You do not have permission to edit this page, for the following reasons:

This IP address has been blocked from editing Wikipedia.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

The IP address or range 44.202.0.0/16 has been blocked (disabled) by ‪Blablubbs‬ for the following reason(s):

The IP address that you are currently using has been blocked because it is believed to be a web host provider or colocation provider. To prevent abuse, web hosts and colocation providers may be blocked from editing Wikipedia.
You will not be able to edit Wikipedia using a web host or colocation provider because it hides your IP address, much like a proxy or VPN.
We recommend that you attempt to use another connection to edit. For example, if you use a proxy or VPN to connect to the internet, turn it off when editing Wikipedia. If you edit using a mobile connection, try using a Wi-Fi connection, and vice versa. If you are using a corporate internet connection, switch to a different Wi-Fi network. If you have a Wikipedia account, please log in.
If you do not have any other way to edit Wikipedia, you will need to request an IP block exemption.

How to appeal if you are confident that your connection does not use a colocation provider's IP address:
If you are confident that you are not using a web host, you may appeal this block by adding the following text on your talk page: ((unblock|reason=Caught by a colocation web host block but this host or IP is not a web host. My IP address is _______. Place any further information here. ~~~~)). You must fill in the blank with your IP address for this block to be investigated. Your IP address can be determined here. Alternatively, if you wish to keep your IP address private you can use the unblock ticket request system. There are several reasons you might be editing using the IP address of a web host or colocation provider (such as if you are using VPN software or a business network); please use this method of appeal only if you think your IP address is in fact not a web host or colocation provider.

Administrators: The IP block exemption user right should only be applied to allow users to edit using web host in exceptional circumstances, and requests should usually be directed to the functionaries team via email. If you intend to give the IPBE user right, a CheckUser needs to take a look at the account. This can be requested most easily at SPI Quick Checkuser Requests. Unblocking an IP or IP range with this template is highly discouraged without at least contacting the blocking administrator.

This block will expire on 21:04, 17 October 2024.

Even when blocked, you will usually still be able to edit your user talk page, as well as email administrators and other editors.

For information on how to proceed, please read the FAQ for blocked users and the guideline on block appeals. The guide to appealing blocks may also be helpful.

Other useful links: Blocking policy · Help:I have been blocked
Your IP address is in a range that has been blocked on all Wikimedia Foundation wikis. The block was made by ‪AntiCompositeNumber‬. The reason given is Open proxy/Webhost: Visit the FAQ if you are affected .
- Start of block: 13:55, 18 August 2022
- Expiry of block: 13:55, 18 September 2024
Your current IP address is 44.202.229.202. The blocked range is 44.202.0.0/16. Please include all above details in any queries you make. If you believe you were blocked by mistake, you can find additional information and instructions in the No open proxies global policy. Otherwise, to discuss the block please post a request for review on Meta-Wiki. You could also send an email to the stewards VRT queue at stewards@wikimedia.org including all above details.

You can view and copy the source of this page:

((Use American English|date = March 2019))
((Short description|Generalization of tensor fields))
In [[differential geometry]], a  '''tensor density''' or '''relative tensor''' is a generalization of the [[tensor field]] concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see [[tensor field]]), except that it is additionally multiplied or ''weighted'' by a power ''W'' of the [[Jacobian determinant]] of the coordinate transition function or its absolute value. A tensor density with a single index is called a '''vector density'''. A distinction is made among (authentic) tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. Sometimes tensor densities with a negative weight ''W'' are called '''tensor capacity.'''<ref>((cite book
 | last = Weinreich
 | first = Gabriel
 | title = Geometrical Vectors
 | date = July 6, 1998
 | pages = 112, 115
 | publisher = University of Chicago Press
 | language = en
 | isbn = 978-0226890487
 ))</ref><ref>((cite book
 | last = Papastavridis
 | first = John G.
 | title = Tensor Calculus and Analytical Dynamics
 | publisher = [[CRC Press]]
 | date = Dec 18, 1998
 | language = en
 | isbn = 978-0849385148
 ))</ref><ref>((cite book
 | last = Ruiz-Tolosa, Castillo
 | first = Juan R., Enrique
 | title = From Vectors to Tensors
 | publisher = Springer Science & Business Media
 | date = 30 Mar 2006
 | language = en
 | isbn = 978-3540228875
 ))</ref> A tensor density can also be regarded as a [[Section (fiber bundle)|section]] of the [[tensor product]] of a [[tensor bundle]] with a [[density bundle]].

