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| {{redirect|Tensor index notation|a summary of tensors in general|Glossary of tensor theory}}
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| In [[mathematics]], '''Ricci calculus''' constitutes the rules of index notation and manipulation for [[tensors]] and [[tensor fields]].<ref>{{cite book |author=Synge J.L., Schild A.|publisher=first Dover Publications 1978 edition |title=Tensor Calculus |pages=6–108|year= 1949}}</ref><ref>{{cite book |pages=85–86, §3.5| author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}</ref><ref>{{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books| year=2007 | isbn=0-679-77631-1}}</ref> It is also the modern name for what used to be called the '''absolute differential calculus''' (the foundation of [[tensor calculus]]), developed by [[Gregorio Ricci-Curbastro]] in 1887–96, and subsequently popularized in a paper <ref>{{citation|title=Méthodes de calcul différentiel absolu et leurs applications|last=Ricci|first=Gregorio|author-link=Gregorio Ricci-Curbastro|last2=Levi-Civita|first2=Tullio|journal=[[Mathematische Annalen]]|publisher=Springer|volume=54|issue=1–2|date=March 1900|pages=125–201|doi=10.1007/BF01454201|url=http://www.springerlink.com/content/u21237446l22rgg7/fulltext.pdf|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0054&DMDID=DMDLOG_0011&L=1}}</ref> written with his pupil [[Tullio Levi-Civita]] in 1900. [[Jan Arnoldus Schouten]] developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to [[general relativity]] and [[differential geometry]] in the early twentieth century.<ref>{{cite book|last=Schouten|first=Jan A.|title=Der Ricci-Kalkül – Eine Einführung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie (Ricci Calculus – An introduction in the latest methods and problems in multi-dimmensional differential geometry)|language=german|year=1924|series=Grundlehren der mathematischen Wissenschaften|volume=10|editor= R. Courant|publisher=Springer Verlag|location=Berlin|url=http://resolver.sub.uni-goettingen.de/purl?PPN373339186}}</ref>
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| A component of a tensor is a [[real number]] which is used as a coefficient of a basis element for the tensor space. The tensor is the sum of its components multiplied by their basis elements. Tensors and tensor fields can be expressed in terms of their components, and operations on tensors and tensor fields can be expressed in terms of operations on their components. The description of tensor fields and operations on them in terms of their components is the focus of the Ricci calculus. This notation allows the most efficient expressions of such tensor fields and operations. While much of the notation may be applied with any tensors, operations relating to a [[differential structure]] are only applicable to tensor fields. Where needed, the notation extends to components of non-tensors, particularly [[multidimensional array]]s.
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| A tensor may be expressed as a linear sum of the [[tensor product]] of [[Vector (mathematics and physics)|vector]] and [[covector]] basis elements. The resulting tensor components are labelled by indices of the basis. Each index has one possible value per [[dimension]] of the underlying [[vector space]]. The number of indices equals the order of the tensor. | |
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| For compactness and convenience, the notational convention implies certain things, notably that of summation over indices repeated within a term and of [[universal quantification]] over free indices (those not so summed). Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions over a manifold, usually more specifically as functions of the coordinates on the manifold. This allows intuitive manipulation of expressions with familiarity of only a limited set of rules.
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| ==Notation for indices==
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| {{see also|Index notation}}
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| ===Basis-related distinctions===
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| ====Space–time split====
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| Where a distinction is to be made between the space-like basis elements and a time-like element in the four dimensional spacetime of classical physics, this is conventionally done through indices as follows:<ref>{{citation | author=C. Møller|title=The Theory of Relativity|year=1952|page=234}} is an example of a variation: 'Greek indices run from 1 to 3, Latin indices from 1 to 4'</ref>
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| *The lowercase [[Latin alphabet]] ''a'', ''b'', ''c''... is used to indicate restriction to 3-dimensional [[Euclidean space]], which take values 1, 2, 3 for the spatial components; and the time-like element, indicated by 0, is shown separately.
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| *The lowercase [[Greek alphabet]] α, β, γ... is used for 4-dimensional [[spacetime]], which typically take values 0 for time components and 1, 2, 3 for the spatial components.
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| Some sources use 4 instead of 0 as the index value corresponding to time; in this article, 0 is used. Otherwise, in general mathematical contexts, any symbols can be used for the indices, generally running over all dimensions of the vector space.
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| ====Coordinate and index notation====
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| The author(s) will usually make it clear whether a subscript is intended as an index or as a label.
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| For example, in 3-D Euclidean space and using [[Cartesian coordinates]]; the [[coordinate vector]] {{nowrap|1='''A''' = (''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>) = (''A''<sub>x</sub>, ''A''<sub>y</sub>, ''A''<sub>z</sub>)}} shows a direct correspondence between the subscripts 1, 2, 3 and the labels x, y, z. In the expression ''A<sub>i</sub>'', ''i'' is interpreted as an index ranging over the values 1, 2, 3, while the x, y, z subscripts are not variable indices, more like "names" for the components. In the context of spacetime, the index value 0 corresponds to the label t.
