Chapter 20. Orbits of Light around the Spinning Black Hole

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1 Septemer 6, :38 SpinLight170906v1 Sheet numer 1 Page numer 19-0 AW Physics Macros 1 2 Chapter 20. Orits of Light around the Spinning Black Hole Introduction: The Purposes of Looking Doran Gloal Equations of Motion for the Stone Effective Potential for Light Exercise What variety of paths does light follow around a spinning lack hole? Can a spinning lack hole reverse the direction of a light eam? Can a light eam go into orit around a spinning lack hole? If so, how many different orits are availale to it? What does a distant spinning lack hole look like? How can I distinguish it visually from a non-spinning lack hole? 13 Download file name: Ch20OritsOfLightAroundSpinningBH170906v1.pdf

2 Septemer 6, :38 SpinLight170906v1 Sheet numer 2 Page numer 20-1 AW Physics Macros C H A P T E Orits of Light around the Spinning Black Hole Edmund Bertschinger & Edwin F. Taylor * Light does strange things near a spinning lack hole; it misleads you aout the locations and shapes of things, so you must grope along as if you are in a haunted house. Seeing is definitely not elieving! The authors New questions aout the spinning lack hole 20.1 INTODUCTION: THE PUPOSES OF LOOKING 21 Who cares what we see? Chapter 11 descried orits of light around the non-spinning lack hole. That earlier chapter focussed on the question, What is the visual size of the lack hole seen y a raindrop diver falling from a great distance? The same question aout the spinning lack hole is of little interest today. Instead, we ask the questions: 27 How can we identify a distant spinning lack hole in the heavens? 28 How can we measure its mass M and spin parameter a? Stone s mass goes to zero To answer these questions, the present chapter does use a method similar to that of these earlier chapters: Start with a stone of mass m. Let the stone s mass go to zero. We egin this program with a review of the stone s equations of motion in gloal Doran coordinates. * Draft of Second Edition of Exploring Black Holes: Introduction to General elativity Copyright c 2017 Edmund Bertschinger, Edwin F. Taylor, & John Archiald Wheeler. All rights reserved. This draft may e duplicated for personal and class use. 20-1

3 Septemer 6, :38 SpinLight170906v1 Sheet numer 3 Page numer 20-2 AW Physics Macros 20-2 Chapter 20 Orits of Light around the Spinning Black Hole DOAN GLOBAL EQUATIONS OF MOTION FO THE STONE 36 Goal: Get rid of wristwatch time Motion of a stone This chapter analyzes the motion of light in the equatorial plane of the spinning lack hole. We derive equations of motion for light y extending equations of motion for a free stone. Begin this process with equations (15), (21), and (20) and (16) for motion of a stone in Section 18.2:. Write them in the form: dr dτ = ± mr (E V + L )1/2 (E V L )1/2 (stone) (1) dφ dτ = L m 2 + sin2 α E ωl ± 1β ] ma (E V +L )1/2 (E V L )1/2 (stone)(2) ( ) 2 dt dτ = 1 E ωl ± β(e V + L rh m )1/2 (E V L )1/2] (stone) (3) ) 1/2 (m 2 + L2 (stone) (4) V ± L (r) ωl ± rh 2 Photon does not age Box 1 in Section 18.2 defines α, β, and ω. To apply these equations to light, we must overcome a fundamental prolem: The differential aging dτ of a light flash along its worldline is automatically zero (Section 1.4), so these equations of motion for the stone have no meaning for the light flash. To give them meaning, we eliminate dτ from these equations of motion, recasting them without dτ. Use the following equations: ( ) ( ) dr dr dτ dt = (stone) (5) dτ dt ( ) ( ) dφ dφ dτ dt = (stone) (6) dτ dt 49 Carry out this comination on equations (1) through (3). esult: 50 equations for dr/dt and dφ/dt. Note that in this process, m disappears from the coefficients ut remains in the expression for V ± L (r) in equation (4) To convert these equations to descrie light, we take the limit of the 53 resulting equation for dr/dt and dφ/dt as m 0. Carry this out first on expressions that contain V + L and V L defined in equation : 54 ( E V + ) ( )] L E V 1/2 L (7) { = E ωl rh ) 1/2 } { (m 2 + L2 2 E ωl + rh ) 1/2 }] 1/2 (m 2 + L2 2 = (E ωl) 2 ( rh ) 2 ) (m ] 1/2 2 + L2 2 (stone) (8)