==Motivation==
In physics and related fields, it is often useful to work with the components of an algebraic object rather than the object itself.  An example would be decomposing a vector into a sum of [[Basis (linear algebra)|basis]] vectors weighted by some coefficients such as 
<math display=block>\vec{v} = c_1 \vec{e}_1 + c_2 \vec e_2 + c_ 3\vec e_3</math>  
where <math>\vec v</math> is a vector in 3-dimensional [[Euclidean space]], <math>c_i \in \R^1 \text{ and } \vec e_i</math> are the usual standard basis vectors in Euclidean space.  This is usually necessary for computational purposes, and can often be insightful when algebraic objects represent complex abstractions but their components have concrete interpretations.  However, with this identification, one has to be careful to track changes of the underlying basis in which the quantity is expanded; it may in the course of a computation become expedient to [[Change of basis|change the basis]] while the vector <math>\vec v</math> remains fixed in physical space.  More generally, if an algebraic object represents a geometric object, but is expressed in terms of a particular basis, then it is necessary to, when the basis is changed, also change the representation.  Physicists will often call this representation of a geometric object a [[tensor]] if it transforms under a sequence of [[linear map]]s given a linear change of basis (although confusingly others call the underlying geometric object which hasn't changed under the coordinate transformation a "tensor", a convention this article strictly avoids).  In general there are representations which transform in arbitrary ways depending on how the geometric invariant is reconstructed from the representation.  In certain special cases it is convenient to use representations which transform almost like tensors, but with an additional, nonlinear factor in the transformation.  A prototypical example is a matrix representing the cross product (area of spanned parallelogram) on <math>\R^2.</math>  The representation is given by in the standard basis by 
<math display=block>
  \vec u \times \vec v =
  \begin{bmatrix} u_1& u_2 \end{bmatrix} \begin{bmatrix}0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix}v_1 \\ v_2 \end{bmatrix} =
  u_1 v_2 - u_2 v_1
</math>

If we now try to express this same expression in a basis other than the standard basis, then the components of the vectors will change, say according to <math display=inline>\begin{bmatrix} u'_1 & u'_2 \end{bmatrix}^\textsf{T} = A \begin{bmatrix} u_1 & u_2 \end{bmatrix}^\textsf{T}</math> where <math>A</math> is some 2 by 2 matrix of real numbers.  Given that the area of the spanned parallelogram is a geometric invariant, it cannot have changed under the change of basis, and so the new representation of this matrix must be:
<math display=block>\left(A^{-1}\right)^\textsf{T} \begin{bmatrix}0 & 1 \\ -1 & 0 \end{bmatrix} A^{-1}</math>
which, when expanded is just the original expression but multiplied by the determinant of <math>A^{-1},</math> which is also <math display=inline>\frac{1}{\det A}.</math>  In fact this representation could be thought of as a two index tensor transformation, but instead, it is computationally easier to think of the tensor transformation rule as multiplication by <math display=inline>\frac{1}{\det A},</math> rather than as 2 matrix multiplications (In fact in higher dimensions, the natural extension of this is <math>n, n \times n</math> matrix multiplications, which for large <math>n</math> is completely infeasible).  Objects which transform in this way are called ''tensor densities'' because they arise naturally when considering problems regarding areas and volumes, and so are frequently used in integration.

==Definition==
((Refimprove|date=September 2012))
Some authors classify tensor densities into the two types called (authentic) tensor densities and pseudotensor densities in this article.  Other authors classify them differently, into the types called even tensor densities and odd tensor densities.  When a tensor density weight is an integer there is an equivalence between these approaches that depends upon whether the integer is even or odd.

Note that these classifications elucidate the different ways that tensor densities may transform somewhat pathologically under orientation-''reversing'' coordinate transformations.  Regardless of their classifications into these types, there is only one way that tensor densities transform under orientation-''preserving'' coordinate transformations.

