UNIVERSITY OF ROCHESTER The Schwatzchild Geomety Byon Osteweil Decembe 21, 2018 1 INTRODUCTION In ou study of geneal elativity, we ae inteested in the geomety of cuved spacetime in cetain special cases such as in the space aound a spheical souce of cuvatue. Einstein s equations in this case wee fist solved by Kal Schwatzchild in 1916. The esulting solution is known as the Schwatzchild geomety, and it is applicable in such cases as the space aound a spheical sta, o aound a spheically symmetic black hole. Seveal significant conclusions can be dawn fom this geomety, such as gavitational edshift of a light ay, o the pecession of a planet s peihelion. 2 THE GEOMETRY The line element of the Schwatzchild geomety is ds 2 = 1 2GM ) c 2 c dt) 2 + 1 2GM c 2 d 2 + 2 dθ 2 + sinθ)) 2 dφ ) 2.1) The above equation is of couse expessed without the assumption of c = G = 1 units. The coodinates t,,θ,φ) ae the Schwatzchild coodinates, and will be always listed in the above ode fom hee on. The metic g αβ associated with the geomety is the Schwatzchild metic. 2.1 PROPERTIES OF THE METRIC 1. Time Independence. The metic has no dependence on the time coodinate t. The Killing vecto associated with this symmety is ξ α = 1,0,0,0). 1
2. Spheical Symmety. If we hold t and constant, the dt 2 and d 2 tems in the above go to 0, and the esulting equation becomes: dσ 2 = 2 dθ 2 + sinθ)) 2 dφ ) 2.2) Which is the typical geomety of an odinay sphee of adius in 3-D Euclidean space. As such the Schwatzchild metic is invaiant unde tansfomations in θ and φ. We can also see this in the φ case simply fom the fact Equation 2.1 has no φ dependence. The φ symmety has Killing vecto η α = 0,0,0,1). 3. Classical Limit. If we assume that GM/c 2 is small i.e. M is small o is lage), then we can make use of the geometic seies: 1 1 x = n=0 x n when x 0,1)) 2.3) We use the fist two tems in place of 1 GM/c 2 in Equation 2.1 to obtain ds 2 1 2GM ) c 2 c dt) 2 + 1 + 2GM ) c 2 d 2 + 2 dθ 2 + sinθ)) 2 dφ ) 2.4) Which is the fom of the static weak field fom a Newtonian potential Φ = GM/. Because of this the constant M is intepeted as being the total mass of the souce of cuvatue. In this case we define "mass" as the ability to poduce cuvatue, which may include both mass in the typical sense as well as EM o nuclea enegy. Futhemoe the geomety is dependent only on total mass of the souce, not on its distibution, a fact which is essentially a elativistic efomulation of Newton s shell theoem. 4. The Schwatzchild Radius. Note that as appoaches 2GM/c 2, the d 2 tem in Eq. 2.1 appoaches 0 1. As such we expect thee must be some sot of exceptional behavio at this adius. In the case of a spheical sta, the adius tuns out to not be teibly impotant; = 2GM/c 2 is always inside the sta itself, whee a diffeent metic applies and the Schwatzchild adius is no longe significant. In the case of a body collapsing into a black hole it is consideably moe impotant, but hee we only deal with the case of a sta. Lastly fo this section we will ewite Eq. 9.1 using units whee c = G = 1, and give an explicit fom fo the metic g αβ : ds 2 = 1 2M ) dt) 2 + 1 2M d 2 + 2 dθ 2 + sinθ)) 2 dφ ) 2.5) 1 2M/ ) 0 0 0 0 1 2M/ 0 0 g αβ = 0 0 2 0 0 0 0 2 sinθ)) 2 Fom hee we exploe vaious consequences of this geomety. 2.6) 2
