MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 9

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics Novembe 17, 2006 Poblem Set 9 Due: Decembe 8, at 4:00PM. Please deposit the poblem set in the appopiate bin, labeled with name and ecitation section numbe and stapled as needed (3 points). Reading: Chaptes 2, 3, 4, 5, and Poject D in the Taylo & Wheele book - Exploing Black Holes, Intoduction to Geneal Relativity. You will be esponsible only fo the coesponding mateial that was actually coveed in the lectues. Poject E should also be undestandable, but this topic will be mentioned only vey biefly in lectue. Poblem 1 Concept questions 1. Accoding to the no hai theoem, which thee physical quantities uniquely chaacteize a black hole? (2 points) 2. To explain why most astophysicists now believe that black holes eally exist, biefly give one piece of evidence fo the existence of supemassive black holes and one piece of evidence fo the existence of stella-mass black holes. (Hint: see the black hole section in the Science handout.)(3 points) 3. Due to a miscalculation, you fiend falls into the supemassive black hole at the cente of ou Galaxy. Assuming that this black hole is neithe chaged no otating, indicate whethe each of the following statements is tue o false.(10 points) (a) He will die just as he entes the event hoizon. (b) He will die only afte he has enteed the event hoizon. (c) He will be killed by tidal foces. (d) He will be killed by the singulaity. (e) He will emege unscathed fom a while hole in anothe Univese. (f) You will see him disappea though the event hoizon. (g) You will see him foeve, seemingly fozen on the event hoizon. (h) You will see him seemingly fozen on the event hoizon until he edshifts out of sight. (i) Most physicists now believe that this black hole will last foeve, since nothing can come out fom it. 1

2 (j) The concepts of space and time as we know them ae no longe valid inside the event hoizon. Poblem 2 (9 points) Show that the Gullstand-Painlevé (GP) metic ( ) dτ 2 2 = dt ff (d + β dt ff ) 2 2 dθ 2 + sin 2 θdϕ 2 and the standad Schwazschild metic ( ) ( ) 1 2M 2M ( ) dτ 2 = 1 dt 2 1 d 2 2 dθ 2 + sin 2 θdϕ 2, ae equivalent. Hee is the escape velocity. ( ) 1/2 2M β 1. fist ewite the Schwazschild metic as ( ) dτ 2 = γ 2 dt 2 γ 2 d 2 2 dθ 2 + sin 2 θdϕ 2, whee 2. Stat with the elation 1 γ. 1 β 2 dt ff = dt + β γ 2 d fom the lectue notes which defines the time coodinate t ff and show that d + β dt ff = β dt + γ 2 d. 3. Show that when you plug these two elations into the GP metic, you get the Schwazschild metic. 2

3 Poblem 3(6 points) Why tidal foces make you tall and slim Fo simplicity, use classical mechanics and Newtonian gavity fo this poblem. Conside a closed capsule that is feely falling adially in the gavitational field outside a spheically symmetic body of mass M. At a given time, the cente of the capsule is at a distance fom the cente of the mass, and the capsule size is vey small compaed with. Define a local Catesian coodinate system with an oigin fixed at the cente of the capsule. The coodinate y is then the distance fom the oigin along the adial diection (), while x is the distance fom the oigin in a diection pependicula to the adial diection. Show that fo x and y, the acceleations of fee test paticles, elative to an obseve at (0,0), ae given by: GM a = ( x xˆ + 2y ŷ) 3 Conside 4 test paticles at locations (0, +y), (0, y), (x, 0), and ( x, 0). Compute the initial motions (befoe the paticle has moved vey fa) fo each paticle elative to (0, 0), and sketch thei tajectoies. If fall feet fist into a Schwazschild black hole, why would you become tall and slim befoe you die? Poblem 4(6 points) Compaison of shell and Compute and plot the shell adius, shell, vs. the coodinate adius. Follow the integation steps outlined in Taylo & Wheele, Sample Poblem 2, page Integate shell fom 0 s, whee 0 is an abitay stating value. Make a plot of ( s 0 ) vs., stating fom = 2M to a sufficiently lage value of to be able to discen the asymptotic behavio. [Note: we use units whee G = 1 and c = 1.] Poblem 5 Gavitational Redhift (3+3+3 points) A adioactive Fe souce emits a 6 kev X-ay line in its est fame. Suppose such a souce is located on the suface of a neuton sta of mass M = kg, and adius 10 km. Assume that the atomic tansition which gives ise to the X-ay line is not significantly physically alteed by the stong gavity o the possible pesence of an intense magnetic field. Ignoe any otation of the neuton sta. (a) Compute the enegy of these X-ays as they would be seen by a vey distant obseve, O, i.e., an astonome on Eath. (Neglect the gavitational potential of the Eath itself.) (b) Suppose that anothe stationay obseve, O, is fixed at adius > 10 km fom the cente of the neuton sta. Find an expession fo the enegy, E, of the X-ay line (coming fom the suface) that would be detected by such an obseve. (c) If the obseve at O sends X-ays of enegy E to the obseve back on Eath, as a epot of what he/she has seen, what enegy would be detected at O? 3

