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Tetrads in General Relativity

I. Metric and Connection

Metric Tensor

   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:

1.9

\[  ds^2= d{\mathbf{x}} \cdot d {\mathbf{x}} = {{\mathbf{g}}_\mu } \cdot {{\mathbf{g}}_\nu } d{x^\mu }d{x^\nu }: = {g_{\mu \nu }} d {x^\mu } d {x^\nu }  \]

Here, the components of the metric tensor field are defined as as the inner product of the coordinate tangent vectors:

1.10

\[  {g_{\mu \nu }}(x): = {{\mathbf{g}}_\mu }(x) \cdot {{\mathbf{g}}_\nu } (x)  \]

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 XI.

   The metric tensor is dimensionless, symmetric in its indices and non-degenerate: $g: = {\text{det}}[{g_{\mu \nu }}] \ne 0$. It has

1.11

\[  {g^{\mu \nu }}(x): = {{\mathbf{g}}^\mu }(x) \cdot {{\mathbf{g}}^\nu }(x)  \]

as its inverse:

1.12

\[  {g^{\mu \lambda }}{g_{\lambda \nu }} = \delta _\nu ^\mu {\quad}{{\mathbf{g}}^\mu } \cdot {{\mathbf{g}}_\nu } = \delta _\nu ^\mu {\quad}{{\mathbf{g}}^\mu } = {g^{\mu \nu }}{{\mathbf{g}}_\nu }  \]

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:

1.13

\[  {\mathbf{v}} \cdot {\mathbf{w}} = {v^\mu }{w^\nu }{{\mathbf{g}}_\mu } \cdot {{\mathbf{g}}_\nu } = {g_{\mu \nu }}{v^\mu }{w^\nu } = {g^{\mu \nu }}{v_\mu }{w_\nu } \]