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1 Chapter 21. Inside the Spinning Black
2 Hole
3 21.1 Escape from the Black Hole 21-1
4 21.2 The Carter-Penrose Diagram for Flat Spacetime 21-2
5 21.3 Topology of the Non-Spinning Black Hole 21-6
6 21.4 Topology of the Spinning Black Hole 21-8
7 21.5 Exercises 21-12
8 21.6 References 21-12
9 • Why can’t I escape from inside the event horizon of the non-spinning
10 black hole?
11 • How can I escape from inside the spinning black hole?
12 • When I do emerge from inside the spinning black hole, where am I?
13 • After I emerge from inside the spinning black hole, can I return to Earth?
14 • What limits does my finite wristwatch lifetime place on my personal
15 exploration of spacetimes?
16 • What limits are there on spacetimes that a group in a rocket can visit?
17 Download file name: Ch21TravelThroughTheSpinningBH170831v1.pdf November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 2 Page number 21-1 AW Physics Macros
CHAPTER
21 18 Inside the Spinning Black Hole
Edmund Bertschinger & Edwin F. Taylor *
19 The non-spinning black hole is like the spinning black hole,
20 but with its gate to other Universes closed. For the spinning
21 black hole, the gate is ajar.
22 —Luc Longtin
21.123 ESCAPE FROM THE BLACK HOLE
24 Exit our Universe; appear in a “remote” Universe!
25 Chapters 18 through 20 examined orbits of stones and light around the
26 spinning black hole. We study orbits to answer the question, “Where do we go
27 near the spinning black hole?” The present chapter shifts from orbits to
Travel to another 28 topology—the connectedness of spacetime. Topology answers the question,
Universe . . . 29 “Where can we go near the spinning black hole?” Astonishing result: We can
30 travel from our Universe to other Universes. These other Universes are
31 “remote” from ours in the sense that from them we can no longer
32 communicate with an observer in our original Universe, nor can an observer in
. . . on a 33 our original Universe communicate with us. Worse: Once we leave our
one-way ticket! 34 Universe, we cannot return to it. Sigh!
35 Figure 1 previews this chapter by examining the r-motion of a free-fall
36 stone—or observer—in the effective potential of the spinning black hole. Free
37 stones with different map energies have different fates as they approach the 38 spinning black hole from far away. Two stones with map energies (E/m)2 and Begin with 39 (E/m)3, for example, enter unstable circular orbits. In contrast, the stone with effective potential. 40 map energy (E/m)4 reaches a turning point where its map energy equals the 41 effective potential, then it reflects outward again into distant flat spacetime. 42 Question: What happens to a stone with map energy (E/m)1? Two question 43 marks label its intersection with the forbidden region inside the Cauchy
44 horizon. Does the stone reflect from this forbidden region? Does it move
45 outward again through the Cauchy and event horizons? Does it emerge into
46 our Universe? into some other Universe? The present chapter marshalls
47 general relativity to answer these questions.
*Draft of Second Edition of Exploring Black Holes: Introduction to General Relativity Copyright c 2017 Edmund Bertschinger, Edwin F. Taylor, & John Archibald Wheeler. All rights reserved. This draft may be duplicated for personal and class use. 21-1 November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 3 Page number 21-2 AW Physics Macros
21-2 Chapter 21 Inside the Spinning Black Hole
a/M = (3/4)1/2 V (r) m ?? L L/(mM) = 5 and (E/m)1 = 5.1 E/m (E/m)2 = 4.347
Our Universe
(E/m)3 = 1.715 (E/m)4 Forbidden r/M
"Remote" Universe r = 0.5M r = 1.5M r = 2M Static limit Event Horizon File: GordonVspinCB Cauchy horizon
FIGURE 1 Effective potential for a stone with L/(mM) = 5 near a spinning black hole with a/M = (3/4)1/2. What happens at the intersection of the horizontal line (E/m)1 with the forbidden region inside the Cauchy horizon? (Adapted from Figure 5 in Section 18.4.)
48 The idea of traveling from our Universe to another Universe is not new. In
49 1964 Roger Penrose devised, and in 1966 Brandon Carter improved, what we
Carter-Penrose 50 now call the Carter-Penrose diagram for spacetime, a navigational tool for
diagram 51 finding one’s way across Universes. This diagram will be the subject of the
52 following sections.