3 GRAVITATIONAL REDSHIFT To an obseve at a fa distance fom a souce of Schwatzchild cuvatue, a light ay emitted fom a souce close to the cente of cuvatue will appea edshifted, i.e. its wavelength will appea to be longe than the wavelength seen fom the souce. This effect, the gavitational edshift, can be appoximated with the equivalence pinciple, but can also be pecisely calculated via the Schwatzchild geomety. The fequency ω of the photon can be diectly elated to its enegy, and as such the edshift can be quantified via the effect of gavity on the photon s enegy. If an obseve has 4-velocity u obs and 4-momentum p, then its enegy can be given as: E = p u obs 3.1) This can of couse be elated to fequency by the familia elation E = ω. If we assume the obseve is stationay, then the spatial coodinates of u obs ae all zeo, and the time component can be detemined fom the nomalization condition: u obs ) u obs ) = g αβ u α obs )uβ ) = 1 3.2) obs With the spatial components all zeo, this gives: g t t ) [ u t obs )] 2 = 1 3.3) Which, substituting in the value of g t t gives: Meaning the full velocity vecto is: u t obs ) = 1 2M u obs ) = 1 2M /2 3.4) /2 ξ 3.5) Whee ξ is the Killing vecto fo the time symmety. So fo a stationay obseve at a adius, we have ω = 1 2M /2 ξ p) 3.6) As, 1 2M/ /2 1 so the photon enegy as seen by a distant obseve is: ω = ξ p) 3.7) Note howeve that ξ p is conseved because of the symmety in t, so it is the same eveywhee. So if ω is the fequency seen by the emitte, the edshift is quantified as: ω = ω 1 2M 3.8) 3
4 PRECESSION OF THE PERIHELION The Schwatzchild geomety is useful in calculating the paths of massive paticles moving within it, in othe wods timelike geodesics. Due to the geometies independence on t and φ, we have associated conseved quantities. These will be defined as: e = ξ u = 1 2M ) dt dτ 4.1) l = η u = 2 sinθ)) 2 dφ dτ These ae associated with enegy and angula momentum, espectively. 4.2) 4.1 EFFECTIVE POTENTIAL As in Newtonian mechanics, we expect that all paticle obits will fall plane, so we will assume that this plane is φ = 0, θ = π/2 fom hee on out. Anothe piece of infomation we can make use of in solving fo these geodesics is the nomalization of the 4-velocity: u u = g αβ u α u β = 1 4.3) Applying this, and the fact θ = π/2 and u θ = 0 to the Schwatzchild metic yields: 1 2M ) u t ) 2 + 1 2M u ) 2 + 2 u φ) 2 = 1 4.4) Making use of the identities given in Eq. 4.1 and Eq. 4.2, we can equivalently ewite this as: O equivalently, 1 2M ) e 2 + 1 2M e 2 1 = 1 ) d 2 + 1 2 2 dτ 2 We make the following definitions: ) d 2 + l 2 = 1 4.5) dτ 2 [ 1 2M )1 + l 2 2 ) ] 1 4.6) V eff ) 1 2 So we can wite Eq. 4.6 as: [ 1 2M E e2 1 2 )1 + l 2 2 4.7) ) ] 1 = M + l 2 2 2 Ml2 3 4.8) ) d 2 +V eff ) 4.9) E = 1 2 dτ 4
Note that the expession used hee fo V eff diffes fom the Newtonian equivalent only by the Ml 2 / 3 tem, making it only significant fo lage M o small. We can solve fo the extema of V eff by solving dv eff /d = 0: max min = l 2 M 1 ± 1 12 2M l ) 2 4.10) If l/m < 12, then both of these values ae complex and V eff is negative eveywhee. If l/m > 12, the maximum and minimum exist and ae both eal. As in Newtonian mechanics, the behavio of the obit depends on the elationship between E and V eff. Tuning points occu when E = V eff. If l/m < 12, this will not occu as long as E is positive, so an inwadly diected paticle will fall diectly into the souce of cuvatue, in contast to Newtonian pedictions. When the effective potential is at an extemum, cicula obits ae possible, with the maximum obit being unstable and the minimum stable. 