Gravity: Newtonian, post-newtonian, Relativistic

Transcription

1 Gavity: Newtonian, post-newtonian, Relativistic A Mini-couse within PH 7608 Cliff Pofesso Cliffod Will Distinguished Pofesso of Physics Univesity of Floida Checheu Associé Institut d Astophysique de Pais cmw@physics.ufl.edu 2071 NPB

2 Outline of the Lectues 1. Newtonian gavity (4 hs) Foundations, Isolated gavitating bodies, Obital dynamics PW, Chaptes Post-Minkowskian theoy (2 hs) Fomulation, Implementation PW, Chaptes Post-Newtonian theoy: Nea-zone physics (3 hs) Implementation, PN celestial mechanics, astomety & timekeeping PW, Chaptes Post-Newtonian theoy: Fa-zone physics (2 hs) Gavitational adiation, Radiation Reaction PW, Chaptes Altenative theoies of gavity (1 h) Metic theoies, PPN famewok, Scala-tenso theoy PW, Chapte 13 Textbook: Gavity: Newtonian, post-newtonian, Geneal Relativistic, by Eic Poisson and Cliffod Will (Cambidge U Pess, 2014)

3 Gound ules and gading Schedule: Jan 14, 21, 28 (oom 2205) Feb 18 Ma 11 Ap 1 Question: Feb 26 (Thu)? Ma 4 (sping beak)? Ma 25? Homewok: N sets of M poblems each, taken fom PW Office hous: Wednesdays afte class, o by appointment Skype calls may wok when I m not at UF

4 Foundations of Newtonian Gavity Newton s 2nd law and the law of gavitation: The pinciple of equivalence: a = m G GM m I 3 If m I a = F F = Gm G M/ 3 m G = m I (1 + ) Then, compaing the acceleation of two diffeent bodies o mateials a = a 1 a 2 = ( 1 2 ) GM 3

5 The Weak Equivalence Pinciple (WEP) 400 CE Ioannes Philiponus: let fall fom the same height two weights of which one is many times as heavy as the othe. the diffeence in time is a vey small one 1553 Giambattista Benedetti poposed equality 1586 Simon Stevin expeiments Galileo Galilei Leaning Towe of Pisa? Newton pendulum expeiments 1889, 1908 Baon R. von Eötvös tosion balance expeiments (10-9 ) UW (Eöt-Wash) Atom intefeometes matte waves vs macoscopic object Bodies fall in a gavitational field with an acceleation that is independent of mass, composition o intenal stuctue

6 Tests of the Weak Equivalence Pinciple 10-8 Eötvös Matte waves 10-9 Renne Fee-fall Pinceton Boulde Fifth-foce seaches Eöt-Wash APOLLO (LLR) Micoscope (2015) Futue: STEP, GG, STE-QUEST Moscow LLR Eöt-Wash a 1 -a 2 (a 1 +a 2 )/ YEAR OF EXPERIMENT

7 Newtonian equations of Hydodynamics Witing Equation of motion Field equation Genealize to multiple souces (sum ove M s) and ma = mu, U = GM/, dv dt = U + ( v) =0, p, 2 U = 4 G, d + v, p = p(, T,... ) Fomal solution of Poisson s field equation: Wite U(t, x) =G G(x, x 0 ) (t, x 0 )d 3 x 0, Eule equation of motion Continuity equation Poisson field equation Total o Lagangian deivative Equation of state Geen function 2 G(x, x 0 )= 4 (x x 0 ) ) G(x, x 0 )=1/ x x 0 U(t, x) =G (t, x 0 ) x x 0 d3 x 0

8 Rules of the oad Consequences of the continuity equation: fo any f(x,t): d (t, x)f(t, x) d 3 x + d 3 f ( v) d 3 I + v f d 3 x f v = df dt d3 x. (t, x 0 )f(t, x, x 0 ) d 3 x 0 + v0 0 f d 3 x 0, (t, x 0 )f(t, x, x 0 ) d 3 x 0 = 0 + v f + v0 0 f d 3 x 0 = 0 df dt d3 x 0

9 Global consevation laws M := (t, x) d 3 x = constant P := (t, x)v d 3 x = constant d( V)+pdV =0 v = V 1 dv/dt E := T (t)+ (t)+e int (t) = constant J := x v d 3 x = constant T (t) := 1 2 (t) := E int (t) := v 2 d 3 x 1 2 G d 3 x 0 x x 0 d3 x 0 d 3 x, R(t) := 1 M (t, x)x d 3 x = P M (t t 0)+R 0 d dt vd 3 x = ( U = G =0 p) d 3 x 0 x x 0 x x 0 3 d3 xd 3 x 0 I pnd 2 S

