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 xi:
(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 gµ (such that gµ g = µ) is used to raise spacetime indices while the inverse 3-metric ij (ij jk = ki) 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 = Aiei where ei is a basis 3-vector obeying the dot product rule ei . ej = ij. A second-rank 3-tensor may be written (using dyadic notation and the tensor product) h = hijei ej. We write the spatial gradient 3-vector operator = ei i (i / xi) where ei . ej = ji. The experts will recognize ei as a basis one-form but we can treat it as a 3-vector ei = ij ej because of the isomorphism between vectors and one-forms. Because the basis 3-vectors in general have nonvanishing gradients, we define the covariant derivative (3-gradient) operator i with i jk = 0. If the space is flat (K = 0) and we use Cartesian coordinates, then ij = ij, i = i, and the 3-tensor index notation reduces to elementary Cartesian notation. If K 0, the 3-tensor equations will continue to look like those in flat space (that is why we use a 3+1 splitting of spacetime!) except that occasionally terms proportional to K will appear in our equations.
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 gµ. These are given by the Einstein equations,
(4.7) |
where Tµ is the stress-energy tensor and Gµ is the Einstein tensor, related to the spacetime Ricci tensor Rµ 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 gµ.
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).