4.2 RADIAL PLUNGE ORBITS Conside a paticle with l = E = 0, falling fom to the cente of gavitational attaction. We then have that: So the 4-velocity of the paticle is: 0 = 1 ) d 2 M 2 dτ 4.11) u α = 1 2M/, 2M/ ) 1/2,0,0) 4.12) By integating Eq. 4.11, we can obtain an expession fo as a function of τ: 3/2) 2/3 2M) 1/3 τ τ) 2/3 4.13) Whee T is an integation constant. We can calculate dt/d fom the definition of e given in Eq. 4.1: dt d = 2M Which we can integate again to obtain: t = t + 2M [ 2 3 2M /2 1 2M ) 3/2 2 2M 4.14) ) 1/2 ] /2M + 1 + ln /2M 1 4.15) Whee t is a second constant of integation. We note a few inteesting facts. Stating fom any fixed value of, it will take only finite pope time to each the Schwatzchild adius of 2M. The same movement, howeve, will take infinite coodinate time. This indicates a flaw in the Schwatzchild coodinates nea the pivotal adius of 2M. 5
4.3 STABLE CIRCULAR ORBITS As discussed peviously, a stable cicula obit exists at the minimum adius min whee the effective potential has its minimum value. As l/m deceases these adii become smalle, but thee is a specific minimum fo l/m = 12: We define angula velocity Ω: ISCO = 6M 4.16) Ω dφ dt = dφ/dτ dt/dτ = 1 2 1 2M ) ) l e 4.17) We know that effective potential must be at a minimum, so we have fom Eq. 4.10 a elationship fo and l. Secondly we know E = V eff min ), giving us: e 2 = 1 2M Meaning that fo a cicula obit we have: Which implies the elation: l e = M 1 2M )1 + l 2 ) 2 4.18) 4.19) Ω 2 = M 3 4.20) Which is anothe fom fo Keple s thid law. The 4-velocity of a paticle in a cicula obit is given by: ) u α = 1 3M/ ) 1,0,0,Ω) 4.21) 4.4 BOUND ORBITS To detemine the shape of an obit we must find eithe a function φ ) o φ). We can do this fom the elations given in Eq. 4.2 and Eq. 4.9 to obtain: dφ d = ± l 1 2 2E Veff )) = ± l [ 2 e 2 1 2M )1 + l 2 )] 1/2 4.22) 2 Which can be integated to obtain φ ) in the fom of an elliptical integal. We will say that an obit closes if the change in angle between successive inne o oute tuning points is 2π. If it does not close, then the angle of pecession is the diffeence between φ and 2π: δφ pec = φ 2π 4.23) If 1 is an inne tuning point and 2 is an oute tuning point, then: 6
2 φ = 2l d 1 1 2 2 = 2l d 1 1 2 [ e 2 1 2M )1 + l 2 2 )] 1/2 [ c 2 e 2 1 ) + 2GM l 2 2 + 2GMl2 c 2 3 ] 1/2 4.24) The second fom of the above is obtained simply by eintoducing c and G explicitly. By expanding this integal we obtain an expession fo δφ pec : ) GM 2 6π = 6πG M cl c 2 a1 ɛ 2 ) 4.25) Whee a is the semimajo axis and ɛ is the eccenticity of the obit. The elativistic effect is popotional to 1/ 3, so in the Sola system it is only significant enough to be obsevable in the case of Mecuy. 5 LIGHT ORBITS The obits of a light ay unde the Schwatzchild geomety shae some significant similaities to the obits of a paticle, although thee ae key diffeences as well. As befoe we have conseved quantities associated with the symmeties in φ and t. The conseved quantities ae: e ξ u = 1 2M ) dt dλ l η u = 2 sinθ)) 2 dφ dλ Whee λ is an affine paamete. The last equiement is also simila to one used in the paticle case: 5.1) 5.2) dx α dx β u u = g αβ dλ dλ = 0 5.3) The deivation followed is also simila. We stat fom Eq. 5.3 and apply the Schwatzchild metic and the condition that θ = π/2. We then substitute in fo e and l. 1 2M We define quantities b and W eff ): e 2 1 2M ) d 2 + l 2 dλ 2 = 0 5.4) b l 5.5) W eff ) 1 2 1 2M ) 5.6) 7
