Black Holes: The Key to Understanding the Universe

Index

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Abbott, Edwin, 42

Abbott, Jacob, 166

absolute zero temperature, 17, 172–6, 189–90

accelerating observer, 64–72, 98–105, 113–14

quantum effects, 197–8

Adams, Douglas, 56

AdS/CFT correspondence, 236–40

age as length of worldline, 33, 62–6

Agnew, Jonathan, 27–8

Akhtar, Shoaib, 25–8

Almheiri, Ahmed, 228, 258

‘AMPS’ paper, 228–31, 245

Andromeda galaxy, 66, 210

anti-de Sitter spacetime (AdS), 236–8

anti-gravity universe, 140

arrangements, number of, 171–8, 221–2

arrow of time, thermodynamic, 177

Astronomical Unit, 7

Atkins, Peter, 168

atomic clocks, 24–25

Bardeen, Carter and Hawking, ‘The Four Laws of Black Hole Mechanics’, 187–90

Barish, Barry, 7

beginning of universe, 2, 5, 155, 177

Bekenstein, Jacob, 163–4, 179

Bekenstein(–Hawking) entropy, 179, 183–6, 232

Bell Burnell, Jocelyn, 4

Bell state, 212

Bell, John, 212

Bertschinger, Edmund, 22, 29

bets

Hawking and Thorne vs. Preskill, 18–19, 238

Hawking vs. Thorne and Preskill, 144

Big Bang, 2, 5, 177

Birkhoff, George David, 157

bit, 179–83

black hole

complementarity, 197–207, 225–31

distorting space and time, 1, 11, 104

early scepticism, 3, 11–15, 149

18th-century ideas, 9–10

energy extraction, 145–8, 195

entanglement with Hawking radiation, 218–25, 229–32, 246–51

entropy, 179–86, 188–90

evaporation, 17, 201–5, 218–25, 247–50

event horizon, 1–2, 9, 11

formation, 2, 7, 14–15, 149–59

geometry, 46, 99, 122, 128–9, 135

hair, 150, 161

inevitability of formation, 14–16, 152–6

information paradox, 17–19, 201–8, 218–53

Kerr, 131–148, 189

observation, 5–8

Penrose diagram, 91–5, 136–9, 157–61

river model, 107–10, 133–4

rotating, see Kerr

Schwarzschild, 88–95, 115–18

shadow, 5–6

supermassive, 1, 5–7, 93, 156

temperature, 17, 186–90, 195–6

thermodynamics, 163, 179–91

‘Black Hole Explosions?’ (Hawking), 16, 186, 201

Black Hole War, The (Susskind), 18, 203

Blandford, Roger, 29

Boltzmann, Ludwig, 171, 175–6

Boltzmann’s equation, 175, 178

brewing, 165–6

bridge, Einstein–Rosen, 114–16, 125, 245–6; see also wormhole

Brief History of Time, A (Hawking), 18

Brownian motion, 167

Buckminster Fuller, R., 76

‘Building up spacetime …’ (Van Raamsdonk), 243–4

caloric, 166

Carter–Penrose diagram, 49

Carter, Brandon, 49, 140, 187

causal influence, 30–1, 37–40, 51–2, 69

cause and effect, 37–40, 45, 215

Chandrasekhar limit, 13

Chandrasekhar, Subrahmanyan, 13–14, 149–50, 156

Character of Physical Law, The (Feynman), 173

Christmas in spacetime, 35–40, 57

chronology protection conjecture, 141

Clausius, Rudolph, 169

clocks and rulers, 37, 55–63

closed timelike curves, 140–41

collapse of stars, 2–4, 15, 149–61

halting, 11–14

inevitability of black hole formation, 14–16, 152–6

realistic black holes, 149–61

spherically symmetric, 151, 157–8

compact object, see black hole, neutron star

complementarity, 197–207, 225–31

conformal field theory (CFT), 236, 240

conformal projection, 49, 51, 235

conformal spacetime diagram, 49, 51

conservation

of energy, 146–7, 168–9, 189

of information, 17–19, 203–7, 219–20, 223–5, 237–8

of momentum, 146

coordinates, 54, 81–3, 96

Kruskal–Szekeres, 115, 117–18

Schwarzschild, 84–90, 93

cosmic censorship conjecture, 142–5, 161

cosmic microwave background radiation temperature, 17

curvature, 80, 84, 120–22

curved spacetime, constructing from flat pieces, 77–80

curved surfaces, constructing from flat pieces, 76–8, 81–3, 96; see also non–Euclidean geometry

