̽��ֱ�� of Cambridge - singularity

‘Saddle-shaped’ universe could undermine general relativity

sc604 — Mon, 22 May 2017 02:00:00 +0000

̽��ֱ��researchers, from the ̽��ֱ�� of Cambridge, have used computer simulations to predict the existence of a so-called naked singularity, which interferes with Einstein’s general theory of relativity. This is the first time that a naked singularity, which causes the laws of physics to break down, has been predicted in three-dimensional space. ̽��ֱ��findings are reported in the journal Physical Review Letters.

Einstein’s general theory of relativity underpins our current understanding of gravity: everything from the estimation of the age of the stars in the universe, to the GPS signals we rely on to help us navigate, is based on his equations. In part, the theory tells us that matter warps its surrounding spacetime, and what we call gravity is the effect of that warp. In the 100 years since it was published, general relativity has passed every test that has been thrown at it, but one of its limitations is the existence of singularities.

A singularity is a point where gravity is so intense that space, time, and the laws of physics, break down. General relativity predicts that singularities exist at the centre of black holes, and that they are surrounded by an event horizon – the ‘point of no return’, where the gravitational pull becomes so strong that escape is impossible, meaning that they cannot be observed from the outside.

For more than 40 years, mathematicians have proposed that whenever singularities form, they will always be hidden from view in this way – this is known as the ‘cosmic censorship conjecture.’ If true, cosmic censorship means that outside of black holes, these singularities have no measurable effect on anything, and the predictions of general relativity remain valid.

In recent years, researchers have used computer simulations to predict the existence of ‘naked singularities’ – that is, singularities which exist outside an event horizon. Naked singularities would invalidate the cosmic censorship conjecture and, by extension, general relativity’s ability to explain the universe as a standalone theory. However, all of these predictions have been modelled on universes which exist in higher dimensions. For example, in 2016, two Cambridge PhD students predicted the existence of a naked singularity, but their predictions were based on a five-dimensional universe.

̽��ֱ��new research, by Toby Crisford and Jorge Santos from Cambridge’s Department of Applied Mathematics and Theoretical Physics, has predicted the existence of a naked singularity in a four-dimensional universe - three spatial dimensions, plus time - for the first time.

Their predictions show that a naked singularity can form in a special kind of curved space known as anti-de Sitter space, in which the universe has a distinctive ‘saddle’ shape. According to general relativity, universes can have various shapes, and anti-de Sitter space is one of these possible shapes.

Anti-de Sitter space has a very different structure to flat space. In particular it has a boundary which light can reach, at which point it is reflected back. “It’s a bit like having a spacetime in a box,” said Crisford. “At the boundary, the walls of the box, we have the freedom to specify what the various fields are doing, and we use this freedom to add energy to the system and eventually force the formation of a singularity.”

While the results are not directly applicable to our universe, as ‘forcing’ a singularity is not a procedure which is possible to simulate in flat space, they do open up new opportunities to study other theories to understand the universe. One such theory could involve quantum gravity, which provides new equations close to a singularity.

“ ̽��ֱ��naked singularity we see is likely to disappear if we were to include charged particles in our simulation – this is something we are currently investigating,” said Santos. “If true, it could imply a connection between the cosmic censorship conjecture and the weak gravity conjecture, which says that any consistent theory of quantum gravity must contain sufficiently charged particles. In anti-de Sitter space, the cosmic censorship conjecture might be saved by the weak gravity conjecture.”

Inset image: Image of (1 + 1)-dimensional anti-de Sitter space embedded in flat (1 + 2)-dimensional space. Credit: Wikimedia Commons.

Reference:
Toby Crisford and Jorge E. Santos. 'Violating the Weak Cosmic Censorship Conjecture in Four-Dimensional Anti–de Sitter Space.' Physical Review Letters (2017). DOI: 10.1103/PhysRevLett.118.181101.

Researchers have shown how singularities – which are normally only found at the centre of black holes and hidden from view – could exist in highly curved three-dimensional space.

It’s a bit like having spacetime in a box.

Toby Crisford

NASA/JPL-Caltech

Artist's concept of a supermassive black hole

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A new twist on soap films

sc604 — Fri, 23 May 2014 15:13:03 +0000

̽��ֱ��way in which soap films collapse and re-form when twisted or stretched could hold the key to predicting the formation and location of mathematical singularities, which can be seen in the motion of solar flares and other natural phenomena.

Research on the processes by which soap films undergo transitions from one stable state to another has led to conjectures on the nature and location of the singular events that occur during the change of form, connecting two previously separate areas in mathematics.

In mathematics, singularities occur when an equation or surface breaks down and ‘explodes’. In surfaces such as soap films, singularities occur when the surface collides with itself, changing shape in the blink of an eye.

Researchers from the ̽��ֱ�� of Cambridge have shown that identifying a special type of curve on the surface can help predict where these singularities are likely to occur in soap films, which could in turn aid in the understanding of singularities in the natural world. ̽��ֱ��results are published in the journal Proceedings of the National Academy of Sciences (PNAS).

We are all familiar with the simplest soap films, which are formed by dipping a wire loop into a soap solution: the flat surface that spans the wire and the bubbles which are formed when we blow on the film. With suitably shaped wires however, much more complex structures can be formed, such as Möbius strips.

All static soap films are ‘minimal surfaces’, for they have the least area of all possible surfaces that span a given wire frame.

What is less understood are the dynamic processes which occur when a minimal surface like a soap film is made unstable by deforming the supporting wire. ̽��ֱ��film typically moves in a fraction of a second to a new configuration through a singular point, at which the surface collides with itself and changes its connectivity.

These kinds of violent events also occur in the natural world – in fluid turbulence and in the motion of solar flares emanating from the sun – and one of the great challenges has been to predict where they will occur.

In research supported by the EPSRC, a team from the Department of Applied Mathematics and Theoretical Physics attempted to understand how to predict where the singularity will occur when soap films are twisted or stretched to a point of instability. For example, it is well-known that the surface spanning two separate wire loops will collapse to a singularity in between the loops.

In previous work, the group had shown that Möbius strip singularity occurs not between the loops but at the wire frame, where there is a complex rearrangement of the surface. “What was unclear was whether there was an underlying mathematical principle by which this striking difference could be explained,” said Professor Raymond Goldstein, who collaborated with Dr Adriana Pesci, Professor Keith Moffatt, and James McTavish, a maths undergraduate, on the research.

̽��ֱ��team recognised that a geometric concept known as a systole might be the key to understanding where singularities will occur. A systole is the length of the shortest closed curve on surface that cannot be shrunk to a point while remaining on the surface. An example of this is found on a bagel, where the shortest such curve encircles the bagel like a handle. Mathematicians have studied the geometric properties of these curves in recent decades, establishing constraints on the relationship between the length of a systole and the area of the surface on which they lie.

Using new laboratory experiments and computations, the researchers found evidence that the ultimate location of the singularities that occur when soap films collapse can be deduced from the properties of the systole. If the systolic curve loops around the wire frame, then the singularity occurs at the boundary, while if there is no such linking the singularity occurs in the bulk.

“This is an example of experimental mathematics, in the sense that we are using laboratory studies to inform conjectures on mathematical connections,” said Professor Goldstein. “While they are certainly not rigorous, we hope they will stimulate further research into this new, developing area.”

Soap films with complex shapes shed light on the formation of mathematical singularities, which occur in a broad range of fields.

This is an example of experimental mathematics, in the sense that we are using laboratory studies to inform conjectures on mathematical connections

Raymond Goldstein

Singularity in a soap film

Raymond Goldstein

Soap film singularity

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