4 Septemer 6, :38 SpinLight170906v1 Sheet numer 4 Page numer 20-3 AW Physics Macros Section 20.2 Doran Gloal Equations of Motion for the Stone 20-3 Let m To convert this expression to light, take the limit as m 0: lim m 0 ( E V + L ) ( E V L )] 1/2 = lim m 0 (E ωl) 2 ( rh ) 2 ) (m ] 1/2 2 + L2 (9) 2 Impact parameter for light = E = E ( 1 ω L ) 2 E (1 ω) ( ) 2 ( ) ] 2 1/2 rh L (10) E ( ) ] 2 1/2 rh (11) E F spin (a,, r) (light) (12) Equation (11) defines a new constant, while (12) defines a new function F spin (a,, r): 58 L E F spin (a,, r) (1 ω) (impact parameter for light) (13) ( ) ] 2 1/2 rh (light) (14) 59 QUEY 1. Motion60of light around the non-spinning lack hole Show that when a 0, then F spin (a,, r) F (, r), 61 defined in equation (16), Section Only L/E matters The spinning lack hole shares an important simplification with the non-spinning lack hole (Section 11.2): the motion of light does not depend on L or E separately, ut only on their ratio, the impact parameter L/E. Comment 1. Both L and can e positive or negative. The angular momentum L of a stone can e positive (prograde orit) or negative (retrograde orit). Equation (13) shows the same to e true of impact parameter. For the non-spinning lack hole, orits in the two directions are simply mirror images of one another. In contrast, for the spinning lack hole counterclockwise and clockwise orits of a stone are quite different, as Chapters 18 and 19 show. Prograde and retrograde orits of light are also different from one another, as Section 20.3 will show. Equations of motion for light These results allow us to write down the equations of motion for light in the equatorial plane of the spinning lack hole:

5 Septemer 6, :38 SpinLight170906v1 Sheet numer 5 Page numer 20-4 AW Physics Macros 20-4 Chapter 20 Orits of Light around the Spinning Black Hole 76 ( ) dr rh 2 dt = ± F spin (light) (15) 1 ω ± βf spin ( ) 2 dφ rh dt = 2 + sin2 α 1 ω ± 1 ] a β F spin (light) (16) 1 ω ± βf spin dr dφ = ± r 2 + sin2 α a F spin 1 ω ± 1 β F spin ] (light) (17) 77 QUEY 2. And when 78 a 0? In Query 1 you showed 79 that as a 0, F spin F for the non-spinning lack hole. Apply the same limit to expressions in equations(15) 80 through (17). Box 1 in Section 18.2 may e useful. As a 0: 81 A. r 82 B. H 2 (1 2M/r) 83 C. ω 0 84 D. β (2M/r) 1/2 85 E. sin 2 α/a 0 86 With these changes, show 87 that equations (15) through (17) for the motion of light around the spinning lack hole reduce to equations 88 (17) through (19) in (Section 11.3) for the non-spinning lack hole EFFECTIVE POTENTIAL FO LIGHT 91 Orits at a glance Derive effective potential for light Now we derive the effective potential for light, which allows us to predict at a glance the r-motion of a light flash that moves in the equatorial plane of the spinning lack hole. This derivation follows a procedure similar to that for the non-spinning lack hole in Section un your finger down that earlier derivation to compare the following derivation for the spinning lack hole. To start, multiply oth sides equation (15) y an expression that leaves only ±F spin on the right side. Then multiply oth sides y M/r and square oth sides of the resulting equation. The result has the form: ( ) 2 ( ) 2 dr M A 2 (a,, r) = F 2 dt spin(a,, r) (light) (18) ( ) 2 ( ) M M = 2Mω + M 2 ω 2 M 2 ( ) 2 rh 2 (19)

6 Septemer 6, :38 SpinLight170906v1 Sheet numer 6 Page numer 20-5 AW Physics Macros Section 20.3 Effective Potential for Light 20-5 FOBIDDEN V + /M 1 a/m = (3/4) 1/ FOBIDDEN r/m FIGUE 1 Effective potential for light with a/m = (3/4) 1/2. The gray region extends from the static limit at r S = 2M down to the event horizon at r EH = 1.5M. The Cauchy horizon is at r CH = 0.5M. There are two foridden regions for light, one outside the event horizon and one inside the Cauchy horizon. Equation (22) shows that inside a foridden region dr/dt is imaginary. 100 where A(a,, r) ( M ) ( rh 2 ) 1 ω ± βf spin (a,, r)] (20) The right side of (19) is quadratic in M/, so factor it using the quadratic equation: M = Mω ± M 2 2 ( ) ] 2 1/2 rh (21) Sustitute this expression for M/ into the second term on the right side of (19) and collect terms, with the result: where ( ) 2 dr A 2 (a,, r) = dt ( ) 2 M V ± ] 2 (a, r) (22) M