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| ====Reference to coordinate systems====
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| Indices themselves may be ''labelled'' using [[diacritic]]-like symbols, such as a hat (^), bar (<sup>–</sup>), tilde (<sup>~</sup>), or prime (′)
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| :<math>X_{\hat{\phi}}\,,Y_{\bar{\lambda}}\,,Z_{\tilde{\eta}}\,,T_{\mu'} \cdots </math>
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| to denote a possibly different [[basis]] (and hence [[coordinate system]]) for that index. An example is in [[Lorentz transformation]]s from one [[frame of reference]] to another, where one frame could be unprimed and the other primed, as in:
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| :<math> v^{\mu'} = v^{\nu}L_\nu{}^{\mu'} .</math>
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| This is not to be confused with [[van der Waerden notation]] for [[spinor]]s, which uses hats and overdots on indices to reflect the chiralty of a spinor.
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| ===Raised and lowered indices===
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| ;[[Covariance and contravariance of vectors|Covariant tensor components]]
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| A ''lower index'' (subscript) indicates covariance of the components with respect to that index: <math>A_{\alpha\beta\gamma \cdots}</math>
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| ;[[Covariance and contravariance of vectors|Contravariant tensor components]]
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| An ''upper index'' (superscript) indicates contravariance of the components with respect to that index: <math>A^{\alpha\beta\gamma \cdots}</math>
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| ;[[Mixed tensor|Mixed-variance tensor components]]
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| A tensor may have both upper and lower indices: <math>A_{\alpha}{}^{\beta}{}_{\gamma}{}^{\delta\cdots}</math>
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| ;[[Einstein summation convention|Summation]]
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| Two indices (one raised and one lowered) with the same symbol within a term are summed over: <math> A_\alpha B^\alpha \equiv \sum_\alpha A_{\alpha}B^\alpha </math> or <math> A^\alpha B_\alpha \equiv \sum_\alpha A^{\alpha}B_\alpha \,.</math>
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| The operation implied by such a summation is called [[tensor contraction]]:
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| :<math> A_\alpha B^\beta \rightarrow A_\alpha B^\alpha \equiv \sum_\alpha A_{\alpha}B^\alpha \,.</math>
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| More than one index may occur twice, but only twice within one term, for example:
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| :<math> A_{\alpha}{}^\gamma B^\alpha C_\gamma{}^\beta \equiv \sum_\alpha \sum_\gamma A_{\alpha}{}^\gamma B^\alpha C_\gamma{}^\beta\,.</math>
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| As for a non-identity,
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| :<math> A_{\alpha\gamma}{}^\gamma B^\alpha C_\gamma{}^\beta \not\equiv \sum_\alpha \sum_\gamma A_{\alpha\gamma}{}^\gamma B^\alpha C_\gamma{}^\beta\,</math>
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| is not considered well-formed, that is, it is meaningless.
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| ;[[Multi-index notation]]
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| If a tensor has a list of indices all raised or lowered, one shorthand is to use a capital letter for the list:<ref>{{citation | author=T. Frankel|page=67| title = The Geometry of Physics| publisher=Cambridge University Press|edition=3rd|year=2012|isbn=978-1107-602601}}</ref>
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| :<math> A_{i_1\cdots i_n}B^{i_1\cdots i_n j_1 \cdots j_m}C_{j_1 \cdots j_m} \equiv A_I B^{IJ} C_J </math>
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| where ''I'' = ''i''<sub>1</sub> ''i''<sub>2</sub> ... ''i<sub>n</sub>'' and ''J'' = ''j''<sub>1</sub> ''j''<sub>2</sub> ... ''j<sub>m</sub>''.
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| ;Sequential summation
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| Two vertical bars | | around a set of indices (with a contraction):<ref>Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0</ref>
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| :<math> A_{|\alpha \beta \gamma|\cdots} B^{\alpha\beta\gamma \cdots} = \sum_\alpha \sum_\beta \sum_\gamma A_{\alpha \beta \gamma\cdots} B^{\alpha\beta\gamma \cdots} </math>
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| denotes the summation in which each preceding index is counted up to (and not including) the value of the next index:
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| :<math> \alpha < \beta < \gamma\,. </math>
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| Only one group of the repeated set of indices has the vertical bars around them (the other contracted indices do not). More than one group can summed in this way:
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| :<math>A_{|\alpha \beta\gamma|}{}^{|\delta\epsilon\cdots\lambda|} B^{\alpha \beta\gamma}{}_{\delta\epsilon\cdots\lambda|\mu \nu \cdots\zeta|} C^{\mu\nu\cdots \zeta}=\sum_\alpha \sum_\beta \sum_\gamma \sum_\delta \sum_\epsilon \cdots \sum_\lambda \sum_\mu \sum_\nu \cdots \sum_\zeta A_{\alpha \beta\gamma}{}^{\delta\epsilon\cdots\lambda} B^{\alpha \beta\gamma}{}_{\delta\epsilon\cdots\lambda\mu \nu\cdots\zeta} C^{\mu\nu\cdots\zeta} </math>
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| where
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| :<math>\alpha < \beta < \gamma \,, \quad \delta < \epsilon < \cdots < \lambda\,,\quad \mu < \nu \cdots < \zeta\,.</math>
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| This is useful to prevent over-counting in some summations, when tensors are [[symmetric tensor|symmetric]] or [[antisymmetric tensor|antisymmetric]].
