============================================ 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 T has a number of slots (say 3) and takes a vector in each slot and returns a real number. It is linear in vectors. \epic{T} ( \alpha \vector{A} + \beta \vector{B}, \vector{C}, \vector{D}) = \alpha \epic{T} (\vector{A}, \vector{C}, \vector{D}) + \beta \epic{T} (\vector{B}, \vector{C}, \vector{D}) The number of "slots" is the rank of the tensor. Even a regular vector is a tensor: pass it a second vector and take the dot product to get a real. Define the metric tensor g(\vector{A}, \vector{B}) = \vector{A} \dot \vector{B} Inner Product: \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 ] Tensor Product: ???????????????? Spacetime -------------- Two types of vectors. Timelike: \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.