Archive for general relativity

Bill Bonnor on Cosmology with Negative Mass

Posted in mathematics, The Universe and Stuff with tags , , , , , on December 10, 2018 by telescoper

My post from Friday about negative mass in cosmology reminded me of my days at Queen Mary and discussions I had at that time with Bill Bonnor, who retired in 1985 but was a regular visitor to the weekly Relativity Seminars. I was sad to discover just now that Bill actually passed away in 2015 (at the age of 94) so I thought I would post a little note as a short tribute.

Bill Bonnor was an old-school mathematical relativist, which I definitely am not, but I recall talking to him quite a lot in the coffee room because we had a shared interest in gambling games. He had a liking for the fixed-odds competition in the football pools, which he played with considerable success.

Anyway, Bill Bonnor published a paper in 1989 about Negative Mass in General Relativity. It’s not all about cosmological implications of negative mass, but I’ve just typed up a quick summary. In fact I used some of this in a university examination question many moons ago!

Before reading this, you might wish to look up active the terms gravitational mass, passive gravitational mass, inertial mass and equivalence principle, which you can find discussed here (for example).

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Gravitational Redshift around the Black Hole at the Centre of the Milky Way

Posted in The Universe and Stuff with tags , , , , , , on July 26, 2018 by telescoper

I’ve just been catching up on the arXiv, and found this very exciting paper by the GRAVITY collaboration (see herefor background on the relevant instrumentation). The abstract of the new paper reads:

The highly elliptical, 16-year-period orbit of the star S2 around the massive black hole candidate Sgr A* is a sensitive probe of the gravitational field in the Galactic centre. Near pericentre at 120 AU, ~1400 Schwarzschild radii, the star has an orbital speed of ~7650 km/s, such that the first-order effects of Special and General Relativity have now become detectable with current capabilities. Over the past 26 years, we have monitored the radial velocity and motion on the sky of S2, mainly with the SINFONI and NACO adaptive optics instruments on the ESO Very Large Telescope, and since 2016 and leading up to the pericentre approach in May 2018, with the four-telescope interferometric beam-combiner instrument GRAVITY. From data up to and including pericentre, we robustly detect the combined gravitational redshift and relativistic transverse Doppler effect for S2 of z ~ 200 km/s / c with different statistical analysis methods. When parameterising the post-Newtonian contribution from these effects by a factor f, with f = 0 and f = 1 corresponding to the Newtonian and general relativistic limits, respectively, we find from posterior fitting with different weighting schemes f = 0.90 +/- 0.09 (stat) +\- 0.15 (sys). The S2 data are inconsistent with pure Newtonian dynamics.

Note the sentence beginning `Over the past 26 years…’!. Anyway, this remarkable study seems to have demonstrated that, although the star S2 has a perihelion over a thousand times the Schwarzschild radius of the central black hole, the extremely accurate measurements demonstrate departures from Newtonian gravity.

The European Southern Observatory has called a press conference at 14.00 CEST (13.00 in Ireland and UK) today to discuss this result.

A Guest Paradox

Posted in Cute Problems, The Universe and Stuff with tags , , , on February 9, 2018 by telescoper

Here’s a short guest post by my old friend Anton. As usual, please feel free to discuss the paradox through the comments box!

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I thought of a physics paradox the other day and Peter has kindly granted me a guest post here about it, as follows. Consider a homogeneous isotropic closed universe as described by general relativity. Let it contain a uniform density of a single species of electrically charged particle, so that this universe has a net charge. The charged particle density is sufficiently low, however, that the perturbation from the regular uncharged metric is negligible. Since this universe is homogeneous and isotropic the electric field in it is everywhere zero. BUT if I consider a conceptual 3-dimensional sphere, small enough for space-time curvature to be neglected, then it contains a finite amount of electric charge, and therefore by Gauss’ theorem a nonzero electric field points out of it at every point on its surface. This contradicts the zero-field conclusion based on the metric.

Here are three responses (one my own) and my further responses to these, in brackets:

  1. In a closed universe it is not clear what is the outside and what is the inside of the sphere, so Gauss’ law is not trustworthy (tell this to a local observer!);
  2. the electric field lines due to the charges inside this (or any) conceptual sphere wrap round the universe an infinite number of times (this doesn’t negate Gauss’ theorem!);
  3. the curved rest of the Universe actually adds a field that cancels out the field in your sphere (neither does this negate Gauss’ theorem!)

The Effect of Gravity on the Muon Magnetic Moment

Posted in The Universe and Stuff with tags , , , , , on February 3, 2018 by telescoper

Only time for a short post today, but I think this may turn out to be an important result. There’s a paper by Morishima et al. on the arXiv with the rather dry title Post-Newtonian effects of Dirac particle in curved spacetime – III : the muon g-2 in the Earth’s gravity, which suggests that the anomalous magnetic dipole moment of the muon.

Here is the abstract of the paper. You can click on it to make it bigger.

In a nutshell the anomaly is that according to basic relativistic quantum theory in the form of the Dirac equation, the muon (and any other charged spin-1/2 fermion) should have a magnetic dipole moment μ of magnitude (given in terms of its mass m and fundamental constants) by μ=geħ/4m with the g-factor g=2 for Dirac fermions. The anomaly is that this can be measured and it appears that g differs from zero by a small but significant amount, i.e. (g-2) is not zero. It has been widely suggested that this discrepancy suggests the existence of physics beyond the Standard Model of Partlce Physics. Well, gravity is not included in the Standard Model so I suppose this could still be right, but the it this calculation may well disappoint those who were hoping that (g-2) might provide evidence for, e.g., supersymmetry when it looks like it might be something rather more mundane, ie the Earth’s gravity!