In this article we have chosen the convention that assigns a weight of +2 to <math>g = \det\left(g_{\rho\sigma}\right)</math>, the determinant of the metric tensor expressed with [[Covariance and contravariance of vectors|covariant]] indices. With this choice, classical densities, like charge density, will be represented by tensor densities of weight +1. Some authors use a sign convention for weights that is the negation of that presented here.<ref>E.g. ((harvnb|Weinberg|1972)) pp 98. The chosen convention involves in the formulae below the [[Jacobian determinant]] of the inverse transition ((math|''x'' → ((overbar|''x'')))), while the opposite convention considers the forward transition ((math|((overbar|''x'')) → ''x'')) resulting in a flip of sign of the weight.</ref>

In contrast to the meaning used in this article, in general relativity "[[pseudotensor]]" sometimes means an object that does not transform like a tensor or relative tensor of any weight.

=== Tensor and pseudotensor densities ===
For example, a mixed rank-two (authentic) tensor density of weight <math>W</math> transforms as:<ref>((Cite book|author1=M.R. Spiegel|author2=S. Lipcshutz|author3=D. Spellman|title=Vector Analysis|edition=2nd|publisher=Schaum's Outline Series|location=New York|year=2009|page=198|isbn=978-0-07-161545-7))</ref><ref>((Cite book|title=McGraw Hill Encyclopaedia of Physics|edition=2nd|author=C.B. Parker|year=1994|page=[https://archive.org/details/mcgrawhillencycl1993park/page/1417 1417]|publisher=McGraw-Hill |isbn=0-07-051400-3|url=https://archive.org/details/mcgrawhillencycl1993park/page/1417))</ref>

:<math>
{\mathfrak{T))^\alpha_\beta =
\left( \det{\left[\frac{\partial \bar{x}^{\iota)){\partial {x}^{\gamma))\right]} \right)^{W} \, \frac{\partial {x}^{\alpha)){\partial \bar{x}^{\delta)) \, \frac{\partial \bar{x}^{\epsilon)){\partial {x}^{\beta)) \, \bar{\mathfrak{T))^{\delta}_{\epsilon}
\,,</math> &nbsp;&nbsp;&nbsp;&nbsp;'''((authentic) tensor density of (integer) weight ''W'')'''
where <math>\bar{\mathfrak{T))</math> is the rank-two tensor density in the <math>\bar{x}</math> coordinate system, <math>{\mathfrak{T))</math> is the transformed tensor density in the <math>{x}</math> coordinate system; and we use the [[Jacobian determinant]].  Because the determinant can be negative, which it is for an orientation-reversing coordinate transformation, this formula is applicable only when <math>W</math> is an integer. (However, see even and odd tensor densities below.)

We say that a tensor density is a pseudotensor density when there is an additional sign flip under an orientation-reversing coordinate transformation.  A mixed rank-two pseudotensor density of weight <math>W</math> transforms as
:<math>
{\mathfrak{T))^\alpha_\beta =
\sgn\left( \det{\left[\frac{\partial \bar{x}^{\iota)){\partial {x}^{\gamma))\right]} \right)
\left( \det{\left[\frac{\partial \bar{x}^{\iota)){\partial {x}^{\gamma))\right]} \right)^{W} \, \frac{\partial {x}^{\alpha)){\partial \bar{x}^{\delta)) \, \frac{\partial \bar{x}^{\epsilon)){\partial {x}^{\beta)) \, \bar{\mathfrak{T))^{\delta}_{\epsilon}
\,,</math> &nbsp;&nbsp;&nbsp;&nbsp;'''(pseudotensor density of (integer) weight ''W'')'''

where [[Sign function|sgn]](&nbsp;) is a function that returns +1 when its argument is positive or −1 when its argument is negative.

=== Even and odd tensor densities ===
The transformations for even and odd tensor densities have the benefit of being well defined even when <math>W</math> is not an integer.  Thus one can speak of, say, an odd tensor density of weight +2 or an even tensor density of weight −1/2.

