The other day I mentioned the forthcoming graduation of a Maynooth PhD student. His name is Aonghus Hunter-McCabe and his main supervisor was Maynooth colleague Brian Dolan, and I just took over when Brian retired to see Aognhus through the latter stages. Anyway, asof yesterday, his thesis is available on arXiv (on hep-th) as well as on the Maynooth University Research Archive Library (MURA) here, so as it is all in the public domain I thought I would advertise it here, as I think it is very good indeed (though I would say that!) and also in case anyone out there is looking to employ a PDRA in a related area…
The abstract is:
This thesis explores the application of differential geometric and general relativistic techniques to deepen our understanding of quantum mechanical systems. We focus on three systems, employing these mathematical frameworks to uncover subtle features within each. First, we examine Unruh radiation in the context of an accelerated two-state atom, determining transition frequencies for a variety of accelerated trajectories via first-order perturbation theory. For harmonic motion of the atom in a vacuum, we derive transition rates with potential experimental realizations. Next, we investigate the quantum Hall effect in a spherical geometry using the Dirac operator for non-interacting fermions in a background magnetic field generated by a Wu-Yang monopole. The Atiyah-Singer index theorem constrains the degeneracy of the ground state, and the fractional quantum Hall effect is studied using the composite fermion model, where Dirac strings associated with the monopole field supply the statistical gauge field vortices. A unique, gapped ground state emerges, yielding fractions of the form ν=1/(2k+1) for large particle numbers. Finally, we examine the AdS/CMT correspondence through a bulk fermionic field in an RN-AdS4 background (with a U(1) gauge field), dual to a boundary fermionic operator. Spherical and planar event horizon geometries are discussed, with the temperature of the RN black hole identified with that of the dual system on the boundary. By numerically solving for the spectral functions of the dual theory, for a spherical event horizon at zero temperature, we identify a shift in the Fermi surface from that which arises in the planar case. Preliminary evidence of a phase transition emerges upon examining these spectral functions, again for the spherical horizon, at non-zero temperature.
The story of the famous 1919 expeditions to measure the bending of light by the Sun as a test of general relativity has featured many times on this blog (e.g. here). I ahve also written elsewhere about it, e.g. here. One way this is often presented is whether the measurements preferred the “Einstein” prediction or one consistent with “Newton”, there being a famous factor of two between the two.
In fact the earliest published calculation of the deflection of light by the Sun was not by Isaac Newton but by Johann Georg von Soldner (Uber die Ablenkung eines Lichstrals von seiner geradlinigen Bewegung, durch die Attraktion eines Weltk¨orpers, and welchem er nahe vorbei geht. Berliner Astronomisches Jahrbuch, 1801: 161-172). This calculation does use Newtonian theory, but as far as I know there is no record tof any calculation of this sort by Newton himself.
There is, however, something very tantalizing in Newton’s 1704 book Opticks, published almost 20 years after his Principia outlined the laws of mechanics and of universal gravitation. Opticks which (unlike the Principia) was written in English, ends with a series of rhetorical questions called “Queries” which present speculative ideas about light and its interactions with matter. The first of these reads:
Query 1. Do not Bodies act upon Light at a distance, and by their action bend its Rays; and is not this action (caeteris paribus) strongest at the least distance?
This looks very much like a speculation about the bending of light by gravitation. But if that’s what it is, he could have done exactly what Soldner did about a century later. Why then did he never publish the result and why was it never found among his unpublished papers?
I’ve spoken to several people about this and there are three main ideas. One is that Newton actual did the Soldner calculation, and that the manuscript was accidentally destroyed in a fire caused by his dog, Diamond. The other is that he just never got round to it, which seems unlikely because it’s not a difficult calculation and Newton lived over 20 years after the publication of the Opticks. The third possibility is that Query 1 wasn’t about gravity at all. If it had been, wouldn’t he have used the word and wouldn’t he have mentioned the inverse-square law specifically? Perhaps what he had in mind was some kind of refraction. This interpretation is consistent with other Queries where he talks about the “aetherial Medium” through which he supposed light to propagate being distorted by the presence of massive bodies and thus causing refraction. For example, from Query 21,
Is not this Medium much rarer within the dense Bodies of the Sun, Stars, Planets and Comets, than in the empty celestial Spaces between them?
I suppose we’ll never know what Newton had in mind. I am split between the first and third explanations above.
It’s worth mentioning that some of the other Queries are very prescient. Take Query 5, for example:
Do not Bodies and Light act mutually upon one another; that is to say, Bodies upon Light in emitting, reflecting, refracting and inflecting it, and Light upon Bodies for heating them, and putting their parts into a vibrating motion wherein heat consists?
