What is a Singularity?

Posted in Education, Maynooth, The Universe and Stuff with tags , , , , , , on November 24, 2022 by telescoper

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:

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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.

R.I.P. Jim Bardeen (1939-2022)

Posted in Biographical, The Universe and Stuff with tags , , , , , on July 4, 2022 by telescoper

I was saddened this morning to hear news of the death at the age of 83 of Jim Bardeen who passed away on June 20th 2022. Jim – the son of John Bardeen, who won two Nobel physics prizes – did important work in theoretical cosmology and general relativity. In my own field of cosmology he is probably best known for his work on perturbation theory where he clarified many longstanding issues about gauge-dependence and as the first author of the famous and heavily cited “BBKS” (Bardeen, Bond, Kaiser & Szalay) paper, published in 1986:

I received this as a very hefty preprint when I started my graduate studies back in 1985 and it scared the hell out of me. I still have the photocopy of the published version I made when it came out (in the days when PhD meant Doctor of Photocopying). You can find the paper on the NASA/ADS system here.

I met Jim Bardeen only once, at an Aspen Summer Workshop back in the 90s. He was a very shy and modest man but very kindly and polite. I remember a couple of times out hiking with him, when a discussion about physics was going on he would keep quiet until he had figured out what he thought and when he did choose to speak it was usually brief and invariably very incisive. He didn’t write all that many papers either, but those he did publish were invariably excellent.

Rest in peace, James Maxwell Bardeen (1939-2022)

New Publication at the Open Journal of Astrophysics

Posted in OJAp Papers, Open Access, The Universe and Stuff with tags , , , , , , on September 16, 2021 by telescoper

Time to announce another publication in the Open Journal of Astrophysics. This one is the ninth paper in Volume 4 (2021) and the 40th in all.

The latest publication is entitled Black Hole Shadow Drift and Photon Ring Frequency Drift. The authors are Emmanuel Frion (Helsinki), Leonardo Giani (Queensland) and  Tays Miranda (Jyväskylä).

Here is a screen grab of the overlay which includes the abstract:

You can click on the image to make it larger should you wish to do so. You can find the arXiv version of the paper here. This one is also in the folder marked Cosmology and Nongalactic Astrophysics; although primarily in general relativity and quantum cosmology (gr-qc) it is cross-listed in astro-ph so it eligible for publication with us.

The end of the summer has been heralded by the arrival at OJAp HQ of a number of revised versions so I expect to be publishing a few more papers in the next few weeks!

Cosmological Non-Linearities as an Effective Fluid

Posted in The Universe and Stuff with tags , , , , , , , on November 27, 2020 by telescoper

We know our Universe is inhomogeneous, comprising regions of high density (galaxies and clusters of galaxies) as well as regions of much lower density (e.g. cosmic voids). Our standard cosmological models are based on exact solutions of Einstein’s equations of general relativity that assume homogeneity and isotropy. The general assumption is that if we confine ourselves to large enough scales the effect of the clumpiness of matter can either be disregarded or treated using perturbation theory. As far as we can tell, that approach works reasonably well but we know it must fail on smaller scales where the structure is in the non-linear regiome where it can’t be described accurately using perturbation theory because the fluctuations are so large.

From time to time I’ve idly wondered whether it might be possible to understand the effect of these non-linearities in general relativity by treating them as a kind of fluid with an energy-momentum tensor that acts as a correction to that of the perfect fluid form of the background cosmological model. This would have to be done via some sort of averaging so it would be an effective, coarse-grained description rather than an exact treatment. It is clear though that non-linearities would generate departures from the perfect fluid form, particularly resulting in off-diagonal terms in the energy-momentum tensor corresponding to anisotropic stresses (e.g. viscosity terms).

Anyway, a recent exchange on Twitter relating to a new paper that has just appeared revealed that far cleverer people than me had looked at this in quite a lot of detail a decade ago:

You can find the full paper here.