21.253 THE CARTER-PENROSE DIAGRAM FOR FLAT SPACETIME
54 Begin around the edges, then fill in.
Global metric 55 As usual, we develop our skills gradually, first with flat spacetime, then with
flat spacetime 56 the non-spinning black hole, and finally with the spinning black hole. Here is a
57 global metric on an [x, t] slice in flat spacetime:
dτ 2 = dt2 − dx2 (global metric, flat spacetime) (1) −∞ < t < ∞, −∞ < x < ∞ (2)
58 The following transformation from [t, x] to [v, u] corrals the infinities in (2)
59 onto a single flat page: November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 4 Page number 21-3 AW Physics Macros
Section 21.2 The Carter-Penrose diagram for flat spacetime 21-3
v = +π/2 u = 0 t = +∞ v FLAT x = 0 A SPACETIME v + u = +π/2 t + x = + ∞ d a v - u = +π/2 ∞ t - x = +
D v=u=0 B u t + x = - t=x=0 v = 0 v + u = -π/2 ∞ v = 0 u = -π/2 ∞ u = +π/2 t - x = - t = 0 v - u = -π/2 t = 0 x = -∞ x = +∞ c C b
v = -π/2 u = 0 t = -∞ x = 0
FIGURE 2 Points and lines on the boundaries in the Carter-Penrose diagram for flat spacetime.
1 t = [tan(u + v) − tan(u − v)] (global coordinates, flat spacetime) (3) 2 1 x = [tan(u + v) + tan(u − v)] (4) 2 −π/2 < v < +π/2, −π/2 < u < +π/2 (5)
60
QUERY 1. Coordinate61 ranges
Show that transformations62 (3) and (4) convert the coordinate ranges of t and x in (2) into the
coordinate ranges of v63 and u in (5). In other words, the Carter-Penrose diagram brings map coordinate
infinities onto a finite64 diagram.
65
66 Figure 2 shows the result of this transformation, which we call the
Carter-Penrose 67 Carter-Penrose diagram. It plots positive infinite t at point A, negative
diagram 68 infinite t at point C, distant positive x at point B, and distant negative x at November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 5 Page number 21-4 AW Physics Macros
21-4 Chapter 21 Inside the Spinning Black Hole
v
FLAT SPACETIME
t = constant
Light
u
x = constant
FIGURE 3 The Carter-Penrose diagram that fills in coordinates of Figure 2 on the [x, t] slice of flat spacetime. These curves plot v vs. u from the inverse of equations (3) through (5). These particular conformal coordinates preserve the ±45o angles for worldlines of light.
69 point D. In Query 2 you use equations (3) through (5) to verify map
70 coordinate values in this figure.
71
QUERY 2. Points72 and boundaries in the Carter-Penrose diagram
Use equations (3) and73 (4) to verify the following statements about points A through D and boundaries
a through d in Figure74 2:
A. Show that when75 u = 0 then x = 0, and when v = 0 then t = 0.
B. Verify the boxed76 values of t and x at points A through D.
C. Verify the values77 of v + u along the two lines labeled a and c.
D. Verify the values78 of v − u along the two lines labeled b and d..
E. Verify the values79 of t + x along the two lines labeled a and c.
F. Verify the values80 of t − x along the two lines labeled b and d.
81
82 The Carter-Penrose diagram is a conformal diagram that brings global
83 coordinate infinities onto the page. A conformal diagram is simply an ordinary
84 spacetime diagram for a metric on which we have performed a particularly November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 6 Page number 21-5 AW Physics Macros
Section 21.2 The Carter-Penrose diagram for flat spacetime 21-5
Conformal diagram 85 clever coordinate transformation. This particular coordinate transformation
86 preserves the causal structure of spacetime defined by the light cone.
87 To find the global metric on the [u, v] slice for flat spacetime, take
88 differentials of (3) and (4) and rearrange the results: 1 du + dv du − dv dx = + (6) 2 cos2(u + v) cos2(u − v) 1 du + dv du − dv dt = − (7) 2 cos2(u + v) cos2(u − v) −π/2 < v < +π/2, −π/2 < u < +π/2 (8)
Global metric in 89 Substitute dx and dt from (6) and (7) into global metric (1) and collect terms.
u, v coordinates 90 Considerable manipulation leads to the global metric on the [u, v] slice:
dv2 − du2 dτ 2 = (9) cos2(u + v) cos2(u − v) −π/2 < v < +π/2 − π/2 < u < +π/2 (10)
91 Equation (9) has the same form as equation (1) except it is multiplied by 2 2 −1 92 [cos (u + v) cos (u − v)] , called the conformal factor. Indeed, equations
93 (9) and (10) are examples of a conformal transformation:
94 DEFINITION 1. Conformal transformation
95 Definiton: Conformal 96 A conformal transformation has two properties: transformation 97 • It transforms global coordinates.