When <math>W</math> is an even integer the above formula for an (authentic) tensor density can be rewritten as
:<math>
{\mathfrak{T))^\alpha_\beta =
\left\vert \det{\left[\frac{\partial \bar{x}^{\iota)){\partial {x}^{\gamma))\right]} \right\vert^{W} \, \frac{\partial {x}^{\alpha)){\partial \bar{x}^{\delta)) \, \frac{\partial \bar{x}^{\epsilon)){\partial {x}^{\beta)) \, \bar{\mathfrak{T))^{\delta}_{\epsilon}
\,.</math> &nbsp;&nbsp;&nbsp;&nbsp;'''(even tensor density of weight ''W'')'''

Similarly, when <math>W</math> is an odd integer the formula for an (authentic) tensor density can be rewritten as
:<math>
{\mathfrak{T))^\alpha_\beta =
\sgn \left( \det{\left[\frac{\partial \bar{x}^{\iota)){\partial {x}^{\gamma))\right]} \right)
\left\vert \det{\left[\frac{\partial \bar{x}^{\iota)){\partial {x}^{\gamma))\right]} \right\vert^{W} \, \frac{\partial {x}^{\alpha)){\partial \bar{x}^{\delta)) \, \frac{\partial \bar{x}^{\epsilon)){\partial {x}^{\beta)) \, \bar{\mathfrak{T))^{\delta}_{\epsilon}
\,.</math> &nbsp;&nbsp;&nbsp;&nbsp;'''(odd tensor density of weight ''W'')'''

=== Weights of zero and one ===
A tensor density of any type that has weight zero is also called an '''absolute tensor'''.  An (even) authentic tensor density of weight zero is also called an '''ordinary tensor'''.

If a weight is not specified but the word "relative" or "density" is used in a context where a specific weight is needed, it is usually assumed that the weight is +1.

=== Algebraic properties ===
#A [[linear combination]] (also known as a [[weighted sum]]) of tensor densities of the same type and weight <math>W</math> is again a tensor density of that type and weight.
#A product of two tensor densities of any types, and with weights <math>W_1</math> and <math>W_2</math>, is a tensor density of weight <math>W_1 + W_2.</math> 
#:A product of authentic tensor densities and pseudotensor densities will be an authentic tensor density when an even number of the factors are pseudotensor densities; it will be a pseudotensor density when an odd number of the factors are pseudotensor densities.  Similarly, a product of even tensor densities and odd tensor densities will be an even tensor density when an even number of the factors are odd tensor densities; it will be an odd tensor density when an odd number of the factors are odd tensor densities.
#The contraction of indices on a tensor density with weight <math>W</math> again yields a tensor density of weight <math>W.</math><ref>((harvnb|Weinberg|1972)) p 100.</ref>
#Using (2) and (3) one sees that raising and lowering indices using the metric tensor (weight 0) leaves the weight unchanged.<ref>((harvnb|Weinberg|1972)) p 100.</ref>

=== Matrix inversion and matrix determinant of tensor densities ===
If <math>{\mathfrak{T))_{\alpha\beta}</math> is a non-singular matrix and a rank-two tensor density of weight <math>W</math> with covariant indices then its matrix inverse will be a rank-two tensor density of weight −<math>W</math> with contravariant indices.  Similar statements apply when the two indices are contravariant or are mixed covariant and contravariant.

If <math>{\mathfrak{T))_{\alpha\beta}</math> is a rank-two tensor density of weight <math>W</math> with covariant indices then the matrix determinant <math>\det {\mathfrak{T))_{\alpha\beta}</math> will have weight <math>N W + 2,</math> where <math>N</math> is the number of space-time dimensions.  If <math>{\mathfrak{T))^{\alpha\beta}</math> is a rank-two tensor density of weight <math>W</math> with contravariant indices then the matrix determinant <math>\det {\mathfrak{T))^{\alpha\beta}</math> will have weight <math>N W - 2.</math>  The matrix determinant <math>\det {\mathfrak{T))^{\alpha}_{~\beta}</math> will have weight <math>N W.</math>

== General relativity ==
((General relativity sidebar))

