Second-order self-force calculation of gravitational binding energy

It is with great pleasure that we present today the first second-order (in the mass-ratio) self-force result. From the monopole piece of the second-order metric perturbation, and specializing to circular orbits in Schwarzschild spacetime, we compute the binding energy of the binary. We have compared the results of our calculation with the result from the first law of binary mechanics (FLBM) and find great agreement — see the figure below.

Screenshot 2019-08-21 09.34.25

In comparing against the FLBM we find a slight discrepancy in the very strong field. This discrepancy is not unexpected as the FLBM only applies to a conservative spacetime (no inspiral) whereas we include the effects of dissipation in our calculation.

This work builds upon almost 7 years of effort since the foundational second-order papers. It’s been a long road with 10+ papers already out and I can think of at least another 5 in preparation.

This first concrete calculation really marks the end of the beginning. As we compute higher modes we will get the binding angular momentum, the radiated flux, gauge invariants, local force, and then the first complete post-adiabatic waveform. Watch this space!


Traveling by train

I think my favourite way to travel is by train so when the opportunity arises, if possible, I will try to take the train there, back or both. A good example of this is that I was fortunate enough to be invited to lecture at a summer school in Beijing in 2016. Whilst I flew out there the only logical choice was to take the train back (or at least as far as St. Petersburg).

One thing I’ve been thinking of doing for a while now is making map of all the rail journey’s I’ve been on. This has resulted in my Rail Journey Map which shows all the intercity rail journeys I have been on totalling ~24,000 km of travel.

Compiling this map was rather challenging and it took me some time to research how to do it. Below is the method I used. It might not be the best, but it worked for me.

Creating the rail journey map

To create the map made use of a piece of open-source software called OpenRailRouting. The instructions are pretty clear on GitHub page about how to install and run it but a key piece of information is missing, at least for those not used to using OpenStreetMap (OSM) data.

To get the OSM data I had to download the 44GB planet PBF file and then use the Osmium tool to extract all the railway information. The command for this is:

osmium tags-filter -o planet-rail.osm.pbf planet.osm.pbf nw/railway

This results in a much more manageable ~350MB file. With this in hand you can run OpenRailRouting as described on the software’s webpage. This will setup a GraphHopper server locally which you can reach at localhost:8989. With this GUI you can select start and end points on a rail journey and the software will provide the route in between. You can also set intermediate points in case the software returns another route you didn’t take. Once you’ve settled on a route, you can export the GPX file by clicking the little ‘gpx’ button.

Finally, to plot all the routes together I used Mathematica as this made it very easy to change the map projection.

One useful thing to note is that the default configuration for OpenRailRouting only routes on standard gauge (1435mm) tracks. Many countries use this rail gauge but some, such as Ireland, Russia and Mongolia, use different gauges. You can modify the config file to add additional gauges easily.

Gravitational waves from bodies orbiting the Galactic Center black hole and their detectability by LISA

Today we put out a paper investigating bodies orbiting Sgr A* as a possible source of gravitational waves for LISA. It just so happens that the 4.1 million solar masses of Sgr A* places the gravitational wave (GW) frequency for bodies on a circular orbit in the strong-field right in the sweet spot for LISA. Our paper then addresses two questions: i) what sort of objects can get close enough to Sgr A* without being tidally disrupted and ii) for objects that are not tidally disrupted what would the resulting GWs and signal-to-noise (SNR) look like in LISA.

For the first question, clear candidates (from EMRI research) are stellar mass black holes, neutron stars and white dwarfs all of which can cross the innermost stable circular orbit (ISCO) without being tidal disrupted. We also investigate the Roche limit for planets (rocky and gaseous), low mass stars, red dwarfs and brown dwarfs. These results are nicely summarised in Fig. 1 of the paper:

LISA sensitivity curve and various gravitational wave frequencies from circular orbits around Sgr A*

This shows that detecting Jupiter-like or rocky planets is unlikely but low-mass stars or brown dwarves seems possible. We then go on to show that such object are all detectable in one year of LISA data with a signal-to-noise ratio above 10. We do not in this work attempt any estimate of event rates for these objects to be in the LISA band but hopefully it helps to motivate those who know how to calculate these things to look into this!

Lots more details can be found in our paper at: arXiv:1903.02049 where we also consider sources from another nearby massive black hole in M32 (see appendix C).

This work was in collaboration with Éric Gourgoulhon, Frederic Vincent and Alexandre Le Tiec.


LISA Waveform Working Group meeting

The first stand-alone meeting of the LISA Waveform working group (WavWG) will take place at the AEI Potsdam from 13 – 15 May 2019.

The WavWG is dedicated to the development of waveform templates for the LISA mission and connects the broader scientific community to the LISA Consortium. As such, the WavWG coordinates efforts to model a broad spectrum of gravitational wave sources, establishing links with LISA’s astrophysics, cosmology, fundamental physics and simulation working groups, and bridge between communities employing different methodologies to prepare waveforms.