98 • The new global metric that results has the same form as the old
99 global metric, multiplied by the conformal factor.
Conformal factor 100 The transformation (3) through (5) has both of these properties. In
101 particular, the resulting metric (9) has the same form (a simple difference
102 of squares) as (1), multiplied by the conformal factor 2 2 −1 103 [cos (u + v) cos (u − v)] .
104 Infinities on the [x, t] slice correspond to finite (non-infinite) values on the
105 [u, v] slice, due to the conformal factor in (9), which goes to x + t = ±∞ or
106 x − t = ±∞ when u + v = ±π/2 or u − v = ±π/2, as shown around the
107 boundaries of Figure 2.
108 For the motion of light, set dτ = 0 in (9). Then the numerator 2 2 109 dv − du = 0 on the right side ensures that dv = ±du, so the worldline of ◦ Worldlines of 110 light remains at ±45 on the [u, v] slice. Therefore a light cone on the [u, v] ◦ light at ±45 111 slice has the same orientation as on the [x, t] slice. We deliberately choose
112 conformal coordinates to make this the case. November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 7 Page number 21-6 AW Physics Macros
21-6 Chapter 21 Inside the Spinning Black Hole
113
QUERY 3. Standing114 still; limits on worldlines
A. Show that when115 dx = 0 in (6), then du = −du, which means that du = 0. Result: The stone with
a vertical worldline116 on the [x, t] slice has a vertical worldline on the [u, v] slice. ◦ B. Show that in117 Figure 2 the worldline of every stone lies inside the light cone ±45 .
118
119
QUERY 4. You cannot120 “reach infinity.”
Show that as x → ±∞121 the global equation of motion dx/dt for a stone takes the form dx/dt → ±0.
Therefore a stone cannot122 reach that limit, any more than it (or you!) can reach infinity.
123
124 Objection 1. Are these predictions real? They sound like science fiction to 125 me!
126 We do not use the word “real” in this book; see the Glossary. These 127 predictions can in principle be validated by future observations carried out 128 by our distant descendents. In that sense they are scientific. They also 129 satisfy Wheeler’s radical conservativism: “Follow what the equations tell 130 us, no matter how strange the results, then develop a new intuition.”
21.3131 TOPOLOGY OF THE NON-SPINNING BLACK HOLE
132 The one-way worldline
133 We move on from flat spacetime to spacetime around the non-spinning black
134 hole. Equations (17) and (18) of Section 8.4 connect the global r-motion of a 135 stone to the effective potential VL(r):
dr 2 E 2 V (r)2 = − L (11) dτ m m
A remote Universe 136 Because all terms in this equation are squared, the effective potential VL(r) 137 and the map energy E/m can be either positive or negative, as shown in
138 Figure 4. our Universe lies above the forbidden region. Below the forbidden
139 region lies a second, “remote” Universe.
140 What does ”forbidden” mean? Equation (11) tells us that global r-motion 2 2 Meaning of 141 dr/dτ becomes imaginary when (E/m) is smaller than (VL/m) . In other “forbidden” 142 words, neither stone nor observer can exist inside the forbidden region.
143 The forbidden region prevents the direct passage from our Universe to this
144 remote Universe. To do so we would have to move inward through the event
145 horizon with positive map energy, then use rocket blasts to re-emerge below November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 8 Page number 21-7 AW Physics Macros
Section 21.3 Topology of the Non-spinning Black Hole 21-7
Non-spinning Event Horizon black hole VL/m and 2.5 r = 2M L/(mM) = +10 E/m 2 Our Universe E/m
1 FORBIDDEN
0 ii 0 2 5 10 15 20 25 30 35 40 r/M FORBIDDEN -1
"remote" Universe -2
-2.5 File: GordonEffPotNonspinC FIGURE 4 Effective potential for the non-spinning black hole, copy of Figure 5 of Section 8.4.
Door to remote 146 the forbidden region with negative map energy. But inside the event horizon
Universe is closed. 147 motion to smaller r is inevitable. Result: For the non-spinning black hole the
148 door to to the remote Universe is closed.
