Lectue VI: Tenso calculus Chistophe M. Hiata Caltech M/C 350-7, Pasadena CA 925, USA (Dated: Octobe 4, 20) I. OVERVIEW In this lectue, we will begin with some examples fom vecto calculus, and then continue to define covaiant deivatives of -foms and tensos. The ecommended eading fo this lectue is: MTW 8.5 8.6. [Note: actually a lot of what we e doing in class is woking though the execises in 8.5.] II. A WORKED EXAMPLE: VECTOR CALCULUS IN POLAR COORDINATES In this section, we will do some examples fom vecto calculus in pola coodinates on R 2. This is a simple case, but should be useful to execise the machiney. Recall that the metic tenso components wee and the invese metic is g =, g θθ = 2, and g θ = g θ = 0, () g =, g θθ = 2, and gθ = g θ = 0. (2) The coodinate basis vectos e and e θ ae not othonomal, but we may define an othonomal basis via eˆ = e and eˆθ = e θ. (3) A. Chistoffel symbols We begin by computing the Chistoffel symbols fo pola coodinates. The only nonzeo deivative of a covaiant metic component is Now etuning to the geneal ule, we can diectly ead off the Chistoffel symbols. They ae: g θθ, = 2. (4) Γ ǫ δη = 2 gǫτ ( g δη,τ + g ητ,δ + g δτ,η ), (5) Γ = 0, Γ θ = Γ θ = 0, Γ θθ = 2 g ( g θθ, ) = ()( 2) =, 2 Γ θ = 0, Γ θ θ = Γ θ θ = 2 gθθ (g θθ, ) = 2 ( 2 )(2) =, and Γ θ θθ = 0. (6) Electonic addess: chiata@tapi.caltech.edu
So of the 6 Chistoffel symbols, only 2 ae nonzeo. This is typical of highly symmetical manifolds (expessed in coodinates that use the symmety). 2 B. Covaiant deivative of a vecto field Let s now find the covaiant deivative of a vecto field v. Using the ule fom the last section, v α ;β = v α,β + Γ α βγv γ. (7) Component by component, this eads v ; = v,, v ;θ = v,θ v θ, v θ ; = v θ, + vθ, and v θ ;θ = v θ,θ + v. (8) The divegence of a vecto field is v = v ; + v θ ;θ = v, + v θ,θ + v. (9) C. Examples of vecto fields and thei popeties Conside the vecto field v that points in the -diection of the oiginal Catesian coodinate system (v = e ). It can be expessed in tems of its components in the othonomal basis o in the coodinate basis, vˆ = cosθ, vˆθ = sinθ; (0) v = cosθ, v θ = sin θ. () You know intuitively that this vecto field is constant, but that is not obvious in the pola coodinate system. We can still pove it, howeve, using Eq. (8): v ; = v, = 0, v ;θ = v,θ v θ = sinθ ( sinθ) = 0, v θ ; = v θ, + vθ = 2 sin θ + ( sin θ) = 0, and v θ ;θ = v θ,θ + v = cosθ + cosθ = 0. (2) As a less tivial example, we can seach fo an axisymmetic adial vecto field E (E θ = 0, E depends only on and not θ) with zeo divegence (except at the oigin): E = 0. Equation (9) tells us that we need Since s depends on, we may then wite 0 = E, + 0 + E. (3) d d (E ) = E, + E = 0. (4) Theefoe Eˆ = E. You may ecognize this as the esult that the electic field of a linea chage scales as /.
3 III. COVARIANT DERIVATIVES OF TENSORS Having found a way to diffeentiate a vecto (i.e. a ank ( ( 0) tenso) in cuved spacetime and geneate a ank ) tenso, we now ask whethe thee is a way to diffeentiate geneal tensos in cuved spacetime. The answe is yes, and fotunately thee is no new messy algeba: the same Chistoffel symbols we ve woked with will suffice fo tensos. A. Covaiant deivative of a -fom To wam up, let s ty taking a -fom i.e. a ank ( 0 ) tenso and finding its covaiant deivative, a ank ( 0 2) tenso. We could epeat the wok we did fo vectos, going to a local Loentz coodinate system, taking the deivatives, and going back to the oiginal space. Thee is howeve an easie way: we may use the vecto-to--fom coespondence. Given a -fom k µ, we know how to associate it with a vecto k α = g αµ k µ. Then we will find the covaiant deivative of the -fom by loweing the indices of k α ;β: Let s evaluate this explicitly: As an aside at this point, we note that k ν;β g να k α ;β. (5) k ν;β g να k α ;β = g να (g αµ k µ ) ;β = g να [(g αµ k µ ),β + Γ α βγg γµ k µ ] = g να [ g αµ,βk µ + g αµ k µ,β + 2 gαδ ( g βγ,δ + g βδ,γ + g γδ,β )g γµ k µ ] = g να g αµ,βk µ + k ν,β + 2 ( g βγ,ν + g βν,γ + g γν,β )g γµ k µ. (6) g να g αµ,β = (g να g αµ ),β g να,β g αµ = (δ µ ν ),β g νγ,β g γµ = g νγ,β g γµ. (7) Theefoe ou covaiant deivative of a -fom satisfies k ν;β = k ν,β + 2 ( g βγ,ν + g βν,γ g γν,β )g γµ k µ = k ν,β Γ µ βνk µ. (8) So we find the emakable esult that the