Sep 17: Volume Element, Differences between Connection and Tensor, Parallel Transport September 21, 2018 1 1 Volume We can write: dnx ! dx0 ^ :: ^ dxn−1 (1) So that the volume element can be defined as: p dV = −gdnx = dV 0 = p−g0dnx0 (2) p Above, the −g takes the "role" of the ^'s in (1). We define the volume element as an n-form by 1 0 = dxµ1 ⊗ ::: ⊗ dxµn = ~ dxµ ^ ::: ^ dxµn (3) µ1...µn n! µ1...µn Where above the ⊗ means tensor product. We provide a proof of how the dV changes to dV', but it uses the fact that: µ µ µ @x µ0 µ0 dx 1 ^ ::: ^ dx n = dx 1 ^ :: ^ dx n (4) @xn0 and the fact that, p @xµ −g = p−g0 (5) @xn0 2 Curvature It is important to note that the Christoffel symbol Γ is NOT a tensor, for the simple reason that it does not transform like a tensor. Only special combinations of the Christoffel Symbols yield a tensor object. So we call the Christoffel symbols connections. ν ν ν σ Theorem if 9 Cµσ such that @µV + CµσV transforms as a tensor (i.e. ν ν rµV ) then Cµσ is a connection. To make this clear, a tensor transforms like: µ ν 0 µ σ 2 ν 0 ν0 @x @x ν @x @x @ x σ0 @ 0 V = @ V + V (6) µ @xµ0 @xν µ @xµ0 @xσ0 @xν @xσ but a connection transforms like this: ν 0 µ σ 2 ν 0 µ σ ν0 @x @x @x ν d x @x @x C 0 0 = C + (7) µ σ @xν @xµ0 @xσ0 µσ @xµ@xσ @xµ0 @xσ0 which shows these objects transform in different ways. We mentioned we can define a tensor via a special combination of these ν ν connections. For example, for two connections Cµσ and Cµσ we can define a ν ν ν ~ν tensor Sµσ as: Sµσ = Cµσ − Cµσ. This object on the left hand side is a tensor, even though the individual objects on the right hand side are connections. An example would be if C and C~ where the christoffel symbols evaluated at xµ 2 µ µ ν and x + δx respectively. This leads us to conclude that the object, d(Γµσ) = ν τ Λµστ dx may be treated as a tensor. For example, the Riemann Curvature tensor looks like: ρ ρ ρ ρ λ ρ λ Rσµν = @µΓνσ − @ν Γµσ + ΓµλΓνσ − ΓνλΓµσ (8) The objects on the right hand side are not by themselves tensors, the special combination written above makes the left hand side a, anti-symmetric in ν, µ tensor. That is, the whole thing behaves as a tensor. For a general Cνσ; rµgτσ 6= 0. Also, we can only act the covariant derivative rτ on tensors, so something like rτ Γ is nonsense. We also define a torsion tensor, although of so far little to no observational ν ν ν evidence, as Tµσ = Cµσ − Cσµ. In order for us to define the Christoffel symbols as "metric compatible", we must require the following conditions: µ i) Tνσ = 0; (9) ii) rµgτσ = 0 8µ, σ; τ (10) 3 Parallel Transport Vectors don't live in the Manifold M, they live in the tangent space at a point P on M. And so we define parallel transport as, for a vector V µ along a curve d D µ dλ , dλ (V ) = 0. So the closest we can get, to having this vector not change is: dxν r V µ = 0 (11) dλ ν µ d µ dxν µ For example, the usual acceleration: a = dτ V = dτ rν V Not every tensor can be parallely transported, only those that satisfy the parallel transport equation above. However if rσgµν = 0, then we can say that the metric tensor can be parallely transported along any curve. 3.