149 Figure 5 displays the double-ended funnel-topology of the non-spinning
150 black hole. The upper and lower flat surfaces represent flat spacetime in our
151 Universe and in the remote Universe, respectively. The pinched connection in
FIGURE 5 Topology of the non-spinning black hole that supplements Figure 4. The upper flat surface represents our Universe. It is connected to a remote Universe (lower flay surface) by the impassable Einstein-Rosen bridge. November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 9 Page number 21-8 AW Physics Macros
21-8 Chapter 21 Inside the Spinning Black Hole
singularity
YOU
light event OUR UNIVERSE "REMOTE" UNIVERSE hori zon horizon light event
File: PenroseNonSpinG FIGURE 6 Carter-Penrose diagram for the non-spinning black hole, which has two event horizons. Heavy dashed lines enclose spacetime spanned by the Schwarzschild Metric, which has access to only one of these event horizons. From our Universe a stone, light flash, or observer cannot reach the “Remote” Universe in Figures 6 and 1 by crossing the second event horizon.
Einstein-Rosen 152 the center, called the Einstein-Rosen Bridge, is too narrow for a stone or
bridge unpassable 153 light flash to pass between the two Universes.
154 Now turn attention to the Carter-Penrose diagram for the non-spinning
155 black hole, displayed in Figure 6. This two-dimensional diagram suppresses the
156 φ-coordinate, leaving t and r global coordinates. The Schwarzschild metric,
157 equation (5) in Section 3.1, becomes:
2M 2M −1 dτ 2 = 1 − dt2 − 1 − dr2 (12) r r −∞ < t < +∞, 0 < r < ∞ (13)
158 In this Carter-Penrose diagram an inward-moving stone or light flash
159 crosses the event horizon, then moves inevitably to the singularity represented
Two event horizons 160 by the spacelike horizontal line. Topologically there is a second event horizon
161 that is not available to this stone or light flash, because their worldlines are
162 corralled within the upward-opening light cones.
21.4163 TOPOLOGY OF THE SPINNING BLACK HOLE
164 No two-way worldline!
165 Figure 1 displays the effective potential for a stone with map angular 1/2 166 momentum L/(mM) = 5 near the spinning black hole with a/M = (3/4) .
Reflect outward 167 The striking new feature of this effective potential is the added forbidden
from inside 168 region inside the Cauchy horizon. This added forbidden region raises the Cauchy horizon? November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 10 Page number 21-9 AW Physics Macros
Section 21.4 Topology of the Spinning Black Hole 21-9
event
OTHER horizon ANOTHER UNIVERSE UNIVERSE event horizon
YOU
Cauchy
horizon
YOU horizon singularity singularity singularity singularity Cauchy
event
"REMOTE" horizon OUR UNIVERSE horizonUNIVERSE event
File: PenroseSpinJ FIGURE 7 Carter-Penrose diagram of the spinning black hole that answers questions posed in the caption to Figure 1. The heavy dashed line shows the boundaries of Doran global coordinates, which enclose one event horizon and one Cauchy horizon. With calibrated rocket blasts, you can choose to enter either the Other Universe or Another Universe at the top of the diagram. The upward orientation of your worldline shows that you cannot return to our Universe once you leave it—according to general relativity.
169 possibility that the stone with, say, (E/m)1 = 5.1 can reflect from this 170 forbidden region and move back outward into a distant region of flat spacetime.
171 Figures 7 and 8 present the topology of such a spinning black hole. You,
Worldline moves 172 the observer who travels along the worldline in Figure 7, start in our Universe,
between Universes. 173 pass inward through the event horizon and the Cauchy horizon, reflect from
174 the forbidden region inside the Cauchy horizon, and emerge from a second
175 Cauchy horizon. Then, with the use of rockets, you can choose which event
176 horizon to cross into one of two alternative Universes at the top of this
177 diagram. November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 11 Page number 21-10 AW Physics Macros
21-10 Chapter 21 Inside the Spinning Black Hole
FIGURE 8 Topology of spacetime around the spinning black hole. In this case the central Einstein-Rosen bridge is wide enough for a traveler to pass through on her one- way trip to another Universe. Indeed, she may use rocket thrusts to choose between two alternative Universes. This figure supplements Figures 1 and 7.
178 To construct Figure 7 suppress the Φ-coordinate of the Doran metric,
179 equation (4) in Section 17.2. The result: " #2 r2 1/2 2M 1/2 dτ 2 = dT 2 − dr + dT (Doran, dΦ = 0) (14) r2 + a2 r − ∞ < T < ∞, 0 < r < ∞
180 Objection 2. Why are the lines labeled “singularity” in Figure 7 vertical, 181 while the line labeled “singularity” in Figure 6 is horizontal?