W eff hee coesponds to the effective potential, and b coesponds to the impact paamete of the light ay. Fom this we ewite the above equation as: W eff ) has a maximum at = 3M: 1 b 2 = 1 ) d 2 l 2 +W eff ) 5.7) dλ W eff 3M) = 1 27M 2 5.8) It is possible fo light to have a cicula obit in the case of b 2 = 27M 2, but these obits ae unstable and cannot occu aound a sun-like sta though it can aound a black hole). If 1/b 2 < 1/27M 2 then the light ay will have a tuning point and then escape the gavity of the cuvatue souce. If 1/b 2 > 1/27M 2, then a plunging obit will be obseved. 5.1 DEFLECTION OF LIGHT When a light ay passes close to a souce of cuvatue, it gets deflected by it. We quantify this in tems of an angle δφ def, defined to be the angle between the initial and final tajectoies of the light ay. As with paticle obits, we do this by solving fo dφ/d fom equations seen ealie: dφ d = ± 1 [ ] 1 1/2 2 b 2 W eff ) 5.9) The light ay comes in fom infinity, passes though a tuning point 1, and goes out to infinity again. So the change in φ is: φ = 2 d 1 [ 1 1 2 b 2 1 2 1 2M Fo small values of M/b, this integal can be appoximately evaluated as: )] 1/2 5.10) φ π + 4M b 5.11) So δφ def 4M/b. Reinseting G and c explicitly gives us: δφ def 4GM c 2 b 5.12) Expeimental obsevation of this effect povided a significant piece of evidence that elativity was tue. 5.2 DELAY OF LIGHT Because of the same physical effect which causes the deflection of light aound souces of cuvatue, light can also be delayed in passing fom one point to anothe and back if its path takes it close to a lage mass. Conside a pulse of light sent fom Eath to a satellite obiting on the fa side of the sun. The satellite eflects the signal back to Eath whee it is detected 8
again. Let E and S denote the obital adii of the Eath and the satellite, espectively, and let 0 denote the adius of closest appoach of the light ay to the sun. The ound-tip tavel time is not long compaed to the length of a yea, so we can safely make the assumption that the Eath emains stationay in the time needed fo the light ay to tavel out and etun. Simila to pevious calculations, we find the time diffeence between the ay s depatue and etun by finding t as a function of. Fom equations 5.1 and 5.7: dt d = ± 1 b 1 2M [ ] 1 1/2 b 2 W eff ) 5.13) Let t 2, 1 ) denote the time needed fo the ay to pass fom a adius 1 to a adius 2. Total time elapsed is then: The equation fo t 2, 1 ) t 2, 1 ) = 2 1 t = 2t e, 0 ) + 2t 0, S ) 5.14) d 1 b 1 2M [ ] 1 1/2 b 2 W eff ) 5.15) And whee 1/b 2 = W eff 1 ). Appoximating this integal in a fashion simila to what was done in pevious sections, we obtain: t 2, 1 ) = 2 2 1 2 + 2M ln 2 + 2 2 1 2 1 + M 2 1 2 + 1 5.16) The fist tem is what we expect fom the non-elativistic case of a ay moving in a staight line though nomal Euclidean space at c. The emaining tems povide elativistic coections. We expess the effect in tems of the diffeence of classical and elativistic pedictions: t) excess = t 2 2E 20 2 2 S 0 2 5.17) If we assume that the ay moves quite close to the sun, i.e. 0 is small, we get the appoximation: With c and G witten explicitly. t) excess 4GM c 3 [ ) ] 4 S E ln + 1 2 0 5.18) 6 SOURCES 1. Hatle, J. B. 2003). Gavity: An Intoduction to Einstein s Geneal Relativity, San Fancisco: Addison Wesley. 9