Next Contents Previous

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(tau) expressed in terms of conformal time. A dot (or partialtau) indicates a conformal time derivative. The comoving expansion rate is written eta(tau) ident adot / a = aH. The scale factor obeys the Friedmann equation,

Equation 4.1 (4.1)

The Robertson-Walker line element is written in the general form using conformal time tau and comoving coordinates xi:

Equation 4.2 (4.2)

Latin indices (i, j, k, etc.) indicate spatial components while Greek indices (µ, nu, lambda, 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 gµnu (such that gµnu gnukappa = deltakappaµ) is used to raise spacetime indices while the inverse 3-metric gammaij (gammaij gammajk = deltaki) is used to raise indices of 3-vectors and tensors. Three-tensors are defined in the spatial hypersurfaces of constant tau with metric gammaij 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

Equation 4.3 (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 = gammaij. A second-rank 3-tensor may be written (using dyadic notation and the tensor product) h = hijei otimes ej. We write the spatial gradient 3-vector operator nabla = ei partiali (partiali ident partial / partialxi) where ei . ej = deltaji. The experts will recognize ei as a basis one-form but we can treat it as a 3-vector ei = gammaij 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 nablai with nablai gammajk = 0. If the space is flat (K = 0) and we use Cartesian coordinates, then gammaij = deltaij, nablai = partiali, and the 3-tensor index notation reduces to elementary Cartesian notation. If Kneq 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, gamma ident det{gammaij}:

Equation 4.4 (4.4)

where epsilonijk = gamma-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 gamma-1/2 ensures that epsilonijk transforms like a tensor; as an exercise one can show that epsilonijk = gamma1/2 [ijk].

The second complication for Kneq 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

Equation 4.5 (4.5)

where [nablaj, nablak] ident (nablaj nablak - nablak nablaj). The commutator involves the spatial Riemann tensor, which for a uniform-curvature space with 3-metric gammaij is simply

Equation 4.6 (4.6)

Finally, we shall need the evolution equations for the full spacetime metric gµnu. These are given by the Einstein equations,

Equation 4.7 (4.7)

where Tµnu is the stress-energy tensor and Gµnu is the Einstein tensor, related to the spacetime Ricci tensor Rµnu by

Equation 4.8 (4.8)

The spacetime Riemann tensor is defined according to the convention

Equation 4.9 (4.9)

where the affine connection coefficients are

Equation 4.10 (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 gµnu.

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).

Next Contents Previous