Dalton, John, 166

dark star, 9–10

Darwin, Charles, 19

Davies–Unruh effect, 198

Davies, Paul, 198

de Kremer, Gerard, 48

degeneracy pressure

electron, 12–14

neutron, 13

determinism, 18, 203, 220

dimensions, higher, 41–4, 119–22, 234–5

Dirac, Paul, 211

distance between events, see interval

distortion of representation of spacetime, 51–3, 59–60

distortion of spacetime by matter and energy, 40–41, 51, 46

Dong, Xi, 258

Doppler effect, 61–2

E = mc2, 27, 183

Eddington, Arthur, 3, 12–13, 46, 164

on thermodynamics, 164

scepticism of black holes, 3, 13

Einstein–Rosen bridge, 114–16, 125, 245–6; see also wormhole

Einstein, Albert

1905 work on molecular motion, 167

1905 work on photons, 180, 183

1905 work on relativity, 24–8, 40, 183

1915 work on gravity, 2, 22–3, 40–41, 70–72, 78–82

1935 scepticism of quantum theory, 209, 245–6

1935 work on wormholes, 114, 125

1939 scepticism of black holes, 3, 11, 14–15, 149

Einstein’s equations, see General Theory of Relativity

electromagnetic wave, electromagnetic, 25, 104–5, 180

electromagnetism, 19, 25, 171

Maxwell’s theory, 25–6

electron degeneracy pressure, 12–14

electrons, 4, 12–13, 17, 194

Ellis, George, 2

embedding diagram, 119–28

emergence of spacetime and gravity from quantum theory, 20, 191, 236, 244–5, 261

energy extraction from black hole, 145–8, 195

entanglement, 20, 208–18, 254–5

between Hawking radiation and black hole, 218–25, 229–32, 246–51

entropy, 222–4, 242–3, 247–51

in quantum computing, 254–61

monogamy of, 230

vacuum, 218, 230–31, 244–5

entropy

black hole, 179–86, 188–90

classical, 164–5, 169–71

entanglement, 222–4, 242–3, 247–51

information-theoretic, 177–9

statistical, 171–7

thermal, 171–7, 221–4

‘EPR’ paper, 245

Equivalence Principle, 71, 89, 198

contradiction, 181, 200, 228

resolution, 231

ER = EPR, 130, 245, 247, 251–2

ergosphere, 132–5, 145–9

escape velocity, 9

Escher, M. C., 47, 235

Euclidean geometry, 43, 81, 121

event, spacetime, 28–40, 51–7

event horizon, 1–2, 9, 11

in river model, 107–8

increasing area rule, 148, 164, 188–9

inner and outer, 132, 135–6, 148

Event Horizon Telescope Collaboration, 5–6

Exclusion Principle, Pauli, 12

Exploring Black Holes (Taylor, Wheeler, Bertschinger), 22, 29

extra dimensions, 41–4, 119–22, 234–5

Feynman, Richard, 150, 173

firewall, 228–32

First Law of Thermodynamics, 168–9, 189

Flamm, Ludwig, 114

Flat Albert, see flatland

Flat Eric, 42

flat rulers on curved surfaces, 76–80, 96; see also metric

flat spacetime, 49–51, 67, 77, 157–8

flatland, 41–4, 76

wormholes in, 119–26

Flatland: A Romance of Many Dimensions (Abbott), 42

floating, 67, 70, 94, 103

fluctuations, vacuum, 16–17, 193–200

‘Four Laws of Black Hole Mechanics, the’ (Bardeen, Carter, Hawking), 187–90

Fowler, R. H., 11–13

free will, 140, 215

freefall, 70–71, 80, 89

fridge, 169–70, 189

Fuller, Robert, 16

Fulling, Stephen, 198

fusion, thermonuclear, 3, 12, 15

General Theory of Relativity, 2, 40–41, 75–83

