## Hawking's 'Escher-verse' could be theory of everything

Stephen Hawking has come up with a way to describe the universe that suggests it may have the same geometry as mind-boggling images by M. C. Escher

THE universe may have the same surreal geometry as some of art's most mind-boggling images. That's the upshot of a study by the world's most famous living scientist, Stephen Hawking of the University of Cambridge.

The finding may delight fans of Dutch artist M. C. Escher, but Hawking's team claim that their study provides a way to square the geometric demands of string theory, a still-hypothetical "theory of everything", with the universe we observe.

Their calculations rely on a mathematical twist that was previously considered impossible. If it stands up, it could explain how the universe emerged from the big bang and unite gravity and quantum mechanics.

"We have a new route towards constructing string theory models of our world," says Hawking's colleague Thomas Hertog of the Institute for Theoretical Physics at the Catholic University of Leuven (KUL) in Belgium.

On the face of it, the idea that Escher's images can describe the layout of the universe seems to contradict what we know about it.

The images in question are tessellations, arrangements of repeated shapes, such as the images of interlocking bats and angels seen in Circle Limit IV. Although these are flat, they serve as "projections" of an alternative geometry called hyperbolic space, rather like a flat map of the world is a projection of a globe. For example, although the bats in the flat projection appear to shrink at an exponential rate at the edges, in hyperbolic space they are all the same size. These distortions in the projection arise because hyperbolic space cannot lie flat. Instead, it resembles a twisting, wiggly landscape of saddle-like hills.

That is not what our universe seems to look like. Measurements of the cosmic microwave background - the echo of the big bang - and distances to supernovae have revealed that our universe is flat, not twisted.

It is also expanding at an accelerating rate, because of a mysterious entity known as dark energy. We don't know what dark energy is or where it came from, but the mathematical language provided by Einstein's theory of general relativity has a way to describe this accelerated expansion. Sticking a constant - known as the cosmological constant - into the general-relativity equations keeps the universe expanding forever, but only if the constant has a positive sign. Until now, saying we live in an ever-expanding universe has been the same as saying our universe has a positive cosmological constant.

There are some outstanding problems, however. General relativity covers this aspect of the universe, but it can't describe the big bang. Nor can it unite gravity, which works on large scales, with quantum mechanics, which works on very small scales. "That means you cannot predict why we live in the universe that we live in," Hertog says.

String theory, in the meantime, offers a beautifully complete picture of the universe's history and connects gravity to quantum mechanics - but is most comfortable in a universe with a negatively curved, Escher-like geometry and with a negative cosmological constant.

This left physicists with a deep chasm to cross: on one side is a universe that works but lacks a complete theory, and on the other is a complete theory that doesn't describe the actual universe.

Now, Hawking, Hertog and James Hartle of the University of California, Santa Barbara, are proposing a bridge. They have found a way to produce expanding, accelerating universes using a negative cosmological constant. This means that string theory may, after all, describe the universe that we observe. The proposal grew from an idea that Hawking and Hartle had in the 1980s to get around general relativity's shortcomings by looking for a quantum picture of cosmology.

In quantum mechanics, a single equation called the wave function describes all the possible states that a quantum object can be in, and assigns each of them a certain probability. Hawking and Hartle sought a similar wave function that can generate the probability of various universes arising from the big bang. It would describe all the possible universes that could have been - including ones in which the solar system never formed, or in which life might have evolved very differently.

Over the past 30 years, Hawking and Hartle have been forcing a positive cosmological constant into their wave function, because that was considered necessary. But that meant sacrificing precision: they just couldn't get these universes to be anything more than clunky approximations of reality.

Tip the balanceString theorists had also been struggling with universes with positive cosmological constants, which tend to be unstable. Building them is a bit like trying to balance a pencil on its tip: it might work for a while, but the pencil's most energetically stable state is lying flat on the table, and eventually it will fall over. The most successful versions of string theory would rather live in the Escher-verse.

"String theory with a negative cosmological constant just goes much better," Hertog says.

But Hawking's latest work suggests that this supposed flaw may actually be the thing that knits string theory back to reality. In a paper posted online, Hawking and colleagues describe how they produced a plethora of universes from wave functions with negative cosmological constants, some of which are expanding and accelerating (arxiv.org/abs/1205.3807).

"Some of those universes are accelerating, just like our universe," Hertog says. "It turns out the quantum state includes both kinds of universes, automatically." For a certain wave function, these accelerating and expanding universes even turn out to be the most likely ones.

The key to this insight was recognising that the universes generated by the team's wave function could evolve to look a lot like a particular formulation of string theory, produced by Juan Maldacena of the Institute for Advanced Study in Princeton, New Jersey in 1997 (arxiv.org/abs/hep-th/9711200). "There was a mathematical connection, a very elegant connection," Hertog says.

Once they had spotted its connection to their wave function, Hawking's team decided to try to stitch the two together by writing a new wave function with a negative cosmological constant. They reasoned that this would allow them to borrow the beautifully complete mathematical picture of the universe provided by string theory and produce universes that accelerate outwards.

What about the observations suggesting that our universe is flat? Similar to how Newton's laws of motion work for everyday objects but give way to the more comprehensive laws of Einstein on cosmological scales, Hawking's team thinks that the universe's apparent flatness may describe it well as far as we can see but ultimately gives way to an underlying Escher-like geometry.

It is too soon to declare the universe solved. Maldacena says the Hawking team's model leaves out aspects of complete versions of string theory, such as provisions for the stability of some particles. "It would be wonderful if it was all we need to do," he says. "But I think it's too simplified. It's hard to see how it can be expanded to a more complete theory."

Hertog agrees that their work isn't finished - but thinks that the negative cosmological constant will eventually lead to a complete, description of the universe we observe. "It's an avenue that is opening up now," he says, "not something we have yet."

