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Notations and definitions

The signature of the metric tensor $g_{\mu \nu}(x)$ is assumed to be (+ - - -). Indices are lowered with $g_{\mu \nu}$ and raised with $g^{\mu \nu}$.

Greek letters run from 0 to 3. Latin letters are used for spatial coordinates only: they run from 1 to 3. A comma (,) denotes an ordinary partial differentiation. A semi-colon (;) denotes a covariant partial differentiation with respect to the metric; so $g_{\mu \nu; \rho} = 0$. Note that for any function F(x), $F_{;\alpha}~=~F_{, \alpha}$.

Any vector field $w^{\rho}$ satisfies the following identity
\begin{displaymath}
w^{\rho}_{\verb*+ +;\mu ;\nu} - w^{\rho}_{\verb*+ +;\nu ;\mu} = -
R^{\rho}_{. \sigma \mu \nu} w^{\sigma} \end{displaymath} (1)
where $R^{\rho}_{. \sigma \mu \nu}$ is the Riemann curvature tensor (note that this identity may be regarded as defining the curvature tensor). The Ricci tensor is defined by
\begin{displaymath}
R_{\mu \nu} = R^{\lambda}_{. \mu \lambda \nu}\end{displaymath} (2)

Given a quantity P, $\overline{P}$ denotes its complex conjugate.

The subscripts em and obs in formulae stand respectively for emitter and observer.

The constant c is the speed of light and $\hbar$ is the Planck constant divided by $2 \pi$.




10/9/1998