringo |author3=W. G. Wagner |title=Feynman Lectures on Gravitation |publisher=Addison-Wesley |year=1995 |isbn=0-201-62734-5 |url-access=registration |url=https://archive.org/details/feynmanlectureso0000feyn_g4q1 }}
* {{cite book | author=Einstein, A. | title=Relativity: The Special and General Theory | location=New York | publisher=Crown | year=1961 | isbn=0-517-02961-8 | url-access=registration | url=https://archive.org/details/relativityspecia00eins_0 }}

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