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| Alternatively, using the capital letter convention for multi-indices, an underarrow is placed underneath the block of indices:<ref>{{citation | author=T. Frankel| title = The Geometry of Physics|page=67| publisher=Cambridge University Press|edition=3rd|year=2012|isbn=978-1107-602601}}</ref>
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| :<math>A_{\underset{\rightharpoondown}{P}}{}^{\underset{\rightharpoondown}{Q}} B^P{}_{Q\underset{\rightharpoondown}{R}} C^R = \sum_\underset{\rightharpoondown}{P} \sum_\underset{\rightharpoondown}{Q} \sum_\underset{\rightharpoondown}{R} A_{P}{}^{Q} B^P{}_{QR} C^R </math>
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| where
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| :<math> \underset{\rightharpoondown}{P} = |\alpha \beta\gamma|\,,
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| \quad \underset{\rightharpoondown}{Q} = |\delta\epsilon\cdots\lambda|\,,
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| \quad\underset{\rightharpoondown}{R} = |\mu \nu \cdots\zeta| </math>
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| ;[[Raising and lowering indices]]
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| By contracting an index with a non-singular [[metric tensor]], the [[mixed tensor|type]] of a tensor can be changed, converting a lower index to an upper index or vice versa:
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| :<math>B^{\gamma}{}_{\beta\cdots} = g^{\gamma\alpha}A_{\alpha\beta\cdots}</math> and <math>A_{\alpha\beta\cdots} = g_{\alpha\gamma}B^{\gamma}{}_{\beta\cdots}</math>
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| The base symbol in many cases is retained (e.g. using ''A'' where ''B'' appears here), and when there is no ambiguity, repositioning an index may be taken to imply this operation.
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| ===Correlations between index positions and invariance===
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| This table summarizes how the manipulation of covariant and contravariant indices fit in with invariance under a [[passive transformation]] between bases, with the components of each basis set in terms of the other reflected in the first column. The barred indices refer to the final coordinate system after the transformation.<ref>{{cite book |pages=61, 202–203, 232| author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}</ref>
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| The [[Kronecker delta]] is used, [[#Notable tensors|see also below]].
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| :{| class="wikitable"
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| |-
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| !
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| ! Basis transformation
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| ! Component transformation
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| ! Invariance
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| |-
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| ! Covector, covariant vector, dual vector, 1-form
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| | <math>e^\bar{\alpha} = L^\bar{\alpha}{}_\beta e^\beta </math>
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| | <math>a_\bar{\alpha} = a_\gamma L^\gamma{}_\bar{\alpha} </math>
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| | <math>a_\bar{\alpha}e^\bar{\alpha} = a_\gamma L^\gamma{}_\bar{\alpha} L^\bar{\alpha}{}_\beta e^\beta = a_\gamma \delta^\gamma{}_\beta e^\beta = a_\beta e^\beta </math>
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| |-
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| ! Vector, contravariant vector
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| | <math>e_\bar{\alpha} = L^\gamma{}_\bar{\alpha} e_\gamma </math>
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| | <math>a^\bar{\alpha} = a^\beta L^\bar{\alpha}{}_\beta </math>
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| | <math>a^\bar{\alpha}e_\bar{\alpha} = a^\beta L^\bar{\alpha}{}_\beta L^\gamma{}_\bar{\alpha} e_\gamma = a^\beta \delta^\gamma{}_\beta e_\gamma = a^\gamma e_\gamma </math>
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| |-
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| |}
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| ==General outlines for index notation and operations==
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| Tensors are equal [[if and only if]] every corresponding component is equal, e.g. tensor ''A'' equals tensor ''B'' if and only if
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| :<math>A^{\alpha}{}_{\beta\gamma} = B^{\alpha}{}_{\beta\gamma} </math>
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| for all α, β and γ. Consequently, there are facets of the notation that are useful in checking that an equation makes sense (an analogous procedure to [[dimensional analysis]]).
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| ;[[Einstein notation#Introduction|Free and dummy indices]]
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| Indices not in contractions are called ''free indices''.
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| Indices in contractions are termed ''dummy indices'', or ''summation indices''.
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| ;A tensor equation represents many ordinary (real-valued) equations
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| The components of tensors (like <math>A^\alpha</math>, <math>B_\beta{}^\gamma</math> etc.) are just real numbers. Since the indices take various integer values to select specific components of the tensors, a single tensor equation represents many ordinary equations. If a tensor equality has ''n'' free indices, and if the dimensionality of the underlying vector space is ''m'', the equality represents ''m<sup>n</sup>'' equations: each has a specific set of index values.
