# Redshift Academy   Wolfram Alpha: Search by keyword: Astronomy

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- Units, Constants and Useful Formulas

- ```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.

xν
/
/
D/--<-C
/    /
v    ^ dxν
/    /
A'    /B
A--->------------- xμ
dxμ

Parallel transport vector, Vα, around the loop and see how deflection angle, δVα,
changes.  The equation that moves the vector around is analagous to the geodesic
equation.

d2xn/dτ2 = -Γnmr(dxm/dτ)(dxr/dτ)

Now dxn/dτ = Vα and dxr/dτ = Vβ Thus, we can write from the above:

dVα/dτ = -Γμαβ(dxm/dτ)Vβ This is the change in V as we move along along the geodesic.

So,

dVα = -ΓμαβdxmVβ

Referring to the above diagram we can write:

δV = VA - VA' = {(VC - VD) - (VB - VA)} - {(VC - VB) - (VD - VA')}

(VC - VD) = ∂V/∂xμdxμV

= dxμDμV

{(VC - VD) - (VB - VA)} = dxνdxμDνDμV

This is the change in V over dxμ multiplied by the change in V over dxν

Likewise,

{(VC - VB) - (VD - VA')} = dxνdxμDμDνV

Thus,

δV = VA - VA' = {(VC - VD) - (VB - VA)} - {(VC - VB) - (VD - VA')}

= {dxνdxμDνDμ}V - {dxνdxμDμDν}V

= {dxνdxμ[Dν,Dμ]}V   [] represents the commutator

δVα = {dxνdxμ(∂ν + Γναβ)(∂μ + Γμαβ) - (∂μ + Γμαβ)(∂ν + Γναβ)}Vβ

= dxνdxμ{∂νΓμαβ - ∂μΓναβ + ΓναδΓμδβ - ΓμαδΓνδβ}Vβ

= dxνdxμRνμαβVβ 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,

Rνμαβ = -Rμναβ

Rνμαβ = -Rνμβα

Rνμαβ = Rβανμ

Ricci Tensor
------------

If we contract the indeces of the Riemann Tensor, we get the Ricci
tensor, Rμβ.

gαγRνμαβ = Rνμβγ

gνμRνμβγ and gβγRνμβγ = 0 because g is symmetric and R is asymmetric
in both νμ and βγ.  The correct contraction is:

gνγRνμβγ = Rμβ

Rμβ 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,

gμβRμβ = R

R corresponds to the trace of the Ricci curvature tensor.``` 