**4.1.2. Summary of essential relativity**

We adopt the following conventions and notations, similar to those of
Misner, Thorne & Wheeler
(1973).
Units are chosen so that *c* = 1.
The metric signature is (- , + , + , +). The unperturbed background
spacetime is Robertson-Walker with scale factor
*a*() expressed in terms of
conformal time. A dot (or
_{}) indicates a conformal
time derivative. The comoving expansion rate is written
()
/ *a* =
*aH*. The scale factor obeys the Friedmann equation,

(4.1) |

The Robertson-Walker line element is written in the general form using
conformal time and comoving
coordinates *x*^{i}:

(4.2) |

Latin indices (*i*, *j*, *k*, etc.) indicate spatial
components while Greek indices (*µ*,
,
, etc.) indicate all
four spacetime
components; we assume a coordinate basis for tensors. Summation is
implied by repeated upper and lower indices. The inverse 4-metric
*g*^{µ}
(such that
*g*^{µ}
*g*_{} =
_{}^{µ})
is used to raise spacetime indices while the inverse 3-metric
^{ij}
(^{ij}
_{jk}
= _{k}^{i})
is used to raise indices of
3-vectors and tensors. Three-tensors are defined in the spatial
hypersurfaces of constant
with metric
_{ij}
and they shall
be clearly distinguished from the spatial components of 4-tensors. We
shall see as we go along how this "3+1 splitting" of spacetime works
when there are metric perturbations.

Many different spatial coordinate systems may be used to cover a
uniform-curvature 3-space. For example, there exist quasi-Cartesian
coordinates (*x*, *y*, *z*) in terms of which the
3-metric components are

(4.3) |

We shall use 3-tensor notation to avoid restricting ourselves to any
particular spatial coordinate system. Three-scalars, vectors, and
tensors are invariant under transformations of the spatial coordinate
system in the background spacetime (e.g., rotations). A 3-vector may
be written
** A** =

Our application is not restricted to a flat Robertson-Walker background
but allows for nonzero spatial curvature. This complicates matters
for two reasons. First, we cannot assume Cartesian coordinates. As a
result, for example, the Laplacian of a scalar and the divergence and
curl of a 3-vector involve the determinant of the spatial metric,
det{_{ij}}:

(4.4) |

where
^{ijk} =
^{-1/2}
[*ijk*] is the three-dimensional
Levi-Civita tensor, with [*ijk*] = + 1 if {*ijk*} is an even
permutation of {123}, [*ijk*] = - 1 for an odd permutation, and 0
if any two indices are equal. The factor
^{-1/2}
ensures that
^{ijk}
transforms like a tensor; as an exercise one can show that
_{ijk} =
^{1/2}
[*ijk*].

The second complication for
*K* 0 is that gradients
do not commute when
applied to 3-vectors and 3-tensors (though they do commute for 3-scalars).
The basic results are

(4.5) |

where [_{j},
_{k}]
(_{j}
_{k} -
_{k}
_{j}). The
commutator involves the spatial Riemann tensor, which
for a uniform-curvature space with 3-metric
_{ij}
is simply

(4.6) |

Finally, we shall need the evolution equations for the full spacetime metric
*g*_{µ}.
These are given by the Einstein equations,

(4.7) |

where *T*^{µ}_{} is the stress-energy tensor and
*G*^{µ}_{} is
the Einstein tensor, related to the spacetime Ricci tensor
*R*_{µ} by

(4.8) |

The spacetime Riemann tensor is defined according to the convention

(4.9) |

where the affine connection coefficients are

(4.10) |

We see that the Einstein tensor involves second derivatives of the metric
tensor components, so that eq. (4.7) provides second-order
partial differential equations for
*g*_{µ}.

The reader who is not completely comfortable with the material summarized above may wish to consult an introductory general relativity textbook, e.g. Schutz (1985).