===Relation of Jacobian determinant and metric tensor===
Any non-singular ordinary tensor <math>T_{\mu\nu}</math> transforms as
<math display=block>T_{\mu\nu} = \frac{\partial \bar{x}^\kappa}{\partial {x}^\mu} \bar{T}_{\kappa\lambda} \frac{\partial \bar{x}^\lambda}{\partial {x}^\nu} \,,</math>

where the right-hand side can be viewed as the product of three matrices.  Taking the determinant of both sides of the equation (using that the determinant of a matrix product is the product of the determinants), dividing both sides by <math>\det\left(\bar{T}_{\kappa\lambda}\right),</math> and taking their square root gives
<math display=block>
  \left\vert \det{\left[\frac{\partial \bar{x}^\iota}{\partial {x}^\gamma}\right]} \right\vert =
  \sqrt{\frac{\det({T}_{\mu\nu})}{\det\left(\bar{T}_{\kappa\lambda}\right)))\,.
</math>

When the tensor <math>T</math> is the [[metric tensor]], <math>{g}_{\kappa\lambda},</math> and <math>\bar{x}^\iota</math> is a locally inertial coordinate system where <math>\bar{g}_{\kappa\lambda} = \eta_{\kappa\lambda} =</math>((nbsp))diag(−1,+1,+1,+1), the [[Minkowski metric]], then <math>\det\left(\bar{g}_{\kappa\lambda}\right) = \det(\eta_{\kappa\lambda}) =</math>((nbsp))−1 and so
<math display=block>
  \left\vert \det{\left[\frac{\partial \bar{x}^{\iota)){\partial {x}^{\gamma))\right]} \right\vert =
  \sqrt{-{g))\,,
</math>

where <math>{g} = \det\left({g}_{\mu\nu}\right)</math> is the determinant of the metric tensor <math>{g}_{\mu\nu}.</math>

===Use of metric tensor to manipulate tensor densities===
Consequently, an even tensor density, <math>\mathfrak{T}^{\mu \dots}_{\nu \dots},</math> of weight ''W'', can be written in the form
<math display=block>\mathfrak{T}^{\mu \dots}_{\nu \dots} = \sqrt{-g}\;^W T^{\mu \dots}_{\nu \dots} \,,</math>

where <math>T^{\mu \dots}_{\nu \dots} \,</math> is an ordinary tensor.  In a locally inertial coordinate system, where <math>g_{\kappa\lambda} = \eta_{\kappa\lambda},</math> it will be the case that <math>\mathfrak{T}^{\mu \dots}_{\nu \dots}</math> and <math>T^{\mu \dots}_{\nu \dots} \,</math> will be represented with the same numbers.

When using the metric connection ([[Levi-Civita connection]]), the [[covariant derivative]] of an even tensor density is defined as
<math display=block>
  \mathfrak{T}^{\mu \dots}_{\nu \dots ; \alpha} =
  \sqrt{-g}\;^W T^{\mu \dots}_{\nu \dots ; \alpha} =
  \sqrt{-g}\;^W \left(\sqrt{-g}\;^{-W} \mathfrak{T}^{\mu \dots}_{\nu \dots}\right)_{;\alpha} \,.
</math>

For an arbitrary connection, the covariant derivative is defined by adding an extra term, namely
<math display=block>-W \, \Gamma^{\delta}_{~\delta \alpha} \, \mathfrak{T}^{\mu \dots}_{\nu \dots}</math>
to the expression that would be appropriate for the covariant derivative of an ordinary tensor.

Equivalently, the product rule is obeyed
<math display=block>
  \left(\mathfrak{T}^{\mu \dots}_{\nu \dots} \mathfrak{S}^{\sigma \dots}_{\tau \dots}\right)_{; \alpha} =
  \left(\mathfrak{T}^{\mu \dots}_{\nu \dots; \alpha}\right) \mathfrak{S}^{\sigma \dots}_{\tau \dots} +
    \mathfrak{T}^{\mu \dots}_{\nu \dots} \left(\mathfrak{S}^{\sigma \dots}_{\tau \dots; \alpha}\right) \,,
</math>

where, for the metric connection, the covariant derivative of any function of <math>g_{\kappa\lambda}</math> is always zero,
<math display=block>\begin{align}
  g_{\kappa\lambda ; \alpha} & = 0 \\
  \left(\sqrt{-g}\;^W\right)_{; \alpha} & = \left(\sqrt{-g}\;^W\right)_{, \alpha} - W \Gamma^{\delta}_{~\delta \alpha} \sqrt{-g}\;^W =
    \frac W2 g^{\kappa\lambda} g_{\kappa\lambda,\alpha} \sqrt{-g}\;^W - W \Gamma^{\delta}_{~\delta \alpha} \sqrt{-g}\;^W = 0 \,.
\end{align}</math>

