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{{Introductory article|廣義相對論中的數學}}
{{General relativity sidebar}}

'''廣義相對論所使用的數學'''很複雜。[[艾萨克·牛顿|牛頓]]的運動理論中，物體做[[加速度]]運動時，其長度和時間流逝的速率保持定值，這表示[[经典力学|牛頓力學]]中的許多問題用[[代數]]就能解決。然而，[[相對論]]中的物體在運動速度接近[[光速]]時，長度和時間流逝的速率會有可觀的改變，這表示要計算物體的運動必須用上更多變數和複雜的數學，如[[向量空間|向量]]、[[張量]]、[[偽張量]]、{{tsl|en|curvilinear coordinates|曲線座標}}等概念。

<!--對於以圓軌道繞著大質量物體運動的粒子的相關介紹，非相對論性和相對論性的處理分別參見：{{tsl|en|Newtonian motivations for general relativity||Newtonian motivations for general relativity}}和[[廣義相對論|Theoretical motivation for general relativity]]。--><!--不翻此段？-->

== 向量與張量 ==
{{main|向量|張量}}

=== 向量 ===
[[Image:Vector by Zureks.svg|right|thumb|典型向量的圖示。]]

在[[數學]]、[[物理學]]及[[工程學]]中，歐幾里得向量（有時也稱為「幾何向量」<ref>{{harvnb|Ivanov|2001}}{{Citation not found}}</ref>、「空間向量」<ref>{{harvnb|Heinbockel|2001}}{{Citation not found}}</ref>，或單稱「向量」）是同時有[[量 (数学)|量值]]（[[范数|長度]]）和方向的幾何對象。一個向量將<math>A</math>點「搬運」至<math>B</math>點；向量的拉丁文「vector」意思為「搬運東西的東西」。<ref>From Latin ''vectus'', [[分词|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 | accessdate = 2007-05-25 | archive-date = 2015-09-05 | archive-url = https://web.archive.org/web/20150905141835/http://jeff560.tripod.com/v.html | dead-url = no }}</ref>向量的量值就是兩點之間的距離，方向則為<math>A</math>到<math>B</math>的位移方向。很多[[實數]]的{{tsl|en|algebraic operation|代數運算}}，像[[加法|加]]、[[減法|減]]、[[乘法|乘]]、[[邏輯非]]，和向量的運算很類似，運算也遵守相似的代數法則，如[[交換律]]、[[結合律]]、[[分配律]]。

=== 張量 ===
[[Image:Components stress tensor cartesian.svg|300px|right|thumb|壓力是二階張量。箱子上各面一系列的向量代表壓力]]

張量將向量的概念延伸至額外的維度。[[标量 (数学)|純量]]是沒有方向的量，是單純的數字，在圖上以點來表示，是零維的物件。向量則有量值和方向，在圖上以線呈現，是一維的物件。張量延伸了向量的概念，一個二維的張量稱為二階張量，可以看成一組相關的向量，在一個平面上的多個方向移動。

=== 應用 ===

向量在物理科學裡很基礎。他們可用來代表所有同時有量值和方向的量，例如[[速度]]。速度的量值為[[速率]]。舉例而言，每秒五公尺向上的速度可以向量（0, 5）表示（在二維以<math>y</math>軸的正方向表示向上）。[[力]]的量也能以向量表示，因為它有量值和方向。向量也能描述很多其他的物理量，如[[位移]]、[[加速度]]、[[動量]]和[[角動量]]。其他物理向量，如[[電場]]和[[磁場]]，以物理空間中所有點的向量系統表示，也就是[[向量場]]。

張量在物理中也有延伸應用：

* [[电磁学]]中的[[電磁張量]]（或法拉第張量）
* 在[[连续介质力学]]中的[[形變]]描述形變和[[应变 (物理学)|應變張量]]的有限應變張量
* [[电容率]]和[[電極化率]]是[[各向异性]]介質中的張量
* [[廣義相對論]]中的[[應力-能量張量]]，用來表示[[动量]]的[[通量]]
* 球張量算符是[[球座標系]]中量子[[角動量]]算符的特徵函數
* 擴散張量是[[擴散磁振造影]]的基礎，代表生物環境中的擴散率

