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============================================
Tensors, Differential Geometry, Manifolds
============================================
.. note:: Most of this content is based on a 2002 Caltech course taught by
Kip Thorn [PH237]_
On a manifold, only "short" vectors exist. Longer vectors are in a space tangent to the manifold.
There are points (:m:`$P$`), separation vectors (:m:`$\Delta \vector P$`),
curves (:m:`$Q(\zeta)$`), tangent vectors (:m:`$\delta P / \delta \zeta \equiv
\lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) 
Q(\zeta) } }{\delta \zeta}$`)
Coordinates: :m:`$\Chi^\alpha (P)$`, where :m:`$\alpha = 0,1,2,3$`;
:m:`$Q(\Chi_0, \Chi_1, ...)$`
there is an isomorphism between points and coordinates
Coordinate basis: :m:`$\vector{e_\alpha} \equiv \left( \frac{\partial
Q}{\partial \Chi^\alpha} \right$`)
for instance, on a sphere with angles :m:`$\omega, \phi$`:
:m:`$\vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta$`
Components of a vector:
:m:`$\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }$`
Directional Derivatives: consider a scalar function defined on a manifold \Psi(P)
:m:`$\partial_\vector{A} \Psi = A^\alpha \frac{\partial \Psi}{\partial \Chi^\alpha}$`
Mathematicians like to say that the coordinate bases are actually directional derivatives
Tensors

A **tensor** :m:`$\bold{T}$` has a number of slots (called it's **rank**), takes a vector in each slot, and returns a real number. It is linear in vectors;
as an example for a rank3 tensor:
:m:`$$\bold{T} ( \alpha \vector{A} + \beta \vector{B}, \vector{C}, \vector{D}) =
\alpha \bold{T} (\vector{A}, \vector{C}, \vector{D}) + \beta \bold{T}
(\vector{B}, \vector{C}, \vector{D}) $$`
Even a regular vector is a tensor: pass it a second vector and take the
inner product (aka dot product) to get a real.
Define the **metric tensor **
:m:`$\bold{g}(\vector{A}, \vector{B}) = \vector{A} \cdot \vector{B}$`. The
metric tensor is rank two and symetric (the vectors A and B could be swapped
without changing the scalar output value) and is the same as the inner product.
:m:`$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2  (AB)^2 ]$$`
Starting with individual vectors, we can construct tensors by taking the
product of their inner products with empty slots; for example
:m:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\_ ,\_ ,\_)$$`
:m:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G}) $$`
Spacetime

Two types of vectors.
Timelike: :m:`$\vector{\Delta P}$`
:m:`$(\vector{\Delta P})^2 = (\Delta \Tau)^2$`
Spacelike: :m:`$\vector{\Delta Q}$`
:m:`$(\vector{\Delta Q})^2 = +(\Delta S)^2$`
Because product of "up" and "down" basis vectors must be a positive Kronecker
delta, and timelikes squared come out negative, the time "up" basis must be negative of the time "down" basis vector.
