Space vs Time

If we want to progress towards a theory of everything, we need to understand how space and time fit together – if they do at all.

To Isaac Newton, space was the “sensorium of God”, the organ through which the deity surveyed His creation. It was absolute, unchanging, infinite. Flowing through it “equably without regard to anything external”, as Newton wrote in his great physical treatise Principia, was another similarly absolute heavenly creation: time.

Not everyone bought that idea. Newton’s perennial antagonist Gottfried Leibniz was notably sniffy of a God who needed an organ to perceive things, and asked pointedly whether a clockmaker deity would need to “wind up his watch from time to time”.

A few centuries on, God features less prominently in the debate, but arguments about the nature of space and time swirl on. Are both basic constituents of reality, or neither – or does one perhaps emerge from the other in some way? We are yet to reach a conclusive answer, but it is becoming clear that if we wish to make further progress in physics, we must. The route to a truly powerful theory of reality passes through an intimate understanding of space and time.

The search for reality’s building blocks goes to the heart of what physics is about. “When we find the simplest equations for everything in the universe, the fundamental quantities would be what appear in those equations,” says theorist Joe Polchinski of the University of California, Santa Barbara.

That makes it all the more embarrassing that the two sets of equations we use to describe the physical world differ so radically in form and content. Einstein’s relativity, which covers gravity, does a stellar job in describing the universe at large. The theory of quantum mechanics, meanwhile, describes all the other forces and paints a peerless picture of the world at the smallest scales.

Problems emerge when the large meets the small: in the first instants after the big bang, for example, when the entire universe was a mere pinprick; or at the immense maw of a black hole, whose gravitational pull is so great that not even photons of light can escape. The contradictions that rear up have, at least in part, a very basic origin. “One of the tensions comes from the fact that the relation between space and time is very, very different in general relativity than it is in quantum mechanics,” says theorist Sean Carroll of the California Institute of Technology in Pasadena.

When Einstein developed the theory of special relativity in 1905, it undid Newton’s notions of a clockwork universe, in which objects in an absolute space followed the beat of a heavenly timepiece. Space and time are intertwined into one four-dimensional fabric called space-time. People moving at different speeds measure space-time differently. Just as “here” does not mean the same thing to people in different places, so it is with “now” in Einstein’s relativistic space-time. “What we call now doesn’t have any unique translation to what a person on Alpha Centauri calls now,” says Carroll.

General relativity, which came along in 1916, muddied the waters still further: massive objects curve space-time, and measurements of ruler lengths and clock ticks depend on the strength of the prevailing gravitational field.

In quantum mechanics, things are even more abstruse. A quantum object’s state is described by a wave function, a mathematical object living in an abstract space, known as Hilbert space, that encompasses all the possible states of the object. We can tell how the wave function evolves in time, moving from one state in its Hilbert space to another, using the Schrödinger equation.

In this picture, time is itself not part of the Hilbert space where everything else physical sits, but somehow lives outside it. When we measure the evolution of a quantum state, it is to the beat of an external timepiece of unknown provenance. “Somebody gave us a clock, a grandfather clock, and we just use this clock,” says Nathan Seiberg of Institute for Advanced Study in Princeton, New Jersey.

As for space, its status depends on what you are measuring. The wave function of an electron orbiting the atomic nucleus will include properties of physical space such as the electron’s distance from the nucleus. But the wave function describing the quantum spin of an isolated electron has no mention of space: according to the mathematics, the picture we often paint of an electron physically rotating is meaningless.

“This is one sense in which there are attributes of physical systems which don’t refer to space, but which change in time,” says Abhay Ashtekar of Pennsylvania State University. “One could say that for those attributes, time is more fundamental than space.”

That is how things stand if you consider general relativity and quantum mechanics in isolation. Relativity says space and time are on the same footing – together they are the fabric of reality. Quantum mechanics, on the other hand, treats time and space differently, with time occasionally seeming more fundamental.

All this starts unravelling, however, whenever we attempt to combine the two theories to find a greater theory capable of describing reality on all scales, large and small.

One of the most ambitious such attempts is string theory. It has an odd relationship with space. One of the theory’s key characteristics is the existence of extra dimensions of space so tightly curled up as to be almost undetectable. It needs at least 10 space-time dimensions to be mathematically consistent. But a celebrated result derived in 1997 by theoristJuan Maldacena suggests mathematical trapdoors exist between these different dimensions. According to his “anti-de Sitter/conformal field theory correspondence” – AdS/CFT for short – under certain circumstances, you can swap the fiendishly complex 10D representations of string theory that include gravity for a more tractable 4D representation that dispenses with gravity.

As you do this, one-dimensional time remains seemingly unchanged, but space is transformed: a point in the 4D world translates to multiple points within the 10D world. “In this example it seems perfectly clear that space is not fundamental. It is very, very different depending on what description of the world you are using,” says Carroll.

Black hole drama

But things may not be so clear-cut. Polchinski, for one, has started to wonder if such insights from Maldacena’s conjecture are entirely justified. We already know that the AdS/CFT correspondence is only valid for a specific sort of space-time that is not quite the space-time of our universe. The fabric of our universe is geometrically almost flat: two light rays that start out parallel stay so. But the space-time for which the AdS/CFT correspondence is valid is negatively curved, such that two initially parallel light rays slowly start to diverge. The intractable mathematics of string theory means that no one has yet worked out an AdS/CFT-like correspondence for our space-time.