=== 維度 ===

廣義相對論需要用到[[四維矢量|四維向量]]，或稱四向量。這四維為長、寬、高、時間。其中的「點」代表事件，因為它同時包含地點和時間。類似向量，相對論中的張量也需要四維。其中一個例子就是[[黎曼曲率張量]]。

=== 座標轉換 ===
{{transH}}
{{multiple image
 | header = 
 | align = 
 | width = 200px
 | image1 = Transformation2polar_basis_vectors.svg
 | caption1 = 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.
 | image2 = Transformation2polar contravariant vector.svg
 | caption2 = 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.
}}

In physics, as well as mathematics, a vector is often identified with a [[多元组|tuple]], or list of numbers, which depend on some auxiliary coordinate system or [[参考系|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 [[共變和反變|''covariant'' or ''contravariant'']] depending on how the transformation of the vector's components is related to the transformation of coordinates. 
* Contravariant vectors are "regular vectors" with units of distance (such as a displacement) or distance times some other unit (such as velocity or acceleration). For example, in changing units from meters to millimeters, a displacement of 1&nbsp;m becomes 1000&nbsp;mm.
* Covariant vectors, on the other hand, have units of one-over-distance (typically such as [[梯度|gradient]]). For example, in changing again from meters to millimeters, a gradient of 1&nbsp;[[开尔文|K]]/m becomes 0.001&nbsp;K/mm.

In [[爱因斯坦求和约定|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 no one correct reference point in the universe. 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, take the signing of the [[美國獨立宣言|Declaration of Independence]]. To a modern observer on [[瑞尼爾山|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.

== 斜軸 ==

An oblique coordinate system is one in which the axes are not necessarily [[正交|orthogonal]] to each other; that is, they meet at angles other than [[直角|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.

== 非張量 ==
{{see also|偽張量}}

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, [[克里斯托费尔符号|Christoffel symbols]] cannot be tensors themselves if the coordinates don't 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 {{tsl|en|Landau–Lifshitz pseudotensor||Landau–Lifshitz pseudotensor}}.

== 曲線座標和彎曲時空 ==

[[Image:Cassini-science-br.jpg|thumb|right|250px|High-precision test of general relativity by the [[卡西尼-惠更斯号|Cassini]] space probe (artist's impression): radio signals sent between the Earth and the probe (green wave) are [[引力时间延迟效应|delayed]] by the warping of [[时空|space and time]] (blue lines) due to the [[太阳|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.]]

{{tsl|en|Curvilinear coordinates||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, gravity has curvature effects on the four dimensions of the universe. 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 [[四维空间|four dimensions]] of curved coordinates instead of three as used to describe a curved 2D surface.

== 平行移動 ==
{{main|平行移動}}

[[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.]]

=== 高維空間中的區間 ===
In a [[欧几里得空间|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 temporal separation. 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 {{tsl|en|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 [[世界线|world line]]s are called [[测地线|geodesic]]s of the spacetime – straight lines in the case of Minkowski space 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 [[大圆|great circle]] on a sphere are geodesics.</ref><ref>{{cite book |title=Principles of Cosmology and Gravitation |first1=Michael V. |last1=Berry |publisher={{tsl|en|CRC Press||CRC Press}} |year=1989 |isbn=0-85274-037-9 |page=58 |url=https://books.google.com/books?id=wTvFxXguod0C}} [https://books.google.com/books?id=wTvFxXguod0C&pg=PA58 Extract of page 58, caption of Fig. 25]</ref> The concept of geodesics becomes central in [[廣義相對論|general relativity]], since geodesic motion may be thought of as "pure motion" ([[慣性力|inertial motion]]) in spacetime, that is, free from any external influences.

