Bitensor

In differential geometry and general relativity, a bitensor is a tensorial object that depends on two points in a manifold, as opposed to ordinary tensors which depend on a single point. Bitensors provide a framework for describing relationships between different points in spacetime and are used in the study of various phenomena in 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. A bitensor field $B$ can be formally defined as a map from the product manifold to an appropriate vector space $B:M\times M\to V$, where $M$ is a smooth manifold and $V$ is the vector space corresponding to the tensor space being considered.

In the language of fiber bundles, a bitensor of type $(r,s,r',s')$ is defined as a section of the exterior tensor product bundle $T_{s}^{r}M\boxtimes T_{s'}^{r'}M$, where $T_{s}^{r}M$ denotes the tensor bundle of rank $(r,s)$ and $\boxtimes$ represents the exterior tensor product $B\in \Gamma (T_{s}^{r}M\boxtimes T_{s'}^{r'}M)$, where $\Gamma$ denotes the space of sections.

The exterior tensor product bundle is constructed as ${\mathcal {V}}_{1}\boxtimes {\mathcal {V}}_{2}=\mathrm {pr} _{1}^{*}{\mathcal {V}}_{1}\otimes \mathrm {pr} _{2}^{*}{\mathcal {V}}_{2}$ where $\mathrm {pr} _{i}$ are projection operators that project onto the respective factors of the product manifold $M\times M$, and $\mathrm {pr} _{i}^{*}$ denotes the pullback of the respective bundles.

In coordinate notation, a bitensor $T$ with components $T_{\alpha \beta '\ldots }^{\mu \nu '\ldots }(x,y)$ has indices associated with two different points $x$ and $y$ in the manifold. By convention, unprimed indices (such as $\mu$, $\alpha$) refer to the first point, while primed indices (such as $\nu '$, $\beta '$) 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, heat kernels, and various Green's functions employed in quantum field theory in curved spacetime.

See also

• Non-Euclidean geometry