182 These diagrams show topology: where you can go, and where you cannot 183 go. Categories “vertical” and “horizontal” in such a diagram carry no 184 prediction for observation. Each case shows that you cannot climb out of 185 the singularity.
186 The heavy dashed line in Figure 7 outlines the spacetime region included
187 in Doran global coordinates. Notice that this included region is only part of
Emerge into 188 available spacetime. Compare the worldline in Figure 7 with the horizontal another Universe November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 12 Page number 21-11 AW Physics Macros
Section 21.4 Topology of the Spinning Black Hole 21-11
189 line (E/m)1 in Figure 1. This comparison shows that the reflected observer 190 does not re-emerges into our Universe, but into one of the alternative
191 Universes at the top of Figure 7. Conclusion: For the spinning black hole, the
192 gate between alternative Universes is ajar (initial quote of this chapter). But
193 your worldline in Figure 7 moves relentless upward; you cannot return to the
194 Universe you have left. You can’t go home again!
195 Objection 3. How can I tell that I have reached the limits of a map, but not 196 the limits of spacetime, when there is uncharted territory ahead.
197 If you reach the boundary of a global coordinate system in finite wristwatch 198 time (so that dT/dτ → ∞) or some other singularity arises, and if you 199 have not reached a singularity, then you have just demonstrated that the 200 original coordinates are incomplete and need to be extended. This limited Geodesic 201 feature of global coordinates is called geodesic incompleteness: There incompleteness 202 exist (portions of) geodesics (or other curves) that reach the edge of your 203 coordinate range and continue beyond your present global 204 coordinates—unless you introduce new global coordinates that cover the 205 new range.
206 Objection 4. How many different Universes are there?
207 In principle the Carter-Penrose diagram of Figure 7 extends indefinitely 208 both upward and downward, embracing an unlimited number of Universes.
209 Figure 8 displays the topology through which you pass along the worldline
Triple funnel for 210 of Figure 7. You enter the funnel from Our Universe, then use rockets to
spinning black hole 211 choose the Universe into which you emerge. Your worldline in Figure 7 shows
212 that you cannot re-enter that funnel in order to return to our Universe. Your
213 trip between Universes is a one-way street!
214 Objection 5. So WHERE are these other Universes? Show them to me!
215 Spacetimes multiply inside the event horizon of the spinning black hole. 216 What does this mean for regions far from the spinning black hole? Big 217 surprise: An observer can use rockets to maneuver inside the event 218 horizon of the spinning black hole in order to choose the remote Universe 219 into which she emerges. Example: Figure 7 shows that the “bouncing” 220 traveler with (E/m)1 in Figure 1 can emerge into either one of two 221 alternative Universes shown in Figure 8. Conclusion: Neither of these 222 alternative Universes “exist” in our Universe in the everyday sense—but 223 you can travel there! November 29, 2017 16:00 InsideSpinBH170831v1 Sheet number 13 Page number 21-12 AW Physics Macros
21-12 Chapter 21 Inside the Spinning Black Hole
21.5224 EXERCISES
225 SUGGESTED EXERCISES, PLEASE!
21.6226 REFERENCES
227 Penrose, Roger, “Conformal treatment of infinity,” in Relativity, Groups and
228 Topology, ed. C.M. de Witt and B.de Witt, Les Houches Summer Schools,
229 1963 Gordon and Breach, New York and London, pages 565-584.
230 Carter, Brandon (1966). ”Complete Analytic Extension of the Symmetry Axis
231 of Kerr’s Solution of Einstein’s Equations,” Physical Review, 1966, Volume
232 141, Number 4, pages 1242–1247. Bibcode:1966PhRv..141.1242C.
233 doi:10.1103/PhysRev.141.1242.
234 Kruskal, M.D, “Maximal Extension of Schwarzschild Metric,” Physical
235 Review, Volume 119, Number 1, September 1, 1960, pages 1743-1745.
236 Figure 5 adapted from From Kip Thorne, Black Holes and Time Warps:
237 Einstein’s Outrageous Legacy, New York, W. W. Norton, 1994, page 487.
238 Figure 8 drawn by Thomas Dahill.
239 Useful Carter-Penrose diagrams reference: William J. Kaufmann, III, Black
240 Holes and Warped Spacetime, W.H. Freeman Co., San Francisco, 1979.
241 Download file name: Ch21TravelThroughTheSpinningBH170831v1.pdf