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| For instance, if
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| :<math>A^\alpha B_\beta{}^\gamma C_{\gamma\delta} + D^\alpha{}_\beta{} E_\delta = T^\alpha{}_\beta{}_\delta </math>
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| is in [[Four-dimensional space|4-dimension]]s (that is, each index runs from 0 to 3 or 1 to 4), then because there are three free indices (α, β, δ), there are 4<sup>3</sup> = 64 equations. Three of these are:
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| :<math>A^0 B_1{}^0 C_{00} + A^0 B_1{}^1 C_{10} + A^0 B_1{}^2 C_{20} + A^0 B_1{}^3 C_{30} + D^0{}_1{} E_0 = T^0{}_1{}_0 </math>
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| :<math>A^1 B_0{}^0 C_{00} + A^1 B_0{}^1 C_{10} + A^1 B_0{}^2 C_{20} + A^1 B_0{}^3 C_{30} + D^1{}_0{} E_0 = T^1{}_0{}_0 </math>
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| :<math>A^1 B_2{}^0 C_{02} + A^1 B_2{}^1 C_{1 2} + A^1 B_2{}^2 C_{2 2} + A^1 B_2{}^3 C_{3 2} + D^1{}_2{} E_2 = T^1{}_2{}_2. </math>
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| This illustrates the compactness and efficiency of using index notation: many equations which all share a similar structure can be collected into one simple tensor equation. | |
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| ;Indices are replaceable labels
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| Replacing any index symbol throughout by another leaves the tensor equation unchanged (provided there is no conflict with other symbols already used). This can be useful when manipulating indices, such as using index notation to verify [[vector calculus identities]] or identities of the [[Kronecker delta]] and [[Levi-Civita symbol#Properties|Levi-Civita symbol]] (see also below). An example of a correct change is:
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| :<math>A^\alpha B_\beta{}^\gamma C_{\gamma\delta} + D^\alpha{}_\beta{} E_\delta \rightarrow A^\lambda B_\beta{}^\mu C_{\mu\delta} + D^\lambda{}_\beta{} E_\delta </math>
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| as for an erroneous change:
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| :<math>A^\alpha B_\beta{}^\gamma C_{\gamma\delta} + D^\alpha{}_\beta{} E_\delta \nrightarrow A^\lambda B_\beta{}^\gamma C_{\mu\delta} + D^\alpha{}_\beta{} E_\delta \,.</math>
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| In the first replacement, λ replaced α and μ replaced γ ''everywhere'', so the expression still has the same meaning. In the second, λ did not fully replace α, and μ did not fully replace γ (incidentally, the contraction on the γ index became a tensor product), which is entirely inconsistent for reasons shown next.
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| ;Indices are the same in every term
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| The same indices on each side of a tensor equation always appear in the same (upper or lower) position throughout every term, ''except'' for indices repeated in a term (which implies a summation over that index), for example:
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| :<math>A^\alpha B_\beta{}^\gamma C_{\gamma\delta} + D^\alpha{}_\beta{} E_\delta = T^\alpha{}_\beta{}_\delta </math>
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| as for an erroneous expression:
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| :<math>A^\alpha B_\beta{}^\gamma C_{\gamma\delta} + D_\alpha{}_\beta{}^\gamma E^\delta. </math>
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| In other words, non-repeated indices must be of the same type in every term of the equation. In the above identity α, β, δ line up throughout and γ occurs twice in one term due to a contraction (correctly once as an upper index and once as a lower index), so it's a valid as an expression. In the invalid expression, while β lines up, α and δ do not, and γ appears twice in one term (contraction) ''and'' once in another term, which is inconsistent.
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| ;Brackets and punctuation used once where implied
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| When applying a rule to a number of indices (differentiation, symmetrization etc., shown next), the bracket or punctuation symbols denoting the rules are only shown on one group of the indices to which they apply.
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| If the brackets enclose ''covariant indices'' – the rule applies only to ''all covariant indices enclosed in the brackets'', not to any contravariant indices which happen to be placed intermediately between the brackets.
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| Similarly if brackets enclose ''contravariant indices'' – the rule applies only to ''all enclosed contravariant indices'', not to intermediately placed covariant indices.
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| ==Symmetric and antisymmetric parts==
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| ;[[Symmetric tensor|Symmetric]] part of tensor
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| [[Bracket#Parentheses ( )|Parentheses ( )]] around multiple indices denotes the symmetrized part of the tensor. When symmetrizing ''p'' indices using σ to range over permutations of the numbers 1 to ''p'', one takes a sum over the [[permutation]]s of those indices <math>\alpha_{\sigma(i)}</math> for ''i'' = 1, 2, 3 ... ''p'', and then divides by the number of permutations:
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| :<math>A_{(\alpha_1\alpha_2\cdots\alpha_p)\alpha_{p+1}\cdots\alpha_q} = \dfrac{1}{p!} \sum_{\sigma} A_{\alpha_{\sigma(1)}\cdots\alpha_{\sigma(p)}\alpha_{p+1}\cdots\alpha_{q}} \,.</math>
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| For example, two symmetrizing indices mean there are two indices to permute and sum over:
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| :<math>A_{(\alpha\beta)\gamma\cdots} = \dfrac{1}{2!} \left(A_{\alpha\beta\gamma\cdots} + A_{\beta\alpha\gamma\cdots} \right)</math>
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| while for three symmetrizing indices, there are three indices to sum over and permute:
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| :<math>A_{(\alpha\beta\gamma)\delta\cdots} = \dfrac{1}{3!} \left(A_{\alpha\beta\gamma\delta\cdots}
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| + A_{\gamma\alpha\beta\delta\cdots}
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| + A_{\beta\gamma\alpha\delta\cdots}
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| + A_{\alpha\gamma\beta\delta\cdots}
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| + A_{\gamma\beta\alpha\delta\cdots}
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| + A_{\beta\alpha\gamma\delta\cdots}
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| \right)</math>
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| The symmetrization is [[Distributive property|distributive]] over addition; | |
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| :<math>A_{(\alpha} \left(B_{\beta)\gamma\cdots} + C_{\beta)\gamma\cdots} \right) = A_{(\alpha}B_{\beta)\gamma\cdots} + A_{(\alpha}C_{\beta)\gamma\cdots}</math>
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| Indices are not part of the symmetrization when they are:
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| *not on the same level, for example;
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| ::<math>A_{(\alpha}B^{\beta}{}_{\gamma)} = \dfrac{1}{2!} \left(A_{\alpha}B^{\beta}{}_{\gamma} + A_{\gamma}B^{\beta}{}_{\alpha} \right)</math>
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| *within the parentheses and between vertical bars (i.e. |···|), modifying the previous example;
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| ::<math>A_{(\alpha}B_{|\beta|}{}_{\gamma)} = \dfrac{1}{2!} \left(A_{\alpha}B_{\beta \gamma} + A_{\gamma}B_{\beta \alpha} \right)</math>
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| Here the α and γ indices are symmetrized, β is not.
