## What is a Singularity?

Following last week’s Maynooth Astrophysics and Cosmology Masterclass, a student asked (in the context of the Big Bang or a black hole) what a singularity is. I thought I’d share my response here in case anyone else was wondering. The following is what I wrote back to my correspondent:

–oo–

In general, a singularity is pathological mathematical situation wherein the value of a particular variable becomes infinite. To give a very simple example, consider the calculation of the Newtonian force due  to gravity exerted by a massive body on a test particle at a distance r. This force is proportional to 1/r2,, so that if one tried to calculate the force for objects at zero separation (r=0), the result would be infinite.

Singularities are not always  signs of serious mathematical problems. Sometimes they are simply caused by an inappropriate choice of coordinates. For example, something strange and akin to a singularity happens in the standard maps one finds in an atlas. These maps look quite sensible until one looks very near the poles.  In a standard equatorial projection,  the North Pole does not appear as a point, as it should, but is spread along straight line along the top of the map. But if you were to travel to the North Pole you would not see anything strange or catastrophic there. The singularity that causes this point to appear is an example of a coordinate singularity, and it can be transformed away by using a different projection.

More serious singularities occur with depressing regularity in solutions of the equations of general relativity. Some of these are coordinate singularities like the one discussed above and are not particularly serious. However, Einstein’s theory is special in that it predicts the existence of real singularities where real physical quantities (such as the matter density) become infinite. The curvature of space-time can also become infinite in certain situations.

Probably the most famous example of a singularity lies at the core of a black hole. This appears in the original Schwarzschild interior solution corresponding to an object with perfect spherical symmetry. For many years, physicists thought that the existence of a singularity of this kind was merely due to the special and rather artificial nature of the exactly spherical solution. However, a series of mathematical investigations, culminating in the singularity theorems of Penrose, showed no special symmetry is required and that singularities arise in the generic gravitational collapse problem.

As if to apologize for predicting these singularities in the first place, general relativity does its best to hide them from us. A Schwarzschild black hole is surrounded by an event horizon that effectively protects outside observers from the singularity itself. It seems likely that all singularities in general relativity are protected in this way, and so-called naked singularities are not thought to be physically realistic.

There is also a singularity at the very beginning in the standard Big Bang theory. This again is expected to be a real singularity where the temperature and density become infinite. In this respect the Big Bang can be thought of as a kind of time-reverse of the gravitational collapse that forms a black hole. As was the case with the Schwarzschild solution, many physicists thought that the initial cosmologcal singularity could be a consequence of the special symmetry required by the Cosmological Principle. But this is now known not to be the case. Hawking and Penrose generalized Penrose’s original black hole theorems to show that a singularity invariably exists in the past of an expanding Universe in which certain very general conditions apply.

So is it possible to avoid this singularity? And if so, how?

It is clear that the initial cosmological singularity might well just be a consequence of extrapolating deductions based on the classical ttheory of general relativity into a situation where this theory is no longer valid.  Indeed, Einstein himself wrote:

The theory is based on a separation of the concepts of the gravitational field and matter. While this may be a valid approximation for weak fields, it may presumably be quite inadequate for very high densities of matter. One may not therefore assume the validity of the equations for very high densities and it is just possible that in a unified theory there would be no such singularity.

Einstein, A., 1950. The Meaning of Relativity, 3rd Edition, Princeton University Press.

We need new laws of physics to describe the behaviour of matter in the vicinity of the Big Bang, when the density and temperature are much higher than can be achieved in laboratory experiments. In particular, any theory of matter under such extreme conditions must take account of  quantum effects on a cosmological scale. The name given to the theory of gravity that replaces general relativity at ultra-high energies by taking these effects into account is quantum gravity, but no such theory has yet been constructed.

There are, however, ways of avoiding the initial singularity in classical general relativity without appealing to quantum effects. First, one can propose an equation of state for matter in the very early Universe that does not obey the conditions laid down by Hawking and Penrose. The most important of these conditions is called the strong energy condition: that r+3p/c2>0 where r is the matter density and p is the pressure. There are various ways in which this condition might indeed be violated. In particular, it is violated by a scalar field when its evolution is dominated by its vacuum energy, which is the condition necessary for driving inflationary Universe models into an accelerated expansion.  The vacuum energy of the scalar field may be regarded as an effective cosmological constant; models in which the cosmological constant is included generally have a bounce rather than a singularity: running the clock back, the Universe reaches a minimum size and then expands again.

Whether the singularity is avoidable or not remains an open question, and the issue of whether we can describe the very earliest phases of the Big Bang, before the Planck time, will remain open at least until a complete  theory of quantum gravity is constructed.

### 4 Responses to “What is a Singularity?”

1. Superb article.
I have that Einstein quote as 1945, in an appendix to the third edition of The Meaning of Relativity. Either way, it’s clear he was still thinking long and hard about relativistic cosmology and the limits of what GR had to say about the universe in his later years. In the same article, AE is greatly perturbed by the timespan problem – I often wonder if he learnt of Baade’s corrections to the galaxial distance scale before he died..

• telescoper Says:

I didn’t have the books with me when I wrote this. I gave the date 1950 because that is the date of the 3rd Edition, but as you say the quote is a bit earlier.

2. Jarle Brinchmann Says:

That is a really nice summary for the student, I hope they are inspired by it!

It reminded me of one my favourite book titles, Earman’s “Bangs, Crunches, Whimpers, and Shrieks – Singularities and Acausalities in Relativistic Spacetimes”, the book is also a great read (or so I seem to recall, admittedly it is a long time ago since I looked at it 🙂 and might be interesting to the student as it is freely available as a (somewhat badly) scanned copy: https://sites.pitt.edu/~jearman/Earman_1995BangsCrunches.pdf which also

3. What about the “Borde–Guth–Vilenkin theorem”?

I’ve seen there had been some controversy around it in the past, but any other references/comments would be welcome! 🙂