UPDATE: It appears there is an error in the paper; see here. You may stand down.

UPDATE: Well, that was pretty fast. There’s now a paper on the arXiv by Matt Visser that gives a detailed refutation of the above claim. Here is the abstract:

In three very recent papers, (an initial paper by Morishima and Futamase, and two subsequent papers by Morishima, Futamase, and Shimizu), it has been argued that the observed experimental anomaly in the anomalous magnetic moment of the muon might be explained using general relativity. It is my melancholy duty to report that these articles are fundamentally flawed in that they fail to correctly implement the Einstein equivalence principle of general relativity. Insofar as one accepts the underlying logic behind these calculations (and so rejects general relativity) the claimed effect due to the Earth’s gravity will be swamped by the effect due to Sun (by a factor of fifteen), and by the effect due to the Galaxy (by a factor of two thousand). In contrast, insofar as one accepts general relativity, then the claimed effect will be suppressed by an extra factor of [(size of laboratory)/(radius of Earth)]^2. Either way, the claimed effect is not compatible with explaining the observed experimental anomaly in the anomalous magnetic moment of the muon.

That’s how science goes!

The Expanding Universe: An Introduction

Posted in The Universe and Stuff with tags , , on January 5, 2018 by telescoper

For those of you reading this blog who feel they need an up-to-date primer for the basics of modern cosmology without too much technical detail, I found a paper on the arXiv that might give you what you want. It’s over a hundred pages long but does not use much complicated mathematics but has some nice illustrations. The author is Markus Pössel; the abstract reads

An introduction to the physics and mathematics of the expanding universe, using no more than high-school level / undergraduate mathematics. Covered are the basics of scale factor expansion, the dynamics of the expanding universe, various distance concepts and the generalized redshift-luminosity relation, among other topics.

This paper focusses on the basics of the standard framework founded on general relativity, especially how cosmological distances are defined and measured, rather than on trendy modern topics like dark energy and the cosmic microwave background. I’d say any first-year physics student should be able to cope with it, but it’s not for someone who hasn’t learned calculus. On the other hand, it’s free to download so you don’t have much to lose by having a look!

You can download a PDF here.

The Story of the 1919 Eclipse Expeditions

Posted in Books, Talks and Reviews, History, The Universe and Stuff with tags , , , , , , on August 21, 2017 by telescoper

Unless you have been living on another planet, you will know that today there will be an eclipse of the Sun although from the UK it will be rather underwhelming, as only about 4% of the Sun’s disk will be covered by the moon; for totality you have to be in the United States.  For the record, however, the eclipse will begin 15:46 GMT on August 21 out over the Pacific. It will reach the coast of Oregon at Lincoln City, just west of Salem, at 16:04 GMT (09:04 local time) where it will reach its maximum  at 17:17 GMT (10:17 local time). The path of totality will then track right across the United States to South Carolina. For more details see here. Best wishes to all who are hoping to see this cosmic spectacle! I saw the total eclipse of August 11, 1999 from Alderney in the Channel Islands, and it was a very special experience.

Here’s a (not very good and slightly damaged) scan of a picture from that eclipse that I found last night in a box of old photographs:

Before starting I can’t resist adding this excerpt from the Times warning about the consequences of a mass influx of people to Cornwall for the 1999 eclipse. No doubt there are similar things going around about today’s eclipse:

I did write a letter to the Times complaining that, as a cosmologist, I felt this was very insulting to druids. They didn’t publish it.

This provides me with a good excuse to repost an old item about the famous expedition during which, on 29th May 1919, measurements were made that have gone down in history as vindicating Einstein’s (then) new general theory of relativity. I’ve written quite a lot about this in past years, including a little book and a slightly more technical paper. I decided, though, to post this 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….

Going NUTs

Posted in The Universe and Stuff, Uncategorized with tags , , , on April 5, 2017 by telescoper

If you’ve studied General Relativity the chances are that you’ve come across the Taub-NUT exact solution discussed in this post. It’s generally regarded as something of an oddity in that it’s a bit contrived, but provides a counter-example to some well-known results. For example, in the context of a Black Hole solution, it violates the No Hair Theorem (by violating the assumption of asymptotic flatness).

When I saw this post at CQG, however, I was reminded of a paper published a few years ago discussing this in a cosmological context, where it can be seen as a special case of the Bianchi IX geometry.

CQG+

By Paul I. Jefremov and Volker Perlick.


Among all known solutions to Einstein’s vacuum field equation the (Taub-)NUT metric is a particularly intriguing one. It is that metric that owing to its counter-intuitive features was once called by Charles Misner “a counter-example to almost anything”. In what follows we give a brief introduction to the NUT black holes, discuss what makes them interesting for a researcher and speculate on how they could be detected should they exist in nature.

paul jefremov-and-volker Volker Perlick and Pavel (Paul) Ionovič Jefremov from the Gravitational Theory group at the University of Bremen in Germany. Volker is a Privatdozent and his research interests are in classical relativity, (standard and non-standard) electrodynamics and Finsler geometry. He is an amateur astronomer and plays the piano with great enthusiasm and poor skills. Paul got his diploma in Physics at the National Research Nuclear University MEPhI in Moscow, 2014. Now he…

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