General Relativity (sort of)

Transcription

1 The Okefenokee Swamp Some book about relativity: Taylor & Wheeler Spacetime Phyic = TW1 (imple preentation, but deep inight) Taylor & Wheeler Introduction to Black Hole = TW (CO-level math, deep inight) Weinberg Gravitation & Comology (hairy-looking math, but example of olving field eqn) Hartle Gravity, An Introduction to Eintein General Relativity (textbook for AST 860) General Relativity (ort of) Gravity Upward acceleration, no gravity. Equivalence Principle: Can t tell difference between gravity & acceleration or between freefall & no gravity. So any experiment hould give ame anwer in either cae. Falling due to gravity No gravity Weak equivalence principle: F ma Gravity = Curved pace-time [CO 17.] F GMm r 1

2 What curve into where? [CO 17.] Gravity = Curved pace-time Metric dx dy dl = dx + dy g xx = g yy = 1 g xy = g yx ( d ) ( rd ) ( r in d ) dl = invariant length dl Metric coefficient g ij : d g dx dx ij i j The et of all ditance between grid of point, along all different coord. direction, pecify hape. dx df de Thi coord. ytem need a more complicated metric, with cro-term.

3 interval Special Relativity [ ] Proper time: Proper ditance: If l : If t : d If d : c dt d ; c light! dt cdt Metric Lorentz tranform [CO pg. 88; TW1 pg. 4] d dt timelike pacelike d c [CO (4.6, 4.9)] Some Metric d g dx dx ij i j Flat pace-time (pecial rel. = no gravity): Flat pace-time, pherical coord: Space filled by uniform matter ditribution: Empty pace around point ource of matter: Unit, etc. -- to c or not to c: ct t (Weinberg) ct t; GM/c M; leave out (Taylor & Wheeler) 3

4 Vague outline of General Relativity Eintein eqn: CO Eqn : Curvature G T Ma-Energy Gravitation & Comology Confuion Alarm!!! The R ued on thi & following lide are NOT the Scale Factor!! T = tre-energy tenor At each point in pace: = Gauian curvature [CO eq ] E p x p y p z p x S xx S xy S xz p y S yx S yy S yz p z S zx S zy S zz E = energy denity p = momentum flux S = tre g = component of metric tenor R = Ricci curvature tenor Full of derivative dg /dx, etc. x g meaure lope So dg /dx meaure curvature. Vague outline of General Relativity Eintein eqn: CO Eqn : Curvature G T Ma-Energy Gravitation & Comology Confuion Alarm!!! The R ued on thi & following lide are NOT the Scale Factor!! T = tre-energy tenor At each point in pace: = Gauian curvature [CO eq ] E p x p y p z p x S xx S xy S xz p y S yx S yy S yz p z S zx S zy S zz E = energy denity p = momentum flux S = tre g = component of metric tenor R = Ricci curvature tenor Full of derivative dg /dx, etc. x g meaure lope So dg /dx meaure curvature. 4

5 Metric for flat pace-time, pherical coord: ( Gravitation & Comology Eintein eqn: In empty pace: Curved pace-time: Unknown function, allow pace to be curved Non-zero component of Eintein Eqn: Where: = d/dr = d /dr Schwarzchild olution (1916): Black Hole & the Schwarzchild Metric Simplified metric, from Taylor & Wheeler (no c, no G, no ): Schwarzchild radiu: R S = M Schwarzchild coordinate recontructed a if een from a point where pace i flat. Metric for oberver itting on a hell at r Proper time: Proper ditance: Oberver meaure thing in locally flat pace-time: Free-falling atronaut: Metric = flat pace-time in local region. Time on writwatch = =. d 5

6 World-Line = path through pace-time Euclidean pace: Straight line = hortet ditance. Space-time: Straight line = longet ditance. TW1, Fig x y 1/ d dt dx t x 1/ Geodeic = traightet poible world line In free-fall, object follow geodeic = extremum of d (= d ) Normally a maximum oberver travelling on Geodeic ee max. time pa Light alway follow null geodeic, with = d d? 0 Geodeic Example in CO (pg ) A atellite in circular orbit. Aume: Circular orbit dr In plane where d Moving at pecified angular peed d = dt Find r that make be an extremum: dt A familiar Newtonian reult! 6

7 Metric for flat pace-time, pherical coord: ( Eintein eqn: Gravitation & Comology In empty pace: Curved pace-time: Unknown function, allow pace to be curved Non-zero component of Eintein Eqn: Where: = d/dr = d /dr Schwarzchild olution (1916): Curved Space & the Roberton-Walker Metric R-W metric: mot general olution for univere obeying Comological Principle. Homogeneou & Iotropic Smooth ditribution of matter. Same everywhere at any given time. [CO] notation, with c The non-zero component of the Eintein equation then reduce to Back to Weinberg notation, without c. dr R = dt etc. Friedmann eqn.[9.10] Form of fluid eqn.[9.50] Gravitation & Comology 7