covaiant deivative of a -fom is given by the patial deivative, but coected by a Chistoffel symbol. This is the same Chistoffel symbol we found fo vectos, but note the sign and the diffeently placed indices. If we take the diectional covaiant deivative of a basis -fom, we find o equivalently ( eβ ω α ) ν = (ω α ) ν;β = Γ α νβ, (9) eβ ω α = Γ α νβω ν. (20) B. Covaiant deivative of a geneal tenso We ae now eady to define the covaiant deivative of a geneal tenso. Since we define covaiant deivatives by efeence to a local coodinate system whee the covaiant deivative becomes a patial deivative, it should satisfy a poduct ule. Fo example, if we take a tenso S of ank ( 2 2), we have To find the covaiant deivative, we emembe that S = S αβ γδe α e β ω γ ω δ. (2) eµ e α = Γ ν αµe ν. and eµ ω α = Γ α νµω ν, (22)
4 so then eµ S = S αβ γδ,µe α e β ω γ ω δ +S αβ γδ eµ e α e β ω γ ω δ + S αβ γδe α eµ e β ω γ ω δ +S αβ γδe α e β eµ ω γ ω δ + S αβ γδe α e β ω γ eµ ω δ. (23) Plugging in the elations fo the covaiant deivatives of vectos and -foms, we find eµ S = S αβ γδ,µe α e β ω γ ω δ +Γ ν αµs αβ γδe ν e α e β ω γ ω δ + Γ ν βµs αβ γδe α e ν ω γ ω δ Γ γ νµs αβ γδe α e β ω ν ω δ Γ δ νµs αβ γδe α e β ω γ ω ν. (24) We may find the components of the ight-hand side by elabeling indices: eµ S = [S αβ γδ,µ + Γ α νµs νβ γδ + Γ β νµs αν γδ Γ ν γµs αβ νδ Γ ν δµs αβ γν]e α e β ω γ ω δ (25) The object in backets is then the αβ γδµ component of the ank ( 2 3) tenso S. Theefoe it is the component of the covaiant deivative of S: S αβ γδ;µ = S αβ γδ,µ + Γ α νµs νβ γδ + Γ β νµs αν γδ Γ ν γµs αβ νδ Γ ν δµs αβ γν. (26) The ules ae quite geneal: one takes the patial deivative, and then wites down a coection tem fo each index. Each coection tem is of the fom ΓS and satisfies the following ules: (i) the diffeentiation index always appeas in the last slot; (ii) the index being coected moves ove to Γ in eithe the up o down position as appopiate; (iii) one fills the two empty slots with a summed dummy index; and (iv) up indices get a + sign and down indices get a sign. Covaiant diffeentiation is somewhat mechanical, just like diffeentiation in feshman calculus. C. Some moe popeties Thee ae a vaiety of useful popeties of covaiant deivatives that one can pove. Hee is a sampling.. The covaiant deivative of the contaction is the same as the contaction of the covaiant deivative In the local Loentz coodinate fame established at any point P, this is obvious. But let s ty poving it explicitly: let s take a ank ( ) tenso S and find the contaction of its covaiant deivative: S α α;β(deivative then contaction) = S α α,β + Γ α γβs γ α Γ γ αβs α γ = S α α,β = (S α α),β = S α α;β(contaction then deivative). (27) The cancellation of the Γ-containing coections is geneal fo tensos of highe ank. So contaction and covaiant deivative commute, and in the futue we will simply wite S α α;β without ambiguity. 2. The covaiant deivative of the metic tenso is zeo This is also obvious in the local Loentz coodinate fame. But we can pove it explicitly: g αβ;γ = g αβ,γ Γ µ αγg µβ Γ µ βγg αµ = g αβ,γ 2 gµν ( g αγ,ν + g αν,γ + g γν,α )g µβ 2 gµν ( g βγ,ν + g βν,γ + g γν,β )g αµ = g αβ,γ 2 ( g αγ,β + g αβ,γ + g γβ,α ) 2 ( g βγ,α + g βα,γ + g γα,β ) = 0. (28)
5 3. The covaiant deivative of the invese-metic tenso is zeo ( ) again: obvious in the local Loentz coodinate fame. But thee is a mathematical poof. Conside the ank Once identity tenso I that takes in a -fom and vecto and etuns thei contaction: I( k, v) = k, v. (29) The components ae easily seen to be I α β = δ α β (they ae the same in any coodinate system!). We know that I α β;γ = δ α β,γ + Γ α µγδ µ β Γ µ βγδ α µ = 0. (30) But since g αβ is the matix invese of g αβ, we have I α β = g αν g νβ. Taking the covaiant deivative, and ecalling that covaiant diffeentiation commutes with contaction: I α β;γ = g αν ;γg νβ + g αν g νβ;γ. (3) The left-hand side is zeo, g νβ;γ = 0, and so we ae left with the conclusion that g αν ;γg νβ = 0. Since g νβ foms an invetible matix, we can conclude that g αν ;γ = 0. 4. The covaiant deivative commutes with aising and loweing of indices Since the aising and loweing of indices involves the oute poduct with the metic tenso (o its invese), followed by contaction, this statement is a coollay of the peceding ones.