# Bitensor

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{{Short description|Tensorial object depending on two points in a manifold}}
{{more citations needed|date=April 2025}}
In [differential geometry](/source/differential_geometry) and [general relativity](/source/general_relativity), a '''bitensor''' (or '''bi-tensor'''<ref>{{cite arXiv |author=Gökler, Can |title=Estimation theory and gravity |eprint=2003.02221 |class=quant-ph |date=2021-02-18}}</ref>) is a [tensorial](/source/tensorial) object that depends on two points in a [manifold](/source/manifold), as opposed to ordinary [tensors](/source/tensors) which depend on a single point.<ref name="Allen-1986">{{cite journal|last1=Allen|first1=Bruce|last2=Jacobson|first2=Theodore|title=Vector Two-Point Functions in Maximally Symmetric Spaces|journal=Communications in Mathematical Physics|volume=103|issue=4|pages=669–692|year=1986|publisher=[Springer-Verlag](/source/Springer-Verlag) |doi=10.1007/BF01211169 |bibcode=1986CMaPh.103..669A |hdl=11858/00-001M-0000-0013-5DC2-0|hdl-access=free}}</ref> Bitensors provide a framework for describing relationships between different points in [spacetime](/source/spacetime) and are used in the study of various phenomena in [curved spacetime](/source/curved_spacetime).

== Definition ==
A ''bitensor'' is a tensorial object that depends on two points in a manifold, rather than on a single point as ordinary tensors do.<ref name="Samuel-Lereah">{{cite web|url=https://samuel-lereah.com/articles/Physics/bitensors|title=Bitensors|access-date=2025-03-22}}</ref>
A ''bitensor field'' <math>B</math> can be formally defined as a [map](/source/map_(mathematics)) from the [product manifold](/source/Cartesian_product) to an appropriate [vector space](/source/vector_space) <math>B: M \times M \to V</math>, where <math>M</math> is a [smooth manifold](/source/smooth_manifold) and <math>V</math> is the vector space corresponding to the [tensor space](/source/tensor_space) being considered.<ref name="Samuel-Lereah"/><ref name="Allen-1986"/>

In the language of [fiber bundle](/source/fiber_bundle)s, a bitensor of type <math>(r,s,r',s')</math> is defined as a [section](/source/Section_(fiber_bundle)) of the [exterior](/source/exterior_product) [tensor product](/source/tensor_product) bundle <math>T^r_s M \boxtimes T^{r'}_{s'} M</math>, where <math>T^r_s M</math> denotes the tensor bundle of [rank](/source/tensor) <math>(r,s)</math> and <math>\boxtimes</math> represents the exterior tensor product <math>B \in \Gamma(T^r_s M \boxtimes T^{r'}_{s'} M)</math>, where <math>\Gamma</math> denotes the space of sections.<ref name="Samuel-Lereah"/>

The exterior tensor product bundle is constructed as <math>\mathcal{V}_1 \boxtimes \mathcal{V}_2 = \mathrm{pr}_1^* \mathcal{V}_1 \otimes \mathrm{pr}_2^* \mathcal{V}_2</math> where <math>\mathrm{pr}_i</math> are [projection operator](/source/projection_operator)s that project onto the respective factors of the product manifold <math>M \times M</math>, and <math>\mathrm{pr}_i^*</math> denotes the [pullback](/source/pullback) of the respective bundles.<ref name="Samuel-Lereah"/>

In coordinate notation, a bitensor <math>T</math> with components <math>T^{\mu\nu'\ldots}_{\alpha\beta'\ldots}(x,y)</math> has indices associated with two different points <math>x</math> and <math>y</math> in the manifold. By convention, unprimed indices (such as <math>\mu</math>, <math>\alpha</math>) refer to the first point, while primed indices (such as <math>\nu'</math>, <math>\beta'</math>) refer to the second point. The simplest example of a bitensor is a ''biscalar field'', which is a scalar function of two points. Applications include [parallel transport](/source/parallel_transport), [heat kernel](/source/heat_kernel)s, and various [Green's function](/source/Green's_function)s employed in quantum field theory in curved spacetime.<ref name="Samuel-Lereah"/><ref name="Allen-1986"/>

== History ==
The concept of bitensors was first formally developed by mathematician [Harold Stanley Ruse](/source/Harold_Stanley_Ruse) in his 1931 paper ''An Absolute Partial Differential Calculus'', published in the [Quarterly Journal of Mathematics](/source/Quarterly_Journal_of_Mathematics). Ruse introduced bitensors as a generalization of [tensor calculus](/source/tensor_calculus) to functions of two sets of variables, drawing an analogy with [partial differentiation](/source/partial_differentiation) in [elementary calculus](/source/calculus). He developed the formalism for bitensor transformations, [covariant derivative](/source/covariant_derivative)s, and scalar connections, establishing the foundation for what he termed an "absolute partial differential calculus."<ref name="Ruse-1931">{{cite journal|last=Ruse|first=Harold|title=An Absolute Partial Differential Calculus|journal=[The Quarterly Journal of Mathematics](/source/The_Quarterly_Journal_of_Mathematics)|volume=os-2|issue=1|year=1931|pages=190–202|doi=10.1093/qmath/os-2.1.190}}</ref><ref>{{cite journal|last1=Procopio|first1=Giuseppe|last2=Giona|first2=Massimiliano|title=Bitensorial formulation of the singularity method for Stokes flows|journal=Mathematics in Engineering|volume=5|issue=2|year=2022|pages=1–34|doi=10.3934/mine.2023046|url=http://www.aimspress.com/journal/mine|hdl=11573/1651830|hdl-access=free}}</ref>

== See also ==
* [Parallel transport](/source/Parallel_transport)
* [Pullback](/source/Pullback)
* [Propagator](/source/Propagator)
* [Riemann curvature tensor](/source/Riemann_curvature_tensor)
* [Stokes flow](/source/Stokes_flow)
* [Synge's world function](/source/Synge's_world_function)

== References ==
{{reflist}}

Category:Concepts in physics
Category:Tensors
Category:General relativity
Category:Differential geometry

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Adapted from the Wikipedia article [Bitensor](https://en.wikipedia.org/wiki/Bitensor) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Bitensor?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.