Tensors, Differential Geometry, Manifolds ============================================ 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 (: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 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:$\bold{g}(\vector{A}, \vector{B}) = \vector{A} \cdot \vector{B}$. 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:$$\Delta P \cdot \Delta P \equiv \Delta P^2 \equiv (length of \Delta P)^2 A \cdot B = 1/4[ (A+B)^2 - (A-B)^2 ]$$ 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:$$\vector{A} \crossop \vector{B} \crossop \vector{C} (\vector{E}, \vector{F}, \vector{G}) = ( \vector{A} \cdot \vector{E})(\vector{B} \cdot \vector{F})(\vecotr{C} \cdot \vector{G})$$ 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.