Theoretical physicists commonly say their biggest challenge is to unite general relativity with quantum theory. But at this month's FQXi conference in Puerto Rico, Carlo Rovelli said they have an even bigger challenge: to unite general relativity with thermodynamics. After all, physicists do have proposed solutions to the first problem, such as string theory and loop quantum gravity, and the rivalry among them is not so bitter that theorists can't find the humor in it. Physicists routinely and uncontroversially work with an effective theory of gravity which, while not the full story, captures quantum corrections to Einstein's theory.
But it's one thing to describe the inner workings of gravity, quite another to understand how gravity behaves in big complicated systems, which are the subject of thermodynamics. "We are totally in the dark about the foundations of statistical thermodynamics and general relativity," Rovelli told his colleagues at the conference. Gravity inverts our usual intuition about thermodynamic concepts such as entropy: a spread-out gas has a high entropy mechanically, but a low entropy gravitationally. Amadeo Balbi remarked to me, "There is no consensus on how to precisely calculate entropy when gravity is important, except in special cases, such as black holes." And for black holes, physicists confront the notorious information paradox, whose latest incarnation as the firewall argument sparked a suitably fiery debate at the conference.
Perhaps most dramatically, Rovelli has been arguing for two decades that space and time themselves are not fundamental to nature, but emerge thermodynamically. He made the case in his prize-winning essay in FQXi's first essay contest, and others have presented similar ideas at past FQXi meetings. To advance this broader program, Rovelli's talk in Puerto Rico went back to basics. He didn't concern himself with the much-discussed second law of thermodynamics, nor the third law, nor the first. He focused on the zeroth law. (You'd think that a discipline--statistical mechanics--that prides itself on counting 1023 molecules at a time would have done a better job of numbering its laws.)
The zeroth law says that a gas or other system that has reached equilibrium has a uniform temperature. This deceptively simple principle underpins the other laws, not least by defining temperature as the collective property that an equilibrium system possesses. Systems that are not in equilibrium do not have well-defined temperatures.
So what's up with gases in a gravitational field? When they reach equilibrium, they do not have a uniform temperature, but cool off with altitude in a phenomenon known as the Tolman-Ehrenfest effect. As Rovelli and Matteo Smerlak have showed, the effect is basically a consequence of gravitational redshift. Light or anything else that climbs away from Earth's center loses energy, which is tantamount to cooling off.
In practice, we never see this effect. The air temperature does drop when you climb a mountain or fly a plane, but that's ultimately because of the disequilibrium between the sun-basked ground and the cold of deep space. Imagine the far future of Earth after the sun goes dark, radioactive ores decay away, and the planet comes into thermal equilibrium with the cosmos. The inhabitants of that sorry world will still feel colder when they climb a mountain. The temperature will fall by one part in 1013 at an altitude of 1 kilometer. Not enough to put on mittens, but more than enough to flout the zeroth law.
To restore the primacy of the law, Rovelli sought a generalized version that holds under all conditions. He noted an interesting fact about temperature. Although we typically measure it in units of degrees or kelvins, according to basic physical laws it really has units of inverse seconds--a rate. One way to see this, Rovelli and Hal Haggard argued last year, is Heisenberg's energy-time uncertainty relation. The time it takes a quantum system to change discernibly is inversely proportional to the spread of its energy, which is proportional to temperature for systems in thermal equilibrium. Thus time is inversely related to temperature, and vice versa. The same goes for classical systems as well.
Specifically, Rovelli argued that temperature is the rate at which systems change their internal state. Air at room temperature, for example, changes state 3 trillion times per second as the molecules feverishly reshuffle themselves. In this spirit, Rovelli proposed defining equilibrium as a condition not of uniform temperature, but of a common rate of changing state. The Tolman-Ehrenfest effect then makes perfect sense. By warping time, gravity mucks with the time standard by which rates are measured. As you climb a mountain, time passes more quickly and rates slow down. The rate at which a gas changes state--and thus its temperature--thus decreases even though the gas molecules are as feverish as ever.
To make the connection to the conference theme, the physics of information, Rovelli suggested thinking of an interaction between two systems in terms of information. Each system gets a glimpse into the internal structure of the other system. If both systems are changing state at the same rate, both gain the same amount of information about each other. And that, Rovelli argued, is what equilibrium really means. The situation is like two poker players reading each other's expressions. If one poker player is expressionless, while the other is an open book, then the two will be out of equilibrium--which translates, quite tangibly, into a flow of money from the second to the first. But if the two are equally able to read each other, they can play equally well and reach a stalemate.
Rovelli has put so much effort into understanding equilibrium because it is crucial to his program of explaining time as emergent. Traditionally physicists have taken time as a given and expressed all physical change with respect to it. A pendulum swings, a clock ticks, and a heart beats once per second. But Rovelli thinks you could eliminate the little t and express all these changes with respect to one another. A pendulum swings once per clock tick or heart beat, and vice versa.
External time vs. relational time
Time here plays the same role that money does in an economy: it provides a convenient medium of exchange, but has no value on its own. In principle, you could shred all the dollar bills and instead perform a complex series of barter transactions. In his charming little autobiographical book What Is Time? What Is Space?, Rovelli wrote: "Time is an effect of our ignorance of the details of the world. If we had complete knowledge of all the details of the world, we would not have the sensation of the flow of time."
Quantitatively, Rovelli solves for t by inverting the equation describing equilibrium states. This is subtle: equilibrium states are not changing macroscopically, by definition, so they seem like an odd choice for defining time. Rovelli argues that time evolution is latent in the statistics of equilibrium states, since a system, if displaced from equilibrium, will return to it. The time evolution given by the second law of thermodynamics builds on how equilibrium states are defined. "I think there's a clear way of getting time out of a timeless theory," Rovelli told me.
That said, he has yet to show conclusively that time is a collective phenomenon. Yasunori Nomura said he shares Rovelli's general aim of recovering time from correlations within a system, but questions his focus on equilibrium states. There seems little risk that theorists will reach equilibrium on equilibrium anytime soon.