==Examples==
((see also|Maxwell's equations in curved spacetime))

The expression <math>\sqrt{-g}</math> is a scalar density.  By the convention of this article it has a weight of +1.  <!-- 
Under the usual convention that <math>\sqrt{x^2} = |x|</math> when ''x'' is a real number, the scalar density <math>\sqrt{-g}</math> transforms as an (even) pseudoscalar density rather than as an (odd) authentic scalar density:
<math display=block>\sqrt{-\det\left(g_{\alpha\beta}\right)}
=\sqrt{-\det\left(\frac{\partial \bar{x}^\mu}{\partial x^\alpha}\bar{g}_{\mu\nu}\frac{\partial \bar{x}^\nu}{\partial x^\beta}\right)}
=\sqrt{-\det\left(\bar{g}_{\mu\nu}\right) \det\left(\frac{\partial \bar{x}^\mu}{\partial x^\alpha}\right)^2}
=\left|\det\left(\frac{\partial \bar{x}^\mu}{\partial x^\alpha}\right)\right| \sqrt{-\det\left(\bar{g}_{\mu\nu}\right)}
\,.</math>
However, this distinction is not made by many authors, especially in discussions focused on orientation-preserving coordinate transformations, in which the distinction is moot.
-->

The density of electric current <math>\mathfrak{J}^\mu</math> (for example, <math>\mathfrak{J}^2</math> is the amount of electric charge crossing the 3-volume element <math>d x^3 \, d x^4 \, d x^1</math> divided by that element — do not use the metric in this calculation) is a contravariant vector density of weight +1.  It is often written as <math>\mathfrak{J}^\mu = J^\mu \sqrt{-g}</math> or <math>\mathfrak{J}^\mu = \varepsilon^{\mu\alpha\beta\gamma} \mathcal{J}_{\alpha\beta\gamma} / 3!,</math> where <math>J^\mu\,</math> and the [[differential form]] <math>\mathcal{J}_{\alpha\beta\gamma}</math> are <!-- (even) authentic --> absolute tensors<!-- , whereby <math>\mathfrak{J}^\mu</math> has the same (even) pseudo vs. (odd) authentic type as <math>\sqrt{-g}</math> -->, and where <math>\varepsilon^{\mu\alpha\beta\gamma}</math> is the [[Levi-Civita symbol]]; see below.

The density of [[Lorentz force]] <math>\mathfrak{f}_\mu</math> (that is, the linear momentum transferred from the electromagnetic field to matter within a 4-volume element <math>d x^1 \, d x^2 \, d x^3 \, d x^4</math> divided by that element — do not use the metric in this calculation) is a covariant vector density of weight +1.  <!-- It has the same (even) pseudo vs. (odd) authentic type as <math>\mathfrak{J}^\mu.</math> -->

In ''N''-dimensional space-time, the [[Levi-Civita symbol]] may be regarded as either a rank-''N'' covariant (odd) authentic tensor density of weight −1 (((math|''ε''<sub>''α''<sub>1</sub>⋯''α''<sub>''N''</sub></sub>))) or a rank-''N'' contravariant (odd) authentic tensor density of weight +1 (((math|''ε''<sup>''α''<sub>1</sub>⋯''α''<sub>''N''</sub></sup>))). Notice that the Levi-Civita symbol (so regarded) does ((em|not)) obey the usual convention for raising or lowering of indices with the metric tensor. That is, it is true that
<math display=block>\varepsilon^{\alpha\beta\gamma\delta}\,g_{\alpha\kappa}\,g_{\beta\lambda}\,g_{\gamma\mu}g_{\delta\nu} \,=\, \varepsilon_{\kappa\lambda\mu\nu}\,g \,,</math>
but in general relativity, where <math>g = \det\left(g_{\rho\sigma}\right)</math> is always negative, this is never equal to <math>\varepsilon_{\kappa\lambda\mu\nu}.</math>

The [[determinant]] of the metric tensor,
<math display=block>g = \det\left(g_{\rho\sigma}\right) = \frac{1}{4!} \varepsilon^{\alpha\beta\gamma\delta} \varepsilon^{\kappa\lambda\mu\nu} g_{\alpha\kappa} g_{\beta\lambda} g_{\gamma\mu} g_{\delta\nu}\,,</math>
is an (even) authentic scalar density of weight +2, being the contraction of the product of 2 (odd) authentic tensor densities of weight +1 and four (even) authentic tensor densities of weight 0.

