Research @ KICP
May 13, 2005
Our Universe and the Forward March of Time
by Jennifer Chen
The fundamental laws of physics exhibit a high degree of symmetry, and yet our world is in many ways very asymmetric. There are many processes that happen, for which the reverse process is never observed. You can mix two colors of paint, for example, but not un-mix them by swirling in the opposite direction. As time goes by, everything only ever gets messier and older, and sadly, nothing ever spontaneously cleans up. All of these unidirectional processes together dictate a direction in time. On a macroscopic scale, rewind is easily distinguishable from fast forward when microscopically there isn't a difference.
Entropy, the 2nd Law, and Initial Conditions.-- Why time exists and only flows in one direction is related to how our observable universe began. It has been known since Boltzmann's time, that in order to have a macroscopic concept of time, a system must start in an orderly state, which then progressively becomes more and more disordered via what is known as the Second Law of Thermodynamics. This evolution is what gives us a sense of time, and the 2nd Law dictates that this evolution only progresses in the direction of disorder. Thus, the mystery of why our universe exhibits this unidirectional evolution, or an arrow of time, reduces to the question of how our universe got to be in such an orderly state in the beginning. You see the difficulty -- if a system can only dynamically evolve to disorder, how did it become orderly in the first place? Your home or office cannot spontaneously clean itself, but the patch of universe we live in appears to have accomplished just that, more than 14 billion years ago.
From the evolution of the gas we can identify a concept and direction of time, but that the gas particles can visibly evolve at all is due to the fact that they started in the particular state of all being in the corner. If all the gas particles started in the equilibrium distribution of uniformly filling the box, nothing would change on a macroscopic level, and there would be no macroscopic concept of time.
Thus, we see that we can pump the entropy out of a subsystem by raising the entropy of the subsystem plus surroundings (#1). Without invoking anything external to our universe, it would be difficult, if not impossible for our universe to dynamically evolve to an orderly state as observed. The fact that the universe is so orderly compared to what it could be, despite having presumably an eternity to equilibrate, is akin to happening upon an ice cube sitting in the open on a summer day, with no one around to have produced it that way. Observing the current universe evolve is like watching the ice cube melt, and wondering how it got frozen to begin with.
Entropy with Gravity.-- To understand why our universe is in such a special state, we must carefully examine just what constitutes "special" and "generic" for an open system with gravity. In the previous example, we have chosen a closed system in which gravitational interactions are negligible, and in such cases it's easy to formally define the entropy -- it's the logarithm of the number of microscopic configurations for a given macroscopic state. As always, "special" is low entropy and "generic" is high. But to be able to count microstates, they must be discrete, and that means quantum. Thus, lacking a quantum description of gravity, we are unable to define the microscopic states of a gravitational system, much less count them. The best we can do is to use the fact that low entropy systems evolve to high entropy systems and say that a "special" state is one that evolves, and a "generic" state is what systems typically evolve to.
We know that in astrophysical systems, stars can collapse into black holes, and thanks to Bekenstein and Hawking, we miraculously know the formula for the entropy of a black hole. It's quite large -- our universe has an entropy of 10 exp(88) in normal matter (in units of Boltzmann's constant), but should we stuff it all into one giant black hole, it would have an entropy of 10 exp (121), a whopping factor of 10 exp (33) larger. Supposing that the box above is large enough for gravity to be important, the most entropic state for the box of gas is in fact a black hole.
In absence of a cosmological constant, the final generic state of an infinitely large universe is nearly empty flat space. A special state can then be defined as a state that bears no resemblance to generic states.
Deviations from Generic Conditions: Fluctuations.-- Our universe is fortunately much more interesting than nearly empty flat space, and the question is how our universe got to be so full of energy and matter. It turns out that with a small vacuum energy, such as a cosmological constant, the picture above changes. Quantum fluctuations, which normally happen everywhere and all the time, can become real, and given the right kind of fluctuation (in energy and wavelength) a small patch of generic state can undergo inflation, which could give rise to the universe we observe. In addition to producing a large flat, homogeneous and monopole-free universe and to seeding the growth of structure, inflation, upon its demise, converts its energy into normal energy and matter, generating all the stuff we have.
Starting with a patch of empty flat space with a small vacuum energy, we calculated the probability of the spontaneous onset of inflation for its simplest model. The probability that inflation will start up in an inflationary Hubble volume (#2) is 10 exp (-10; 10; 56). It's interesting that this number is so small, but the important thing is that it's nonzero. Given an infinitely large universe and an infinite amount of time to wait, such a fluctuation into inflation will occur infinitely many times. Occasionally, a pocket universe will exit inflation in some of these inflating regions, and form universes that look like ours. Thus, in this picture, the initial conditions for our universe could naturally arise from a generic state, and we'd have a mechanism for generating an arrow of time.
There are two more compelling reasons why it may be more likely that our universe arose out of a fluctuation into inflation over anything else. The first is the exponentially large volumes inflation generates. One fluctuation into inflation (or rather the eternal version of it) may give birth to zillions of universes like ours, whereas a fluctuation into a FRW universe or a brain only generates one. The other factor is that a small patch of generic universe more closely resembles a patch of inflationary universe than a patch of our current FRW universe. A universe of nearly empty space may have high total entropy, but like an inflationary region, it has low entropy density. The inflating patch differs in that it has high energy density, and the fluctuation required to start inflation is a high-energy fluctuation but over a very small region. Thus, if inflation arose out of a rare fluctuation and the FRW universe subsequently evolved from a part of that, a small inflationary universe is entropically closer to a small patch of generic state than a larger FRW universe is to a larger patch of generic state.
A fluctuation into a solipsistic brain is not necessarily more likely than a fluctuation into inflation, and most fluctuations into brains don't have an internal universe that makes physical sense (our dream-worlds never seem to follow a rigorous set of rules). Since our brains perceive a remarkably sensible world, it's hard to imagine that our brains made it up.
Ultra-Large-Scale Picture.-- We have argued that pretty much any state will evolve to a generic state of nearly empty flat space, and with a cosmological constant, fluctuations from generic states into inflationary universes are possible, which could then produce universes like ours. A remarkable result of this picture is that on ultra-large scales, one again recovers a time symmetry. Starting with a generic hypersurface, space can evolve in one direction to smooth out, with patches inflating and smaller patches converting to universes that may form galaxies and such, but in the other direction, it can also evolve to produce a similar fractal-like structure on ultra-large scales.
#1. We are in fact, such living pumps ourselves.
#2. A Hubble volume is the volume of spacetime in which everything is in causal contact. There are about 10 exp (214) inflationary Hubble volumes in our observable universe.
Reference: Spontaneous Inflation and the Origin of the Arrow of Time, Sean M. Carroll and Jennifer Chen, hep-th/0410270
KICP Students: Jennifer Chen