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The Arrow of Time. LeCosPA Group Meeting 19 October 2010. The Arrow of Time. The past is different from the future. E.g. We don’t remember the future. [ Duh ?] Entropy tends to increase in a closed system [2 nd Law of Thermodynamics]. The Arrow of Time.

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The Arrow of Time

LeCosPAGroup Meeting

19 October 2010


The Arrow of Time

  • The past is different from the future. E.g. We don’t remember the future. [Duh?]
  • Entropy tends to increase in a closed system [2nd Law of Thermodynamics]

The Arrow of Time

“So far as we know, all the fundamental laws of physics, such as Newton’s equations, are reversible. Then were does irreversibility come from? It comes from order going to disorder, but we do not understand this until we know the origin of the order. Why is it that the situations we find ourselves in every day are always out of equilibrium?”



  • Entropy measures volume in phase space.
  • Entropy increases because there are more high-entropy states than low-entropy ones.

Arrow of Time as

Cosmological Problem

  • Why are there eggs in the first place?

Arrow of Time as

Cosmological Problem

  • Why did the entropy start out so low?
  • Question about initial condition of the universe.

Boltzmann’s Idea:

Fluctuation in Equilibrium

“There must then be in the universe, which is in thermal equilibrium as a whole and therefore dead, here and there relatively small regions of the size of our galaxy (which we call worlds), which during the relatively short time of eons deviate significantly from thermal equilibrium. Among these worlds the state probability increases as often as it decreases.” (1895)


Boltzmann’s Brain

  • We cannot be thermal fluctuation:

What is the Entropy of the Universe?

Penrose’s approximation:

Early Universe [Thermal gas]:

Approximate by CMB photon energy density: S ~ N

If all matter is made into a black hole:


The Unnaturalness of

The Initial Condition


The Unnaturalness of

The Initial Condition

  • Past Hypothesis: The universe was born in a very non-generic state. This corresponds to low entropy.
  • Why does a soup of hot particles in low entropy?!

The Unnaturalness of

The Initial Condition


Does Inflation Offer An Solution?

  • Maybe accelerated expansion smoothes out inhomogeneities that would otherwise be susceptible to gravitational collapse?
  • Not good enough!

Sean Carroll and Jennifer Chen, Gen. Relativ. Gravit. (2005) 37(10): 1671–1674


Does Inflation Offer An Solution?

  • Inflation itself requires extremely special initial conditions to get started. It demands an extremely smooth scalar field:
  • T dominated by potential term which mimics cosmological constant provided it is much larger than kinetic term, which involves the field derivatives.

Does Inflation Offer An Solution?

  • The matter and radiation of the early Big Bang era derived its low entropy from a single source: Inflation. But inflation has even lower entropy!
  • Question: Why was the inflation itself in a special state in the beginning?

Possible Solutions?

  • Assume fundamental laws of physics are reversible and the space of allowed states remains fixed. How to account for initial low entropy condition?
  • Beware of the Double Standard: If you have natural explanation for initial low entropy condition, and claim not to invoke time asymmetry, why shouldn’t the late universe look the same way?

Cosmology, Time’s Arrow, and That Old Double Standard, Huw Price, gr-qc/9310022v1


Gold Universe

  • Suggested by Thomas Gold.

 The arrow of time, T. Gold, American Journal of Physics30, pp. 403–410, doi:10.1119/1.1942052.


Possible Solution?

  • Wave hand and declare it a law of nature 
  • Penrose:Weyl Curvature Hypothesis: Past singularities have to be smooth, but future singularities can be arbitrarily messy and complicated.
  • Amount to introducing time asymmetry by hand.

Look for Solution before the Big Bang?

Since inflation doesn’t solve the problem, let’s push it further back!

We accept that the entropy was low at Big Bang, but deny the Big Bang was the beginning of the universe – and look for explanation of low entropy before then.

Pre-Big Bang:

Gasperini and Veneziano, 1993.

Cyclic Universe: Steinhardt and Turok, 2007.


Infinite Fine Tuning!

  • Entropy was small at the bounce, and so as small, if not smaller just before the bounce, even though it has been steadily increasing since infinitely far past!
  • Even if this is possible, we have not explained why this has to be so.

Find a time symmetric universe?

Perhaps the very far past is not different from the far future: they are both high-entropystate. If so, the hot, dense state we have been calling “the early Universe” is actually not the true beginning of the Universe but rather just a transitional state between stages of its history.


Find a time symmetric universe?

The universe is time symmetric by having high entropy at both ends. [c.f. Gold Universe]

But why does the entropy is so low in the middle?


Find a time symmetric universe?

  • Most generic universe (high entropy): empty space. In the context of positive cosmological constant, de Sitter space.
  • de Sitter space: high entropy, eternal, no arrow of time (thermal equilibrium), small but nonzero temperature.
  • So why don’t we live in de Sitter space? Boltzmann brain in de Sitter?!

Babies in de Sitter Space?

  • Perhaps de Sitter space is not so utterly boring, and there are ways to have more ordinary observers than Boltzmann brains?
  • Creation of baby universes?

Adjusting Gravitational Strength?

In essence, as gravity is turned on, the system is forced from a region of large entropy (with respect to the non-gravitational dynamics) into a region with low entropy (with respect to gravity)

Smooth Initial Conditions from Weak Gravity, Brian Greene, Kurt Hinterbichler, Simon Judes, Maulik Parikh, hep-th/0911.0693v1.


Creation on a Torus?

  • Idea: Perhaps there is a natural way to select some extremely smooth geometry from the vast space of possible geometries and topologies; that the special initial conditions were set up by the same process that leads to the “emergence of time” from some TIMELESS [irreducibly stringy? Euclidean? Both?] state.

Arrow of Time in String Theory,

Brett McInnes, Nucl.Phys.B782:1-25,2007


Creation on a Torus?

  • Aim: We should explain why the earliest spatial sections be locally isotropic at the quasi-classical level. That is, explainingWeyl Curvature Hypothesis.

Creation on a Torus?

  • In 3-dimension, a vector is dual to a two-form, so a locally isotropic 3-manifold must have sectional curvature at each point independent of direction.
  • That means, it is a space of constant curvature throughout the (connected) region in which it is locally isotropic. Schur’s Theorem.

Creation on a Torus?

Hirosi Ooguri, Cumrun Vafa, Erik Verlinde, Hartle-Hawking Wave-Function for

Flux Compactifications: The Entropic Principle, Lett.Math.Phys. 74 (2005) 311,



Creation on a Torus?

  • Ooguri et al: This defines Lorentzian metric via Euclidean time. It is the Euclidean version of de Sitter spacetime with flat but compact spatial sections parametrized by
  • Creation from nothing in String Theory.

Kazdan-Warner Classification

Thus torus must be in class Z.


What does this say about babies?

String theory allows for a very large variety of possible universes, brought into existence by the birth of baby universes. This ensemble of many universes is called the String Landscape. The Landscape is actuallyvery small:there is no hope whatever of finding a universe like ours in the Landscape by mere chance. 

The Arrow Of Time In The Landscape, Brett McInnes, To appear in R. Vaas (ed.): Beyond the Big Bang. Springer: Heidelberg 2008, arXiv:0711.1656v2 [hep-th].


What does this say about babies?

Baby universe, by definition, are not “creation from nothing”!