=== 協變導數 ===

{{main|協變導數}}

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.

=== 平行移動 ===

Given the covariant derivative, one can define the [[平行移动|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.

=== 克里斯多福符號 ===
{{main|克里斯多福符號}}

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 [[廣義相對論|general relativity]], where [[时空|spacetime]] is represented by a curved 4-dimensional [[伪黎曼流形|Lorentz manifold]] with a [[列维-奇维塔联络|Levi-Civita connection]]. The [[爱因斯坦场方程|Einstein field equations]] – which determine the geometry of spacetime in the presence of matter – contain the [[里奇曲率張量|Ricci tensor]]. Since the Ricci tensor is derived from the Riemann 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 {{tsl|en|solving the geodesic equations||solving the geodesic equations}} in which the Christoffel symbols explicitly appear.

== 測地線 ==
{{main|廣義相對論中的測地線}}

In [[廣義相對論|general relativity]], a '''geodesic''' generalizes the notion of a "straight line" to curved [[时空|spacetime]]. Importantly, the [[世界线|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 [[應力-能量張量|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 {{tsl|en|tangent vector||tangent vector}} of the curve at any point is equal to the [[平行移动|parallel transport]] of the {{tsl|en|tangent vector||tangent vector}} of the base point.

== 曲率張量 ==
{{main|黎曼曲率張量}}

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 [[黎曼曲率張量|Riemann curvature tensor]]: {{math|''R<sup>ρ</sup><sub>σμν</sub>''}}, which gives the most information on the curvature of a space and is derived from derivatives of the [[度量张量|metric tensor]]. In flat space this tensor is zero. 
#The [[里奇曲率張量|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 [[数量曲率|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 [[張量|tensor]] defined over [[伪黎曼流形|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 [[里奇曲率張量|Ricci tensor]], {{math|'''g'''}} is the [[度量张量|metric tensor]] and {{math|''R''}} is the [[数量曲率|scalar curvature]]. It is used in the [[爱因斯坦场方程|Einstein field equations]].

== 應力-能量張量 ==
{{main|應力-能量張量}}
[[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 [[張量|tensor]] quantity in [[物理学|physics]] that describes the [[密度|density]] and [[通量|flux]] of [[能量|energy]] and [[动量|momentum]] in [[时空|spacetime]], generalizing the [[應力|stress tensor]] of Newtonian physics. It is an attribute of [[物质|matter]], [[辐射|radiation]], and non-gravitational {{tsl|en|force field (physics)||force field}}s. The stress–energy tensor is the source of the [[引力場|gravitational field]] in the [[爱因斯坦场方程|Einstein field equations]] of [[廣義相對論|general relativity]], just as mass density is the source of such a field in [[牛顿万有引力定律|Newtonian gravity]].

== 愛因斯坦方程 ==

{{main|愛因斯坦場方程}}
The '''Einstein field equations''' ('''EFE''') or '''Einstein's equations''' are a set of 10 [[方程|equation]]s in [[阿尔伯特·爱因斯坦|Albert Einstein's]] [[廣義相對論|general theory of relativity]] which describe the [[基本相互作用|fundamental interaction]] of [[引力|gravitation]] as a result of [[时空|spacetime]] being [[曲率|curved]] by [[物质|matter]] and [[能量|energy]].<ref name=ein>{{cite journal|last=Einstein |first=Albert |title=The Foundation of the General Theory of Relativity |journal=[[物理年鑑|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=[[可移植文档格式|PDF]] |id= |accessdate= |bibcode=1916AnP...354..769E |deadurl=yes |archiveurl=https://web.archive.org/web/20060829045130/http://www.alberteinstein.info/gallery/gtext3.html |archivedate=2006-08-29 |df= }}</ref> First published by Einstein in 1915<ref name=Ein1915>{{cite journal| last=Einstein| first=Albert| authorlink=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| accessdate=2006-09-12| archive-date=2016-10-27| archive-url=https://web.archive.org/web/20161027044950/http://nausikaa2.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.x.cgi?dir=6E3MAXK4| dead-url=no}}</ref> as a [[張量|tensor equation]], the EFE equate local spacetime [[曲率|curvature]] (expressed by the [[爱因斯坦张量|Einstein tensor]]) with the local energy and [[动量|momentum]] within that spacetime (expressed by the [[應力-能量張量|stress–energy tensor]]).<ref>{{Cite book
 | last1=Misner |first1=Charles W. |authorlink1=Charles W. Misner
 | last2=Thorne |first2=Kip S. |authorlink2=Kip Thorne
 | last3=Wheeler |first3=John Archibald |authorlink3=John Archibald Wheeler
 | year=1973
 | title={{tsl|en|Gravitation (book)||Gravitation}}
 | url= 
 | publisher={{tsl|en|W. H. Freeman||W. H. Freeman}} |location=San Francisco
 | isbn=978-0-7167-0344-0
 