equivalence principle, 70–71; see also Equivalence Principle

gravitational waves, 7, 151, 156

Kerr solution, 131–3, 149–51

Schwarzschild solution, 75–7, 80, 83–7

Wheeler’s maxim, 46, 75

Genzel, Reinhard, 7

geodesic, greedy, 260–61

geodesic dome, see Montreal Biosphere

geometry

distortion by matter and energy, 40–41, 51, 46

Euclidean, 43, 81, 121

Kerr, 131–9, 142–5, 149–51

non-Euclidean, 44, 67, 76–7, 119, 235

Schwarzschild, 75–7, 80, 83–95

Geons, Black Holes and Quantum Foam (Wheeler), 150

Ghez, Andrea, 7

Gibbs, Josiah Willard, 185

Goodstein, David L., 172

Gravitation (Misner, Thorne, Wheeler), 22, 147

gravitational collapse, see collapse of stars

‘Gravitational Collapse and Space-Time Singularities’ (Penrose), 16, 152

gravitational wave, 7, 151, 156

gravity, 1–5, 23, 40–41, 70–71

as emergent from quantum theory, 191, 236, 245, 261

as geometry, 41, 46, 72, 79, 81

great circle, 59–60, 76

greedy geodesic, 260–61

Gullstrand, Allvar, 107

Hafele–Keating experiment, 24, 27–8

Hafele, Joseph, 24

Hamilton, Andrew, 107

happiest thought, Einstein’s, 70, 78

HaPPY code, 259–61

Hardy, Lucien, 212

Harlow, Daniel, 258–9

Hawking radiation, 17–18, 187, 191–5

entanglement with black hole, 218–25, 229–32, 246–51

information content, 17–19, 220–5, 253

Hawking temperature, 17, 186–90, 195–6

Hawking, Stephen

1971 work, 148

1973 work, 2, 187–90

1974 work, 16–18, 186, 190, 201

1975 work, 22, 190, 194, 234

area theorem, 148, 188–9

bets, 18–19, 144, 238

on time travel, 141

Hayden, Patrick, 229

heat, 165–70

Heisenberg Uncertainty Principle, 12, 191–2

Heisenberg, Werner, 12

Hewish, Antony, 4

higher-dimensional spaces, 41–4, 119–22, 234–5

Hollowood, Tim, 246–7

hologram, 232

holographic principle, 182, 227, 232–8

horizon, 69

event, 1–2, 9, 11

inner and outer, 132, 135–6, 148

Rindler, 69, 114, 231

hyperspace, 119

inertial reference frame, 27, 55

infinity on Penrose diagrams, 46–74

future lightlike, ℑ⁺, 53, 73–4

future timelike, i ⁺, 56–7, 73–4

past lightlike, ℑ⁻, 53, 73–4

past timelike, i ⁻, 56–7, 73–4

spacelike, i ⁰, 56, 73–4

information

and area, 179–82, 189, 232–3

and entanglement, 20, 217–19, 224

and reality, 18, 20, 255, 262

apparent absence in Hawking radiation, 17–18, 202

emergence in Hawking radiation, 18–19, 220–25, 253

encoding, 20, 226, 231–3, 254–61

information paradox, 17–19, 201–8

and entanglement, 218–20

and holography, 237–8

and quantum wormholes, 251–3

Page curve vs. Hawking curve, 224–7

information theory, 177–9

iNSPIRE database, 234

Internal Constitution of Stars, The (Eddington), 12

International Space Station, 41, 70

Interstellar (film), 97

interval (spacetime), 27–9, 32–40, 43

as fundamental, 29, 37

as invariant, 28–9, 37–40, 54

in curved spacetimes, 77–8

Schwarzschild, 83–6

three sorts, 33

invariant quantity, 29

island, quantum extremal, 248–53

Jenkinson, Ross, 97

jiggling, 167, 171, 191

quantum, 5, 12–14

Joule, James Prescott, 165–7

Keating, Richard, 24

Kelvin, Lord, 166

Kerr black hole, 131–48, 189

