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GR Lecture 5 Covariant Derivatives, Christoffel Connection, Geodesics

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GR Lecture 5 Covariant Derivatives, Christoffel Connection, Geodesics GR lecture 5 Covariant derivatives, Christoffel connection, geodesics, electromagnetism in curved spacetime, local conservation of 4-momentum I. PARALLEL TRANSPORT, AFFINE CONNECTIONS, COVARIANT DERIVA- TIVES µ The Lie derivative Lu is all fine and well, but it requires a vector field u (x) to define ν the relevant flow. It would be nice to have a tensorial derivative operator similar to @µv ν µ ν { let us call it rµv { from which we can construct directional derivatives u rµv by µ ν simply contracting with a vector u , without needing to know its derivatives @µu . As we've seen, such an operator would require some prescription for relating the vector bases at adjacent points, or, equivalently, for deciding how to move the vector's \head", given some µ motion (along u ) of its \tail". In other words, such an operator rµ requires some extra geometric structure on our spacetime. When it exists, rµ is called a covariant derivative. The corresponding geometric structure is called parallel transport, because it generalizes the flat notion of moving a vector's head in parallel to the motion of its tail. Let us then postulate this extra structure! What we need is a basis transformation ν µ matrix Mµ corresponding to motion along an infinitesimal vector u . Since we're dealing with infinitesimals, this basis transformation should be linear in uµ: ν ν ρ ν −1 ν ν ρ ν Mµ = δµ + u Γρµ ;(M )µ = δµ − u Γρµ : (1) Thus, the recipe for parallel-transporting vectors and covectors reads: µ µ ν µ ρ vtransported = v − u Γνρv ; (2) transported ν ρ wµ = wµ + u Γνµwρ : (3) µ The object Γνρ is known as an affine connection. As we'll see, it is not a tensor, but can be used in the construction of tensors. We normally don't try to raise and lower its indices. Let µ us work out how Γνρ should transform under a change of coordinates. It is slightly easier to do this starting from the covector transport rule (3). Under a change of coordinates, a µ covector wµ at a point x transforms via the matrix: @xν w0 = w : (4) µ @x0µ ν 1 The RHS of the transport rule (3) transforms as: 0transported 0 0ν 0ρ 0 wµ = wµ − u Γνµwρ ; (5) 0µ µ where Γνρ is the sought-after transformation of Γνρ. To find it, we should notice that the transported 0 parallel-transported covector wµ should actually transform according to the @x=@x matrix evaluated not at xµ (or x0µ), but at the new point xµ + uµ (or x0µ + u0µ): @xν @2xν w0transported = + u0ρ wtransported µ @x0µ @x0ρ@x0µ ν ν 2 ν @x 0ρ @ x ρ σ = 0µ + u 0ρ 0µ wν + u Γρνwσ @x @x @x (6) @xν @xν @2xν ≈ w + uρΓσ w + u0ρ w @x0µ ν @x0µ ρν σ @x0ρ@x0µ ν @xν @xρ @x0κ @2xν @x0σ = w0 + u0λΓσ w0 + u0ρ w0 : µ @x0µ @x0λ @xσ ρν κ @x0ρ@x0µ @xν σ where we neglected a piece quadratic in the inifinitesimal uµ. Comparing (5) and (6), we obtain the transformation rule: @x0µ @xκ @xλ @x0µ @2xσ Γ0µ = Γσ + : (7) νρ @xσ @x0ν @x0ρ κλ @xσ @x0ν@x0ρ µ The first term is the standard tensor transformation rule; the second tells us that Γνρ is not µ a tensor. Nevertheless, we are now ready to use Γνρ to construct the covariant derivative µ ν rµ. To take the covariant derivative u rµv of e.g. a vector, we must simply take the ν µ ν transported value (2), and subtract it from the actual value v + u @µv at the new point: µ ν µ ν µ ν ρ u rµv ≡ u @µv + u Γµρv ; (8) µ and similarly for a covector wµ. Since everything is just proportional to u , we can get rid of it. We end up with the following expressions for the covariant derivative: ν ν ν ρ ρ rµv = @µv + Γµρv ; rµwν = @µwν − Γµνwρ : (9) When performing a coordinate transformation, the \unwanted" terms in the transformation ν µ ν of e.g. @µv and Γνρ cancel. Thus, the covariant derivative rµv (as well as rµwν) is an honest tensor! The covariant derivative can now be defined for tensors with any number of indices. For scalars, we define simply rµf ≡ @µf. Exercise 1. Demonstrate the Leibniz rules: ν ν ν ν ν ν rµ(fv ) = v @µf + frµv ; @µ(uνv ) = v rµuν + uνrµv : (10) 2 Just as we did for Lie derivatives, we use the Leibniz rule to define the covariant derivative of arbitrary tensors, e.g.: λ λ κ λ κ λ κ λ λ κ rµTνρσ = @µTνρσ − ΓµνTκρσ − ΓµρTνκσ − ΓµσTνρκ + ΓµκTνρσ : (11) The construction above may be familiar if you've encountered electromagnetism at a suf- ficiently high level. The wavefunction (xµ) of a charged non-spinning particle is a com- plex number with a phase. The