ouroboros @wikipedia. 2 black holes and the end of time 1. because of the singularity – s. hawking...
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ouroboros @wikipedia
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Black Holes and the End of Time
1. Because of the Singularity – S. Hawking
2. Far future of the Universe – F. Adams & G. Laughlin
3. Black Holes as the End of Thermodynamic Time
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End of time at Black Hole’s Singularity
• “Time” itself, beginning and ending at singularities (infinities, discontinuities) – hence subtitle
• If inflation explains Big Bang...
– Universe born from pre-existing “parent” universe
– No singularity at start
• In most theories of quantum gravity...
– New physics at Planck scale prevents infinities
– No singularity in black holes
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Black Holes at the End of TimeFive Ages of the Universe by Fred Adams & Greg Laughlin
I. Primordial: Age < 105 yr
– Inflation, Baryogenesis, Nucleosynthesis, Plasma Universe
– Ends with release of CMB
II. Stelliferous: 106 yr < Age < 1014 yr
– Stars, galaxies, planets... life
– Ends when lowest-mass M dwarf stars exhaust hydrogen fuel
III. Degenerate: 1015 yr < Age < 1049 yr
– White dwarfs, brown dwarfs, neutron stars, black holes
– “Evaporation” via proton decay
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Black Holes at the End of TimeFive Ages of the Universe by Fred Adams & Greg Laughlin
IV. Black Holes: 1050 yr < Age < 10100 yr
- Black holes, radiation, Dark Energy
- Evaporation via Hawking radiation
V. Dark: Age > 10101 yr
- Radiation and Dark Energy in thermal equilibrium
- Near-perfect de Sitter space
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Black Holes and the End of Time
1. Because of the Singularity – S. Hawking
2. Far future of the Universe – F. Adams & G. Laughlin
3. Black Holes as the End of Thermodynamic Time
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What is Time?
“Time is what prevents everything from happening at once.”
—John Wheeler (original to Ray Cummings)
The Nature of Time: Time is a Dimension
Vision #1: Time is a dimension:
Time is what we measure with clocks.
“Clock” = any predictably periodic process.
•Motions of Sun, Moon, planets•Heart beat or pulse•Pendulum (Galileo)•Springs and ratchets•Vibrations of a quartz crystal•Atomic or nuclear transitions•Pulsars
katerha@flickr
#1. Time is a Dimension
What we know about clock time:
•Some clocks are better than others
•All clocks can be synchronized, to the limits of their accuracy
•Psychological or subjective time, as we experience it, has some odd effects – stress, age (not our focus here)
•Various effects observed thanks to Einstein’s relativity theories, and the finite speed of light
katerha@flickr
#1. Time is a Dimension
• Special Relativity:– Clock time varies with observer
motion – Simultaneity of distant events varies
with observer motion• General Relativity:
– Clock time varies with gravity
Examples:
• GPS satellites (observed)– Slower by 0.007 ms/day due to motion– Faster by 0.046 ms/day due to gravity– Net effect: 0.039 ms/day, ~12 km
• Black holes (theoretical)– Extreme corrections near event horizon– “Freezing” of time at horizon
• Can correct our measurements for all of these effects without contradiction
katerha@flickr
#1. Time is a Dimension
Copernican Solar [email protected]
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Reversibility of clock time:
•Equations of physics do not distinguish past from future
•Since time is just another dimension, past vs. future is the same as left vs. right, forward vs. backward…
•Example: Solar System could be run in reverse
•“No up or down in outer space”
A Simple Physical System…
Returns to initial state every ~10 sec: “Reversible”
A Somewhat Less Simple Physical System…
20 balls recurrence time: 4 days
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Even Less Simple…
50 balls recurrence time: 11 million years
Reversible & Irreversible
Copernican Solar [email protected]
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What’s going on here?
•Increasing number of elements leads to exponentially-increasing number of possible states
•Split n balls between two sides of the box in 2n ways; with two states “satisfactory”
•Realize a new arrangement of balls every ~1 sec – time for average ball to travel one box length
•Each added ball doubles recurrence time
•As for the Solar System...– Many bodies– But each can be treated as interacting
with the Sun only– Chief exception: Minor bodies that
get close to one of the planets...
From Reversible to Irreversible
• 5 balls: 10 sec
• 20 balls: 4 days
• 50 balls: 11 million years
• 100 balls (50+50):
– Multiply 107 years by 250 ~ 1015
– 1022 years ~ 1012 * Age of Universe
• 1023 molecules:
– Multiply 1022 years by 2(10^23 – 100)
– 1030,000,000,000,000,000,000,022 years
– 1030,000,000,000,000,000,000,012 * Age of the Universe
– 1030,000,000,000,000,000,000,039 nanosec
– A Very Large Number
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Extremely, Extremely Not Simple…
into15 @youtube
The Nature of Time – Time is Change
Vision #2. Time is change:
The passage of time is what changes an unknown future into an unchanging past.