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| ;[[Antisymmetric tensor|Antisymmetric]] or alternating part of tensor
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| [[Bracket#Square brackets [ ]|Square brackets [ ]]] around multiple indices denotes the ''anti''symmetrized part of the tensor. For ''p'' antisymmetrizing indices – the sum over the permutations of those indices <math>\alpha_{\sigma(i)}</math> multiplied by the [[signature (permutation)|signature of the permutation]] <math>\sgn(\sigma)</math> is taken, then divided by the number of permutations:
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| :<math>\begin{align}
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| A_{[\alpha_1\cdots\alpha_p]\alpha_{p+1}\cdots\alpha_q} & = \dfrac{1}{p!} \sum_{\sigma}\sgn(\sigma) A_{\alpha_{\sigma(1)}\cdots\alpha_{\sigma(p)}\alpha_{p+1}\cdots\alpha_{q}} \\
| |
| & = \dfrac{1}{(n-p)!} \varepsilon_{\alpha_1 \dots \alpha_p\,\beta_1 \dots \beta_{n-p}} \dfrac{1}{p!} \varepsilon^{\gamma_1 \dots \gamma_p\,\beta_1 \dots \beta_{n-p}} A_{\gamma_1 \dots \gamma_p\alpha_{p+1}\cdots\alpha_q} \\
| |
| \end{align} </math>
| |
| | |
| where ''n'' is the dimensionality of the underlying vector space and <math>\varepsilon_{\alpha_1 \dots \alpha_n} \,</math> is the [[Levi-Civita symbol]].
| |
| | |
| For example – two antisymmetrizing indices imply:
| |
| | |
| :<math>A_{[\alpha\beta]\gamma\cdots} = \dfrac{1}{2!} \left(A_{\alpha\beta\gamma\cdots} - A_{\beta\alpha\gamma\cdots} \right)</math>
| |
| | |
| while three antisymmetrizing indices imply:
| |
| | |
| :<math>A_{[\alpha\beta\gamma]\delta\cdots} = \dfrac{1}{3!} \left(A_{\alpha\beta\gamma\delta\cdots}
| |
| + A_{\gamma\alpha\beta\delta\cdots}
| |
| + A_{\beta\gamma\alpha\delta\cdots}
| |
| - A_{\alpha\gamma\beta\delta\cdots}
| |
| - A_{\gamma\beta\alpha\delta\cdots}
| |
| - A_{\beta\alpha\gamma\delta\cdots}
| |
| \right)</math>
| |
| | |
| as for a more specific example, if ''F'' represents the [[electromagnetic tensor]], then the equation | |
| | |
| :<math>0 = F_{[\alpha\beta,\gamma]} = \dfrac{1}{3!} \left(
| |
| F_{\alpha\beta,\gamma}
| |
| + F_{\gamma\alpha,\beta}
| |
| + F_{\beta\gamma,\alpha}
| |
| - F_{\beta\alpha,\gamma}
| |
| - F_{\alpha\gamma,\beta}
| |
| - F_{\gamma\beta,\alpha}
| |
| \right) \,</math>
| |
| | |
| represents [[Gauss's law for magnetism]] and [[Faraday's law of induction]].
| |
| | |
| As before, the antisymmetrization is distributive over addition;
| |
| | |
| :<math>A_{[\alpha} \left(B_{\beta]\gamma\cdots} + C_{\beta]\gamma\cdots} \right) = A_{[\alpha}B_{\beta]\gamma\cdots} + A_{[\alpha}C_{\beta]\gamma\cdots}</math>
| |
| | |
| As with symmetrization, indices are not antisymmetrized when they are:
| |
| | |
| *not on the same level, for example;
| |
| | |
| ::<math>A_{[\alpha}B^{\beta}{}_{\gamma]} = \dfrac{1}{2!} \left(A_{\alpha}B^{\beta}{}_{\gamma} - A_{\gamma}B^{\beta}{}_{\alpha} \right)</math>
| |
| | |
| *within the square brackets and between vertical bars (i.e. |···|), modifying the previous example;
| |
| | |
| ::<math>A_{[\alpha}B_{|\beta|}{}_{\gamma]} = \dfrac{1}{2!} \left(A_{\alpha}B_{\beta \gamma} - A_{\gamma}B_{\beta \alpha} \right)</math>
| |
| | |
| Here the α and γ indices are antisymmetrized, β is not.
| |
| | |
| ;Symmetry and antisymmetry sum
| |
| | |
| Any tensor can be written as the sum of its symmetric and antisymmetric parts on two indices:
| |
| | |
| :<math>A_{\alpha\beta\gamma\cdots}=A_{(\alpha\beta)\gamma\cdots}+A_{[\alpha\beta]\gamma\cdots}</math>
| |
| | |
| as can be seen by adding the above expressions for <math>A_{(\alpha\beta)\gamma\cdots}</math> and <math>A_{[\alpha\beta]\gamma\cdots}</math>. This does not hold for other than two indices.