THE universe may have the same surreal geometry as some of art's most mind-boggling images. That's the upshot of a study by the world's most famous living scientist, Stephen Hawking of the University of Cambridge.

The finding may delight fans of Dutch artist M. C. Escher, but Hawking's team claim that their study provides a way to square the geometric demands of string theory, a still-hypothetical "theory of everything", with the universe we observe.

Their calculations rely on a mathematical twist that was previously considered impossible. If it stands up, it could explain how the universe emerged from the big bang and unite gravity and quantum mechanics.

"We have a new route towards constructing string theory models of our world," says Hawking's colleague Thomas Hertog of the Institute for Theoretical Physics at the Catholic University of Leuven (KUL) in Belgium.

On the face of it, the idea that Escher's images can describe the layout of the universe seems to contradict what we know about it.

The images in question are tessellations, arrangements of repeated shapes, such as the images of interlocking bats and angels seen in Circle Limit IV. Although these are flat, they serve as "projections" of an alternative geometry called hyperbolic space, rather like a flat map of the world is a projection of a globe. For example, although the bats in the flat projection appear to shrink at an exponential rate at the edges, in hyperbolic space they are all the same size. These distortions in the projection arise because hyperbolic space cannot lie flat. Instead, it resembles a twisting, wiggly landscape of saddle-like hills.

That is not what our universe seems to look like. Measurements of the cosmic microwave background - the echo of the big bang - and distances to supernovae have revealed that our universe is flat, not twisted.

It is also expanding at an accelerating rate, because of a mysterious entity known as dark energy. We don't know what dark energy is or where it came from, but the mathematical language provided by Einstein's theory of general relativity has a way to describe this accelerated expansion. Sticking a constant - known as the cosmological constant - into the general-relativity equations keeps the universe expanding forever, but only if the constant has a positive sign. Until now, saying we live in an ever-expanding universe has been the same as saying our universe has a positive cosmological constant.

There are some outstanding problems, however. General relativity covers this aspect of the universe, but it can't describe the big bang. Nor can it unite gravity, which works on large scales, with quantum mechanics, which works on very small scales. "That means you cannot predict why we live in the universe that we live in," Hertog says.

String theory, in the meantime, offers a beautifully complete picture of the universe's history and connects gravity to quantum mechanics - but is most comfortable in a universe with a negatively curved, Escher-like geometry and with a negative cosmological constant.

This left physicists with a deep chasm to cross: on one side is a universe that works but lacks a complete theory, and on the other is a complete theory that doesn't describe the actual universe.

Now, Hawking, Hertog and James Hartle of the University of California, Santa Barbara, are proposing a bridge. They have found a way to produce expanding, accelerating universes using a negative cosmological constant. This means that string theory may, after all, describe the universe that we observe. The proposal grew from an idea that Hawking and Hartle had in the 1980s to get around general relativity's shortcomings by looking for a quantum picture of cosmology.

In quantum mechanics, a single equation called the wave function describes all the possible states that a quantum object can be in, and assigns each of them a certain probability. Hawking and Hartle sought a similar wave function that can generate the probability of various universes arising from the big bang. It would describe all the possible universes that could have been - including ones in which the solar system never formed, or in which life might have evolved very differently.

Over the past 30 years, Hawking and Hartle have been forcing a positive cosmological constant into their wave function, because that was considered necessary. But that meant sacrificing precision: they just couldn't get these universes to be anything more than clunky approximations of reality.

Tip the balanceString theorists had also been struggling with universes with positive cosmological constants, which tend to be unstable. Building them is a bit like trying to balance a pencil on its tip: it might work for a while, but the pencil's most energetically stable state is lying flat on the table, and eventually it will fall over. The most successful versions of string theory would rather live in the Escher-verse.

"String theory with a negative cosmological constant just goes much better," Hertog says.

But Hawking's latest work suggests that this supposed flaw may actually be the thing that knits string theory back to reality. In a paper posted online, Hawking and colleagues describe how they produced a plethora of universes from wave functions with negative cosmological constants, some of which are expanding and accelerating (arxiv.org/abs/1205.3807).

"Some of those universes are accelerating, just like our universe," Hertog says. "It turns out the quantum state includes both kinds of universes, automatically." For a certain wave function, these accelerating and expanding universes even turn out to be the most likely ones.

The key to this insight was recognising that the universes generated by the team's wave function could evolve to look a lot like a particular formulation of string theory, produced by Juan Maldacena of the Institute for Advanced Study in Princeton, New Jersey in 1997 (arxiv.org/abs/hep-th/9711200). "There was a mathematical connection, a very elegant connection," Hertog says.

Once they had spotted its connection to their wave function, Hawking's team decided to try to stitch the two together by writing a new wave function with a negative cosmological constant. They reasoned that this would allow them to borrow the beautifully complete mathematical picture of the universe provided by string theory and produce universes that accelerate outwards.

What about the observations suggesting that our universe is flat? Similar to how Newton's laws of motion work for everyday objects but give way to the more comprehensive laws of Einstein on cosmological scales, Hawking's team thinks that the universe's apparent flatness may describe it well as far as we can see but ultimately gives way to an underlying Escher-like geometry.

It is too soon to declare the universe solved. Maldacena says the Hawking team's model leaves out aspects of complete versions of string theory, such as provisions for the stability of some particles. "It would be wonderful if it was all we need to do," he says. "But I think it's too simplified. It's hard to see how it can be expanded to a more complete theory."

Hertog agrees that their work isn't finished - but thinks that the negative cosmological constant will eventually lead to a complete, description of the universe we observe. "It's an avenue that is opening up now," he says, "not something we have yet."

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