There are quite a lot of subtleties in this – how to do the spatial averaging, how to do the time-slicing, etc – which I don’t fully understand but at least I’m reassured that it isn’t a daft idea to try thinking of things this way!

P.S. The relativistic simulations reported in this paper could in principle be used to estimate the parameters mentioned in the abstract above, if that hasn’t been done before!

Lights all askew in the Heavens – the 1919 Eclipse Expeditions (Updated)

Posted in History, Talks and Reviews, The Universe and Stuff with tags , , , , , on June 3, 2019 by telescoper

Here is a video of my talk at the Open Meeting of the Royal Astronomical Society on April 12 2019. Was it really so long ago?

You can find the slides here:

The 1919 Eclipse: That Was The Talk That Was…

Posted in History, The Universe and Stuff with tags , , , , , on May 29, 2019 by telescoper

Well, I did my talk this afternoon to mark the centenary of the 1919 Eclipse Experiment that was performed on May 29th 1919. It’s a good job we changed the venue to a bigger lecture theatre than originally booked because even the new one was full! Thanks to everyone who came, and I hope you enjoyed the talk!

Anyway, here are the slides if you’d like to see them:

Here is a picture of me about to start:

Now that the centenary has passed I promise to post a bit less about this topic, although there are still a few things coming up that I might mention…

The Centenary of the 1919 Eclipse Expeditions

Posted in History, The Universe and Stuff with tags , , , , on May 29, 2019 by telescoper

Well, the big day has arrived. Today, 29th May 2019, is the centenary of the 1919 Solar Eclipse during which an experiment was carried out to test Einstein’s theory of general relativity. I’m giving a public talk this afternoon and will post the slides afterwards.

In the meantime, however, I’ll just re-post his little piece which is based on an article I wrote some years ago for Firstscience.

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The Eclipse that Changed the Universe

A total eclipse of the Sun is a moment of magic: a scant few minutes when our perceptions of the whole Universe are turned on their heads. The Sun’s blinding disc is replaced by ghostly pale tentacles surrounding a black heart – an eerie experience witnessed by hundreds of millions of people throughout Europe and the Near East last August.

But one particular eclipse of the Sun, eighty years ago, challenged not only people’s emotional world. It was set to turn the science of the Universe on its head. For over two centuries, scientists had believed Sir Isaac Newton’s view of the Universe. Now his ideas had been challenged by a young German-Swiss scientist, called Albert Einstein. The showdown – Newton vs Einstein – would be the total eclipse of 29 May 1919.

Newton’s position was set out in his monumental Philosophiae Naturalis Principia Mathematica, published in 1687. The Principia – as it’s familiarly known – laid down a set of mathematical laws that described all forms of motion in the Universe. These rules applied as much to the motion of planets around the Sun as to more mundane objects like apples falling from trees.

At the heart of Newton’s concept of the Universe were his ideas about space and time. Space was inflexible, laid out in a way that had been described by the ancient Greek mathematician Euclid in his laws of geometry. To Newton, space was the immovable and unyielding stage on which bodies acted out their motions. Time was also absolute, ticking away inexorably at the same rate for everyone in the Universe.

Sir Isaac Newton, painted by Sir Godfrey Kneller. Picture Credit: National Portrait Gallery,

For over 200 years, scientists saw the Cosmos through Newton’s eyes. It was a vast clockwork machine, evolving by predetermined rules through regular space, against the beat of an absolute clock. This edifice totally dominated scientific thought, until it was challenged by Albert Einstein.

In 1905, Einstein dispensed with Newton’s absolute nature of space and time. Although born in Germany, during this period of his life he was working as a patent clerk in Berne, Switzerland. He encapsulated his new ideas on motion, space and time in his special theory of relativity. But it took another ten years for Einstein to work out the full consequences of his ideas, including gravity. The general theory of relativity, first aired in 1915, was as complete a description of motion as Newton had prescribed in his Principia. But Einstein’s description of gravity required space to be curved. Whereas for Newton space was an inflexible backdrop, for Einstein it had to bend and flex near massive bodies. This warping of space, in turn, would be responsible for guiding objects such as planets along their orbits.

