Homework Math 80: Introduton to GR Temple-Wnter 208 (3) Summarze the artle: https://www.udas.edu/news/dongwthout-dark-energy/ (4) Assume only the transformaton laws for etors. Let X P = a = a α y = Y α P, and assume f s a salar funton on the spaetme manfold M. Let f x and f y be the x- and y-oordnate representatons of f, respetely. Proe that a f x = a α y αf y. Explan why ths makes sense as funtons from oordnates to oordnates een though f s defned on spaetme, and you an t dfferentate spaetme dretly. Let f be a funton on spaetme, f : M R. Let X P = a be a etor at x(p ), and let df = f dx be the -form whh orresponds to the dfferental of f. Explan how X operates on f to ge the gradent of f n dreton X: And then explan how, alternately, we an ew df as operatng on X to ge the gradent of f n dreton X. Fnally, say n words why ths ges an nterpretaton of a etor as somethng ndependent of oordnates. (5) Show that f g j are the omponents of a gratatonal metr n x-oordnates, and g j transforms lke a (0, 2) tensor oer to
y-oordnates as ḡ αβ, that s, ḡ αβ = j y αg j y, β then g j s symmetr (g j = g j ) f and only f ḡ αβ s symmetr, and g j, as a 4 4 matrx, has an nerse f and only f ḡ αβ has an nerse as a 4 4 matrx. (Hnt: Wrte the tensor transformaton laws as matrx multplaton, and use propertes of matres, lke A matrx s nertble f and only f ts determnant s nonzero. The produt of nonsngular matres s nonsngular, et.) (6) (I) Let g and J be 3 3 matres, and let gj = A. Wrte out the rows of all three matres, and explan matrx multplaton by the rows of G ontrat wth the rows of J to reate the rows of A. Then wrte out the olumns of all three matres, and explan how the olumns of J ontrat wth the olumns of G to reate the olumns of A. (II) Now let g j and Jj be (0, 2) and (, ) tensors ewed as 3 3 matres, the row and j the olumn, and onsder the two matrx multplatons g k Jj k = A j and g kj J k = A j (the latter beng equalent to J k g kj = A j beause the order n whh you lst the tensors doesn t matter when you use the summaton onenton to express matrx multplaton). (a) Usng the olumn nterpretaton of matres, explan why g k J k j = A j expresses matrx multplaton gj = A. (b) Usng the row nterpretaton of matres, explan why g kj J k = A j expresses matrx multplaton J t g = A. 2
(7) Proe that f Sj and Tj are the omponents of a (, )-tensors n x-oordnates, then A j = SkT j k (sum repeated up-down ndes from 0 to 3) transforms lke a (, )-tensor. What would the general theorem about tensors arbtrary tensors S, T be? (8) Let Tj be the omponents of a (, )-tensor T at a pont n x-oordnates, and assume there s a etor X = a suh that Tja j = λa. (That s, λ s an egenalue of the 4 4 matrx Tj.) Proe that λ s ndependent of oordnates. (Hnt: See what s true n y-oordnates, for any other oordnate system y.) (9) Let X, Y, Z be three ndependent etors n T P (M) (the tangent spae of M at P ) gen n x-oordnates by X = a, Y = b j,z = k j. Let n = (n k 0, n, n 2, n 3 ) be the x-oordnate unt normal to the hyperplane spanned by X, Y, Z, so that n a = n b = n = 0, and n n =, where dot s the dot produt n x-oordnates. (a) Show that f we assume n transform o-arantly as a down ndex to n = ( n 0, n, n 2, n 3 ) n y-oordnates, then n ā = n b = n = 0. (b) Is n n = n y-oordnates? Explan. (0) Lete {X 0, X } be a postely orented orthonormal frame n + speal relatty. Defne the x oordnate system n terms of the gen x-oordnate system by spefyng that P has 3
x-oordnates (x 0, x ) and x-oordnates ( x 0, x ) f and only f x 0 Argue that n ths ase, + 0 x = x0 X 0 + x X. X 0 = (Hnt: What defnes x x 0, X = x. n the frst plae?) () In the feld of asymptots, we say f(t) s O(t n ) as t 0 to mean that there exsts a onstant C > 0 suh that f(t) C t n for t suffently small. (Often we omt to add as t 0, but ths s always mpled.) (a) Use Taylor s theorem to show that: + x = x + O(x2 ), and + x = + 2 x + O(x2 ), (b) Use these to show that x = + x + O(x2 ), x = 2 x + O(x2 ). ( ) 2 = ( ) 2 ( + O 2 ) 4, and ( ) 2 = + 2 ( ) 2 ( + O ) 4. 4
(2) Twn Paradox: Imagne two twn obserers who start out fxed wth respet to a gen Mnkowsk oordnate system. One twn remans at rest n that frame, whle the other goes off n a roket shp and returns to see the frst twn at a later tme. Aordng to our tme dlaton, the seond traelng twn wll age less than the twn who remans at rest n the Mnkowsk frame, beause mong obserers appear to age slower. The Paradox s, to the seond twn, the frst twn appears to hae gone off on a trael and returned to hm, so by symmetry, why wouldn t the seond twn be older than the frst, a ontradton? Resole the paradox n words. (3) Show L(θ)L( θ) = L(θ + θ), and use ths to proe the relatst addton of elotes formula = + +, 2 where s the eloty of the x-frame wth respet to the unbarred frame, and s the eloty of a thrd x-frame wth respet to the barred frame. (4) Assume X s non-lghtlke, so < X, X > 0. Dere the relatst erson of the orthogonal projeton of a etor Y onto a etor X gen by P roj X Y = < X, Y > < X, X > X, and nterpret t geometrally. (Hnt: Start wth X, wrte Y as a lnear ombnaton of X and the unt etor othogonal to X, and use the nner produt to sole for the omponents.) 5