Kerr solution, 131–3, 149–51

Kerr wonderland, 139–42

Kerr, Roy, 131, 149

Knight, Nick, 25–8

Kruskal–Szekeres coordinates, 115, 117–18

Kruskal, Martin, 117

Kumar, S. Prem, 246

Kwiat, Paul, 212

Landau, Lev, 13

Laplace, Pierre-Simon, 9–10

Large Hadron Collider, 25, 234

‘Large N Limit of Superconformal Field Theories …, the’ (Maldacena), 234

Large Scale Structure of Space-Time, The (Hawking, Ellis), 2

Laws of Thermodynamics, The (Atkins), 168

Legramandi, Andrea, 246

Lemaître, Georges, 5, 107

light cone, 31, 34–40, 45

Kerr, 134–5

on Penrose diagrams, 51–2, 74

Schwarzschild, 89–93

lightlike separation of events, 34

LIGO, gravitational wave detector, 7

Lisle, Jason, 107

‘Little Green Men-1’, 4

M87 galaxy, 5–6, 93–7, 103

Mach, Ernst, 185

Maldacena, Juan, 233–40, 245–6

Marolf, Donald, 228

mathematics, ‘The miracle of the appropriateness of …’, 20

Mathur, Samir, 226

matter, rigidity of, 4

maximal entanglement, 223, 242, 259

maximal spacetime, 117

Kerr, 138–42

Schwarzschild, 115–18, 122–30

Maxwell, James Clerk, 25, 171

Maxwell’s equations, 25–6

Mercator projection, 48–54, 59–60

metric, 81–3

Earth, 96

Schwarzschild, 83–4, 128

visual analogy, 120, 235

Michell, John, 9–10

Milky Way, 1

minimal surface, 244

Minkowski spacetime, 49–51, 67, 157–8

Minkowski, Herman, 49–51

Misner, Thorne, Wheeler, Gravitation, 22, 147

Modern Classical Physics (Thorne, Blandford), 29

monogamy of entanglement, 230

Montreal Biosphere, 76–7

Morris, Michael, 129

naked singularity, 142–5, 156, 161

neutrinos, 14

neutron degeneracy pressure, 13

neutron star, 4–5, 7–8, 14

density, 11

escape velocity, 9

mass limit, 5, 14

neutrons, 5, 14

Newton, Isaac

absolute space, time and motion, 23

theory of gravity, 23, 157

Newtonian argument for a black hole, 10

no-cloning theorem, 227

Nobel Prizes, 7, 14, 16, 107, 166–7, 203

Noether, Amalie Emmy, 146

non-Euclidean geometry, 44, 67, 76–7, 119, 235

nuclear fusion, thermonuclear, 3, 12, 15

number of arrangements, 171–8, 221–2

Nurse, Paul, 166

‘On Dense Matter’ (Fowler), 12

Oppenheimer, J. Robert, 14–15, 151

order of events, 37–40

Page curve, 223–5, 228–30, 246–51

Page time, 223–4

Page, Don, 22, 220, 225

Painlevé, Paul, 107

Paley, William, 19–20

paradox, 208

crossing event horizon, 11, 15

information, 17–19, 201–8, 218–53

time machine, 140–41

twin, 64–6, 79

particle accelerator, 25, 57

‘Particle Creation by Black Holes’ (Hawking), 22, 190, 194, 234

particles, 4, 12, 19–20, 191–2

Hawking, 16–17; see also Hawking radiation

virtual, 17, 193–4; see also vacuum fluctuations

Pastawski, Fernando, 259

Pauli Exclusion Principle, 12

Pauli, Wolfgang, 12

Penrose diagram, 45, 49–56

evaporating black hole, 203–5

exploring with immortal observers, 57–69, 94–105

Kerr black hole, 136–40, 142–3

real black hole formation, 157–61

Schwarzschild black hole, 91–5, 112–18

with two space dimensions, 72–74

Penrose process, 145–8, 195

Penrose–Hawking singularity theorem, 155–6

Penrose, Roger