10 Spheical and nealy spheical bodies Spheical symmety @ = 4 G = Gm(t, ) 2 m(t, ) := 0 4 (t, 0 ) 02 d 0 U(t, ) = Gm(t, ) +4 G R (t, 0 ) 0 d 0. Outside the body U = GM/

11 Spheical and nealy spheical bodies Non-spheical bodies: the extenal field x 0 < x Taylo expansion: 1 x x 0 = 1 1X = `=0 1 x j x0j x k ( 1)` 1 x L `! Then the Newtonian potential outside the body becomes 1X ( 1)` 1 U ext (t, x) =G I hli, `! `=0 I hli (t) := (t, x 0 )x 0hLi d 3 x 0 x L := x i x j...(l L j...(l times) h...i := symmetic tacefee poduct

12 Symmetic tacefee (STF) tensos A hijk...i Symmetic on all indices, and ija hijk...i =0 Example: gadients of j 1 = n j jk 1 = 3n j n k jk 3, jkn 1 = h15n j n k n n 3 n j kn + n k jn + n n jk L 1 hli 1 =( 1)`(2` 1)!! n hli `+1 Geneal fomula fo n <L> : n hli = [`/2] X p (2` 2p 1)!! h ( 1) 2P n +sym(q)i L 2P (2` 1)!! p=0 q := `!/[(` 2p)!(2p)!!]

13 Symmetic tacefee (STF) tensos Link between n <L> and spheical hamonics e hli n hli = n hli := `! (2` 1)!! P`(e n) 4 `! (2` + 1)!! `X m= ` Y hli `m Y`m(, ) Y hzi 10 = 3 Y hxxi 20 = 4, 5 16, Yhxi 3 11 = 8, Yhyyi 20 = 5 16, 3 Yhyi 11 = i 8, Yhzzi 20 =2 5 16, Aveage of n <L> ove a sphee: 8 I < hhn L ii := 1 4 n L d = : 1 (2`+1)!! L/2 +sym[(` 1)!!] ` =even 0 ` =odd

14 Spheical and nealy spheical bodies Example: axially symmetic body I hli A = m ARÀ(J`) A e hli J` := 4 2` +1 1 MR` (t, x)`y `0(, e ) d 3 x U ext (t, x) = GM " 1 1X `=2 J` R ` P`(cos ) # Note that: J 2 = C A MR 2 C A

15 Measuing the Eath s Newtonian hai Gavity Recovey And Climate Expeiment (GRACE) Eath: j (5) X = X 10, j (1) X = 9.6 X 10-7, j (4) X = 5.4 X 10-7, j 360,0 = 3.2 (5) X 10-10

16 Motion of extended fluid bodies Main assumptions: Bodies small compaed to typical sepaation (R << ) isolated -- no mass flow T int ~ (R 3 /Gm) 1/2 << T ob ~ ( 3 /Gm) 1/2 -- quasi equilibium adiabatic esponse to tidal defomations -- nealy spheical Extenal poblem: detemine motions of bodies as functions (o functionals) of intenal paametes Intenal poblem: given motions, detemine evolution of intenal paametes Solve the two poblems self-consistently o iteatively Example: Eath-Moon system -- obital motion aises tides, tidally defomed fields affect motions

17 Motion of extended fluid bodies Basic definitions m A := (t, x) d 3 x A A (t) := 1 (t, x)x d 3 x m A A dm A /dt =0 v A (t) := d A dt a A (t) := dv A dt = 1 m A = 1 m A Is the cente of mass unique? pue convenience, should not wande outside the body not physically measuable almost impossible to define in GR m A a A = G G A A 0 x x 0 A x x 0 3 d3 xd 3 x X 0 x x 0 B6=A B x x 0 3 d3 x 0 5 d 3 x A A v d 3 x dv dt d3 x Define: x := A (t)+ x x 0 := B (t)+ x 0 AB := A B

18 N-body point mass tems Motion of extended fluid bodies a j A = G X B6=A + 1X `=2 + 1 m A ( m B 2 AB n j AB 1 h ( 1)`I hli B `! 1X 1X `=2 `0=2 ( 1)`0 `!`0! Moments of othe bodies + m i B I hli 1 jl A m A AB I hli i 1 A IhL0 jll 0 AB ) Effect of body s own moments Two-body system with only body 2 having non-zeo I <L> := 1 2, := R := (m m 2 2 )/m Moment-moment inteaction tems m := m 1 + m 2 µ := m 1 m 2 /m a j = Gm 2 nj + Gm 1X `=2 ( 1)` `! I hli 2 m jl 1