==See also==
((cols|colwidth=26em))
* ((annotated link|Action (physics)))
* ((annotated link|Conservation law))
* ((annotated link|Noether's theorem))
* ((annotated link|Pseudotensor))
* ((annotated link|Relative scalar))
* ((annotated link|Variational principle))
((colend))

==Notes==
((reflist))
((reflist|group=note))

==References==
* ((citation|first=Michael|last=Spivak|author-link=Michael Spivak|title=A Comprehensive Introduction to Differential Geometry, Vol I|edition=3rd|year=1999|page=134)).
* ((springer|id=t/t092390|title=Tensor Density|first=L.P.|last=Kuptsov)).
* ((cite book | author=[[Charles Misner]]; [[Kip S Thorne]] & [[John Archibald Wheeler]] | title=[[Gravitation (book)|Gravitation]]  | publisher=[[W. H. Freeman]] | year=1973 | isbn=0-7167-0344-0|page=501ff))
*((citation|last=Weinberg|first=Steven|year=1972|title=Gravitation and Cosmology|isbn=0-471-92567-5|publisher=John Wiley & sons, Inc|author-link=Steven Weinberg|url=https://archive.org/details/gravitationcosmo00stev_0))

((Tensors))
((Manifolds))

[[Category:Differential geometry]]
[[Category:Tensors|Density]]
[[Category:Tensors in general relativity|Density]]

Pages transcluded onto the current version of this page (help):

