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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

\end{displaymath} (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
\end{displaymath} (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{}
\end{displaymath} (3)

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

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

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

\left\langle{{\bf v},{\bf v}}\right\rangle{}\geq 0,
\end{displaymath} (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
\end{displaymath} (8)

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

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

where $\eta_{\alpha\beta}$ is the Minkowski Metric. This can also be written
g=D^{\rm T}\eta D,
\end{displaymath} (11)

$\displaystyle D_{\alpha\mu}$ $\textstyle \equiv$ $\displaystyle {\partial\xi^\alpha\over\partial x^\mu}$ (12)
$\displaystyle {D_{\alpha\mu}}^{\rm T}$ $\textstyle \equiv$ $\displaystyle D_{\mu\alpha}.$ (13)

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

g_{il} {\partial g^{lk}\over\partial x^m} =-g^{lk}{\partial g_{il}\over\partial x^m}.
\end{displaymath} (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.
\end{displaymath} (16)

The Orthogonality of Contravariant and Covariant metrics stipulated by
\end{displaymath} (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,

$\displaystyle g^{11}$ $\textstyle =$ $\displaystyle {g_{22}\over g}$ (18)
$\displaystyle g^{12}$ $\textstyle =$ $\displaystyle g^{21}=-{g_{12}\over g}$ (19)
$\displaystyle g^{22}$ $\textstyle =$ $\displaystyle {g_{11}\over g}.$ (20)

If $g$ is symmetric, then
$\displaystyle g_{\alpha\beta}$ $\textstyle =$ $\displaystyle g_{\beta \alpha}$ (21)
$\displaystyle g^{\alpha\beta}$ $\textstyle =$ $\displaystyle g^{\beta \alpha}.$ (22)

In Euclidean Space (and all other symmetric Spaces),
g_\alpha^\beta =g^\beta_\alpha =\delta_\alpha^\beta,
\end{displaymath} (23)

g_{\alpha\alpha}={1\over g^{\alpha\alpha}}.
\end{displaymath} (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},
\end{displaymath} (25)

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

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

The Line Element can be written

ds^2={dx_i}\,{dx_i}= g_{ij}\,dq_i\,dq_j
\end{displaymath} (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,
\end{displaymath} (29)

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

For Orthogonal coordinate systems, $g_{ij}=0$ for $i\not=j$, and the Line Element becomes (for 3-space)
$\displaystyle ds^2$ $\textstyle =$ $\displaystyle g_{11}\,{dq_1}^2+g_{22}\,{dq_2}^2+g_{33}\,{dq_3}^2$  
  $\textstyle =$ $\displaystyle (h_1\,dq_1)^2+(h_2\,dq_2)^2+(h_3\,dq_3)^2,$ (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

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© 1996-9 Eric W. Weisstein