Metric Tensor

A Tensor, also called a Riemannian Metric, which is symmetric and Positive Definite. Very roughly, the metric tensor $g_{ij}$ is a function which tells how to compute the distance between any two points in a given Space. Its components can be viewed as multiplication factors which must be placed in front of the differential displacements $dx_i$ in a generalized Pythagorean Theorem

$$ds^2=g_{11}{dx_1}^2+g_{12}\,dx_1\,dx_2+g_{22}\,{dx_2}^2+\ldots.$$ (1)

In Euclidean Space, $g_{ij}=\delta_{ij}$ where $\delta$ is the Kronecker Delta (which is 0 for $i\not=j$ and 1 for $i=j$), reproducing the usual form of the Pythagorean Theorem

$$ds^2={dx_1}^2+{dx_2}^2+\ldots.$$ (2)

The metric tensor is defined abstractly as an Inner Product of every Tangent Space of a Manifold such that the Inner Product is a symmetric, nondegenerate, bilinear form on a Vector Space. This means that it takes two Vectors ${\bf v}, {\bf w}$ as arguments and produces a Real Number $\left\langle{{\bf v}, {\bf w}}\right\rangle{}$ such that

$$\left\langle{k{\bf v},w}\right\rangle{}=k\left\langle{{\bf v... ...}\right\rangle{}=\left\langle{{\bf v},k{\bf w}}\right\rangle{}$$ (3)

$$\left\langle{{\bf v}+{\bf w},{\bf x}}\right\rangle{}=\left\l... ...}}\right\rangle{}+\left\langle{{\bf w},{\bf x}}\right\rangle{}$$ (4)

$$\left\langle{{\bf v},{\bf w}+{\bf x}}\right\rangle{}=\left\l... ...}}\right\rangle{}+\left\langle{{\bf v},{\bf x}}\right\rangle{}$$ (5)

$$\left\langle{{\bf v},{\bf w}}\right\rangle{}=\left\langle{{\bf w},{\bf v}}\right\rangle{}$$ (6)

$$\left\langle{{\bf v},{\bf v}}\right\rangle{}\geq 0,$$ (7)

with equality Iff ${\bf v}=0$.

In coordinate Notation (with respect to the basis),

$$g^{\alpha \beta }=\vec e^\alpha \cdot \vec e^\beta$$ (8)

$$g_{\alpha \beta}=\vec e_\alpha \cdot \vec e_\beta.$$ (9)

$$g_{\mu\nu} \equiv {\partial\xi^\alpha\over\partial x^\mu}{\partial \xi^\beta\over\partial x^\nu} \eta_{\alpha\beta},$$ (10)

where $\eta_{\alpha\beta}$ is the Minkowski Metric. This can also be written

$$g=D^{\rm T}\eta D,$$ (11)

where

$$D_{\alpha\mu} \equiv {\partial\xi^\alpha\over\partial x^\mu}$$ (12)

$${D_{\alpha\mu}}^{\rm T} \equiv D_{\mu\alpha}.$$ (13)

$${\partial\over\partial x^m} g_{il}g^{lk} = {\partial\over\partial x^m} \delta^k_i$$ (14)

gives

$$g_{il} {\partial g^{lk}\over\partial x^m} =-g^{lk}{\partial g_{il}\over\partial x^m}.$$ (15)

The metric is Positive Definite, so a metric's Discriminant is Positive. For a metric in 2-space,

$$g\equiv g_{11}g_{22}-{g_{12}}^2>0.$$ (16)

The Orthogonality of Contravariant and Covariant metrics stipulated by

$$g_{ik}g^{ij}=\delta_k^j$$ (17)

for $i=1$, ..., $n$ gives $n$ linear equations relating the $2n$ quantities $g_{ij}$ and $g^{ij}$. Therefore, if $n$ metrics are known, the others can be determined.

In 2-space,

$$g^{11} = {g_{22}\over g}$$ (18)

$$g^{12} = g^{21}=-{g_{12}\over g}$$ (19)

$$g^{22} = {g_{11}\over g}.$$ (20)

If $g$ is symmetric, then

$$g_{\alpha\beta} = g_{\beta \alpha}$$ (21)

$$g^{\alpha\beta} = g^{\beta \alpha}.$$ (22)

In Euclidean Space (and all other symmetric Spaces),

$$g_\alpha^\beta =g^\beta_\alpha =\delta_\alpha^\beta,$$ (23)

so

$$g_{\alpha\alpha}={1\over g^{\alpha\alpha}}.$$ (24)

The Angle $\phi$ between two parametric curves is given by

$$\cos\phi=\hat{\bf r}_1\cdot\hat {\bf r}_2={{\bf r}_1\over g_1}\cdot {{\bf r}_2\over g_2} = {g_{12}\over g_1 g_2},$$ (25)

so

$$\sin\phi={\sqrt{g}\over g_1g_2}$$ (26)

and

$$\vert{\bf r}_1\times{\bf r}_2\vert=g_1g_2\sin\phi=\sqrt{g}.$$ (27)

The Line Element can be written

$$ds^2={dx_i}\,{dx_i}= g_{ij}\,dq_i\,dq_j$$ (28)

where Einstein Summation has been used. But

$$dx_i={\partial x_i\over\partial q_1}\,dq_1+{\partial x_i\ove... ...r\partial q_3}\,dq_3 = {\partial x_i\over\partial q_j}\,dq_j,$$ (29)

so

$$g_{ij} = \sum_k {\partial^2 x_k\over\partial q_i\partial q_j}.$$ (30)

For Orthogonal coordinate systems, $g_{ij}=0$ for $i\not=j$, and the Line Element becomes (for 3-space)

$$ds^2 = g_{11}\,{dq_1}^2+g_{22}\,{dq_2}^2+g_{33}\,{dq_3}^2$$

$$= (h_1\,dq_1)^2+(h_2\,dq_2)^2+(h_3\,dq_3)^2,$$ (31)

where $h_i\equiv \sqrt{g_{ii}}$ are called the Scale Factors.

See also Curvilinear Coordinates, Discriminant (Metric), Lichnerowicz Conditions, Line Element, Metric, Metric Equivalence Problem, Minkowski Space, Scale Factor, Space