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Mathematics for Physics

192 Tensor Field

A mapping that assigns a tensor to each point of a domain that is used to describe geometric quantities that vary from point to point.

definition [d] (Tensor Field = Section) A mapping that assigns to each point of a manifold a tensor of fixed type: for \(U \subset M\),

  • \(T : U \rightarrow T^{r}_{\ s}(M)\) ,
  • \(T(P) \equiv T_{P} \in T^{r}_{\ s,P}(M)\) .

where

  • \(M\) is a smooth manifold.
  • \(U\) is a subset of \(M\).
  • \(T\) is the tensor field.
  • \(P\) is a point of \(M\).
  • \(T^{r}_{\ s}(M)\) is the bundle of type-\((r,s)\) tensors on \(M\).
  • \(T^{r}_{\ s,P}(M)\) is the space of type-\((r,s)\) tensors at \(P\).
  • \(r\) is the contravariant degree.
  • \(s\) is the covariant degree.

Note:

  • a smooth tensor field is a smooth such assignment, that is, a smooth section.

definition [d] (Tensor Field = Section) A coordinate-independent field on a manifold whose local components vary differentiably: at each point, a multilinear map

  • \(T_{P} : \underbrace{(T_{P}M)^{*}\times\cdots\times(T_{P}M)^{*}}_{r} \times \underbrace{T_{P}M\times\cdots\times T_{P}M}_{s} \rightarrow \mathbb{R}\) .

where

  • \(T_{P}\) is the value of the tensor field at the point \(P\).
  • \(T_{P}M\) is the tangent space at \(P\).
  • \((T_{P}M)^{*}\) is the cotangent space at \(P\).
  • \(r\) is the number of covector arguments.
  • \(s\) is the number of vector arguments.
  • \(\mathbb{R}\) is the set of real numbers.

Note:

  • examples: vector fields are type \((1,0)\); covector fields type \((0,1)\); the metric type \((0,2)\).
  • equivalently, a section of the tensor bundle \(T^{r}_{\ s}(M)\).

192.1 Examples

example 1 [d] (Metric Tensor Field — Frankel) The type-\((0,2)\) metric tensor field on \(\mathbb{R}^{3}\) in spherical coordinates \((r, \theta, \phi)\), with line element

  • \(ds^{2} = dr^{2} + r^{2}\, d\theta^{2} + r^{2}\sin^{2}\theta\, d\phi^{2}\) .

The nonzero components as functions of position are

  • \(g_{rr}(r, \theta, \phi) = 1\) ,
  • \(g_{\theta\theta}(r, \theta, \phi) = r^{2}\) ,
  • \(g_{\phi\phi}(r, \theta, \phi) = r^{2}\sin^{2}\theta\) .

In matrix form,

  • \((g_{ij}) = \operatorname{diag}(1,\, r^{2},\, r^{2}\sin^{2}\theta)\) .

where

  • \((r, \theta, \phi)\) are spherical coordinates.
  • \(g_{ij}\) are the components of the metric tensor field.
  • \(ds\) is the infinitesimal arc length.

Note:

  • off-diagonal components vanish.
  • this is a smooth assignment of a symmetric type-\((0,2)\) tensor to each point.

example 2 [d] (Electromagnetic Field Strength — Carroll, Hassani) The antisymmetric type-\((0,2)\) tensor field \(F_{\mu\nu}\) on Minkowski spacetime with coordinates \(x^{\mu} = (t, x, y, z)\):

  • \(F_{\mu\nu} = \begin{pmatrix} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & -B_{z} & B_{y} \\ E_{y} & B_{z} & 0 & -B_{x} \\ E_{z} & -B_{y} & B_{x} & 0 \end{pmatrix}\) ,

where \(E_{i}(t,x,y,z)\) and \(B_{i}(t,x,y,z)\) are the electric and magnetic field components at each spacetime point.

As a differential form,

  • \(F = -E_{x}\, dt\wedge dx - E_{y}\, dt\wedge dy - E_{z}\, dt\wedge dz + B_{z}\, dx\wedge dy - B_{y}\, dx\wedge dz + B_{x}\, dy\wedge dz\) .

where

  • \(F_{\mu\nu}\) are the components of the electromagnetic field strength tensor field.
  • \(E_{x}, E_{y}, E_{z}\) are the electric field components.
  • \(B_{x}, B_{y}, B_{z}\) are the magnetic field components.
  • \(x^{\mu} = (t, x, y, z)\) are spacetime coordinates.

Note:

  • \(F\) assigns an antisymmetric bilinear form to each event in spacetime.
  • Hassani identifies \(F_{j0} = E_{j}\), \(F_{12} = B_{3}\), \(F_{13} = -B_{2}\), \(F_{23} = B_{1}\).

192.2 Elementary Example

192.2.1 Simple

A tensor field assigns one tensor to each point of a domain.

\[ U = \{ p,\ q,\ r \} \]

\[ T(p) = e_{1},\quad T(q) = e_{2},\quad T(r) = 0 \]

where

  • \(T\) is the tensor field.
  • \(T(p)\) is the tensor at the point \(p\).

192.2.2 General

A type-\((0,2)\) tensor field can assign a \(3 \times 3\) matrix at each point.

\[ U = \{ p,\ q \} \]

\[ T(p) = I_{3},\quad T(q) = \operatorname{diag}(1,2,3) \]

where

  • \(T(p)\) is a type-\((0,2)\) tensor at \(p\).
  • \(I_{3}\) is the \(3 \times 3\) identity matrix.

192.3 References

  1. Hassani, S. Mathematical Physics, 2nd ed. Springer. — tensor field \(T:U\to T^{r}_{s}(M)\); electromagnetic field strength \(F\).
  2. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — metric tensor field in spherical coordinates; \(ds^{2}=dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}\).
  3. Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — electromagnetic field strength tensor \(F_{\mu\nu}\).
  4. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — tensor fields via component transformation laws.