# Beyond space-time: Welcome to phase space

08 August 2011 by Amanda Gefter Magazine issue 2824. Subscribe and save For similar stories, visit the Cosmology Topic Guide

*A theory of reality beyond Einstein’s universe is taking shape – and a mysterious cosmic signal could soon fill in the blanks*

IT WASN’T so long ago we thought space and time were the absolute and unchanging scaffolding of the universe. Then along came Albert Einstein, who showed that different observers can disagree about the length of objects and the timing of events. His theory of relativity unified space and time into a single entity – space-time. It meant the way we thought about the fabric of reality would never be the same again. "Henceforth space by itself, and time by itself, are doomed to fade into mere shadows," declared mathematician Hermann Minkowski. "Only a kind of union of the two will preserve an independent reality."

But did Einstein’s revolution go far enough? Physicist Lee Smolin at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada, doesn’t think so. He and a trio of colleagues are aiming to take relativity to a whole new level, and they have space-time in their sights. They say we need to forget about the home Einstein invented for us: we live instead in a place called phase space.

If this radical claim is true, it could solve a troubling paradox about black holes that has stumped physicists for decades. What’s more, it could set them on the path towards their heart’s desire: a "theory of everything" that will finally unite general relativity and quantum mechanics.

So what is phase space? It is a curious eight-dimensional world that merges our familiar four dimensions of space and time and a four-dimensional world called momentum space.

Momentum space isn’t as alien as it first sounds. When you look at the world around you, says Smolin, you don’t ever observe space or time – instead you see energy and momentum. When you look at your watch, for example, photons bounce off a surface and land on your retina. By detecting the energy and momentum of the photons, your brain reconstructs events in space and time.

The same is true of physics experiments. Inside particle smashers, physicists measure the energy and momentum of particles as they speed toward one another and collide, and the energy and momentum of the debris that comes flying out. Likewise, telescopes measure the energy and momentum of photons streaming in from the far reaches of the universe. "If you go by what we observe, we don’t live in space-time," Smolin says. "We live in momentum space."

And just as space-time can be pictured as a coordinate system with time on one axis and space – its three dimensions condensed to one – on the other axis, the same is true of momentum space. In this case energy is on one axis and momentum – which, like space, has three components – is on the other (see diagram).

Simple mathematical transformations exist to translate measurements in this momentum space into measurements in space-time, and the common wisdom is that momentum space is a mere mathematical tool. After all, Einstein showed that space-time is reality’s true arena, in which the dramas of the cosmos are played out.

Smolin and his colleagues aren’t the first to wonder whether that is the full story. As far back as 1938, the German physicist Max Born noticed that several pivotal equations in quantum mechanics remain the same whether expressed in space-time coordinates or in momentum space coordinates. He wondered whether it might be possible to use this connection to unite the seemingly incompatible theories of general relativity, which deals with space-time, and quantum mechanics, whose particles have momentum and energy. Maybe it could provide the key to the long-sought theory of quantum gravity.

Born’s idea that space-time and momentum space should be interchangeable – a theory now known as "Born reciprocity" – had a remarkable consequence: if space-time can be curved by the masses of stars and galaxies, as Einstein’s theory showed, then it should be possible to curve momentum space too.

At the time it was not clear what kind of physical entity might curve momentum space, and the mathematics necessary to make such an idea work hadn’t even been invented. So Born never fulfilled his dream of putting space-time and momentum space on an equal footing.

That is where Smolin and his colleagues enter the story. Together withLaurent Freidel, also at the Perimeter Institute, Jerzy Kowalski-Glikman at the University of Wroclaw, Poland, and Giovanni Amelino-Camelia at Sapienza University of Rome in Italy, Smolin has been investigating the effects of a curvature of momentum space.

The quartet took the standard mathematical rules for translating between momentum space and space-time and applied them to a curved momentum space. What they discovered is shocking: observers living in a curved momentum space will no longer agree on measurements made in a unified space-time. That goes entirely against the grain of Einstein’s relativity. He had shown that while space and time were relative, space-time was the same for everyone. For observers in a curved momentum space, however, even space-time is relative (see diagram).

This mismatch between one observer’s space-time measurements and another’s grows with distance or over time, which means that while space-time in your immediate vicinity will always be sharply defined, objects and events in the far distance become fuzzier. "The further away you are and the more energy is involved, the larger the event seems to spread out in space-time," says Smolin.

For instance, if you are 10 billion light years from a supernova and the energy of its light is about 10 gigaelectronvolts, then your measurement of its location in space-time would differ from a local observer’s by a light second. That may not sound like much, but it amounts to 300,000 kilometres. Neither of you would be wrong – it’s just that locations in space-time are relative, a phenomenon the researchers have dubbed "relative locality".

Relative locality would deal a huge blow to our picture of reality. If space-time is no longer an invariant backdrop of the universe on which all observers can agree, in what sense can it be considered the true fabric of reality?

That is a question still to be wrestled with, but relative locality has its benefits, too. For one thing, it could shed light on a stubborn puzzle known as theblack hole information-loss paradox. In the 1970s, Stephen Hawking discovered that black holes radiate away their mass, eventually evaporating and disappearing altogether. That posed an intriguing question: what happens to all the stuff that fell into the black hole in the first place?

