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I. Metric and Connection
Spacetime is locally Lorentz invariant, i.e. the vector space
${V_4}(x)$, $x = \{ {x^\mu }\} $, is a Minkowski space. Thus, an infinitesimal
displacement gives the line element:
\[ ds^2=\delta {\mathbf{x}} \cdot
\delta {\mathbf{x}} = \delta {x^\mu }\delta {x^\nu }{{\mathbf{g}}_\mu } \cdot
{{\mathbf{g}}_\nu }: = {g_{\mu \nu }}\delta {x^\mu }\delta {x^\nu } \]
Here, the components of the metric tensor field are defined as as
the inner product of the coordinate tangent vectors:
\[ {g_{\mu \nu }}(x): =
{{\mathbf{g}}_\mu }(x) \cdot {{\mathbf{g}}_\nu } (x) \]
In GRT the metric is not given apriori, except for special cases in which
some features of the metric are known (static gravitational field,
presence of symmetries, etc.) or postulated. In general the metric tensor
must be found from Einstein's field equation(s); see section X.
The
metric tensor is dimensionless, symmetric in its indices
and non-degenerate: $g: = {\text{det}}[{g_{\mu \nu }}] \ne 0$. It has
\[ {g^{\mu \nu }}(x): =
{{\mathbf{g}}^\mu }(x) \cdot {{\mathbf{g}}^\nu }(x) \]
as its inverse:
\[ {g^{\mu \lambda
}}{g_{\lambda \nu }} = \delta _\nu ^\mu {\quad}{{\mathbf{g}}^\mu } \cdot
{{\mathbf{g}}_\nu } = \delta _\nu ^\mu {\quad}{{\mathbf{g}}^\mu } =
{g^{\mu \nu }}{{\mathbf{g}}_\nu } \]
The covector ${{\mathbf{g}}^\mu }$ thus implicitly contains a directed
measure on the manifold that defines the inner product between
vectors in the tangent space:
\[ {\mathbf{v}} \cdot
{\mathbf{w}} = {v^\mu }{w^\nu }{{\mathbf{g}}_\mu } \cdot {{\mathbf{g}}_\nu
} = {g_{\mu \nu }}{v^\mu }{w^\nu } \]
The inner product is bi-linear and symmetric.