physics (i.e. the Schrodinger equation and Born's rule) is invariant under rotating this phase by a constant ! eiθ . However, once we allow spacetime-dependent phase transformations θ(xµ), the Schrodinger equation is no longer invariant, since the derivative @µ now has the messy transformation law: iθ iθ @µ ! @µ(e ) = e (@µ + i @µθ) : (12) The wavefunction in this story is analogous to our tensors. The phase transformation θ is analogous to the basis transformation matrix @xµ=@x0ν. The second term in (12) is analogous to the unwanted @2xµ=@x0ν@x0ρ term in the transformation of a partial derivative ν @µv . In electromagnetism, the way to make the local phase transformation (12) a symmetry µ after all is to introduce the electromagnetic potential Aµ, analogous to Γνρ in the GR story. We then postulate the gauge transformation: Aµ ! Aµ − @µθ ; (13) which is analogous to the second term in (7). The presence of the first term in (7) signals that the relevant \force field” is self-interacting; this is true for GR (gravitational waves have energy) and Yang-Mills (gluons have color), but not for electromagnetism (photons have no charge). Anyway, now that we have the new field Aµ, we can use it to replace the partial derivative (12) with a covariant derivative: rµ ≡ @µ + iAµ ; (14) iθ which transforms nicely as rµ ! e , and is analogous to (9) in the GR story. Note that this modern understanding of electromagnetism came only after GR, and was inspired by it! µ In the geometric jargon, both Aµ and Γνρ are called connections. The more specific name µ “affine connection" for Γνρ indicates its more specialized role in transforming the tangent space (rather than some internal space, such as the space of wavefunction values). 3 II. COMPARING CONNECTIONS; TORSION µ Our discussion so far of the connection Γνρ has been very abstract, and completely dis- connected from the fact that spacetime has a metric gµν. As we will see, the metric actually chooses a connection for us: parallel transport, just like all other geometric structures, is already encoded in gµν. Nevertheless, it's often useful to consider parallel transport as an µ independent geometric concept, and the connection as Γνρ as an independent variable. One µ ~µ useful observation is that if we have two different connections, then the difference Γνρ − Γνρ between them is a tensor: the \unwanted" second transformation term in (7) cancels. This also makes sense from the point of view of the corresponding covariant derivatives r and ~ ~ ν ν ~ν ρ r: the explicitly tensorial quantity (rµ − rµ)v can be written also as (Γµρ − Γµρ)v , since the partial-derivative pieces in (9) cancel. A related observation is that the \unwanted" second term in (7) is symmetric in the two µ lower indices. Thus, we can obtain a tensor from Γνρ by simply antisymmetrizing these two indices: µ µ µ µ Tνρ ≡ 2Γ[νρ] = Γνρ − Γρν : (15) µ This tensor is known as the torsion of the connection Γνρ. A slightly fancier way to define it is through the commutator of covariant derivatives: [rµ; rν] ≡ rµrν − rνrµ : (16) Note that partial derivatives always commute: [@µ;@ν] = 0, but covariant derivatives don't have to: in fact, we'll gradually see how their non-commutativity essentially defines the curvature of spacetime. µ Exercise 2. Show that the torsion Tνρ defines the commutator of covariant derivatives on a scalar field f(xµ): ρ [rµ; rν]f = −Tµν@ρf : (17) For yet another definition of torsion, recall that the \exterior derivative" @[µvν] is a tensor, which doesn't require a connection to define it, and can be compared against r[µvν]: Exercise 3. Show that the torsion tensor can be defined via: 1 r v − @ v = − T ρ vρ ; (18) [µ ν] [µ ν] 2 µν 4 and that, more generally, when torsion vanishes, we have: r[µ1 fµ2...µn] = @[µ2 fµ2...µn] : (19) Clearly, life is simpler when the torsion vanishes: the antisymmetrized r then agrees with the antisymmetrized @, and inherits its interpretation of measuring circulations around closed loops. We then accordingly have [rµ; rν]f = 0, which expresses the fact that, upon traveling around a closed loop, f returns to itself. In fact, even if we have a connection with torsion, we can always just replace it by one without: µ µ Exercise 4. If Γνρ is a connection, show that Γ(νρ) is also a connection, and is torsionless. From now on, we will assume by default that our connection is torsion-free, i.e. that µ µ Γνρ = Γρν. III. GEODESICS AND THE CHRISTOFFEL CONNECTION µ It is time to show how the connection Γνρ can be derived from the metric gµν.
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