String theory, spatial expansion and the cosmological constant – the flatness problem revisited [1]
Whilst on the subject of of the cosmological constant, there are already a few conventional areas where string theory may yet be making a worthwhile contribution. We saw earlier that based on measurements of the cosmic background microwave radiation and, as a separate and distinct factor, measurements of the distance to supernova explosions, the universe is generally supposed to be flat (not two dimensional flat like a piece of paper, a tabletop or a pancake, but three dimensional flat, meaning that light travels in straight lines; parallel lines do not intersect and the rules of Euclidean geometry apply).
However, recent theoretical work by a team led by Stephen Hawking suggests that the universe may instead resembles a twisting, wiggly landscape of saddle-like hills. A useful model is the tessellated shapes in the woodcuts of the 1930’s Dutch artist M.C. Escher, which appear to shrink exponentially at the edges, whereas in hyperbolic space they are all the same size[2]. Another useful analogy is the distortion wrought by a flat map of the world such as Mercator’s projection as a projection of the globe.
Thanks to the equations of general relativity with their inbuilt positive cosmological constant and our understanding of dark energy, we also know that the universe is expanding. However, general relativity has nothing to say about the big bang, nor can it unite gravity, which works on large scales, with quantum mechanics, which works on very small scales, but string theory can and is most comfortable with a negatively curved, Escher-like geometry with a negative cosmological constant. Physicists were therefore left with a problem: on one side a universe that works, but lacks a complete theory, and on the other a complete theory that doesn’t describe the actual universe.
String theory with its negative cosmological constant may provide the answer. Hawking and his team profess to have found a way to produce expanding, accelerating universes using a negative cosmological constant by seeking a wave function[3] that can generate the probability of various universes arising from the big bang. This method has also been able to produce a plethora of universes from wave functions with negative cosmological constants, some of which are expanding and accelerating, just like ours, and for a certain wave function, these accelerating and expanding universes even turn out to be the most likely ones. This allowed the team to borrow the beautifully complete picture of the universe provided by string theory and produce universes that expanded outwards.
But how does this square with astronomical observations suggesting that our universe is in flat? The answer may be that just as Newton’s laws of motion work for large everyday objects but give way to the more comprehensive laws of Einstein on cosmological scales, so Hawking’s team thinks that the universe’s apparent flatness may describe it well as far as we can see but must ultimately give way to an underlying Escher-like geometry. The jury is still out, but some take heart from the fact that the concept of a negative cosmological constant may eventually lead to a complete description of the universe we observe, uniting general relativity and quantum mechanics in the process.
[1] For this segment, see “Welcome to the Escher-verse”, New Scientist, 9 June 2012, 8-9. Cf the segment on quantum gravity.
[2] Hyperbolic space is a type of non-Euclidean geometry with a negative curvature. This contrasts with Euclidean geometry, where parallel lines are a unique pair, and spherical geometry, where parallel lines do not exist since all lines (which are great circles) cross each other.
[3] Recall that in quantum terms a wave function describes all the possible states a quantum object can be in.
However, recent theoretical work by a team led by Stephen Hawking suggests that the universe may instead resembles a twisting, wiggly landscape of saddle-like hills. A useful model is the tessellated shapes in the woodcuts of the 1930’s Dutch artist M.C. Escher, which appear to shrink exponentially at the edges, whereas in hyperbolic space they are all the same size[2]. Another useful analogy is the distortion wrought by a flat map of the world such as Mercator’s projection as a projection of the globe.
Thanks to the equations of general relativity with their inbuilt positive cosmological constant and our understanding of dark energy, we also know that the universe is expanding. However, general relativity has nothing to say about the big bang, nor can it unite gravity, which works on large scales, with quantum mechanics, which works on very small scales, but string theory can and is most comfortable with a negatively curved, Escher-like geometry with a negative cosmological constant. Physicists were therefore left with a problem: on one side a universe that works, but lacks a complete theory, and on the other a complete theory that doesn’t describe the actual universe.
String theory with its negative cosmological constant may provide the answer. Hawking and his team profess to have found a way to produce expanding, accelerating universes using a negative cosmological constant by seeking a wave function[3] that can generate the probability of various universes arising from the big bang. This method has also been able to produce a plethora of universes from wave functions with negative cosmological constants, some of which are expanding and accelerating, just like ours, and for a certain wave function, these accelerating and expanding universes even turn out to be the most likely ones. This allowed the team to borrow the beautifully complete picture of the universe provided by string theory and produce universes that expanded outwards.
But how does this square with astronomical observations suggesting that our universe is in flat? The answer may be that just as Newton’s laws of motion work for large everyday objects but give way to the more comprehensive laws of Einstein on cosmological scales, so Hawking’s team thinks that the universe’s apparent flatness may describe it well as far as we can see but must ultimately give way to an underlying Escher-like geometry. The jury is still out, but some take heart from the fact that the concept of a negative cosmological constant may eventually lead to a complete description of the universe we observe, uniting general relativity and quantum mechanics in the process.
[1] For this segment, see “Welcome to the Escher-verse”, New Scientist, 9 June 2012, 8-9. Cf the segment on quantum gravity.
[2] Hyperbolic space is a type of non-Euclidean geometry with a negative curvature. This contrasts with Euclidean geometry, where parallel lines are a unique pair, and spherical geometry, where parallel lines do not exist since all lines (which are great circles) cross each other.
[3] Recall that in quantum terms a wave function describes all the possible states a quantum object can be in.