4 Poblem 6 Global Positioning Satellite System (GPS) (9 3 points) Stat with equation (3) of Poject A in Taylo & Wheele, page A 3, which we will deive in lectue. Poceed to answe Queies 1 though 9. Poblem 7 A Dilute Black Hole (3 points) Taylo & Wheele, Poblem 2 5, page Poblem 8 Obital Peiods Aound Black Holes (6 points) Conside two black holes of mass M 1 = 1 M and M 2 = 10 6 M (whee M = kg). a. Find the Schwazschild adius (R S 2GM/c 2 ) fo each object. b. Keple s 3d law (fo cicula obits, at least) woks exactly fo obits in the Schwazschild metic if the bookkeepe s coodinates and t ae used: (2π/P ) 2 = GM 3, whee P is the obital peiod. Find the obital peiod in seconds fo a cicula obit at just outside the Schwazschild adius fo an abitay mass (expessed in units of M ). What ae the coesponding obital peiods fo the two black holes given in this poblem? Poblem 9 (6 3 points) Falling into a Black Hole You fall adially into a black hole with Ẽ = E/m = 1, i.e., stating with negligible velocity fa away. 1. Wite down you total aging Δτ using the GP metic fom Poblem 2 as an integal along an abitay tajectoy (t ff ). 2. Pove that the motion given by d = β (1) dt ff is a geodesic, i.e., maximizes you aging. Hint: a simple agument suffices no need to use the vaiational calculus. 3. What is the elation between Δτ and the fee-fall time inteval Δt ff? 4. Integate equation (1) above to compute the time elapsed on you wistwatch between passing a adius and when it gets destoyed at What is the bookkeepe time inteval Δt between the stat of you fall and you cossing the event hoizon? (No calculation needed.) 4

5 6. To get used to woking with the obital equations of motion, use them to edeive equation (1) above. Specifically, use dτ = dt ff and these two equations as you stating point: ( ) 2 d = Ẽ 2 Ṽ ( L, ) 2, dτ ( ) ( ) 2M L 2 Ṽ ( L, ) 2 = Poblem 10(9 points) Dopping in on a Black Hole Execise 7, Chapte 3, page 3 30 of Taylo & Wheele. Poblem 11(6 points) Obit of a Satellite With Same Clock Speed As One On Eath Find the obital adius,, of a satellite whose clock will be found to un at the same ate as that fo an Eath-bound clock. Take the obital speed to be v = (GM/) 1/2, whee M is the mass of the Eath. Why would you guess that the GPS satellites ae placed in highe obits? Othe possibly useful pieces of infomation: R Eath = 6378 km; M Eath = kg; otation speed of the Eath s suface equals 6% of the obital speed at R Eath. Poblem 12 Time Tavel Using the Black Hole (6+3+3 points) 1. Do execise 7, Chapte 4, page 4 32 of Taylo & Wheele. 2. A moe ealistic cicula obit to use is the touist obit with E/m = 1 that can be eached with essentially no use expenditue of ocket fuel. Deive fo this obit using Taylo & Wheele equations [30] and [43], then compute the time dilation facto dτ/dt using the same fomula as you did above fo the = 6M case. Hint: If you ve found the coect -value in the lectue notes, you can simply veify that it satisfies equations [30] and [43] fo an appopiate L-value. 3. Compute dτ/dt fo the = 3M obit. How much enegy is equied to each this obit? 5

6 Optional Poblem A Without knowing it, you have almost leaned how to do advanced classical mechanics with Lagangeans, whee a paticle moves along a tajectoy x(t) such that the action S t 0 t 1 (T V )dt is minimized. This is called the pinciple of least action. Hee V = V (x(t)) is the potential enegy and T = 1 2 mẋ(t)2 is the kinetic enegy. Use the Eule-Lagange equation to deive the law of motion F = ma, i.e., mẍ = V (x). Optional Poblem B Heuistic Deivation of the Schwazschild Metic (a) Read the fist 4 pages of the aticle by Matt Visse on the web site. (b) Follow Visse s simple deivation of the invaiant inteval leading to: [ ] dτ 2 2M = dt 2 2M ( ddt + d dθ 2 + sin θ 2 dφ ) 2 whee we have set c = G = 1. (c) To tansfom this fom of the metic to the usual Schwazschild fom, make the following tansfomation of vaiables: d = d dt = dt + αd, whee α will tun out to be a function of. (d) Set the coefficient of the dt d tem equal to zeo to find α. (e) Finally, use the functional fom of α to deive the coefficients in font of the d 2 and dt 2 tems, and theeby show that ds 2 takes the standad fom of the Schwazschild metic. Big hint: The metic above is simply the GP metic fom poblem 2 if you ename the time coodinate t ff. Optional poblem C Moe igoous deivation of the Schwazschild metic As indicated in the lectue notes and optional handout, the Einstein field equations fo GR ae 8πG G αβ = 4 T αβ, (1) c whee G αβ is the Einstein tenso and T αβ is the stess enegy tenso, and G (with no indices) is Newton s gavitational constant. Fo the case of tying to find the space-time 6

