Metric tensor

The metric tensor (also metric tensor or Maßtensor ) serves mathematical spaces , in particular differentiable manifolds , provide a measure of distances and angles.

This measure does not necessarily have to meet all the conditions that are placed on a metric in the definition of a metric space : in the Minkowski space of the special theory of relativity , these conditions only apply to distances that are either uniformly spacelike or uniformly temporal .

It is important for differential geometry and general relativity that the metric tensor, unlike a metric defined by an inner product and norm , can depend on the location.

Definition and meaning

The metric tensor over an affine point space with real displacement vector space is a mapping of into the space of the scalar products on . That is, for each point it is ${\displaystyle g}$ ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle A}$ ${\displaystyle V}$ ${\displaystyle P\in A}$

${\displaystyle g(P)\colon V\times V\to \mathbb {R} }$

a positively definite , symmetrical bilinear form .

Based on the distinction between metrics and pseudometrics , the case is sometimes considered that for some or all points there is only positive semidefinite, i.e. H. the requirement of definiteness ${\displaystyle g(P)}$ ${\displaystyle P}$

${\displaystyle g(P)\left({\vec {x}},\,{\vec {x}}\right)>0}$ for all ${\displaystyle 0\neq {\vec {x}}\in V}$

is weakened too

${\displaystyle g(P)\left({\vec {x}},\,{\vec {x}}\right)\geq 0}$for everyone . ${\displaystyle {\vec {x}}\in V}$

Such a tensor is then called a pseudometric tensor . ${\displaystyle g}$

A metric tensor defines a (point- dependent) length ( norm ) on the vector space : ${\displaystyle P}$ ${\displaystyle \|{\vec {x}}\|_{P}}$ ${\displaystyle V}$

${\displaystyle \|{\vec {x}}\|_{P}={\sqrt {g(P)\left({\vec {x}},\,{\vec {x}}\right)}}}$

Analogous to the standard scalar product , the angle at the point between two vectors is defined by: ${\displaystyle \theta \in [0,\pi ]}$ ${\displaystyle P}$ ${\displaystyle {\vec {x}},{\vec {y}}\in V}$

${\displaystyle \cos \theta ={\frac {g(P)({\vec {x}},{\vec {y}})}{{\sqrt {g(P)({\vec {x}},{\vec {x}})}}\,{\sqrt {g(P)({\vec {y}},{\vec {y}})}}}}}$

Coordinate representation

When a local coordinate system on with base from is selected to write the components of a . Using Einstein's summation convention , for the vectors and ${\displaystyle (x^{i})}$ ${\displaystyle V}$ ${\displaystyle (e_{i})}$ ${\displaystyle V}$ ${\displaystyle g}$ ${\displaystyle g_{ij}(P)=g(P)(e_{i},e_{j})}$ ${\displaystyle {\vec {x}}=x^{i}{\vec {e}}_{i}}$ ${\displaystyle {\vec {y}}=y^{i}{\vec {e}}_{i}}$

${\displaystyle g(P)\left({\vec {x}},\,{\vec {y}}\right)=g_{ij}(P)\,x^{i}\,y^{j}}$.

In terms of the category theory of variant metric tensor is contraindicated, as in (affine) linear injective pictures naturally from a metric tensor on a metric tensor on constructed can be, ${\displaystyle \varphi \colon (A,V)\to (B,W)}$ ${\displaystyle (B,W)}$ ${\displaystyle (A,V)}$

${\displaystyle (\varphi ^{*}g)(P)({\vec {x}},{\vec {y}})=g{\bigl (}\varphi (P){\bigr )}{\Bigl (}\varphi _{*}({\vec {x}}),\varphi _{*}({\vec {y}}){\Bigr )}}$.

In physics , the metric tensor, or rather its coordinate representation, is called covariant , since its components transform under a coordinate change in every index like the base. Is a coordinate change as ${\displaystyle g_{ij}}$

${\displaystyle x^{k}=A^{k}{}_{i}\;{\tilde {x}}^{i}}$ or. ${\displaystyle {\tilde {x}}^{i}=(A^{-1})^{i}{}_{k}\;x^{k}}$

given, basis vectors transform as

${\displaystyle {\tilde {e}}_{i}=A^{k}{}_{i}\;e_{k}=(A^{T})_{i}{}^{k}\;e_{k}}$

and it holds for the metric tensor

${\displaystyle {\tilde {g}}_{ij}=g(P)({\tilde {e}}_{i},\,{\tilde {e}}_{j})=(A^{T})_{i}{}^{k}\,(A^{T})_{j}{}^{l}\;g_{kl}.}$

Length of curves

If there is a differentiable curve in the affine point space, it has a tangential vector at every point in time ${\displaystyle \gamma \colon [a,b]\to A}$ ${\displaystyle t}$

${\displaystyle {\vec {x}}(t)={\dot {\gamma }}(t)={\frac {\mathrm {d} }{\mathrm {d} t}}\gamma (t)}$.

