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These class notes by w. Zach korth cover the topics of geodesics and covariant derivatives in the context of spherical polar coordinates. The metric of flat space, the geodesic equation, parameterized curves, acceleration vectors, and parallel transport. The notes also discuss the concept of covariant derivatives and provide examples of their application to scalars, vectors, one-forms, and higher-rank tensors.
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∫ L dλ (1)
m
dxi(λ) dλ
dxj^ (λ) dλ
m
( r ˙^2 + r^2 θ˙^2 + r^2 sin^2 θ φ˙^2
) (2)
The metric of flat space in spherical polar coordinates is
ds^2 = dr^2 + r^2 dθ^2 + r^2 sin^2 θdφ^2 (3) On the surface of a sphere, r = a, dr = 0, so
ds^2 = a^2
( dθ^2 + sin^2 θdφ^2
) (4)
The geodesic equation is given by
d^2 xi dλ^2
dxj dλ
dxk dλ
where
Γijk =
gil^ (glk,j + gjl,k − gjk,l) (6)
is known as the affine connection.
If we consider a parameterized curve xi(λ) with parameter λ, the proper length between points A and B is given by
lAB =
∫ (^) B
A
√ gij dxidxj^ (7)
But what does it mean to integrate over these infinitesimals? This is why we choose a parameter, λ over which to integrate, as
lAB =
∫ (^) B
A
√ gij
dxi dλ
dxj dλ
dλ (8)
We note that this form is reparameterization invariant, as any changes in the parameter λ → f (λ) leave the physics unchanged.
3 Acceleration vector
We have what we defined as the tangent vector
ui^ =
dxi dλ
We now define an“acceleration vector”
ai^ =
Dui Dλ
dui dλ
Here, we define (^) ( ∂wi ∂xj^
) ≡ ∇j wk^ (16)
to be the covariant derivative, which is necessary to ensure that tensors remain as tensors under differentiation. This arises because a true vector is
w = wiei (17)
so dw = d(wiei) = (dwi)ei + wi(dei) (18)
Hence, we have two terms in the covariant derivative: one for component changes, and one for changes in the coordinate bases.
6 Christoffel symbols
We can now give a proper definition of the Christoffel symbols (Γijk):
Γijk ≡ ei∂kej (19)
Note that this is not a tensor, but rather a “tensor-like object”. Good tensors can be constructed from the Christoffel symbols, however; for example, the torsion tensor T (^) jki ≡ Γijk − Γikj (20)
In GR, we will be dealing with the Einstein Equations, in which manifolds have no torsion, thus
T (^) jki ≡ Γijk − Γikj = 0 −→ Γijk = Γikj (21)
7 Covariant derivative examples
Let us examine the form of what results from a covariant derivative of...
a scalar
∇iφ(xi) =
∂φ ∂xi^
a vector
∇j V k^ =
∂V k ∂xj^
a one-form
∇j Wk =
∂Wk ∂xj^
− Γikj Wi (24)
some higher-rank tensor
∇j T ik^ = ∂j (T ik) + Γilj T lk^ + Γklj T il^ (25)
Recall some operators
gradient (scalar) ∂iφ → ∇iφ (26)
divergence (vector) ∇iV i^ = ∂iV i^ + ΓijiV j^ (27) Noting the indices, we see that this must contract into a scalar. Let’s look at the connection:
Γiji =
gil^ (gli,j + gjl,i − gji,l) (28)
The last two terms in the parentheses cancel, leaving
Γiji =
gilgli,j =
√ |g|
√ |g|),j (29)
where g is the determinant of the metric. Thus,
∇iV i^ = ∂iV i^ +
√ |g|
√ |g|),j V j^ =
√ |g|
∂i(
√ |g|V i) (30)
curl (vector) ijk∇iBj (31) This equation works well for 3-D, but for higher dimensions it is useful to define Xij = ∇iBj − ∇j Bi = ∂iBj − ∂j Bi (32) where the last step follows from the fact that our connections Γijk are symmetric in their lower indices.
