Introduction to the mathematics of general relativity: Difference between revisions

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{{Short description|none}}

{{Introductory article|Mathematics of general relativity}}

{{General relativity sidebar}}

The '''mathematics of general relativity''' areis complexcomplicated. 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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===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' vector''<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 | accessdateaccess-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]].

===Tensors===

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In general relativity, four-dimensional vectors, or [[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_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.

</gallery></div>

In physics, as well as mathematics, a vector is often identified with a [[tuple]], or list of numbers, which depend on some auxiliarya coordinate system or [[frame of reference|reference frame]]. WhenIf the coordinates are transformed, forsuch exampleas by rotation or stretching of the coordinate system, then the components of the vector also transform. The vector itself hasdoes not changedchange, but the reference frame has,does. This means sothat the components of the vector (or measurements taken with respecthave 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 in 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]]/ per m becomes 0.001&nbsp;K/Kelvin per mm.

In [[Einstein notation]], contravariant vectors and components of tensors are shown with superscripts, e.g. {{math|''xⁱ''}}, and covariant vectors and components of tensors with subscripts, e.g. {{math|''x_i''}}. 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 nonot one correct 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 [[United States Declaration of Independence|Declaration of Independence]]. To a modern observer on [[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]]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==

{{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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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, gravityenergy hasand 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==

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:<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''²}} above follows the [[sign convention#Relativity|space-like convention (−+++)]]. A notation like {{math|Δ''r''²}} means {{math|(Δ''r'')²}}. The reason {{math|''s''²}} is called the interval and not {{math|''s''}} is called the interval is that {{math|''s''²}} can be positive, zero or negative.

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

Certain types of [[world line]]s are called [[geodesic]]s of the spacetime – straight lines in the case of flat Minkowski spacespacetime 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]] 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 [[general relativity]], since geodesic motion may be thought of as "pure motion" ([[Fictitious force|inertial motion]]) in spacetime, that is, free from any external influences.

===The covariant derivative===

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===Christoffel symbols===

{{main|Christoffel symbolsymbols}}

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 [[general relativity]], where [[spacetime]] is represented by a curved 4-dimensional [[Lorentz manifold]] with a [[Levi-Civita connection]]. The [[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]] in which the Christoffel symbols explicitly appear.

==Geodesics==

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{{main|Riemann curvature tensor}}

The [[Riemann curvature tensor]] {{math|''R^ρ_σμν''}} tells us, mathematically, how much curvature there is in any given region of space. ContractingIn theflat tensorspace producesthis 3tensor differentis mathematical objects:zero.

Contracting the tensor produces 2 more 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 [[scalarRicci curvaturetensor]]: {{math|''R_σν''}}, comes from the simplestneed measurein ofEinstein's curvature,theory assignsfor a singlecurvature scalartensor valuewith toonly each2 point in a spaceindices. It is obtained by averaging certain portions of the RicciRiemann 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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==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 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]]. 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|Einstein field equations}}

The '''Einstein field equations''' ('''EFE''') or '''Einstein's equations''' are a set of 10 [[equation]]s in [[Albert Einstein|Albert Einstein's]] [[general relativity|general theory of relativity]] which describe the [[fundamental interaction]] of [[gravitation]] as a result of [[spacetime]] being [[curvature|curved]] by [[matter]] and [[energy]].<ref name=ein>{{cite journal|last=Einstein |first=Albert |title=The Foundation of the General Theory of Relativity |journal=[[Annalen der Physik]] |volume=354 |issue=7 |pages=769 |year=1916 |publisher= |url=http://www.alberteinstein.info/gallery/gtext3.html |doi=10.1002/andp.19163540702 |format=[[PDF]] |id= |accessdate= |bibcode=1916AnP...354..769E |deadurlurl-status=yesdead |archiveurlarchive-url=https://web.archive.org/web/20060829045130/http://www.alberteinstein.info/gallery/gtext3.html |archivedatearchive-date=2006-08-29 |df= }}</ref> First published by Einstein in 1915<ref name=Ein1915>{{cite journal|last=Einstein| first=Albert| authorlinkauthor-link = Albert Einstein| 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 | accessdateaccess-date=2006-09-12}}</ref> as a [[tensor equation]], the EFE equate local spacetime [[curvature]] (expressed by the [[Einstein tensor]]) with the local energy and [[momentum]] within that spacetime (expressed by the [[stress–energy tensor]]).<ref>{{Cite book

| last1=Misner |first1=Charles W. |authorlink1author-link1=Charles W. Misner

| last2=Thorne |first2=Kip S. |authorlink2author-link2=Kip Thorne

| last3=Wheeler |first3=John Archibald |authorlink3author-link3=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 Fieldfield Equationsequations can be written as

:<math>G_{\mu \nu}= {8 \pi G \over c^4} T_{\mu \nu} ,</math>

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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.

==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}}

*{{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

}}.

{{sfn whitelist |CITEREFIvanov2001}}

* {{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 (Fourth Revised English Edition) | ___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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