The entire curve or a segment of it can now be given a length with the help of the metric tensor

${\displaystyle L_{[a,b]}(\gamma )=\int _{a}^{b}{\sqrt {g{\bigl (}\gamma (t){\bigr )}{\Bigl (}\,{\vec {x}}(t),\,{\vec {x}}(t)\,{\Bigr )}}}\,\mathrm {d} t=\int _{a}^{b}\|{\dot {\gamma }}(t)\|_{\gamma (t)}\,\mathrm {d} t}$

assign.

Line element

The expression

${\displaystyle \mathrm {d} s^{2}=g_{ij}\mathrm {d} x^{i}\mathrm {d} x^{j}}$,

again using the summation convention, is called a line element . One substitutes according to the chain rule

${\displaystyle \mathrm {d} x^{i}={\frac {\mathrm {d} x^{i}}{\mathrm {d} t}}\mathrm {d} t}$and , ${\displaystyle \mathrm {d} x^{j}={\frac {\mathrm {d} x^{j}}{\mathrm {d} t}}\mathrm {d} t}$

so it turns out

${\displaystyle \mathrm {d} s^{2}=g_{ij}{\frac {\mathrm {d} x^{i}}{\mathrm {d} t}}{\frac {\mathrm {d} x^{j}}{\mathrm {d} t}}\mathrm {d} t^{2}}$.

${\displaystyle \mathrm {d} s}$is therefore the integrand of the above integral for determining a curve length .

Induced metric tensor

If one has a -dimensional submanifold of a Riemannian space with the metric , which by means of the parametric representation ${\displaystyle p}$ ${\displaystyle (g_{ij})}$

${\displaystyle q^{i}=q^{i}(t^{1},t^{2},\dots ,t^{p}),\qquad i=1,\dots ,n}$

is given, a metric is induced. The called coordinates induced . Looking at a curve ${\displaystyle (a_{\alpha \beta })}$ ${\displaystyle t^{\alpha }}$

${\displaystyle t^{\alpha }=t^{\alpha }(t),\qquad a\leq t\leq b,\qquad \alpha =1,\dots ,p}$

on this sub-manifold, one obtains for the arc length according to the chain rule

${\displaystyle s=\int _{a}^{b}{\sqrt {g_{ij}{\frac {\mathrm {d} q^{i}}{\mathrm {d} t}}{\frac {\mathrm {d} q^{j}}{\mathrm {d} t}}}}\mathrm {d} t=\int _{a}^{b}{\sqrt {g_{ij}{\frac {\partial q^{i}}{\partial t^{\alpha }}}{\frac {\mathrm {d} t^{\alpha }}{\mathrm {d} t}}{\frac {\partial q^{j}}{\partial t^{\beta }}}{\frac {\mathrm {d} t^{\beta }}{\mathrm {d} t}}}}\mathrm {d} t=\int _{a}^{b}{\sqrt {g_{ij}{\frac {\partial q^{i}}{\partial t^{\alpha }}}{\frac {\partial q^{j}}{\partial t^{\beta }}}{\frac {\mathrm {d} t^{\alpha }}{\mathrm {d} t}}{\frac {\mathrm {d} t^{\beta }}{\mathrm {d} t}}}}\mathrm {d} t}$.

The size

${\displaystyle a_{\alpha \beta }:=g_{ij}{\frac {\partial q^{i}}{\partial t^{\alpha }}}{\frac {\partial q^{j}}{\partial t^{\beta }}}}$

is the induced metric tensor . With this, the curve length finally results as

${\displaystyle s=\int _{a}^{b}{\sqrt {a_{\alpha \beta }{\frac {\mathrm {d} t^{\alpha }}{\mathrm {d} t}}{\frac {\mathrm {d} t^{\beta }}{\mathrm {d} t}}}}\mathrm {d} t}$.

