Archive for general relativity

Lights all askew in the Heavens – the 1919 Eclipse Expeditions

Posted in History, Talks and Reviews, The Universe and Stuff with tags , , , , , on April 23, 2019 by telescoper

I completely forgot to upload the slides from my talk at the Open Meeting of the Royal Astronomical Society on April 12 2019 so here they are now!

Just a reminder that the centenary of the famous 1919 Eclipse Expeditions is on 29 May 2019.


R. I. P. Wolfgang Rindler (1924-2019)

Posted in Books, Talks and Reviews, Education, The Universe and Stuff with tags , , , , on March 5, 2019 by telescoper

A recent comment on this blog drew my attention to the sad news of the death, at the age of 94, of Wolfgang Rindler. He passed away almost a month ago, in fact, but I have only just heard. My condolences to his family, friends and colleagues.

Wolfgang Rindler was a physicist who specialized in relativity theory and especially its implications for cosmology. Among other things he is attributed with the first use of the phrase `Event Horizon‘ as well as elucidating the nature of horizons in general relativity, both in the context of black holes and in cosmology. I never met him personally but to me, and I think to many other people, Wolfgang Rindler will be familiar through his textbooks on relativity theory. I have two in my collection:

I bought the one on the right on recommendation when I was an undergraduate over thirty years ago and the other (shorter) one I acquired second-hand some years later. Both are still very widely used in undergraduate courses.
I found the first one then (as I do now) rather idiosyncratic in approach and notation but full of deep insights and extremely effective from a pedagogical point of view. I still recommend it to students, to balance more conventional modern texts which tend to be far more conventional. It’s no easy thing to write textbooks and Wolfgang Rindler deserves high praise for having devoted so much of his time, and considerable talent, into writing ones whose impact has been so widespread and lasted so long.

Rest in peace, Wolfgang Rindler (18th May 1924 – 8th February 2019).

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).

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!


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.