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.