2.1 The metric and spacetime curvature

Our strategy is to provide a “working understanding” of the mathematical objects that enter the Einstein equations of general relativity. We assume that the metric is the fundamental “field” of gravity. For four-dimensional spacetime it determines the distance between two spacetime points along a given curve, which can generally be written as a one parameter function with, say, components . As we will see, once a notion of parallel transport is established, the metric also encodes information about the curvature of its spacetime, which is taken to be of the pseudo-Riemannian type, meaning that the signature of the metric is (cf. Equation (2) below).

In a coordinate basis, which we will assume throughout this review, the metric is denoted by . The symmetry implies that there are in general ten independent components (modulo the freedom to set arbitrarily four components that is inherited from coordinate transformations; cf. Equations (8) and (9) below). The spacetime version of the Pythagorean theorem takes the form

and in a local set of Minkowski coordinates (i.e. in a local inertial frame, or small patch of the manifold) it looks like
This illustrates the signature. The inverse metric is such that
where is the unit tensor. The metric is also used to raise and lower spacetime indices, i.e. if we let denote a contravariant vector, then its associated covariant vector (also known as a covector or one-form) is obtained as

We can now consider three different classes of curves: timelike, null, and spacelike. A vector is said to be timelike if , null if , and spacelike if . We can naturally define timelike, null, and spacelike curves in terms of the congruence of tangent vectors that they generate. A particularly useful timelike curve for fluids is one that is parameterized by the so-called proper time, i.e.  where

The tangent to such a curve has unit magnitude; specifically,
and thus

Under a coordinate transformation , contravariant vectors transform as

and covariant vectors as
Tensors with a greater rank (i.e. a greater number of indices), transform similarly by acting linearly on each index using the above two rules.

When integrating, as we need to when we discuss conservation laws for fluids, we must be careful to have an appropriate measure that ensures the coordinate invariance of the integration. In the context of three-dimensional Euclidean space the measure is referred to as the Jacobian. For spacetime, we use the so-called volume form . It is completely antisymmetric, and for four-dimensional spacetime, it has only one independent component, which is

where is the determinant of the metric (cf. Appendix A for details). The minus sign is required under the square root because of the metric signature. By contrast, for three-dimensional Euclidean space (i.e. when considering the fluid equations in the Newtonian limit) we have
where is the determinant of the three-dimensional space metric. A general identity that is extremely useful for writing fluid vorticity in three-dimensional, Euclidean space – using lower-case Latin indices and setting , and in Equation (361) of Appendix A – is
The general identities in Equations (360, 361, 362) of Appendix A will be used in our discussion of the variational principle.