d 2 x 0a d d =0. Relative to an arbitrary (accelerating frame) specified by x a = x a (x 0b ), the latter becomes: d 2 x a d 2 + a dx b dx c

Transcription

1 Chapte 6 Geneal Relativity 6.1 Towads the Einstein equations Thee ae seveal ways of motivating the Einstein equations. The most natual is pehaps though consideations involving the Equivalence Pinciple. In gavitational fields thee exist local inetial fames in which Special Relativity is ecoveed. The equation of motion of a fee paticle in such fames is: d 2 x 0a 2 =0. Relative to an abitay (acceleating fame) specified by x a = x a (x 0b ), the latte becomes: whee d 2 x a 2 + a dx b dx c bc =0, a 2 x c. Hee the a bc ae the fictitious tems that aise due to the non-inetial natue of the fame. Now, due to the Equivalence Pinciple the latte implies that locally gavity is equivalent to acceleation and this in tun gives ise to non-inetial fames. The main idea of Geneal elativity is to ague that gavitation as well as inetial foces should be descibed by appopiate a bc s! The simplest way to do this is by means of a Loentzian manifold the latte is endowed with geodesics of the equied type: d 2 x a 2 + a dx b dx c bc =0. Now, if the a0 bc s ae associated with gavitational foces, then the metic g ab may be associated with a potential. Note that the gavitational potential in the Newtonian theoy satisfies 2 =4 G, the density. The elativistic analogue of this equation should be tensoial and of second ode in the metic. To take this analogy futhe, conside two neighbouing paticles with coodinates 65

2 x (t), x (t)+ (t), with (t) small =1, 2, 3, moving in a gavitational field with a potential. the equations of motion ae then given: and ẍ + = Subtacting the two last equations: ẍ (x) (x) 2 + O( 2 This is the elative acceleation of two test paticles sepaated by by a 3-vecto the second deivative of the potential gives the tidal foces. This is in analogy to the geodesic deviation equation: V V = R a cdbv c V d b, povided that one 2 and Ra cdbv c V d b. This identification would make clea the elation between gavity and geomety note that the Riemann tenso involves second deivatives of the metic tenso. 6.2 Pinciples employed in Geneal Relativity The main idea undelying Geneal Relativity is that matte including enegy cuves spacetime (assumed to be a Loentzian manifold). This in tun a ects the motion of paticles and light ays, postulated to move on timelike and null geodesics of the manifold, espectively. These ideas ae undestood in conjunction with the main pinciples of Geneal Relativity, listed below. (1) Equivalence Pinciple. (2) Pinciple of Geneal Covaiance. This states that laws of Natue should have tensoial fom. (3) Pinciple of minimal gavitational coupling. This is used to deive the Geneal Relativity analogues of Special Relativity esults. Fo this change ab! g Fo example in Special Relativity the equations fo a pefect fluid ae given by: T ab =( + p)v a V b p ab, T ab,b =0. In Geneal Relativity these should be changed to: T ab =( + p)v a V b pg ab, T ab ;b =0. (4) Coespondence pinciple. Geneal elativity must agee with Special Relativity in absence of gavitation and with Newtonian gavitational theoy in the case of weak gavitational fields and in the non-elativistic limit (slow speed). 66

3 6.2.1 The Einstein equations in vacuum In vacuum, such as in the outside of a body in empty space, one has that the density vanishes and the equation fo the Newtonian potential becomes: 2 =0. The Laplace equation involves an object with two indices (@ 2 /@x j ). As a esult, what one needs is an object with two indices a contaction of the Riemann tenso, like the Ricci tenso: R bc =0. (6.1) The latte ae called the Einstein vacuum field equations. The vacuum equations fom a set of ten nonlinea, second ode patial di eential equations fo the components of the metic tenso g ab. These ae had to solve, apat fom simple settings. Remak 1. One of the simplest solutions to the vacuum equations is Minkowski spacetime. Expessing the metic ab locally as ds 2 = 2 + dx 2 + dy 2 + dz 2, we see that all the Chisto el symbols vanish, fom which R ab = 0 is tivially satisfied. Remak 2. The most geneal fom of the vacuum equations which is tensoial and depends linealy on second deivatives of the metic is: R bc = g ab, whee is the so-called Cosmological constant. Outside Cosmology, is usually taken to be zeo The (full) Einstein Equations Matte in elativity is descibed by a (0,2) tenso T ab called the enegy-momentum tenso. Consevation of mass-enegy is descibed by a T ab = 0, consistent with the calculation done in the Special Relativity case. Equating T ab with a cuvatue tenso with vanishing divegence, one has G ab = applet ab. Note, howeve, that since a g ab = 0, we could also have witten G ab + g ab = applet ab. (6.2) These ae the complete Einstein field equations fo the metic g ab of a spacetime. Note that the Einstein equations ae the simplest compatible with the Equivalence Pinciple, but they ae not the only ones. In geneal, the Einstein field equations ae extemely complicated set of non-linea patial di eential equations. In some simple settings, analytic solutions may be found. These include: (i) The vacuum spheically symmetic static case (the Schwazschild spacetime). (ii) The weak field case (gavitational waves). (iii) The isotopic and homogeneous case (Cosmology). We will discuss the fist case now. 67