7 metic in the egion outside of a spheically symmetic mass distibution (e.g., a neuton sta o black hole), the Einstein tenso educes to: whee the metic has been taken to be of the fom: G 00 = 1 e 2Φ d [(1 e 2Λ )] ; 2 d 1 2 dφ G 11 = e 2Λ (1 e 2Λ ) +, (2) 2 d ds 2 = e 2Φ d(ct) 2 + e 2Λ d dθ sin θ 2 dφ 2 (3) and e 2Φ and e 2Λ ae convenient ways of witing the two unknown functions of (only). In the egion outside the mass distibution, take the stess enegy tenso to be equal to zeo, and use G 00 and G 11 to solve fo e 2Φ and e 2Λ. Take the constant of integation fom the G 00 equation to be 2GM/c 2, and take the constant fom the G 11 equation to be 0. Optional poblem D Non-Relativistic Kepleian Obits In the vey weak-field limit (the Keple poblem), the equation fo the conseved enegy becomes: ( ) 2 1 d GMm L 2 E = m + 2 dt 2m, (4) 2 whee L is the angula momentum constant associated with the obit, E is a negative quantity fo a bound obit, and m is the mass of the obiting paticle. The distances of closest and fathest appoach of the obiting body occu when d/dt = 0. These ae the points whee a line of constant E intesects the effective potential cuve. Solve the esulting quadatic equation to find: [ ( ) ] 1/2 GMm 2L 2 E max,min = 1 ± 1 +. (5) 2E G2 M 2 m 3 The leading tem is defined as the semimajo axis, a, of the binay obit, while the squae oot tem is the obital eccenticity, i.e., max = a(1 + e) and min = a(1 e). Show that: and L 2 = a(1 e 2 )GMm 2 (6) GMm E =. (7) 2a Note that the enegy of a Kepleian obit depends only on the semimajo axis, and not on the obital eccenticity. The physical paametes L and E theeby uniquely detemine the obital shape which tuns out to be an ellipse. 7

8 Optional poblem E Simple Gavitational Lens System Conside an astonomical object, S, sufficiently distant (D S ) that it appeas pointlike to an obseve at O. Now intoduce a pointlike gavitational lens at a distance D L fom the obseve. The unpetubed angula sepaation between the souce and the lens is β as indicated in the sketch. In the pesence of the gavitational lens, light fom the souce can tavel the heavy-line path shown in the sketch, pass a distance of closest appoach b to the lens, and then be deflected by an angle δ so that it subsequently passes though O. The appaent angula distance between the lens and the image of the souce I, is θ (also indicated on the sketch). Thee ae two convenient appoximations that one can make in doing this poblem. (1) All angles ae taken to vey small such that tan x sin x x. (2) The light path fom the souce to the obseve may be consideed as two staight-line segments with a deflection of angle δ taking place at the point of closest appoach to the lens. Utilize the following steps to deive the elation between the angles θ and β: (a) In paticula, show that: 4GM h b h δ = ; α = ; θ = ; δ =. c 2 b D S D L D LS β = θ θ2 E θ, whee θ 2 = 4GM D LS E, c 2 D L D S whee θ E is defined as the angle of the Einstein Ring, i.e., the value of θ when β 0 (the lens and the souce ae along a line). (b) Show that thee ae two solutions fo θ, the appaent position of the souce (i.e., the image) fo each value of β, and find these two solutions. Make a sketch, analogous to the one above, to indicate what the geomety of these two solutions looks like. Optional poblem F Gavitational Acceleation on the Spheical Shell Execise 9, Chapte 3, page 3 31 of Taylo & Wheele. Optional poblem G Enegy Measued by a Shell Obseve 8

9 Stat with the geneal expession fo dτ/dt in the Schwazschild metic: [ ( ) ( ) 1 ( ) 2 ( ) ] 2 1/2 dτ 2M 2M d 2 dφ dt = 1 1 dt dt, and show that { ( ) [ ( ) 2 ( ) ]} 2 1/2 dτ 2M d s 2 dφ = 1 1. dt dt s dt s Caefully examine all the tems and ague that the quantity in squae backets ( [ ] ) is, in fact, γ 2 shell, and theefoe: ( ) 1/2 dτ 2M 1 = 1. dt γ shell Finally, combine this elation with ou expession fo the conseved quantity, E: ( ) E 2M dt = 1, m 0 dτ to elate E and E shell as follows: ( ) 1/2 2M E = 1 E shell, whee E shell m 0 γ s, the elativistic enegy measued by the shell obseve. Optional poblem H Speed of Light in a Schwazschild Metic Follow the deivation of equations (14) and (15) in the Boxed Execise Motion of Light in Schwazschild Geomety, page 5 8 of the Taylo & Wheele book. (a) Fom these two equations deive equation 5 16 on page 5 7. (b) Fo puely adial motion of the photon, show that the speed of light as eckoned by the bookkeepe is: ( ) d 2GM c. dt = ± 1 c 2 (c) In the weak field limit, find the bookeepe s time fo a photon to move adially (with espect to a cental 1 M sta) fom a distance of = 10 8 m to = m. How much longe does this take than fo a photon making the same tip in the absence of the sta? 9