Albert Einstein (left), pictured with Arthur Stanley Eddington (right). Picture Credit: Royal Greenwich Observatory.

By the time he developed his general theory, Einstein was back in Germany, working in Berlin. But a copy of his general theory of relativity was soon smuggled through war-torn Europe to Cambridge. There it was read by Arthur Stanley Eddington, Britain’s leading astrophysicist. Eddington realised that Einstein’s theory could be tested. If space really was distorted by gravity, then light passing through it would not travel in a straight line, but would follow a curved path. The stronger the force of gravity, the more the light would be bent. The bending would be largest for light passing very close to a very massive body, such as the Sun.

Unfortunately, the most massive objects known to astronomers at the time were also very bright. This was before black holes were seriously considered, and stars provided the strongest gravitational fields known. The Sun was particularly useful, being a star right on our doorstep. But it is impossible to see how the light from faint background stars might be bent by the Sun’s gravity, because the Sun’s light is so bright it completely swamps the light from objects beyond it.

A scientific sketch of the path of totality for the 1919 eclipse. Picture Credit: Royal Greenwich Observatory.

Eddington realised the solution. Observe during a total eclipse, when the Sun’s light is blotted out for a few minutes, and you can see distant stars that appear close to the Sun in the sky. If Einstein was right, the Sun’s gravity would shift these stars to slightly different positions, compared to where they are seen in the night sky at other times of the year when the Sun far away from them. The closer the star appears to the Sun during totality, the bigger the shift would be.

Eddington began to put pressure on the British scientific establishment to organise an experiment. The Astronomer Royal of the time, Sir Frank Watson Dyson, realised that the 1919 eclipse was ideal. Not only was totality unusually long (around six minutes, compared with the two minutes we experienced in 1999) but during totality the Sun would be right in front of the Hyades, a cluster of bright stars.

But at this point the story took a twist. Eddington was a Quaker and, as such, a pacifist. In 1917, after disastrous losses during the Somme offensive, the British government introduced conscription to the armed forces. Eddington refused the draft and was threatened with imprisonment. In the end, Dyson’s intervention was crucial persuading the government to spare Eddington. His conscription was postponed under the condition that, if the war had finished by 1919, Eddington himself would lead an expedition to measure the bending of light by the Sun. The rest, as they say, is history.

The path of totality of the 1919 eclipse passed from northern Brazil, across the Atlantic Ocean to West Africa. In case of bad weather (amongst other reasons) two expeditions were organised: one to Sobral, in Brazil, and the other to the island of Principe, in the Gulf of Guinea close to the West African coast. Eddington himself went to Principe; the expedition to Sobral was led by Andrew Crommelin from the Royal Observatory at Greenwich.

British scientists in the field at their observing site in Sobral in 1919. Picture Credit: Royal Greenwich Observatory

The expeditions did not go entirely according to plan. When the day of the eclipse (29 May) dawned on Principe, Eddington was greeted with a thunderstorm and torrential rain. By mid-afternoon the skies had partly cleared and he took some pictures through cloud.

Meanwhile, at Sobral, Crommelin had much better weather – but he had made serious errors in setting up his equipment. He focused his main telescope the night before the eclipse, but did not allow for the distortions that would take place as the temperature climbed during the day. Luckily, he had taken a backup telescope along, and this in the end provided the best results of all.

After the eclipse, Eddington himself carefully measured the positions of the stars that appeared near the Sun’s eclipsed image, on the photographic plates exposed at both Sobral and Principe. He then compared them with reference positions taken previously when the Hyades were visible in the night sky. The measurements had to be incredibly accurate, not only because the expected deflections were small. The images of the stars were also quite blurred, because of problems with the telescopes and because they were seen through the light of the Sun’s glowing atmosphere, the solar corona.

