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Curvature and Parallel Transport
--------------------------------
Consider a flattened out cone. Parallel transport a vector around the edge
as shown. Keep the vector parallel to itself in every respect as you move
around the loop. If the angle, θ, between the vector and the surface at the
starting point and the ending point is the same then the surface is flat.
If there is a difference in the angles then the surface is curved.
Now consider a loop drawn on a surface.
x^{ν}
/
/
D/--<-C
/ /
v ^ dx^{ν}
/ /
A' /B
A--->------------- x^{μ}
dx^{μ}
Parallel transport vector, V^{α}, around the loop and see how deflection angle, δV^{α},
changes. The equation that moves the vector around is analagous to the geodesic
equation.
d^{2}x^{n}/dτ^{2} = -Γ^{n}_{mr}(dx^{m}/dτ)(dx^{r}/dτ)
Now dx^{n}/dτ = V^{α} and dx^{r}/dτ = V^{β} Thus, we can write from the above:
dV^{α}/dτ = -Γ^{μ}_{αβ}(dx^{m}/dτ)V^{β} This is the change in V as we move along along the geodesic.
So,
dV^{α} = -Γ^{μ}_{αβ}dx^{m}V^{β}
Referring to the above diagram we can write:
δV = V_{A} - V_{A'} = {(V_{C} - V_{D}) - (V_{B} - V_{A})} - {(V_{C} - V_{B}) - (V_{D} - V_{A'})}
(V_{C} - V_{D}) = ∂V/∂x^{μ}dx^{μ}V
= dx^{μ}D_{μ}V
{(V_{C} - V_{D}) - (V_{B} - V_{A})} = dx^{ν}dx^{μ}D_{ν}D_{μ}V
This is the change in V over dx^{μ} multiplied by the change in V over dx^{ν}
Likewise,
{(V_{C} - V_{B}) - (V_{D} - V_{A'})} = dx^{ν}dx^{μ}D_{μ}D_{ν}V
Thus,
δV = V_{A} - V_{A'} = {(V_{C} - V_{D}) - (V_{B} - V_{A})} - {(V_{C} - V_{B}) - (V_{D} - V_{A'})}
= {dx^{ν}dx^{μ}D_{ν}D_{μ}}V - {dx^{ν}dx^{μ}D_{μ}D_{ν}}V
= {dx^{ν}dx^{μ}[D_{ν},D_{μ}]}V [] represents the commutator
δV^{α} = {dx^{ν}dx^{μ}(∂_{ν} + Γ_{ν}^{α}_{β})(∂_{μ} + Γ_{μ}^{α}_{β}) - (∂_{μ} + Γ_{μ}^{α}_{β})(∂_{ν} + Γ_{ν}^{α}_{β})}V^{β}
= dx^{ν}dx^{μ}{∂_{ν}Γ_{μ}^{α}_{β} - ∂_{μ}Γ_{ν}^{α}_{β} + Γ_{ν}^{α}_{δ}Γ_{μ}^{δ}_{β} - Γ_{μ}^{α}_{δ}Γ_{ν}^{δ}_{β}}V^{β}
= dx^{ν}dx^{μ}R_{νμ}^{α}_{β}V^{β} where R is the RIEMANN TENSOR
The Riemann tensor is comprised of Christoffel symbols and their
derivatives. In turn, the Christoffel symbols are comprised of the
metric tensor and its derivatives.
The tensor is asymmetric with the interchange of μ and ν. It is also
asymmetric with the interchange of α and β. The tensor is, however,
symmetric with the interchange of the pair μ and ν with the pair α
and β. Therefore,
R_{νμ}^{α}_{β} = -R_{μν}^{α}_{β}
R_{νμ}^{α}_{β} = -R_{νμ}^{β}_{α}
R_{νμ}^{α}_{β} = R^{β}_{α}_{νμ}
Ricci Tensor
------------
If we contract the indeces of the Riemann Tensor, we get the Ricci
tensor, R_{μβ}.
g_{αγ}R_{νμ}^{α}_{β} = R_{νμβγ}
g^{νμ}R_{νμβγ} and g^{βγ}R_{νμβγ} = 0 because g is symmetric and R is asymmetric
in both νμ and βγ. The correct contraction is:
g^{νγ}R_{νμβγ} = R_{μβ}
R_{μβ} corresponds to a trace of the Riemann curvature tensor.
In 3-D space the Ricci and Riemann tensors are equivalent.
Scalar Tensor
-------------
Contract indeces of the Ricci tensor,
g^{μβ}R_{μβ} = R
R corresponds to the trace of the Ricci curvature tensor.