“The Arrow of Time” – not merely subjective!
Deep distinctions between the two directions of time:
•We remember the past and don’t know the future
•Complex systems evolve toward states of greater disorder
•Second Law of Thermodynamics – entropy always increasing
•Enabling life as we know it: Increasing disorder elsewhere (that we don’t care about) allows us to increase order here (that we care about) – make memories, process data, create a civilization, ponder the nature of time…
•How do we reconcile the two visions of time?
Reconciling the Two Visions of Time
Ludwig Boltzmann (1844–1906)
•Explored gas dynamics and the approach to equilibrium (mixing of gases, changes in heat/volume, etc.)•Key insight:
• Low-entropy states are special, rare – low probability
• High-entropy states are generic – high probability
• Equilibrium (maximum entropy) states the most generic = most probable
•Boltzmann’s Law: Entropy is nothing but counting states•Connects atomic description to macroscopic measurements•No Arrow of Time for simple systems, nor for complex systems once they reach equilibrium
S = k log Ω
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Entropy = Counting Equivalent Microstates
System temporarily exhibits Arrow of Time as it evolves from high to low entropy, and vice versa
Reconciling the Two Visions of Time
Boltzmann’s Law and Time:
•No Arrow of Time for simple systems, nor for complex systems once they reach equilibrium•The Second Law of Thermodynamics is a probabilistic law: Entropy can decrease even for large systems (just extremely, extremely rare)•Reconciles the two visions of time, as long as entropy was very low in the past – Boltzmann’s “Past Hypothesis”•Boltzmann’s Cosmology: We live in a temporary, low-entropy “bubble” within a vast, near-equilibrium Universe
Ludwig Boltzmann(martinroell @flickr)
The Galaxy as a Low-Entropy Fluctuation
Form Galaxy here
Time and the Universe
x30,000 contrast
!
In fact, our Universe was born in a state of extremely low entropy – high uniformity – as seen in the cosmic microwave background radiation.
Time and the Universe
• Boltzmann’s “Past Hypothesis” confirmed: Universe had a definite, low-entropy origin in the observable past
• Low-entropy Big Bang explains the Arrow of Time, which enables life... and all of Universe’s complexity
• Reconciles physics-level reversibility with large-scale irreversibility
• Equilibrium for our Universe lies in the far, far future (whew!)
• In a Multiverse picture, the “parent” Universe that gave birth to our Universe could have been (probably was?) at equilibrium – also as per Boltzmann
Entropy and Gravity
• Gravity reverses the usual sense in which “entropy” corresponds to “homogeneity”
• Gases & mechanical systems: Maximum entropy at maximum homogeneity
• But a homogeneous gravitating system has maximum potential energy – maximum amount of work to be done
• For Universe at large, gravitational energies dominate over all other forces’
• Universe’s minimum entropy state: Pure homogeneity – very close to what we see in CMB
• Universe’s maximum entropy state?
– Black Holes
Minimum entropy
Maximum entropy
Entropy and Black Holes
• Entropy for gases given by number of interacting particles
• Entropy for black holes: Number of Planck-scale patches on event horizon
• Earth consists of ~1051 particles
• Earth-mass black hole (R=1 cm): ~1066 Planck-scale patches
• Since black hole area A~M2, entropy/mass increases with M
• Maximum entropy for Universe: all mass in a single black hole, S~10123
• Compare to entropy at CMB, S~1088
• As number of states: Universe born within one of lowest-entropy 1010^88 out of 1010^123 total possible states
Entropy and Black Holes
• Ultimate fate of matter, including dark matter
• Ultimate Thermodynamic Equilibrium State: The end of the arrow of time
• Transient configurations of baryons and dark matter...
– Humans
– Galaxy
– Everything we observe
Black Holes and the End of Time• Time, as a physical dimension, does not
distinguish past and future
• “Time as change,” thermodynamic time, derives from the irreversible nature of many-body interactions, occurring in systems far from equilibrium
• As Boltzmann predicted with his Past Hypothesis, we inhabit a transient low-entropy region (the Universe) that exhibits a strong “arrow of time”
• In particular, the CMB shows the Universe’s origins as extraordinarily low entropy
• Since our Universe is dominated by gravity on its largest scales, the ultimate destiny of nearly all matter lies within gravity’s most extreme objects, black holes
• The present interesting phase in the life of the Universe thus ends in black holes – the end of time