Action (physics) (edit)
Conservation law (edit)
Noether's theorem (edit)
Pseudotensor (edit)
Relative scalar (edit)
Tensor density (edit)
Variational principle (edit)
Template:Ambox (view source) (template editor protected)
Template:Annotated link (view source) (template editor protected)
Template:Category-inline (view source) (template editor protected)
Template:Citation (view source) (protected)
Template:Cite book (view source) (protected)
Template:Colend (view source) (template editor protected)
Template:Cols (view source) (semi-protected)
Template:DMCA (view source) (template editor protected)
Template:Dated maintenance category (view source) (template editor protected)
Template:Dated maintenance category (articles) (view source) (template editor protected)
Template:Div col (view source) (template editor protected)
Template:Div col/styles.css (view source) (template editor protected)
Template:Div col end (view source) (template editor protected)
Template:Em (view source) (template editor protected)
Template:FULLROOTPAGENAME (view source) (template editor protected)
Template:Find sources mainspace (view source) (template editor protected)
Template:General relativity sidebar (edit)
Template:Harvard citation no brackets (view source) (template editor protected)
Template:Harvnb (view source) (template editor protected)
Template:Hlist (view source) (template editor protected)
Template:Hlist/styles.css (view source) (protected)
Template:Icon (view source) (template editor protected)
Template:Main other (view source) (protected)
Template:Manifolds (edit)
Template:Math (view source) (template editor protected)
Template:More citations needed (view source) (template editor protected)
Template:Navbox (view source) (template editor protected)
Template:Nbsp (view source) (protected)
Template:Nowrap (view source) (protected)
Template:Ns has subpages (view source) (protected)
Template:Overbar (edit)
Template:Overline (view source) (template editor protected)
Template:Pagetype (view source) (protected)
Template:Plainlist/styles.css (view source) (protected)
Template:Portal-inline (view source) (template editor protected)
Template:Refimprove (view source) (template editor protected)
Template:Reflist (view source) (protected)
Template:Reflist/styles.css (view source) (protected)
Template:Replace (view source) (protected)
Template:SDcat (view source) (template editor protected)
Template:See also (view source) (template editor protected)
Template:Short description (view source) (protected)
Template:Short description/lowercasecheck (view source) (protected)
Template:Sidebar (view source) (template editor protected)
Template:Sidebar with collapsible lists (view source) (template editor protected)
Template:Small (view source) (protected)
Template:Smaller (view source) (template editor protected)
Template:Spaces (view source) (protected)
Template:Springer (view source) (semi-protected)
Template:SpringerEOM (view source) (semi-protected)
Template:Subinfobox bodystyle (view source) (template editor protected)
Template:Subsidebar bodystyle (view source) (template editor protected)
Template:Tensors (edit)
Template:Use American English (view source) (template editor protected)
Module:Annotated link (view source) (template editor protected)
Module:Arguments (view source) (protected)
Module:Category handler (view source) (protected)
Module:Category handler/blacklist (view source) (protected)
Module:Category handler/config (view source) (protected)
Module:Category handler/data (view source) (protected)
Module:Category handler/shared (view source) (protected)
Module:Check for unknown parameters (view source) (protected)
Module:Citation/CS1 (view source) (protected)
Module:Citation/CS1/COinS (view source) (protected)
Module:Citation/CS1/Configuration (view source) (protected)
Module:Citation/CS1/Date validation (view source) (protected)
Module:Citation/CS1/Identifiers (view source) (protected)
Module:Citation/CS1/Utilities (view source) (protected)
Module:Citation/CS1/Whitelist (view source) (protected)
Module:Citation/CS1/styles.css (view source) (protected)
Module:Disambiguation (view source) (protected)
Module:Disambiguation/templates (view source) (protected)
Module:Find sources (view source) (template editor protected)
Module:Find sources/config (view source) (template editor protected)
Module:Find sources/links (view source) (template editor protected)
Module:Find sources/templates/Find sources mainspace (view source) (template editor protected)
Module:Footnotes (view source) (template editor protected)
Module:Footnotes/anchor id list (view source) (template editor protected)
Module:Footnotes/anchor id list/data (view source) (template editor protected)
Module:Footnotes/whitelist (view source) (template editor protected)
Module:Format link (view source) (template editor protected)
Module:GetShortDescription (view source) (template editor protected)
Module:Hatnote (view source) (template editor protected)
Module:Hatnote/styles.css (view source) (template editor protected)
Module:Hatnote list (view source) (template editor protected)
Module:Icon (view source) (template editor protected)
Module:Icon/data (view source) (template editor protected)
Module:Labelled list hatnote (view source) (template editor protected)
Module:Lang (view source) (template editor protected)
Module:Lang/ISO 639 synonyms (view source) (template editor protected)
Module:Lang/data (view source) (template editor protected)
Module:Language/data/iana languages (view source) (template editor protected)
Module:Language/data/iana regions (view source) (template editor protected)
Module:Language/data/iana scripts (view source) (template editor protected)
Module:Language/data/iana suppressed scripts (view source) (template editor protected)
Module:Language/data/iana variants (view source) (template editor protected)
Module:List (view source) (protected)
Module:Message box (view source) (protected)
Module:Message box/ambox.css (view source) (protected)
Module:Message box/configuration (view source) (protected)
Module:Namespace detect/config (view source) (protected)
Module:Namespace detect/data (view source) (protected)
Module:Navbar (view source) (protected)
Module:Navbar/configuration (view source) (protected)
Module:Navbar/styles.css (view source) (protected)
Module:Navbox (view source) (template editor protected)
Module:Navbox/configuration (view source) (template editor protected)
Module:Navbox/styles.css (view source) (template editor protected)
Module:Ns has subpages (view source) (template editor protected)
Module:Pagetype (view source) (protected)
Module:Pagetype/config (view source) (protected)
Module:Portal (view source) (template editor protected)
Module:Portal-inline (view source) (template editor protected)
Module:Portal/images/p (view source) (template editor protected)
Module:Redirect (view source) (protected)
Module:SDcat (view source) (template editor protected)
Module:Sidebar (view source) (template editor protected)
Module:Sidebar/configuration (view source) (template editor protected)
Module:Sidebar/styles.css (view source) (template editor protected)
Module:String (view source) (protected)
Module:TableTools (view source) (protected)
Module:Template wrapper (view source) (template editor protected)
Module:Unicode data (view source) (template editor protected)
Module:Unsubst (view source) (protected)
Module:Yesno (view source) (protected)

Return to Tensor density.

Retrieved from "https://en.wikipedia.org/wiki/Tensor_density"