| |
| | |
| ==Differentiation==
| |
| | |
| {{see also|Four-gradient|D'Alembertian|Intrinsic derivative}}
| |
| | |
| For compactness, derivatives may be indicated by adding indices after a comma or semicolon.<ref>{{cite book | author=G. Woan| title=The Cambridge Handbook of Physics Formulas| publisher=Cambridge University Press| year=2010 | isbn=978-0-521-57507-2}}</ref><ref>[http://mathworld.wolfram.com/CovariantDerivative.html Covariant derivative] – Mathworld, Wolfram</ref>
| |
| | |
| ;[[Partial derivative]]
| |
| | |
| To indicate partial differentiation of a tensor field with respect to a coordinate variable <math>x^\gamma</math>, a ''[[comma]]'' is placed before an added lower index of the coordinate variable.
| |
| | |
| :<math>A_{\alpha\beta\cdots,\gamma} = \partial_\gamma A_{\alpha\beta\cdots} = \dfrac{\partial}{\partial x^\gamma} A_{\alpha\beta\cdots}</math>
| |
| | |
| This may be repeated (without adding further commas):
| |
| | |
| :<math>A_{\alpha_1\alpha_2\cdots\alpha_p\,,\,\alpha_{p+1}\cdots\alpha_q} = \partial_{\alpha_q}\cdots\partial_{\alpha_{p+2}}\partial_{\alpha_{p+1}} A_{\alpha_1\alpha_2\cdots\alpha_p} = \dfrac{\partial^{q-p}}{\partial x^{\alpha_q}\cdots\partial x^{\alpha_{p+2}}\partial x^{\alpha_{p+1}}} A_{\alpha_1\alpha_2\cdots\alpha_p}.</math>
| |
| | |
| These components do ''not'' transform covariantly. This derivative is characterized by the [[product rule]] and the derivatives of the coordinates
| |
| | |
| :<math> x^{\alpha}{}_{, \gamma} = \delta^{\alpha}{}_\gamma </math>
| |
| | |
| where δ is the [[Kronecker delta]].
| |
| | |
| ;[[Covariant derivative]]
| |
| | |
| To indicate covariant differentiation of any tensor field, a ''[[semicolon]]'' ( ; ) is placed before an added lower (covariant) index. Less common alternatives to the semicolon include a ''[[Slash (punctuation)|forward slash]]'' ( / )<ref>{{citation | author=T. Frankel|page=298| title = The Geometry of Physics| publisher=Cambridge University Press|edition=3rd|year=2012|isbn=978-1107-602601}}</ref> or in three-dimensional curved space just ''one'' vertical bar ( | ).<ref>{{cite book |pages=510, §21.5| author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}</ref>
| |
| | |
| For a contravariant vector: <math>A^{\alpha}{}_{;\beta} = A^{\alpha}{}_{,\beta} + \Gamma^{\alpha} {}_{\gamma\beta}A^\gamma</math> where <math>\Gamma^{\alpha}{}_{\beta\gamma} \,</math> is a [[Christoffel symbol]] of the second kind.
| |
| | |
| For a covariant vector: <math>A_{\alpha ;\beta} = A_{\alpha,\beta} - \Gamma^{\gamma} {}_{\alpha\beta}A_\gamma \,.</math>
| |
| | |
| For an arbitrary tensor:<ref>{{citation | author=T. Frankel|page=299| title = The Geometry of Physics| publisher=Cambridge University Press|edition=3rd|year=2012|isbn=978-1107-602601}}</ref>
| |
| :<math> \begin{align}
| |
| T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; \gamma} = T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s , \gamma} & + \, \Gamma^{\alpha_1}{}_{\delta \gamma} T^{\delta \alpha_2 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} + \cdots + \Gamma^{\alpha_r}{}_{\delta \gamma} T^{\alpha_1 \cdots \alpha_{r-1} \delta}{}_{\beta_1 \cdots \beta_s} \\
| |
| & - \, \Gamma^\delta{}_{\beta_1 \gamma} T^{\alpha_1 \cdots \alpha_r}{}_{\delta \beta_2 \cdots \beta_s} - \cdots - \Gamma^\delta{}_{\beta_s \gamma} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_{s-1} \delta} \,.
