Sep 17: Volume Element, Differences Between Connection and Tensor, Parallel Transport

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: √ dV = −gdnx = dV 0 = p−g0dnx0 (2) √ 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, √ ∂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 ∃ Cµσ such that ∂µV + CµσV transforms as a tensor (i.e. ν ν ∇µV ) 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, ∇µgτσ 6= 0. Also, we can only act the covariant derivative ∇τ on tensors, so something like ∇τ 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) ∇µgτσ = 0 ∀µ, 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 ∇ V µ = 0 (11) dλ ν

µ d µ dxν µ For example, the usual acceleration: a = dτ V = dτ ∇ν V Not every tensor can be parallely transported, only those that satisfy the parallel transport equation above. However if ∇σgµν = 0, then we can say that the metric tensor can be parallely transported along any curve.