}} 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 [[爱因斯坦张量|Einstein tensor]] and {{math|''T<sub>μν</sub>''}} is the [[應力-能量張量|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).

== 史瓦西解與黑洞 ==
{{main|史瓦西度規}}
In [[阿尔伯特·爱因斯坦|Einstein]]'s theory of [[廣義相對論|general relativity]], the '''Schwarzschild metric''' (also '''Schwarzschild vacuum''' or '''Schwarzschild solution'''),  is a solution to the [[爱因斯坦场方程|Einstein field equations]] which describes the [[引力場|gravitational field]] outside a spherical mass, on the assumption that the [[電荷|electric charge]] of the mass, the [[角动量|angular momentum]] of the mass, and the universal [[宇宙學常數|cosmological constant]] are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many [[恒星|star]]s and [[行星|planet]]s, including Earth and the Sun. The solution is named after [[卡爾·史瓦西|Karl Schwarzschild]], who first published the solution in 1916.

According to {{tsl|en|Birkhoff's theorem (relativity)||Birkhoff's theorem}}, the Schwarzschild metric is the most general {{tsl|en|rotational symmetry||spherically symmetric}}, {{tsl|en|Vacuum solution (general relativity)||vacuum solution}} of the [[爱因斯坦场方程|Einstein field equations]]. A '''Schwarzschild black hole''' or '''static black hole''' is a [[黑洞|black hole]] that has no [[荷 (物理)|charge]] or [[角动量|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.
{{transF}}
== 參見 ==

*[[微分流形]]
*[[克里斯托费尔符号]]
*[[黎曼几何]]
*{{tsl|en|Ricci calculus|瑞奇微積分}}
*[[微分几何]]
*{{tsl|en|List of differential geometry topics|微分幾何話題列表}}
*[[廣義相對論]]
*{{tsl|en|Gauge gravitation theory|規範重力論}}
*{{tsl|en|General covariant transformations|廣義協變}}
*{{tsl|en|Derivations of the Lorentz transformations|勞侖茲變換的推導}}

== 註釋 ==

{{reflist}}

== 參考文獻 ==

* {{cite book | author=P. A. M. Dirac | title=General Theory of Relativity | publisher=[[普林斯頓大學出版社|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 |url=https://archive.org/details/gravitation0000misn | 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) |url=https://archive.org/details/classicaltheoryo0000land_k6k2  | location=Oxford | publisher=Pergamon | year=1975 | isbn=0-08-018176-7}}
* {{cite book |author1=R. P. Feynman |author2=F. B. Moringo |author3=W. G. Wagner | title=Feynman Lectures on Gravitation |url=https://archive.org/details/feynmanlectureso0000feyn_g4q1 | publisher=[[艾迪生韦斯利|Addison-Wesley]] | year=1995 | isbn=0-201-62734-5}}
* {{cite book | author=Einstein, A.  | title=Relativity: The Special and General Theory | url=https://archive.org/details/relativityspecia00eins_0  | location= New York | publisher=Crown| year=1961 | isbn=0-517-02961-8}}

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