on black hole formation, 7, 16, 152–5

on naked singularities, 142

Perrin, Jean Baptiste, 167

photons, 17, 179–80, 183, 212

as fish, 108–11, 133

energy–wavelength relation, 180

pixels, Planckian, 181

Planck length, 181, 184

Planck, Max, 184

Planckian pixels, 181

Podolsky, Boris, 245

Poincaré disk, 235, 255, 260–61

Polchinski, Joseph, 226, 228, 231

Preskill, John, 229, 259

bets, 19, 144, 238

Price, Richard H., 151

Principia Mathematica (Newton), 23, 157

Principle of Maximal Ageing, 66, 79–80

probability, quantum, 191–2, 211

protons, 14, 20, 25

pulsar, 4

Pythagoras’ theorem, 76–7, 80–82

quantum (mechanics/physics/theory), 12, 191–3, 210–18

book, 218, 224

computing, 20–21, 255–63

conformal field, 236, 240

entanglement, 20, 208–18, 254–5

entanglement entropy, 222–4, 242–51

error correction, 255–62

extremal surface (QES), 248–9

field, 17, 130, 192–3

gravity, 161–2, 209, 239, 262–3

jiggling, 5, 12–14

kitchen, 212–18

probability, 191–2, 211

state, 192, 211–18, 227

tickling, 194, 197

vacuum, 16–17, 192–200, 218

Quantum Universe, The, 211

quarks, 17, 236

qubit, 210–18, 227, 255–61

redshift, 104–5

of fish, 109

reductionism, 20

redundant encoding, 256–9

Reeh–Schlieder theorem, 218

relativity

general, 2, 40–41, 75–83; see also General Theory of Relativity

special, 24, 26–40, 49, 61–2

Rindler horizon, 69, 114, 231

Rindler trajectory, 67

Rindler, Wolfgang, 67

Rosen, Nathan, 114, 125, 245

rulers, 55–63

Ryu, Shinsei 242

Ryu–Takayanagi (RT) conjecture, 242–5, 251–3, 260–61

S2 and ‘S stars’, 6

Sagittarius A*, 1, 6

Hawking temperature, 17

Schrödinger’s cat, 130, 210

Schwarzschild black hole, 88–95, 115–18

Schwarzschild coordinates, 84–90, 93

Schwarzschild radius, 10–16, 83, 89–90

of Sun, 87

Schwarzschild solution, 75–7, 80, 83–95

maximally extended, 115–18, 122–30

metric equation, 83, 87

Schwarzschild, Karl, 10, 75

Second Law of Thermodynamics, 163–4, 168–9

separation

lightlike, 34

spacelike, 33–4

timelike, 33–5

Shannon, Claude, 177–8

signals, 16, 37, 214–15

to an accelerating observer, 69, 114

singularity, 2, 16, 208

Big Bang, 5, 142

disappearing, 204–5

naked, 142–5, 156, 161

Penrose’s theorem, 16, 152–5

ring (Kerr), 131, 136, 139–40, 142

Schwarzschild, 91–5, 116–17

white hole, 117

Sirius B, 12

slices of space, 55–6, 121–6

Small. Far away. 109–11

Snyder, Hartland, 14–15, 151

space

absolute, 23, 25–6

observer dependence, 24–25

space diagram, 84–5, 88, 132

spacelike separation of events, 33–4

spacetime

anti-de Sitter, 236–8

as arena for laws of physics, 20

as collection of all events, 30

as emergent from quantum theory, 20, 244, 261

distortion by matter and energy, 40–41, 51, 46

flat (Minkowski), 49–51, 67, 77, 157–8

Kerr, 131–9, 142–5, 149–51

maximal, 117

maximal Kerr, 138–42

maximal Schwarzschild, 112–18, 122–30

Schwarzschild, 75–7, 80, 83–95

spacetime diagrams, 30–31, 36–9, 89–90, 135, 152; see also Penrose diagrams

spacetime event, 28–40, 51–7

spacetime interval, 27–9, 32–40, 43

as fundamental, 29, 37