Looking at the title of this paper you might be tempted to dismiss it on the grounds that warp drives are the stuff of science fiction (which they are), but this paper is really a rigorous technical study of the dynamical evolution and stability of spacetimes that violate the null energy condition, inspired by the idea of a warp drive. As soon as I announced this paper on social media it started to get attention. That will probably increase because there is now a press release to accompany the paper. I’ve taken the liberty of reproducing the text of the press release here:
–o–
Imagine a spaceship driven not by engines, but by compressing the spacetime in front of it. That’s the realm of science fiction, right? Well, not entirely. Physicists have been exploring the theoretical possibility of “warp drives” for decades, and a new study published in the Open Journal of Astrophysics takes things a step further – simulating the gravitational waves such a drive might emit if it broke down.
Warp drives are staples of science fiction, and in principle could propel spaceships faster than the speed of light. Unfortunately, there are many problems with constructing them in practice, such as the requirement for an exotic type of matter with negative energy. Other issues with the warp drive metric include the potential to use it to create closed time-like curves that violate causality and, from a more practical perspective, the difficulties for those in the ship in actually controlling and deactivating the bubble.
This new research is the result of a collaboration between specialists in gravitational physics at Queen Mary University of London, the University of Potsdam, the Max Planck Institute (MPI) for Gravitational Physics in Potsdam and Cardiff University. Whilst it doesn’t claim to have cracked the warp drive code, it explores the theoretical consequences of a warp drive “containment failure” using numerical simulations.
Dr Katy Clough of Queen Mary University of London, the first author of the study explains: “Even though warp drives are purely theoretical, they have a well-defined description in Einstein’s theory of General Relativity, and so numerical simulations allow us to explore the impact they might have on spacetime in the form of gravitational waves.”
Co-author Dr Sebastian Khan, from Cardiff University’s School of Physics and Astronomy, adds: “Miguel Alcubierre created the first warp drive solution during his PhD at Cardiff University in 1994, and subsequently worked at the MPI in Potsdam. So it’s only natural that we carry on the tradition of warp drive research in the era of gravitational wave astronomy .”
The results are fascinating. The collapsing warp drive generates a distinct burst of gravitational waves, a ripple in spacetime that could be detectable by gravitational wave detectors that normally target black hole and neutron star mergers. Unlike the chirps from merging astrophysical objects, this signal would be a short, high-frequency burst, and so current detectors wouldn’t pick it up. However, future higher-frequency instruments might, and although no such instruments have yet been funded, the technology to build them exists. This raises the possibility of using these signals to search for evidence of warp drive technology, even if we can’t build it ourselves.
Dr Khan cautions “In our study, the initial shape of the spacetime is the warp bubble described by Alcubierre. While we were able to demonstrate that an observable signal could in principle be found by future detectors, given the speculative nature of the work this isn’t sufficient to drive instrument development.”
The study also delves into the energy dynamics of the collapsing warp drive. The process emits a wave of negative energy matter, followed by alternating positive and negative waves. This complex dance results in a net increase in the overall energy of the system, and in principle could provide another signature of the collapse if the outgoing waves interacted with normal matter.
This research pushes the boundaries of our understanding of exotic spacetimes and gravitational waves. Prof Dietrich comments: “For me, the most important aspect of the study is the novelty of accurately modelling the dynamics of negative energy spacetimes, and the possibility of extending the techniques to physical situations that can help us better understand the evolution and origin of our universe, or the avoidance of singularities at the centre of black holes.”
Dr Clough adds: “It’s a reminder that theoretical ideas can push us to explore the universe in new ways. Even though we are sceptical about the likelihood of seeing anything, I do think it is sufficiently interesting to be worth looking!”
The researchers plan to investigate how the signal changes with different warp drive models and explore the collapse of bubbles travelling at speeds exceeding the speed of light itself. Warp speed may be a long way off, but the quest to understand the universe’s secrets continues, one simulated crash at a time.
I was intrigued to see this graphic accompanying an article about hurling. Notice that the left hand side shows the field equations of Einstein’s General Theory of Relativity and some expressions to do with quantum mechanics. Hurling is indeed an extraordinary – and extraordinarily fast – sport but is the article implying that classical physics is inadequate to describe it? Perhaps it is implying that through hurling we will at last arrive at a Theory of Everything?
It’s New Year’s Eve and I just remembered that there was a paper at the Open Journal of Astrophysics site that we published before Christmas but that I haven’t yet announced on here, so for the sake of completeness here it is. It takes us to 50 papers published in 2023.
The paper in question is the 50th and final paper in Volume 6 (2023) and it’s the 115th altogether. This one was actually published on Friday 22nd December 2023 but owing to the vacations we had to wait a bit to get the metadata registered.
The title of this one is “What are the parities of photon-ring images near a black hole?” and is a discussion of the Fermat potential (also known as the arrival-time surface) in the context of gravitational lensing by strong gravitational fields and the implication for image parities thereby produced. This one is actually listed in General Relativity and Quantum Cosmology (gr-qc, on arXiv) but is cross-listed as Cosmology and Non-galactic Astrophysics so is eligible for publication here in the appropriate folder.
