The covariant derivative operator ∇j will replace the ordinary derivative opera-tor ∂j when we generalize many of the concepts from special relativity in inertial coordinates to either curvilinear coordinates or curved spacetime. We shall describe several mathematical aspects of this operator and the resulting expressions in this section.
The derivatives of the metric tensor can be expressed in terms of the Christoffel symbols. From the definition, it follows that
∂lgik= Γkil+ Γikl, (4.99) which expresses the partial derivatives in terms of Christoffel symbols. From the equation ∇lgik = 0, we can also express the derivatives of gik in terms of Christoffel symbols
∂lgik =−Γimlgmk− Γkmlgim. (4.100) The calculation of the covariant derivatives can be simplified by using two identities which we now derive. From the definition of Γikmit follows that
Γaba= 1
2gad∂bgad = 1
2g∂bg = ∂b(ln√
−g), (4.101)
where we have usedEq. (4.34). Similarly, gbcΓabc= gbcgad
∂cgdb− 1 2∂dgbc
=− 1
√−g∂b(√
−ggab). (4.102)
These results allow us to express covariant derivatives of antisymmetric and symmetric tensors in a simple form. For an antisymmetric tensor Qab, we have
∇bQab= ∂bQab+ ΓadbQdb+ ΓbdbQad= ∂bQab+ ΓbdbQad (4.103) since ΓadbQdb= 0 for an antisymmetric Qdb. UsingEq. (4.101), we get
∇bQab = 1
√−g∂b(√
−gQab). (4.104)
This shows that one can compute the covariant derivatives of an antisymmetric tensor directly from the metric without first having to compute the Christoffel sym-bols. The symmetry of the Christoffel symbols in the lower two indices also shows that, for any vector field Va,
∇bVa− ∇aVb = ∂bVa− ∂aVb (4.105) since the term involving Γs cancels out in this expression. So the antisymmet-ric part of the covariant derivative of any vector field is the same as the ordinary derivative. In fact, these results generalize to any completely antisymmetric tensor Qabc...= Q[abc...]. This is because, in the computation of∇aQabc..., only one term with Christoffel symbols (the one with ΓajaQjbc...) will survive since all others will involve the contraction of two symmetric indices of the Christoffel symbols with antisymmetric indices in the tensor.
Consider next a symmetric tensor Tabfor which we have the result,
∇kTki = ∂kTik+ ΓklkTil− ΓlikTlk = 1
√−g ∂k
√−g Tik
− ΓlkiTlk. (4.106)
Expanding out Γlkiand using the symmetry of Tkl, we get
∇kTki = 1
√−g ∂k
√−g Tik
−1
2(∂igkl) Tkl. (4.107) This expression is not as simple as the one for antisymmetric tensors but is still easier to use than the basic definition because we do not need to compute the Christoffel symbols.
We shall now take up the generalization of the concept of covariant divergence of a vector field to curved spacetime. In special relativity, in Cartesian coordinates, the covariant divergence is defined as ∂iAi, which generalizes to a curvilinear coordinates as∇iAi. From the definition of the covariant derivative we get
∇iAi = ∂iAi+ ΓiciAc= ∂iAi+ Ac∂c(ln√
−g) = 1
√−g∂i(√
−gAi), (4.108) where we have usedEq. (4.101). This structure is identical to the covariant diver-gence for an antisymmetric tensor obtained inEq. (4.104). Further, if Ai was a
4.6 Covariant derivative 169 gradient of some function φ, so that Ai = ∂iφ, Ai = gik∂kφ≡ ∂iφ, then∇aAa will represent the covariant Laplacian∇a∇aφ of the scalar φ. We see that
∇i∇iφ = 1
√−g∂i√
−ggik∂kφ
. (4.109)
The above considerations remain valid in any dimension and for metrics with any signature with√
−g interpreted as
|g|. In fact, these results are routinely used in three-dimensional vector analysis in the context of spherical polar coordinates. If the metric is taken to be
ds2= dr2+ r2(dθ2+ sin2θ dφ2) (4.110) the Laplacian operator inEq. (4.109)becomes
∇α∇αf = 1 r2sin θ∂α
r2sin θ gαβ ∂βf
= 1
r2∂r(r2∂rf ) + 1
r2sin θ∂θ(sin θ ∂θf ) + 1
r2∂φ2f, (4.111) which should be a result familiar from standard vector analysis. (The same is true as regardsEq. (4.108)except that the θ and φ components of a vector in spherical polar coordinates (vθ, vφ) are usually defined with extra factors of r and r sin θ by convention; such a complication does not arise in the defintion of a Laplacian.)
Using the definition of divergence, we can generalize the Gauss theorem to the curved spacetime in a straightforward manner. Consider a region of spacetimeV with a boundary ∂V. Since the proper volume element is now√−gd4x, we have the result
V
√−g d4x (∇iJi) =
∂V |h|1/2d3y (niJi), (4.112) where h is the determinant of the induced metric (seeEq. (4.31)) on the surface
∂V, which is given in parametric form, as xi= xi(yα), and niis the normal to the surface.
As we mentioned in the context of special relativity (see page 28), the above result holds even if the Jis are not components of a four-vector but just a set of four functions. In that case, the integral over the coordinates d4x (without the√−g fac-tor) of the quantity√
−g Jiwill be given by√
−g niJievaluated on the boundary of region of integration. Of course, when the Jis are not components of a vector, the result of integration will not be generally covariant.
Exercise 4.13
Covariant derivative of tensor densities The objects of the form√
−g T, where T is tensor of arbitrary rank, are called a tensor densities. Show that abcd and√−g have vanishing
covariant derivatives. Therefore one has, for example,∇a(√−g vi) = √−g ∇avi, etc.
When one thinks of√
−g vias a single entity, it is sometimes useful to write the expansion of∇a(√−g vi) in terms of ∂a(√−g vi) and correction terms. Find a suitable rule for cal-culating the covariant derivatives of such tensor densities in terms of the partial derivative and correction terms.
In later chapters, we will need the notion of the spacetime being foliated by a family of hypersurfaces. One simple example of this could be a sequence of spacelike surfaces parametrized by a variable t that can generalize the notion of constant time slice in special relativity. We can define the normal to this sequence of hypersurfaces everywhere in spacetime, thereby obtaining a vector field ni(x).
Such vector fields are called hypersurface-orthogonal vector fields.
We can show that such a hypersurface-orthogonal vector field, ni(x), will satisfy the condition n[l∇nnm] = 0. If a vector field na is orthogonal to a family of hypersurfaces given by Φ(x) = constant, it can be expressed in the form
na=−μ∂aΦ (4.113)
with some function μ. Differentiating this relation gives∇bna = −μ∇b∇aΦ−
∂aΦ∂bμ. We now construct the combination:
n[c∇bna] ≡ 1
3!(nc∇bna+ nb∇anc+ na∇cnb− nc∇anb− nb∇cna− na∇bnc) (4.114) which, on explicit evaluation usingEq. (4.113)and the result∇b∇aΦ =∇a∇bΦ, will vanish. Thus we conclude that any vector field which is hypersurface orthogonal must satisfy the condition
n[c∇bna]= 0 (4.115)
This result and its converse, which is also true, are called the Frobenius theorem.