Relativity prevents anything that falls into a black hole from escaping, because it would have to travel faster than light to do so – a cosmic speed limit that is strictly enforced. But quantum mechanics enforces its own strict law: things, or more precisely the information that they contain, cannot simply vanish from reality. Black hole evaporation put physicists between a rock and a hard place.

According to Smolin, relative locality saves the day. Let’s say you were patient enough to wait around while a black hole evaporated, a process that could take billions of years. Once it had vanished, you could ask what happened to, say, an elephant that once succumbed to its gravitational grip. But as you look back to the time at which you thought the elephant had fallen in, you would find that locations in space-time had grown so fuzzy and uncertain that there would be no way to tell whether the elephant actually fell into the black hole or narrowly missed it. The information-loss paradox dissolves.

Big questions still remain. For instance, how can we know if momentum space is really curved? To find the answer, the team has proposed several experiments.

One idea is to look at light arriving at the Earth from distant gamma-ray bursts. If momentum space is curved in a particular way that mathematicians refer to as "non-metric", then a high-energy photon in the gamma-ray burst should arrive at our telescope a little later than a lower-energy photon from the same burst, despite the two being emitted at the same time.

Just that phenomenon has already been seen, starting with some unusual observations made by a telescope in the Canary Islands in 2005 (*New Scientist*, 15 August 2009, p 29). The effect has since been confirmed by NASA’s Fermi gamma-ray space telescope, which has been collecting light from cosmic explosions since it launched in 2008. "The Fermi data show that it is an undeniable experimental fact that there is a correlation between arrival time and energy – high-energy photons arrive later than low-energy photons," says Amelino-Camelia.

Still, he is not popping the champagne just yet. It is not clear whether the observed delays are true signatures of curved momentum space, or whether they are down to "unknown properties of the explosions themselves", as Amelino-Camelia puts it. Calculations of gamma-ray bursts idealise the explosions as instantaneous, but in reality they last for several seconds. While there is no obvious reason to think so, it is possible that the bursts occur in such a way that they emit lower-energy photons a second or two before higher-energy photons, which would account for the observed delays.

In order to disentangle the properties of the explosions from properties of relative locality, we need a large sample of gamma-ray bursts taking place at various known distances (arxiv.org/abs/1103.5626). If the delay is a property of the explosion, its length will not depend on how far away the burst is from our telescope; if it is a sign of relative locality, it will. Amelino-Camelia and the rest of Smolin’s team are now anxiously awaiting more data from Fermi.

The questions don’t end there, however. Even if Fermi’s observations confirm that momentum space is curved, they still won’t tell us what is doing the curving. In general relativity, it is momentum and energy in the form of mass that warp space-time. In a world in which momentum space is fundamental, could space and time somehow be responsible for curving momentum space?

Work by Shahn Majid, a mathematical physicist at Queen Mary University of London, might hold some clues. In the 1990s, he showed that curved momentum space is equivalent to what’s known as a noncommutative space-time. In familiar space-time, coordinates commute – that is, if we want to reach the point with coordinates (*x*,*y*), it doesn’t matter whether we take *x*steps to the right and then *y* steps forward, or if we travel *y* steps forward followed by *x* steps to the right. But mathematicians can construct space-times in which this order no longer holds, leaving space-time with an inherent fuzziness.

In a sense, such fuzziness is exactly what you might expect once quantum effects take hold. What makes quantum mechanics different from ordinary mechanics is Heisenberg’s uncertainty principle: when you fix a particle’s momentum – by measuring it, for example – then its position becomes completely uncertain, and vice versa. The order in which you measure position and momentum determines their values; in other words, these properties do not commute. This, Majid says, implies that curved momentum space is just quantum space-time in another guise.

What’s more, Majid suspects that this relationship between curvature and quantum uncertainty works two ways: the curvature of space-time – a manifestation of gravity in Einstein’s relativity – implies that momentum space is also quantum. Smolin and colleagues’ model does not yet include gravity, but once it does, Majid says, observers will not agree on measurements in momentum space either. So if both space-time and momentum space are relative, where does objective reality lie? What is the true fabric of reality?

Smolin’s hunch is that we will find ourselves in a place where space-time and momentum space meet: an eight-dimensional phase space that represents all possible values of position, time, energy and momentum. In relativity, what one observer views as space, another views as time and vice versa, because ultimately they are two sides of a single coin – a unified space-time. Likewise, in Smolin’s picture of quantum gravity, what one observer sees as space-time another sees as momentum space, and the two are unified in a higher-dimensional phase space that is absolute and invariant to all observers. With relativity bumped up another level, it will be goodbye to both space-time and momentum space, and hello phase space.

"It has been obvious for a long time that the separation between space-time and energy-momentum is misleading when dealing with quantum gravity," says physicist João Magueijo of Imperial College London. In ordinary physics, it is easy enough to treat space-time and momentum space as separate things, he explains, "but quantum gravity may require their complete entanglement". Once we figure out how the puzzle pieces of space-time and momentum space fit together, Born’s dream will finally be realised and the true scaffolding of reality will be revealed.

### Bibliography

*The principle of relative locality*by Giovanni Amelino-Camelia and others (arxiv.org/abs/1101.0931)

**Amanda Gefter** is a consultant for *New Scientist* based in Boston