| |
| \end{align}</math>
| |
| | |
| The components of this derivative of a tensor field transform covariantly, and hence form another tensor field. This derivative is characterized by the product rule and the fact that the derivative of the metric <math>g_{\mu \nu} \,</math> is zero:
| |
| | |
| :<math>g_{\mu \nu ; \gamma} = 0 \,.</math>
| |
| | |
| The covariant formulation of the [[directional derivative]] of any tensor field along a vector <math>v^\gamma</math> may be expressed as its contraction with the covariant derivative, e.g.:
| |
| | |
| :<math>v^\gamma A_{\alpha ;\gamma} \,.</math>
| |
| | |
| One alternative notation for the covariant derivative of any tensor is the subscripted nabla symbol <math>\nabla_\beta</math>. For the case of a vector field <math>A^\alpha</math>:<ref>{{cite book|title=Relativity|series=Demystified|isbn=0-07-145545-0|year=2006|author=D. McMahon|publisher=McGraw Hill|page=67}}</ref>
| |
| | |
| :<math>\nabla_\beta A^\alpha = \frac{\partial A^\alpha}{\partial x^\beta} + \Gamma^\alpha{}_{\gamma\beta}A^\gamma .</math>
| |
| | |
| ;[[Lie derivative]]
| |
| | |
| The Lie derivative is another derivative that is covariant, but which should not be confused with the ''covariant derivative''. It is defined even in the absence of a metric. The Lie derivative of a type (''r'',''s'') tensor field <math>T</math> along (the flow of) a contravariant vector field <math>X^\rho</math> [[Lie derivative#Coordinate expressions|may be expressed as]]<ref>{{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=S.I.| year=1968| title = Tensor Analysis on Manifolds|page=130}}</ref>
| |
| | |
| :<math> \begin{align}
| |
| (\mathcal{L}_X T)^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} = X^\gamma T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s , \gamma} & - \, X^{\alpha_1}{}_{, \gamma} T^{\gamma \alpha_2 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} - \cdots - X^{\alpha_r}{}_{, \gamma} T^{\alpha_1 \cdots \alpha_{r-1} \gamma}{}_{\beta_1 \cdots \beta_s} \\
| |
| & + \, X^{\gamma}{}_{, \beta_1} T^{\alpha_1 \cdots \alpha_r}{}_{\gamma \beta_2 \cdots \beta_s} + \cdots + X^{\gamma}{}_{, \beta_s} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_{s-1} \gamma} \,.
| |
| \end{align}</math>
| |
| | |
| This derivative is characterized by the product rule and the fact that the derivative of the given contravariant vector field <math>X^\rho</math> is zero.
| |
| | |
| :<math> (\mathcal{L}_X X)^{\rho} = [X, X]^{\rho} = 0 \,.</math>
| |
| | |
| The Lie derivative of a type (''r'',''s'') [[relative tensor]] field <math>\Lambda</math> of weight <math>w\,</math> along (the flow of) a contravariant vector field <math>X^\rho</math> [[Lie derivative#Coordinate expressions|may be expressed as]]<ref>{{cite book|last = Lovelock| first = David|coauthors = Hanno Rund|year= 1989|title = Tensors, Differential Forms, and Variational Principles|page=123}}</ref>
| |
| | |
| :<math> \begin{align}
| |
| (\mathcal{L}_X \Lambda)^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} = X^\gamma \Lambda^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s , \gamma} & - \, X^{\alpha_1}{}_{, \gamma} \Lambda^{\gamma \alpha_2 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} - \cdots - X^{\alpha_r}{}_{, \gamma} \Lambda^{\alpha_1 \cdots \alpha_{r-1} \gamma}{}_{\beta_1 \cdots \beta_s} \\
| |
| & + \, X^{\gamma}{}_{, \beta_1} \Lambda^{\alpha_1 \cdots \alpha_r}{}_{\gamma \beta_2 \cdots \beta_s} + \cdots + X^{\gamma}{}_{, \beta_s} \Lambda^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_{s-1} \gamma} \\
| |
| & + \, wX^{\gamma}{}_{, \gamma} \Lambda^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_{s}}\,.
| |
| \end{align}</math>
| |
| | |
| ==Notable tensors==
| |
| ;[[Kronecker delta]]
| |
| The Kronecker delta is like the [[identity matrix]]
| |
| :<math>\delta^{\alpha}_{\beta} \, A^{\beta} = A^{\alpha} \,</math>
| |
| :<math>\delta^{\mu}_{\nu} \, B_{\mu} = B_{\nu} \,</math>
| |
| when multiplied and contracted. The components <math>\delta^{\alpha}_{\beta}\,</math> are the same in any basis and form an invariant tensor of type (1,1), i.e. the identity of the [[tangent bundle]] over the [[identity mapping]] of the [[base manifold]], and so its trace is an invariant.<ref>{{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=S.I.| year=1968| title = Tensor Analysis on Manifolds|page=85}}</ref>
| |
| The dimensionality of [[spacetime]] is its [[Trace (linear algebra)|trace]]:
| |
| :<math>\delta^{\rho}_{\rho} = \delta^{0}_{0} + \delta^{1}_{1} + \delta^{2}_{2} + \delta^{3}_{3} = 4 \,</math>
| |
| in [[four-dimensional space]]time.
| |
| ;[[Metric tensor]]
| |
| | |
| The metric tensor gives the length of any [[space-like]] curve
| |
| :<math>\text{Length} = \int^{y_2}_{y_1} \sqrt{ g_{\alpha \beta} \frac{d x^{\alpha}}{d y} \frac{d x^{\beta}}{d y} } \, d y \,</math>
| |
| where ''y'' is any [[Smooth function|smooth]] [[Monotonic function|strictly monotone]] [[parameterization]] of the path. It also gives the duration of any [[time-like]] curve
| |
| :<math>\text{Duration} = \int^{t_2}_{t_1} \sqrt{ \frac{-1}{c^2} g_{\alpha \beta} \frac{d x^{\alpha}}{d t} \frac{d x^{\beta}}{d t} } \, d t \,</math>
| |
| where ''t'' is any smooth strictly monotone parameterization of the trajectory. See also [[line element]].