The authors are Ashish Kumar Meena (Ben Gurion University of the Negev, Israel) and Prasenjit Saha (University of Zurich, Switzerland).
Here is the overlay of the paper containing the abstract:
You can click on the image of the overlay to make it larger should you wish to do so. You can find the officially accepted version of the paper on the arXiv here.
And that concludes Volume 6 of the Open Journal of Astrophysics. Roll on Volume 7.
This morning found me in Renehan Hall in St Patrick’s House in Maynooth for ‘DonalFest’, a meeting to mark the retirement of former colleague (now Emeritus) Professor Brian P. Dolan, who retired a couple of years ago in the midst of the pandemic, which delayed his leaving do.
Today’s meeting involved a number of talks given by Brian’s past and present collaborators in the splendid surroundings of the old college (and, I might add, in glorious weather). Unfortunately I had to leave before the end in order to attend to some logistical matters to do with my impending departure on sabbatical, but I’m sure the rest of it was as enjoyable as the bit I was able to be at.
All of which gives me an excuse to plug again this textbook (left), based on the lecture notes Brian used to teach a final-year undergraduate course in General Relativity to Mathematical Physics students here in Maynooth.
The book’s description reads:
Einstein’s general theory of relativity can be a notoriously difficult subject for students approaching it for the first time, with arcane mathematical concepts such as connection coefficients and tensors adorned with a forest of indices. This book is an elementary introduction to Einstein’s theory and the physics of curved space-times that avoids these complications as much as possible. Its first half describes the physics of black holes, gravitational waves and the expanding Universe, without using tensors. Only in the second half are Einstein’s field equations derived and used to explain the dynamical evolution of the early Universe and the creation of the first elements. Each chapter concludes with problem sets and technical mathematical details are given in the appendices. This short text is intended for undergraduate physics students who have taken courses in special relativity and advanced mechanics.
You can order the book and/or recommend a copy to your library here.
Anyway, let me end with some personal wishes to Brian for a long and happy retirement!
Just a very quick note to advertise a new book by former colleague (now Emeritus) Professor Brian P. Dolan, who retired a couple of years ago, but is still active in research.This textbook (left) is based on the lecture notes he used to teach a final-year undergraduate course in General Relativity to Mathematical Physics students here in Maynooth.
The book’s description reads:
Einstein’s general theory of relativity can be a notoriously difficult subject for students approaching it for the first time, with arcane mathematical concepts such as connection coefficients and tensors adorned with a forest of indices. This book is an elementary introduction to Einstein’s theory and the physics of curved space-times that avoids these complications as much as possible. Its first half describes the physics of black holes, gravitational waves and the expanding Universe, without using tensors. Only in the second half are Einstein’s field equations derived and used to explain the dynamical evolution of the early Universe and the creation of the first elements. Each chapter concludes with problem sets and technical mathematical details are given in the appendices. This short text is intended for undergraduate physics students who have taken courses in special relativity and advanced mechanics.
You can order the book and/or recommend a copy to your library here.
It’s another one of the occasions on which I have to use this blog to pass on some sad news. Renowned physicist James B. Hartle has passed away.
Jim Hartle’s scientific work was concerned with the application of Einstein’s theory of general relativity to astrophysics, especially gravitational waves, relativistic stars, black holes, and cosmology, specifically the theory of the wave function of the universe. For much of his career he was interested in the earliest moments of the big bang where the subjects of quantum mechanics, gravity theory and cosmology overlap, leading among other things to the Hartle-Hawking conjecture.
Jim Hartle was one of the speakers at the very first scientific conference I attended in Cargèse, Corsica way back in 1986. I remember his lectures very well after all these years, not least because he was so witty. I remember his response when someone asked him about the existence of large dimensionless numbers in cosmology: “…it’s a property that numbers have that some of them are larger than others.”
Condolences to his family, friends and colleagues. Rest in peace, Jim Hartle (1939-2023).
Today, 16th February 2023, sees the official publication of a special 50th anniversary edition classic monograph on the large scale structure of space-time by Stephen Hawking and George Ellis. My copy of a standard issue of the book is on the left; the special new edition is on the right. The book has been reprinted many times, which testifies to its status as an authoritative treatise. I don’t have the new edition, actually. I just stole the picture from the Facebook page of George Ellis, with whom I have collaborated on a book (though not one as significant as the one shown above).
This book is by no means an introductory text but is full of interesting insights for people who have studied general relativity before. Stephen Hawking left us some years ago, of course, but George is still going strong so let me take this opportunity to congratulate him on the publication of this special anniversary edition!
P.S It struck me while writing this post that I’ve been working as a cosmologist in various universities for getting on for about 35 years and I’ve never taught a course on general relativity. As I’ll be retiring pretty soon it’s looking very likely that I never will…
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.
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In the Dark 