4 6.3 The Schwazschild solution This is the basis fo nealy all the tests of Geneal Relativity. The solution coesponds to the metic coesponding to a static, spheically symmetic gavitational field in the empty spacetime suounding a cental mass (like the Sun). Choosing coodinates (t,,, '), it can be shown that a metic of this type is of the fom: ds 2 = e A() 2 + e B() d (d 2 +sin 2 d' 2 ), (6.3) whee A() and B() descibe deviation of the metic fom Minkowski spacetime. Note that fo constant t and the metic educes to the standad metic fo the suface of a sphee. As one is dealing with vacuum, one is poised to solve R ab =0. (6.4) Substituting (6.3) in (6.4), and afte some algeba, the only non-zeo components of (6.4) have the fom: R = R 11 = 1 2 A A0 B B 0 4 A02,, (6.5a) R = R 22 = e B (A0 B 0 ) 1,, (6.5b) R '' = R 33 = R 22 sin 2, (6.5c) R tt = R 00 = e A B 1 2 A A0 B A02 + A0, (6.5d) whee 0 denotes di eentiation with espect to. To solve R ab = 0, we stat by looking at the combination: Integating one obtains R + e B A R tt = 1 2 (B0 + A 0 )=0. A = B. One can without loss of geneality change t to absob the constant of integation. Substituting in (6.5b): e A (1 + A 0 ) 1=0. The latte can be ewitten as which can be integated to give (e A ) 0 =1, e A = +, a constant so that e A =1+, so that the metic one obtains is given by 1 ds 2 = d (d 2 +sin 2 d' 2 ). To fix, we use the fact that in the Newtonian limit of a cental mass M, 2GM g 00 = 1. 68

5 We shall pove this statement in the next subsection. Next, compaing with 1+, one finds that Hence, at the end of the day one has ds 2 2GM = = 2GM. 2GM 1 d (d 2 +sin 2 d' 2 ). (6.6) The latte is called the Schwazschild metic. Remak 1. This solution demonstates how the pesence of mass cuves flat spacetime. Remak 2. The metic (6.6) is asymptotically flat. That is, it becomes Minkowskian as!1. Remak 3. The solution only applies to the exteio of a sta. Remak 4. The Bikho Theoem: a spheically symmetic solution in vacuum is necessaily static. That is, thee is no time dependence is spheically symmetic solutions Newtonian limit In this subsection, we etun to the statement above which fixed the value of by using 2GM the Newtonian limit of a cental mass M via g 00 = 1. Conside a slowly moving paticle in a weak stationay gavitational field. Recall the geodesic equation: d 2 x a 2 + a dx b dx c bc =0. (6.7) Fo a slow moving paticle dx / ( =1, 2, 3) may be neglected elative to /, so that (6.7) implies that d 2 x a a 00 =0. (6.8) Since the gavitational field is assumed to be stationay, all t-deivatives of g ab vanish and theefoe a 00 = 1 2 d. (6.9) Futhemoe, since the field is weak, one may adopt a local coodinate system in which Substitution into (6.9) one has that g ab = ab + h ab, h ab 1. (6.10) Substituting in (6.8): a 00 = 1 2 d. 2 = (6.11a) d 2 t 2 =0, as h 00,0 =0. (6.11b) 69

6 Fom (6.11a) it follows that / is a constant. Also, fom it follows that which in ou case educes to 2 dx = d2 x 2 = dx, 2 + dx d 2 t 2, 2 = d2 x 2 2. Combining the latte with (6.11a) The coesponding Newtonian esult is = 1 2 h 00. (6.12) = (6.13) whee is the gavitational potential which fa fom a cental body of mass M at a distance is given by = GM. Compaing (6.12) and (6.13) one finds then that h 00 = 2 + constant. Howeve, at lage distances fom M one has that! 0 (gavity becomes negligible) and g ab! ab (the space becomes flat). Theefoe the constant must be zeo so that Substituting in (6.10) on finds h 00 = 2. g 00 = (1 + 2 ). Now, ecall that has dimensions of (velocity) 2,[ ]=[GM/R] =L 2 /T 2. Theefoe one has that /c 2 at the suface of the Eath is 10 9, one the suface of the Sun 10 6 and at the suface of a white dwaf It follows that in most cases the distotion poduced by gavity is in g ab vey small. 70