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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 (P), separation vectors (\Delta \vector P), curves ( Q(\zeta) ), tangent vectors ( \delta P / \delta \zeta \equiv \lim_{\Delta \zeta \rightarrow 0} \frac{ \vector{ Q(\zeta+\delta \zeta) - Q(\zeta) } }{\delta \zeta} )
Coordinates: \Chi^\alpha (P), where \alpha = 0,1,2,3; Q(\Chi_0, \Chi_1, ...)
there is an isomorphism between points and coordinates
Coordinate basis: \vector{e_\alpha} \equiv \left( \frac{\partial Q}{\partial \Chi^\alpha} \right)
for instance, on a sphere with angles \omega, \phi:
\vector{e_\phi} = \left( \frac{\partial Q(\phi, \theta)}{\partial \phi}\right)_\theta
Components of a vector:
\vector{A} = \frac{\partial P}{\partial \Chi^\alpha }
Directional Derivatives: consider a scalar function defined on a manifold \Psi(P)
\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 rank-3 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:`$\bold{g}(\vector{A}, \vector{B}) = \vector{A} \dot \vector{B}$`. 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:`$$\Delta P \dot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \dot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$`
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:`$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \dot \vector{E})(\vector{B} \dot \vector{F})(\vecotr{C} \dot \vector{G}) $$`
Spacetime
--------------
Two types of vectors.
Timelike: :m:`$\vector{\Delta P}$`
(\vector{\Delta P})^2 = -(\Delta \Tau)^2
Spacelike: \vector{\Delta Q}
(\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.
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