NOTE PHZ 6607 Lecture Notes 1. Lecture 2 1.1. Defntons Books: ( Tensor Analyss on Manfols ( The mathematcal theory of black holes ( Carroll (v Schutz Vector: ( In an N-Dmensonal space, a vector s efne as a lnear combnaton of tangental rectons wthn a bass of that space; enotes recton. ( Vectors are not free to sle from pont to pont outse the context of that spaces. A vector s attache to a sngle pont. A=A µ ê µ ( We efne a vector whch reles only on nfntesmal neghbourhoos. = x u u Tensors: ( Wthn a vector space V, a tensor s a scalar-value multlnear functon wth varables all n ether V or V*. ( A famly of tangent vector spaces over the curve of a manfol. ( A tensor T of type (k,l s a multlnear map from a collecton of ual-vectors an vectors to R. (v A tensor of type (N,N at P s efne to be a lnear functon whch takes a arguements N one-forms an N vectors an whose value s a real number. Manfol: ( A topologcal space n whch some neghbourhoo of each pont amts a coornate system consstng of real coornate functons on ponts of the neghbourhoo whch etermne the poston of ponts an the topology of that neghbourhoo: aka t s locally cartesan. ( When a Euclean space s strppe of ts vector space only ts fferentable structure s retane; by percng together omans of the Euclean space n a smooth manner we obtan a manfol. x
PHZ 6607 Lecture Notes 2 ( A manfol s a space that may be curve an have complcate topology but n local regons look just le R n. The local regons of same menson patche together create the manfol. (v A set of ponts M s efne to be a manfol f each pont of M has an open neghbourhoo whch has a contnuous 1-1 map onto an open set of R n for some n. Tangent vector: ( Drectonal ervatve on an n-mensonal manfol. ( The set of numbers xl on the curve xl (λ ( The tangent vector V (λ has components V µ = xµ (v For a parametrze curve f(x (λ the tangent vector s one-form: = x x ( A lnear mappng of the tangent space onto the reals. ( The ual mappng of vectors onto a real number space. ( One forms are elements of the ual vector space T p W = W µˆθµ (v A one-form s a lnear real value functon of vectors. A one form W at P assocates wth a vector V at P a real number whch we call W ( V Notes: Answer 1.3 Two ponts P1 an P2 are (x 1, y 1 (x 2, y 2 n ol coornate system an are (u 1, v 1 (u 2, v 2 n new coornate system l 2 = x 2 + y 2 = A D u2 + 2B D u v + A D v2 f u,v are orthogonal u v coeffcent = 0 1.2. METRIC s 2 = g ab (x a (x b g ab s calle metrc Example, For sphercal polar coornates θ, ph, R = a s 2 = a 2 (θ 2 + Sn 2 θ φ 2 Lookng close to top of sphere(on a tangent plane, sn θ θ s 2 = a 2 (θ 2 + θ 2 φ 2 but ths correspons to coornate transformaton x = aθ cos φ an y = aθ sn φ whch mples that metrc for sphere becomes metrc for cartesan coornates uner the assumpton of small θ. Ths s an example of gong to local cartesan coornates
PHZ 6607 Lecture Notes 3 Relaton between e a an e a :: Let bass vectors for e a space be A, B, C Let bass vectors for e a space be D, E, F Snce these are two fferent vector spaces, hence A. D s not necessarly 0,1 but can be any number α an smlarly wth ther other combnatons. Usng these bass vectors we can construct another set of vectors such that A. D = 1. Smlarly, B. E = 1 an C. F = 1. In Q1.3, u=ax+by an v=cx+y. How to fn e u an e u? Example of e x s. x e u = = xe u u x + y e u y Example of e x s x. e u = u = u x ex + u y ey Note that î s e x e u e u = 1 e u e v = 0 Parametrze curve s efne as: X = X 1 (λ Y = X 2 (λ Z = X 3 (λ At each pont we can efne a vector whch s tangent to curve V (X = X x Scalar create by tangent vector : g 1.3. Relatonshp between upper an lower nex bass vectors u = ax + by v = cx + y for x,y space we have e x, e y, e x, e y. What s e u, e u? A possble bass for ornary vectors n x,y space s gven by the fferental operators an. We can wrte x y u = x u x + y u y We can generalze to e u = x u e x + y u e y
PHZ 6607 Lecture Notes 4 an e u = u x ex + u y ey Orthogonalty between upper an lower nces s by ecree. 1.4. Defne a tangent vector In space x, λ s the arbtrary parameterzaton of the curve. The tangent vector on the curve s then v = x L = 1 2 mv2 = 1 2 mg x fn the varaton of acton wth respect to fferent paths 1 S = L = 2 mg x Frst path: x (λ Secon path: x (λ + δx (λ For the secon path 1 S = L = we get To frst orer δs = 1 ( 2 m g x xk 2 mg (x k + δx k (x + δx + g δx + g x We are at lberty to change labels wthn the summaton Usng δs = m 2 (x j + δx j δs = 1 ( 2 m g x xk + g δx k kj + g x (g kj δx k xj ( g 2 x j jk g jk x 2 + m 2 δx j δx k = g kj xj + g δx k kj + g kjδx k 2 x j 2 g 2 x (g kj δx k xj + g x g 2 x j x + g x x k δx k
PHZ 6607 Lecture Notes 5 δs = m [ g 2 x j jk then + m 2 multply by ( g kl 1 2 2 ( g g x x k x j (g kj δx k xj + g x 2 x j g jk 1 ( g 2 2 x j 2 x l 2 + 1 2 glk ( g x j ] δx k g x x k = 0 g x x k = 0 the bounary term goes away f we keep the en ponts fxe