7 Septemer 6, :38 SpinLight170906v1 Sheet numer 7 Page numer 20-6 AW Physics Macros 20-6 Chapter 20 Orits of Light around the Spinning Black Hole a/m = 0 V/M FOBIDDEN 0 r/m FIGUE 2 For comparison, effective potential for light for the non-spinning lack hole (a = 0), to the same scale as Figure 1. Track r-motion of light. Foridden region for light V ± ] 2 (a, r) M 2 M 2 ( ) 2 rh ± 2M 2 ω ( ) rh M 2 ω 2 (23) The superscript ± on the left side of (23) is the same as the ± on the right side. Equation (22) tracks the r-motion of a light flash in the equatorial plane: The first term on the right side is a function of ut not a function of a or r, while the second term the square of the effective potential is a function of a and r, ut not a function of. Equation (22) then permits us to plot on the same diagram the effective potential function (23) Figure 1 and a horizontal line that represents any given value of (M/) 2 in (22). Equation (22) tells us why the region etween V and V + in Figure 1 is foridden: If (M/) 2 is less than (V + /M) 2 ut greater than (V /M) 2, then dr/dt is imaginary, which is indeed foridden. Figure 2 reminds us of the corresponding effective potential for the non-spinning lack hole. 120 QUEY 3. Effective 121 potential for the non-spinning lack hole Show that when a 122 0, equations (22) and (23) reduce to equations (25) and (26) in Section 11.3, shown in Figure

8 Septemer 6, :38 SpinLight170906v1 Sheet numer 8 Page numer 20-7 AW Physics Macros Section 20.4 Exercise QUEY 4. Mass vs. 126 massless Compare Figure 1 for 127light with the corresponding Figure 1 in Section 18.3 for a stone, oth for a/m = (3/4) 1/2. Is the 128 following statement true or false: Outside the event horizon, the foridden region for light is entirely 129 enclosed in the foridden region for the stone Where light goes Gloal radial motion dr/dt must e real. Therefore the right side of (22) must e positive. esult: The light flash that moves along a horizontal line at (M/) 2 in Figure 1, for example, cannot enter either foridden region etween the two effective potential curves. A light flash from far away that reaches one of these curves either reverses the sign of its r-motion or holds its r-value at the curve s maximum or minimum EXECISE Does a spinning ack hole appear smaller or larger than a non-spinning lack hole? Think of two lack holes. The first is a non-spinning lack hole of a particular given mass M. The second is a spinning lack hole of the same mass M. Can a distant oserver tell them apart? Specifically, (1) How large does a lack hole look to an Earth oserver? (2) Can the Earth oserver determine whether or not the lack hole is spinning or non-spinning? (3) Since most lack holes in Nature spin, can the Earth oserver determine how fast the spinning lack hole rotates that is, what is its value of a/m? When you finish this exercise, you should e ale to answer these three questions clearly and definitively. Non-spinning lack hole: eview Figures 3 and 12 in Chapter 11. They show that r-coordinate of the knife-edge circular orit for light around a non-spinning lack hole r = 3M corresponds to the maximum of the effective potential curve V (r), and that this knife-edge orit determines the visual size of the non-spinning lack hole through the critical impact parameter critical : the visual diameter of the non-spinning lack hole is 2 critical. Spinning lack hole: The visual size of the spinning lack hole in the equatorial plane is determined y two critical impact parameters + critical and critical that 156 elong to the prograde and retrograde knife-edge orits of light, respectively, as shown in Figure 3 for the special case a = (3/4) 1/2 M A. Figure 1 in this chapter plots the effective potentials V ± (r) for a light eam that moves near a lack hole with a/m = (3/4) 1/2. From Figure 1, show that

9 Septemer 6, :38 SpinLight170906v1 Sheet numer 9 Page numer 20-8 AW Physics Macros 20-8 Chapter 20 Orits of Light around the Spinning Black Hole FIGUE 3 Figure for Exercise 1. Schematic diagram showing the visual size of a spinning lack hole with a/m = (3/4) 1/2 = = 1 1 V max + = V + (r + knife edge ) prograde knife-edge orit (24) = 1 1 Vmax = V (r knife edge ) retrograde knife-edge orit (25) 162 B. Derive the following expressions for ± critical and r± knife edge : ( ) ( ) ± critical = r ± knife edge + 3M r ± 1/2 knife edge (26) 4M r + knife edge = 4M cos2 Ψ ± where Ψ = 1 ( 3 arccos a ) M (27) C. Evaluate equations (26) and (27) for a = 0 and show that oth results agree with equation (28) in Chapter 11. D. Find the values of + and for a spinning lack hole with (a) a/m = (3/4) 1/2 and () a/m = 1 (maximum-spin lack hole). E. Optional: Plot the visual size ( + critical + critical ) of the spinning lack hole as a function of its spin parameter a/m F. Answer the question posed in the title of this exercise. Include a sentence that starts, This depends on.....

10 Septemer 6, :38 SpinLight170906v1 Sheet numer 10 Page numer 20-9 AW Physics Macros Section 20.4 Exercise Download file name: Ch20OritsOfLightAroundSpinningBH170906v1.pdf