| |
| | |
| The [[inverse matrix]] (also indicated with a ''g'') of the metric tensor is another important tensor
| |
| :<math> g^{\alpha \beta} g_{\beta \gamma} = \delta^{\alpha}_{\gamma} \,.</math>
| |
| | |
| ;[[Riemann curvature tensor]]
| |
| If this tensor is defined as
| |
| :<math>R^\rho{}_{\sigma\mu\nu} = \Gamma^\rho{}_{\nu\sigma,\mu}
| |
| - \Gamma^\rho_{\mu\sigma,\nu}
| |
| + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma}
| |
| - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma} \,,</math>
| |
| then it is the [[commutator]] of the covariant derivative with itself:<ref>{{cite book |author=Synge J.L., Schild A.|publisher=first Dover Publications 1978 edition |title=Tensor Calculus |pages=83, p. 107|year= 1949}}</ref><ref>{{cite book |author=P. A. M. Dirac|pages=20–21| title=General Theory of Relativity| publisher= | year=| isbn=}}</ref>
| |
| :<math>A_{\nu ; \rho \sigma} - A_{\nu ; \sigma \rho} = A_{\beta} R^{\beta}{}_{\nu \rho \sigma} \,,</math>
| |
| since the [[Connection (mathematics)|connection]] <math>\Gamma^\alpha{}_{\beta\mu}\,</math> is torsionless, which means that the [[torsion tensor]] <math>\Gamma^\lambda{}_{\mu\nu}-\Gamma^\lambda{}_{\nu\mu}\,</math> vanishes.
| |
| | |
| ;Ricci identities
| |
| | |
| This can be generalized to get the commutator for two covariant derivatives of an arbitrary tensor as follows
| |
| :<math> \begin{align}
| |
| T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; \gamma \delta} - T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; \delta \gamma} = \, & - R^{\alpha_1}{}_{\rho \gamma \delta} T^{\rho \alpha_2 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s} - \cdots - R^{\alpha_r}{}_{\rho \gamma \delta} T^{\alpha_1 \cdots \alpha_{r-1} \rho}{}_{\beta_1 \cdots \beta_s} \\
| |
| & + \, R^\sigma{}_{\beta_1 \gamma \delta} T^{\alpha_1 \cdots \alpha_r}{}_{\sigma \beta_2 \cdots \beta_s} + \cdots + R^\sigma{}_{\beta_s \gamma \delta} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_{s-1} \sigma} \,
| |
| \end{align}</math>
| |
| which are often referred to as the ''Ricci identities''.<ref>{{cite book| last = Lovelock| first = David| coauthors = Hanno Rund| year= 1989|title = Tensors, Differential Forms, and Variational Principles|page=84}}</ref>
| |
| | |
| ==See also==
| |
| | |
| *[[Regge calculus]]
| |
| *[[Tensor (intrinsic definition)#Basis|Tensor (intrinsic definition), Section: Basis]]
| |
| *[[Holonomic basis]]
| |
| *[[Exterior algebra]]
| |
| *[[Exterior calculus]]
| |
| *[[Differential form]]
| |
| *[[Hodge dual]]
| |
| *[[Penrose graphical notation]]
| |
| *[[Ricci decomposition]]
| |
| | |
| ==References==
| |
| | |
| {{reflist}}
| |
| | |
| ==Books==
| |
| * {{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=S.I. | title = Tensor Analysis on Manifolds| publisher=The Macmillan Company | year=1968|edition=First Dover 1980|isbn=0-486-64039-6}}
| |
| *{{cite book
| |
| | last = Danielson
| |
| | first = Donald A.
| |
| | title = Vectors and Tensors in Engineering and Physics
| |
| | edition = 2/e
| |
| | year= 2003
| |
| | publisher = Westview (Perseus)
| |
| | isbn = 978-0-8133-4080-7
| |
| }}
| |
| *{{cite book
| |
| | last = Dimitrienko
| |
| | first = Yuriy
| |
| | title = Tensor Analysis and Nonlinear Tensor Functions
| |
| | year= 2002
| |
| | publisher = Kluwer Academic Publishers (Springer)
| |
| | url = http://books.google.com/books?as_isbn=140201015X
| |
| | isbn = 1-4020-1015-X
| |
| }}
| |
| *{{cite book
| |
| | last = Lovelock
| |
| | first = David
| |
| | coauthors = Hanno Rund
| |
| | title = Tensors, Differential Forms, and Variational Principles
| |
| | year= 1989
| |
| | publisher = Dover
| |
| | isbn = 978-0-486-65840-7
| |
| | origyear = 1975
| |
| }}
| |
| * {{citation | author=C. Møller| title = The Theory of Relativity| publisher=Oxford University Press|edition=3rd|year=1952|url=http://archive.org/details/theoryofrelativi029229mbp |isbn=}}
| |
| * {{cite book |author=Synge J.L., Schild A. |title=Tensor Calculus |publisher=first Dover Publications 1978 edition |year= 1949
| |
| |isbn=978-0-486-63612-2}}
| |
| * {{citation | author=J.R. Tyldesley| title = An introduction to Tensor Analysis: For Engineers and Applied Scientists| publisher=Longman| year=1975|isbn=0-582-44355-5}}
| |
| * {{citation | author=D.C. Kay| title = Tensor Calculus| publisher=Schaum’s Outlines, McGraw Hill (USA)|year=1988|isbn=0-07-033484-6}}
| |
| * {{citation | author=T. Frankel| title = The Geometry of Physics| publisher=Cambridge University Press|edition=3rd|year=2012|isbn=978-1107-602601}}
| |
| | |
| {{tensors}}
| |
| | |
| [[Category:Differential geometry]]
| |
| [[Category:Tensors]]
| |