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The preferred time direction
in the dynamics of the full universe
carlo rovelli

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ta
Time
tb

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ta
tb
Time tb < ta

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: subset of the set of balls in the box
Observable: A =P
bxbPb
1
Macrostates: labeled by A
Entropy (microcanonical): number of microstates states in a
given macro stateA
Ss,An = log
Z
ds0Yn
(An(
s0
)A
n(s
)).
Second lawis verified:Ss,A(t)
dt) 0.

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ta
tb
Time tb < ta

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Time
ta
tb

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ta
tb
Time ta < tb

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: subset of the set of balls in the box
Observable:
Macrostates: labeled by
Entropy: number of microstates states in a macro state
Second lawis verified:
A
A =P
bxbPb
1
A
Ss,A(t)
dt) 0. =

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Time
ta
tb

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Message #1:
Entropy increase (passing of time)
can depend on the macroscopical observable
defining the coarse graining
rather than the microstate of the world.
Strong conjecture:
For almost every microscopic history
of a sufficiently rich ergodic system,there is always a macroscopic observable
that defines an increasing entropy,
in either time direction.

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Does this mean that
entropy increase is arbitrary:depends only on definition
and is therefore unphysical?
No, of course
Because the coarse graining
and the choice of the relevant macroscopical observables
is not free: it is dictated by the physics.

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N heavy particles
2N small watcher particles
One watcher particle for each subset of heavy particles
The watcher particle is attracted by the subset of heavy particles

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N heavy particles
2N small watcher particles
One watcher particle for each subset of heavy particles
The watcher particle is attracted by the subset of heavy particles

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Time
ta
tb

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ta
tb
Time tb < ta

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ta
tb
Time ta < tb

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Observable:
The macrostates of the heavy particles
are objectively described by thecoarse graining defined by
for what concerns their interactions
with
Therefore interacts with a world
where the second law is true:
A
A =
Pb
xbP
b1
Ss,A(t)dt
0.
The quantity is precisely the macroscopicobservable that governs the interaction of
the heavy balls with
A

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Therefore interacts with a world
where the second law is true:
Ss,A(t)
dt 0.
While interacts with a world
where a different second law is true:Ss,A(t)
dt 0.

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The sameworld is seen as having
increasingor decreasingentropy
by different subsystems having
different couplingswith the rest.

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Message #2:
Entropy increase (passing of time)
experienced by a subsystem depend
on the coupling of this subsystem
rather than the microstate of the world.
Strong conjecture:
For almost every microscopic history
of a sufficiently rich ergodic system,there is always a subsystem
which experiences entropy increase.

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Message #1: Entropy increase (passing of time)depend on the macroscopic observables.
For almost every microscopic history of a
sufficiently rich ergodic system, there isalways a macroscopic observablethat definesan increasing entropy, in either time direction.
Conjecture 1:
Message #2: Entropy increase (passing of time) experiencedby a subsystem depend on its coupling.
For almost every microscopic history of asufficiently rich ergodic system, there is always asubsystemwhich experiences entropy increase.
Conjecture 2:

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Why is the future different from the past?
There are numerous arrows of time in science:
Second law of thermodynamics
Retarded potential in electromagnetism
Biological (individuals go from birth to death)
Biological (species evolution in one time direction)
Psychological
All of these can be traced
to the second law of thermodynamics.
See Lebowitz, Boltzmannsentropy and times arrow, (1993)

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How can the second law of thermodynamics be compatiblewith the microscopic time (CPT) reversal invariance?
There is no contradiction: time reversal invarianceis broken anytime entropy is low in the past.
Why was entropy low in the past?
Because it was low in the cosmologicalpast.
Therefore the problem of the difference between
past and future is a problem for cosmology.

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This is puzzling
The initial (micro) state of the universe was very special (low entropy)
Why spacetime is not crumpled in the past?
See Penrose, Singularitiesand TimeAsymmetry, 1979.
Ex: a generic spacetime is crumpled.
Inflation does not help: it onlydisplaces the problem.

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The initial (micro) state of the universe was very special (low entropy)?
I suggest that there is a more interesting possibility:
i) The initial microstateof the universe is generic.
ii) We describe it is in terms of macroscopic variables withrespect to whichentropy is low.
iii) We do so because we belong to a subsystem that couples tosuch variables.
iv) The reason we couple with such variables is that this fact(living in time) is precisely what characterises us.
THE SECOND LAW OF THERMODYNAMICS IS TRUE, BUT IT IS PERSPECTIVAL

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Ein = Eout
in > out
Sin < Sout
With respect to themacroscopic variables
energy and frequency,
there is entropyproduction in the
biosphere

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It is not the (microstate) of the universe to be peculiar.
It is ourself (the system we belong to), to be peculiar observers,
we couple to macroscopic variablesgiving low entropy at one end.
The low initial entropy is not due to the state.
It is due to the coarse graining observables.

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The real world:
Cosmology: It is not the dynamics of everything. It is the dynamics of a few verycoarse grained degrees of freedom (see J Barrows talk). Comology is a feast of coarsegraining!
General relativity: The dynamics does not have a single preferred time variable. Evenless a direction.
Quantum mechanics: Entropy includes von Neumann entropy, which event morethan in the classical case, depends on the splitting in subsystems and the couplingbetween these.
Quantum Gravity: Loop Quantum Gravity: Space and time emerge in thesemiclassical approximation. Cosmology is a violently coarse grained description ofreality.
H = L2[SU(2)L/SU(2)N]
Lia, L
jb
= iab
2ijk L
ka
Wv = PSL(2,C) Yv 1I

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The scale factor may be like
The variable.A

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ta
tb
Time ta < tb

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Related ideas and issues:
Simon Saunders David Wallace: Branching of the universal wave function is timeoriented. Why? Could it be that it is with respect to a subsystem with suitablecouplings that the branching is time oriented?
or ?
Jenann Ismael: Which aspects of the world depend on our own perspective on
them?
Max Tegmark: How unitary cosmology generalizes thermodynamics and solves theinflationary entropy problem (2012).
David Albert: The low entropy initial condition is what allows us to reconstruct thepast.
Jim Hartle: Predictability emerges in the quantum state from quasi classical realms.The relevant observables are averages of densities of conserved quantities oversmall volumes. In quantum gravity, does locality emerges after the establishment ofa semiclassical approximation? Is coarse graining determined or determininglocality?

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 Entropy increase (passage of time) depend on the coarse graining, hence the
subsystem, not the microstate of the world.
For almost every microscopic history of a sufficiently
rich ergodic system, there is always a subsystem whichexperiences entropy increase.
 Conjecture:
Summary
 Time asymmetry, and time flow, might be a feature of a subsystem towhich we belong, (needed for information gathering creatures like us to be
what we are), not features of the universe at large.
The second law of thermodynamics is true,but it is perspectival.
CR: Why do we remember the past and not the future?The time oriented coarse graining hypothesis arXiv: 1407.3384.