Polchinski’s team has raised a more fundamental objection, however – by peering into the dark heart of a black hole. Black holes have a history of testing theories to the limit. Predicted by general relativity and thought to exist where massive stars once lived, as well as at the heart of every galaxy, they have insatiable appetites that appear to gobble up even information, something forbidden by quantum theory. If you try to sidestep the issue by allowing information to escape from a black hole, quantum theory says that a blazing “firewall” of high-energy radiation appears just inside the event horizon, the black hole’s point of no return.

That in turn goes against the predictions of general relativity, which says that anything going past a black hole’s event horizon should encounter nothing but gently curved space-time – no theatrics, no drama. If you want to keep quantum mechanics intact and avoid a firewall too, something else about space-time needs to give, such as the dictum that nothing can travel faster than the speed of light. “This does point to the fact that we may be missing something in our conceptual description,” says Steve Giddings, also at the University of California, Santa Barbara.

It was to crack this conundrum that Polchinski’s team turned to Maldacena’s conjecture. They started by chucking a black hole into a volume of negatively curved space-time. If the conjecture were to hold, then the 4D physics of an observer on the surface of the volume should be able to account for the physics of an observer deep inside the 10D bulk of a black hole held within – only with easier mathematics. Instead, what the two observers see is described by two different quantum theories (arxiv.org/abs/1304.6483).

The AdS/CFT correspondence is beloved among string theorists, and Polchinski knows he is going against the grain. “I want to shake people’s faith in AdS/CFT,” he says. If he succeeds, any conclusions drawn from it and string theory about the status of space and time may not be the last word.

But string theory is just one as-yet-unproven approach to unifying relativity and quantum theory. Another, known as loop quantum gravity, has its origins in the mid-1980s when Ashtekar rewrote Einstein’s equations of general relativity using a quantum mechanical framework. Working with physicistsLee Smolin and Carlo Rovelli, he used these equations to arrive at a picture in which space-time is a smooth fabric – until, as with any fabric, you look at it very closely. In the case of the space-time of loop quantum gravity, if you zoom in to tiny scales of about 10-35 metres, a distance known as the Planck length, you see a warp and weft of loops of gravitational field lines.

As with string theory, the equations of loop quantum gravity have proved difficult to work with, and have produced little in terms of verifiable experimental predictions. Still, they provide a different perspective on the primacy of space and time. Chunks of space, one Planck length to a side, appear first in the theory, while time pops up only later as an expression of the relationships between other observable physical properties. For instance, you can define the tick of a clock in terms of changes in the gravitational field and then observe how another field, say the electromagnetic field, changes with respect to the “ticking” of the gravitational field. Here, both space and time seem to emerge from something deeper, says Ashtekar. “But somehow space might emerge first, and time is born by observing relations between various physical subsystems.”

In a sense, that’s coming full circle to how Newton viewed some aspects of time. While seeing absolute time as something God-given, he recognised that we measure a “common” time relationally – that is, by keeping track of other properties. Earth’s motion around the sun in space represents a unit of time, for example – the year – which we subdivide or multiply to calculate the duration of anything else, like the length of a season.

We do not know as yet what the deeper thing might be from which space and time emerge, but Ashtekar is not the only one thinking along these lines. Seiberg has a similar intuition based on findings in string theory, which suggest space is emergent, and general relativity, which says that space and time are interwoven. “We have many examples where we have emergent space and, given that space and time mix, there’s no doubt that time will also be emergent,” he says.

Relaxing restrictions

Giddings is also exploring the idea that neither space nor time is fundamental, using that most uncomfortable of theoretical test beds. He has been trying to describe a black hole, from its interior to way outside its event horizon, using a network of interconnected quantum-mechanical Hilbert spaces that do not presuppose the existence of space or time. This allowed him, for example, to relax the restriction that nothing can travel through space-time faster than light – imposed by Einstein’s relativity – and see what would happen as a result.

By observing how one attribute of a quantum system changes with respect to another in this set-up, Giddings showed last year how time can emerge relationally, in another nod to Newton’s “common” time. A concept of space also emerges from his calculations by loosely specifying different Hilbert spaces to correspond to different physical locations, such as the interior of a black hole (arxiv.org/abs/1201.1037). Ultimately, Giddings thinks that even this notion of space can be linked to the dynamics of the system, just like time. In this case, neither would be primary.

It is early days for the idea. To be taken seriously, Giddings must show that general relativity and the normal picture of space-time can be derived from his network of Hilbert spaces for situations that are not as extreme as black holes. “We have a way to go to fully understand this,” he says.

Polchinski is also tugging at that constant speed of light in relativity. In a sense, this provides a reference of both space and time. A light ray always moves at one unit of space per unit of time – a constant diagonal on any graph of space against time.

“The direction that light rays travel in is neither space nor time; we call it ‘null’. It’s on the edge between space and time,” says Polchinski. “A lot of people have this intuition that in some sense the existence of these null directions might be more fundamental than space or time.”

Will that or any other intuition lead us anywhere in our quest towards a greater theory? The current impasse in physics is not unlike the situation in the early 20th century before Einstein’s master stroke replaced space and time with one space-time. Today we appear to be facing a similar obstruction to progress. Many potential ways around lead to different worlds of space and time – and we have as yet little clue which route to follow. Space, time, both or neither? Perhaps only time will tell.

(Article written by Anil Ananthaswamy and first appeared in New Scientist Magazine).

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