as invariant, 28–9, 37–40, 54

in curved spacetimes, 77–8

Schwarzschild, 83–6

three sorts, 33

‘Spacetime tells matter how to move …’ 46

spaghettification, 102–3, 197–201, 206

Special Theory of Relativity, 24, 26–40, 56

speed of light, 5, 25

limit of cause and effect, 31, 37–8, 210, 217

near a black hole, 9–10, 106–9, 132–4

universality, 25–6, 37, 184

spontaneous emission, 194

star

catastrophic collapse, 2–4, 15, 149–61

dark, 9–10

phases of life, 3

States of Matter (Goodstein), 172

statistical nature

of gravity, 185, 187, 190–91

of thermodynamics, 171–5, 177, 185

steam engine, 168, 170

Sully, James, 228

Sun, 3

Schwarzschild radius, 10, 87

spin, 131

thermodynamics, 170

supermassive black hole, 1, 5–7, 93, 156

image, 5–6

surface gravity, 188–90

Susskind, Leonard, 18, 203, 232–3, 245–6

Szekeres, George, 118

Takayanagi, Tadashi, 242

Talwar, Neil, 246

Taylor, Edwin F., 22, 29, 120

temperature, 172–4, 165–76

black hole, 17, 186–90, 195–6

thermodynamics, 17, 22

black hole, 163, 179–91

classical, 163–71, 187–9

statistical, 171–7

Third Law of Thermodynamics, 188–90

’t Hooft, Gerard, 203, 232–3

Thomson, William, 166

Thorne, Kip

bets, 144, 238

on the spacetime interval, 29

work on LIGO, 7

work on wormholes, 130

tidal effects, 71, 101–3, 108–9, 199

time

absolute or universal, 23, 25, 36, 62

as fourth dimension, 41–4

observer dependence, 24

order (before and after), 37–40

time machines, 130, 140–42

timelike separation of events, 33–5

Tolman, Richard, 13

trapped surface, 154–6

Twin Paradox, 64–6, 79

Uncertainty Principle, Heisenberg, 12, 191–2

universe, 1–2

beginning, 2, 5, 155, 177

designer, 20–21, 262

diagram, see Penrose diagram

entropy, 163–4, 170–71, 177

holographic, 247, 263

model of, 234–6, 240

multiple, 97, 117–18, 125–9, 139–42

temperature and age, 202

Unruh, Bill, 198

vacuum

entanglement, 218, 230–31, 244–5

fluctuation, 16–17, 193–200

integrity, 230–31, 245, 254

quantum, 16–17, 192–200, 218

vacuum solutions to Einstein’s equations, 88, 141

van Gogh, Vincent, 263

Van Raamsdonk, Mark, 244–5, 258

Virgo, gravitational wave detector, 7

virtual particles, 17, 193–4; see also vacuum fluctuations

Volkov, George, 13

waves, 104

electromagnetic, 25, 104–5, 180

gravitational, 7, 151, 156

weightlessness, 67, 70, 94, 103

Weiss, Rainer, 7

Westminster Abbey, 196

Wheeler, John Archibald, 15–16

central dogma of general relativity, 46

on black hole hair, 150

on entropy, 163–5, 174–5

on the interval, 29

on the metric, 120, 122

on thermodynamics, 163–5, 174–5

Wheeler’s iceberg, 120

Wheeler’s teacups, 163

white dwarf, 12–14

white hole, 117, 139, 158

Wigner, Eugene, 20

worldline, 29–35

length, 32–3, 64

wormhole

evolution, 126–9

in collapsing star, 155–9

keeping open, 130

Kerr, 139, 145

microscopic, 130, 239–46, 251–3

Schwarzschild, 116, 124–30

Wowbagger the Infinitely Prolonged, 56

Yoshida, Beni, 259

Yurtsever, Ulvi, 130

Zeroth Law of Thermodynamics, 173, 188

Zwicky, Fritz, 13