Examples

Euclidean space

In a Euclidean space with Cartesian coordinates , the metric tensor is through the identity matrix

${\displaystyle g_{ij}=\delta _{ij}}$

given. In the Euclidean space the scalar product is given and according to the assumption the metric tensor should correspond to this scalar product. So it applies to this in local coordinates where the vectors are the standard basis . For arbitrary vectors and Euclidean space holds ${\displaystyle \textstyle \langle x,y\rangle =\sum _{i=1}^{n}x^{i}y^{i}}$ ${\displaystyle g_{ij}=\langle e_{i},e_{j}\rangle =\delta _{ij},}$ ${\displaystyle e_{1},\dots ,e_{n}}$ ${\displaystyle x=x^{i}e_{i}}$ ${\displaystyle y=y^{j}e_{j}}$

${\displaystyle g_{ij}\,x^{i}y^{j}=\delta _{ij}x^{i}y^{j}=\sum _{i=1}^{n}x^{i}y^{i}.}$

Here is the Einstein notation used.

For the curve length

${\displaystyle L=\int _{a}^{b}{\sqrt {\left(\mathrm {d} x\right)^{2}}}}$

and the angle

${\displaystyle \cos \theta ={\frac {\mathbf {u} \,\mathbf {v} }{|\mathbf {u} |\cdot |\mathbf {v} |}}}$

the usual formulas for vector analysis are obtained .

If a manifold is embedded in a Euclidean space with Cartesian coordinates, then its metric tensor results from the Jacobi matrix of the embedding as ${\displaystyle J}$

${\displaystyle g=J^{T}J.}$

In some other coordinate systems, the metric tensor and the line element of Euclidean space are as follows:

• In polar coordinates : ${\displaystyle (x^{1},x^{2})=(r,\theta )}$

${\displaystyle g={\begin{bmatrix}1&0\\0&r^{2}\end{bmatrix}}}$, or.

${\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}}$

• In cylindrical coordinates : ${\displaystyle (x^{1},x^{2},x^{3})=(r,\theta ,z)}$

${\displaystyle g={\begin{bmatrix}1&0&0\\0&r^{2}&0\\0&0&1\end{bmatrix}}}$, or.

${\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}+\mathrm {d} z^{2}}$

• In spherical coordinates : ${\displaystyle (x^{1},x^{2},x^{3})=(r,\theta ,\varphi )}$

${\displaystyle g={\begin{bmatrix}1&0&0\\0&r^{2}&0\\0&0&(r\sin \theta )^{2}\end{bmatrix}}}$, or.

${\displaystyle \mathrm {d} s^{2}=\mathrm {d} r^{2}+r^{2}\,\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}\theta \;\mathrm {d} \varphi ^{2}}$

Minkowski space (special theory of relativity)

→ Main article : Minkowski space

The flat Minkowski space of the special theory of relativity describes a four-dimensional space-time without gravity . Spatial distances and periods of time depend in this space on the choice of an inertial system; If one describes a physical process in two different inertial systems that are moving uniformly against one another, they can assume different values.

On the other hand, the so-called four - digit distance, which combines spatial and temporal distances, is invariant under Lorentz transformations . Using the speed of light c , this distance of four is calculated from spatial distance and time span as ${\displaystyle \mathrm {d} \mathbf {r} }$ ${\displaystyle \,\mathrm {d} t}$

${\displaystyle \mathrm {d} s^{2}=c^{2}\,\left(\mathrm {d} t\right)^{2}\,-\left(\mathrm {d} \mathbf {r} \right)^{2}}$

In the Minkowski space the contravariant position four-vector is defined by . ${\displaystyle \,x^{\mu }=(x^{0},x^{1},x^{2},x^{3})=(ct,x,y,z)}$

The metric (more precisely: pseudometric) tensor reads in a convention that is mainly used in quantum field theory ( signature −2, i.e. +, -, -, -)

${\displaystyle \eta _{\mu \nu }={\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}}\equiv \operatorname {diag} (1,-1,-1,-1)}$.

In a convention that is mainly used in general relativity (signature +2, i.e. -, +, +, +), one writes

${\displaystyle \eta _{\mu \nu }={\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\equiv \operatorname {diag} (-1,1,1,1)}$.

In the general theory of relativity, the metric tensor is location-dependent and therefore forms a tensor field , since the curvature of space-time is usually different at different points.

literature

• Rainer Oloff: Geometry of space-time: A mathematical introduction to the theory of relativity . Springer-Verlag, 2013, ISBN 3-322-94260-0 .
• Chris Isham: Modern Differential Geometry for Physicists . Allied Publishers, 2002, ISBN 81-7764-316-9 .