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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 xμ(λ). 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 (2View Equation) below).

In a coordinate basis, which we will assume throughout this review, the metric is denoted by gμν = gνμ. 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 (8View Equation) and (9View Equation) below). The spacetime version of the Pythagorean theorem takes the form

ds2 = gμν dxμ dxν, (1 )
and in a local set of Minkowski coordinates {t,x,y, z} (i.e. in a local inertial frame, or small patch of the manifold) it looks like
ds2 = − (dt )2 + (dx)2 + (dy)2 + (dz)2. (2 )
This illustrates the − + ++ signature. The inverse metric μν g is such that
g μρg = δ μ , (3 ) ρν ν
where δμν is the unit tensor. The metric is also used to raise and lower spacetime indices, i.e. if we let V μ denote a contravariant vector, then its associated covariant vector (also known as a covector or one-form) V μ is obtained as
ν μ μν V μ = gμνV ⇔ V = g Vν. (4 )

We can now consider three different classes of curves: timelike, null, and spacelike. A vector is said to be timelike if gμνV μVν < 0, null if g μνVμV ν = 0, and spacelike if gμνVμV ν > 0. 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. μ x (τ) where

dτ2 = − ds2. (5 )
The tangent μ u to such a curve has unit magnitude; specifically,
dx μ uμ ≡ ----, (6 ) dτ
and thus
μ ν dx-μdx-ν -ds2 g μνu u = gμν dτ dτ = d τ2 = − 1. (7 )

Under a coordinate transformation -- x μ → x μ, contravariant vectors transform as

-- -μ ∂xμ- ν V = ∂xν V (8 )
and covariant vectors as
ν V- = ∂x--V . (9 ) μ ∂xμ ν
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 ε μνρτ. It is completely antisymmetric, and for four-dimensional spacetime, it has only one independent component, which is

√ --- 1 ε0123 = − g and ε0123 = √----, (10 ) − g
where g 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
√ -- 123 -1-- ε123 = g and ε = √g--, (11 )
where g 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 s = 0, n = 3 and j = 1 in Equation (361View Equation) of Appendix A – is
mij i j j i ε εmkl = δ kδ l − δ kδ l. (12 )
The general identities in Equations (360View Equation, 361View Equation, 362View Equation) of Appendix A will be used in our discussion of the variational principle.
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