Before long the results were ready. Britain’s premier scientific body, the Royal Society, called a special meeting in London on 6 November. Dyson, as Astronomer Royal took the floor, and announced that the measurements did not support Newton’s long-accepted theory of gravity. Instead, they agreed with the predictions of Einstein’s new theory.

The final proof: the small red line shows how far the position of the star has been shifted by the Sun’s gravity. Each star experiences a tiny deflection, but averaged over many exposures the results definitely support Einstein’s theory. Picture Credit: Royal Greenwich Observatory.

The press reaction was extraordinary. Einstein was immediately propelled onto the front pages of the world’s media and, almost overnight, became a household name. There was more to this than purely the scientific content of his theory. After years of war, the public embraced a moment that moved mankind from the horrors of destruction to the sublimity of the human mind laying bare the secrets of the Cosmos. The two pacifists in the limelight – the British Eddington and the German-born Einstein – were particularly pleased at the reconciliation between their nations brought about by the results.

But the popular perception of the eclipse results differed quite significantly from the way they were viewed in the scientific establishment. Physicists of the day were justifiably cautious. Eddington had needed to make significant corrections to some of the measurements, for various technical reasons, and in the end decided to leave some of the Sobral data out of the calculation entirely. Many scientists were suspicious that he had cooked the books. Although the suspicion lingered for years in some quarters, in the end the results were confirmed at eclipse after eclipse with higher and higher precision.

In this cosmic ‘gravitational lens,’ a huge cluster of galaxies distorts the light from more distant galaxies into a pattern of giant arcs. Picture Credit: NASA

Nowadays astronomers are so confident of Einstein’s theory that they rely on the bending of light by gravity to make telescopes almost as big as the Universe. When the conditions are right, gravity can shift an object’s position by far more than a microscopic amount. The ideal situation is when we look far out into space, and centre our view not on an individual star like the Sun, but on a cluster of hundreds of galaxies – with a total mass of perhaps 100 million million suns. The space-curvature of this immense ‘gravitational lens’ can gather the light from more remote objects, and focus them into brilliant curved arcs in the sky. From the size of the arcs, astronomers can ‘weigh’ the cluster of galaxies.

Einstein didn’t live long enough to see through a gravitational lens, but if he had he would definitely have approved….

Statistical Analysis of the 1919 Eclipse Measurements

Posted in Bad Statistics, The Universe and Stuff with tags , , , , on May 27, 2019 by telescoper

So the centenary of the famous 1919 Eclipse measurements is only a couple of days away and to mark it I have a piece on RTÉ Brainstorm published today in advance of my public lecture on Wednesday.

I thought I’d complement the more popular piece by posting a very short summary of how the measurements were analyzed for those who want a bit more technical detail.

The idea is simple. Take a photograph during a solar eclipse during which some stars are visible in the sky close enough to the Sun to be deflected by its gravity. Take a similar photograph of the same stars at night at some other time when the Sun is elsewhere. Compare the positions of the stars on the two photographs and the star positions should have shifted slightly on the eclipse plates compared to the comparison plate. This gravitational shift should be radially outwards from the centre of the Sun.

One can measure the coordinates of the stars in two directions: Right Ascension (x) and Declination (y) and the corresponding (small) difference between the positions in each direction are Dx and Dy on the right hand side of the equations above.

In the absence of any other effects these deflections should be equal to the deflection in each component calculated using Einstein’s theory or Newtonian value. This is represented by the two terms Ex(x,y) and Ey(x,y) which give the calculated components of the deflection in both x and y directions scaled by a parameter α which is the object of interest – α should be precisely a factor two larger in Einstein’s theory than in the `Newtonian’ calculation.

