A Tensor, also called a Riemannian Metric, which is symmetric and Positive Definite. Very roughly, the metric tensor
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
in a generalized Pythagorean Theorem
![\begin{displaymath}
ds^2=g_{11}{dx_1}^2+g_{12}\,dx_1\,dx_2+g_{22}\,{dx_2}^2+\ldots.
\end{displaymath}](m_1092.gif) |
(1) |
In Euclidean Space,
where
is the Kronecker Delta (which is 0 for
and
1 for
), reproducing the usual form of the Pythagorean Theorem
![\begin{displaymath}
ds^2={dx_1}^2+{dx_2}^2+\ldots.
\end{displaymath}](m_1097.gif) |
(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
as arguments and produces a Real Number
such that
![\begin{displaymath}
\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}](m_1100.gif) |
(3) |
![\begin{displaymath}
\left\langle{{\bf v}+{\bf w},{\bf x}}\right\rangle{}=\left\l...
...}}\right\rangle{}+\left\langle{{\bf w},{\bf x}}\right\rangle{}
\end{displaymath}](m_1101.gif) |
(4) |
![\begin{displaymath}
\left\langle{{\bf v},{\bf w}+{\bf x}}\right\rangle{}=\left\l...
...}}\right\rangle{}+\left\langle{{\bf v},{\bf x}}\right\rangle{}
\end{displaymath}](m_1102.gif) |
(5) |
![\begin{displaymath}
\left\langle{{\bf v},{\bf w}}\right\rangle{}=\left\langle{{\bf w},{\bf v}}\right\rangle{}
\end{displaymath}](m_1103.gif) |
(6) |
![\begin{displaymath}
\left\langle{{\bf v},{\bf v}}\right\rangle{}\geq 0,
\end{displaymath}](m_1104.gif) |
(7) |
with equality Iff
.
In coordinate Notation (with respect to the basis),
![\begin{displaymath}
g^{\alpha \beta }=\vec e^\alpha \cdot \vec e^\beta
\end{displaymath}](m_1106.gif) |
(8) |
![\begin{displaymath}
g_{\alpha \beta}=\vec e_\alpha \cdot \vec e_\beta.
\end{displaymath}](m_1107.gif) |
(9) |
![\begin{displaymath}
g_{\mu\nu} \equiv {\partial\xi^\alpha\over\partial x^\mu}{\partial \xi^\beta\over\partial x^\nu} \eta_{\alpha\beta},
\end{displaymath}](m_1108.gif) |
(10) |
where
is the Minkowski Metric. This can also be written
![\begin{displaymath}
g=D^{\rm T}\eta D,
\end{displaymath}](m_1110.gif) |
(11) |
where
![\begin{displaymath}
{\partial\over\partial x^m} g_{il}g^{lk} = {\partial\over\partial x^m} \delta^k_i
\end{displaymath}](m_1115.gif) |
(14) |
gives
![\begin{displaymath}
g_{il} {\partial g^{lk}\over\partial x^m} =-g^{lk}{\partial g_{il}\over\partial x^m}.
\end{displaymath}](m_1116.gif) |
(15) |
The metric is Positive Definite, so a metric's Discriminant is Positive. For a metric in 2-space,
![\begin{displaymath}
g\equiv g_{11}g_{22}-{g_{12}}^2>0.
\end{displaymath}](m_1117.gif) |
(16) |
The Orthogonality of Contravariant and Covariant metrics stipulated by
![\begin{displaymath}
g_{ik}g^{ij}=\delta_k^j
\end{displaymath}](m_1118.gif) |
(17) |
for
, ...,
gives
linear equations relating the
quantities
and
. Therefore, if
metrics are known, the others can be determined.
In 2-space,
If
is symmetric, then
In Euclidean Space (and all other symmetric Spaces),
![\begin{displaymath}
g_\alpha^\beta =g^\beta_\alpha =\delta_\alpha^\beta,
\end{displaymath}](m_1130.gif) |
(23) |
so
![\begin{displaymath}
g_{\alpha\alpha}={1\over g^{\alpha\alpha}}.
\end{displaymath}](m_1131.gif) |
(24) |
The Angle
between two parametric curves is given by
![\begin{displaymath}
\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}](m_1132.gif) |
(25) |
so
![\begin{displaymath}
\sin\phi={\sqrt{g}\over g_1g_2}
\end{displaymath}](m_1133.gif) |
(26) |
and
![\begin{displaymath}
\vert{\bf r}_1\times{\bf r}_2\vert=g_1g_2\sin\phi=\sqrt{g}.
\end{displaymath}](m_1134.gif) |
(27) |
The Line Element can be written
![\begin{displaymath}
ds^2={dx_i}\,{dx_i}= g_{ij}\,dq_i\,dq_j
\end{displaymath}](m_1135.gif) |
(28) |
where Einstein Summation has been used. But
![\begin{displaymath}
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}](m_1136.gif) |
(29) |
so
![\begin{displaymath}
g_{ij} = \sum_k {\partial^2 x_k\over\partial q_i\partial q_j}.
\end{displaymath}](m_1137.gif) |
(30) |
For Orthogonal coordinate systems,
for
, and the Line Element becomes (for 3-space)
where
are called the Scale Factors.
See also Curvilinear Coordinates, Discriminant (Metric), Lichnerowicz Conditions, Line Element,
Metric, Metric Equivalence Problem, Minkowski Space, Scale Factor, Space
© 1996-9 Eric W. Weisstein
1999-05-26