The problem is that there are several other things that can cause differences between positions of stars on the photographic plate, especially if you remember that the eclipse photographs have to be taken out in the field rather than at an observatory.  First of all there might be an offset in the coordinates measured on the two plates: this is represented by the terms c and f in the equations above. Second there might be a slightly different magnification on the two photographs caused by different optical performance when the two plates were exposed. These would result in a uniform scaling in x and y which is distinguishable from the gravitational deflection because it is not radially outwards from the centre of the Sun. This scale factor is represented by the terms a and e. Third, and finally, the plates might be oriented slightly differently, mixing up x and y as represented by the cross-terms b and d.

Before one can determine a value for α from a set of measured deflections one must estimate and remove the other terms represented by the parameters a-f. There are seven unknowns (including α) so one needs at least seven measurements to get the necessary astrometric solution.

The approach Eddington wanted to use to solve this problem involved setting up simultaneous equations for these parameters and eliminating variables to yield values for α for each plate. Repeating this over many allows one to beat down the measurement errors by averaging and return a final overall value for α. The 1919 eclipse was particularly suitable for this experiment because (a) there were many bright stars positioned close to the Sun on the sky during totality and (b) the duration of totality was rather long – around 7 minutes – allowing many exposures to be taken.

This was indeed the approach he did use to analyze the data from the Sobral plates, but tor the plates taken at Principe during poor weather he didn’t have enough star positions to do this: he therefore used estimates of the scale parameters (a and e) taken entirely from the comparison plates. This is by no means ideal, though he didn’t really have any choice.

If you ask me a conceptually better approach would be the Bayesian one: set up priors on the seven parameters then marginalize over a-f  to leave a posterior distribution on α. This task is left as an exercise to the reader.

Revolution in the Skies: The Experiment that made Einstein Famous

Posted in History, The Universe and Stuff with tags , , on May 14, 2019 by telescoper

At the risk of being a complete bore about the 1919 Eclipse Expeditions, here is a plug for a public talk I am giving in Maynooth on 29 May 2019, the centenary of the event itself.

Here is the blurb:

Albert Einstein is the undisputed genius whose insights have revolutionised the way we think about the Universe. He is also a cultural icon whose fame extends far beyond the realm of theoretical physics.

Einstein’s transition to global stardom can be dated precisely to 29th May 1919, the date of a total solar eclipse at which the first measurements were made of the bending of light by the Sun’s gravity that tested Einstein’s then new general theory of relativity. The announcement of the results created an unprecedented media sensation: news of Einstein and his revolutionary theory made front-page news around the world.

To mark the centenary of this historic event, Peter Coles will describe the historical and scientific background to an experiment that changed the world, and explain why it was such an important event both for Einstein the physicist and Einstein the celebrity.

The event will be on the North Campus of Maynooth University. It is free, but please register at the Eventbrite site here if you want to attend so we can get an idea of numbers. If, for some reason, you can’t get to Maynooth, we are planning to do a live feed of the talk too, so please watch this blog for more details.

The Observer and the Eclipse

Posted in Books, Talks and Reviews, History, The Universe and Stuff with tags , , on May 12, 2019 by telescoper

Not surprisingly, given that the centenary is fast approaching, pieces are appearing in the mainstream media about the 1919 Eclipse Expeditions that first measured the deflection of light by the Sun’s gravitational field. One such article, by Robin McKie, appears in today’s Observer. It’s a nice piece, though it concentrates almost entirely on Eddington’s measurements taken at Principe. In fact it was Crommelin’s measurements from Sobral that proved decisive.

Anyway, the article gives me a (very brief) mention courtesy of the piece I wrote in Nature a few weeks ago:

For many years at Cardiff I ran an undergraduate project in which the students had to reanalyze the measurements from the eclipse expeditions. That is possible because all the necessary star positions are tabulated in the paper by Dyson et al. (1920). It is undoubtedly the case that Eddington had to improvise a bit because of the unexpected problems that arose in the field, but this is actually quite normal. As a famous general put it `No plan of battle survives first contact with the enemy’. I remain convinced that Eddington didn’t do anything dodgy, but you don’t have to take my word for it: if you don’t believe me then go ahead and look at the data yourself! At the very least you will then understand what a difficult experiment this was!