This meeting is the first stand-alone meeting of the WavWG and aims at getting together leading international experts and young scientists to identify pressing tasks concerning LISA waveform modelling in its broadest sense and to foster new international collaborations. The meeting will have few talks per day and provide ample time for discussions.

While the WavWG is part of the LISA Consortium, you do not need to be in the Consortium to attend.

You can find additional information here:

The registration will end on *30 April 2019*.

We have limited funds available to support PhD students. Should you require financial support, please follow the instructions on our website and apply until *30 March 2019*.

Virtual reality black holes (update)


Two of our PhD students, Phil and Josh, recently made a nice video (above) showing off their VR work. In the video you can see their code has two modes: one for visualising the orbital motion of a test body about a rotating black hole and another for visualising the distortion of light by a Schwarzschild black hole. You can also see the nice menu system they set up so the user can change various settings (black hole spin, orbital paramters,  background sky, etc).


They’ve now shown off their work at a couple of public events at UCD, included at the end of a public lecture by Scott Hughes. Looking to the future we are planning to take their black hole virtual reality experience to public engagement events across Ireland. We’re already signed up to present their work at the Mayo Dark Sky Festival and have a few other things in the pipeline. I’ll keep this site updated as we confirm events.



I am pleased to announce a major update to the KerrGeodesics module of the Black Hole Perturbation Toolkit. This module calculates the properties of bound, timelike geodesics in Kerr spacetime and this major update brings the ability to work with fully generic (non-equatorial) geodesics. Key to the release of this update was the donation of code by Maarten van de Meent (who implemented the analytic formula of Fujita and Hikida) and Charles Evans, Zac Nasipak and Thomas Osburn (who implemented their own algorithm). This mixing of the best codes out there is exactly how I hoped the Toolkit would grow so a big thank you for these contributions. In addition to a slew of bug fixes, the documentation and website have also been updated.

It is now extremely easy to work with generic geodesics in Kerr spacetime. For instance, the geodesic plotted at the top of this post is generated using:

orbit = KerrGeoOrbit[0.998, 3, 0.6, Cos[π/4]];
{t, r, θ, φ} = orbit["Trajectory"];

Once you have the trajectory it is easily plotted via, e.g.,

   {r[λ] Sin[θ[λ]] Cos[φ[λ]], r[λ] Sin[θ[λ]] Sin[φ[λ]], r[λ] Cos[θ[λ]]}, {λ, 0, 20}, ImageSize -> 700, Boxed -> False, Axes -> False, 
   PlotStyle -> Red, PlotRange -> All],
 Graphics3D[{Black, Sphere[{0, 0, 0}, 1 + Sqrt[1 - 0.998^2]]}ß

In addition to computing the orbital trajectory there are many quantities that can be calculated. A complete list of the currently available functions is

KerrGeoEnergy[a, p, e, x]
KerrGeoAngularMomentum[a, p, e, x]
KerrGeoCarterConstant[a, p, e ,x]
KerrGeoConstantsOfMotion[a, p, e, x]
KerrGeoFrequencies[a, p, e, x]
KerrGeoOrbit[a, p, e, x]
KerrGeoPhotonSphereRadius[a, x]
KerrGeoISCO[a, x]
KerrGeoIBSO[a, x]
KerrGeoISSO[a, x]
KerrGeoSeparatrix[a, e, x]
KerrGeoBoundOrbitQ[a, p, e, x]

Many of these have additional options that can be set. For instance, the orbital frequencies can be computed w.r.t. Boyer-Lindquist or Mino time (see the documentation for more detail). Each orbit is parametrized by the black hole spin ‘a’, the semi-latus rectum ‘p’, the orbital eccentricity ‘e’ and the inclination angle ‘x_inc’. This is defined via

\( x_{inc} = Cos(\theta_{inc})\)    where     \( \theta_{inc} = \pi/2 – Sign(L_z)\theta_{min}  \)

where \( \theta_{min} \) is the minimum theta angle obtained during the orbital motion. Prograde orbits occur for \(x_{inc} > 0\), retrograde ones for \( x_{inc} <0 \).

So please download and use the KerrGeodesics package and give us feedback. There is a tutorial built into the Mathematica documentation and you can also find example notes books in the Mathematica Toolkit Examples section of the Toolkit. If you find any bugs or there’s a feature missing you would like to see implemented, please report them to the issue tracker.

Generic Kerr Orbit Parameters

Virtual Reality Black Hole Orbits

Two of our PhD students, Josh Mathews and Philip Lynch, recently got to show off the black hole virtual reality visualization project they’ve been working on over the summer. They’ve done a really amazing job at bringing both the orbit of a particle and the motion of light around a black hole to life using virtual reality technology and it was great to hear their demonstration received so much interest at the UCD open day.

Both students worked with me on their final year theoretical physics projects where they explored the motion of particles and photons around black holes. They both did an excellent job and with support from my Royal Society – Science Foundation Ireland grant I was able to support them working to bring their projects to the virtual world over the summer.

The motivation for me to encourage them to do this came from the beautiful videos of Steve Drasco showing the evolution of extreme mass ratio inspirals (EMRIs) due to the emission of gravitational waves. I encountered Steve’s videos early in my PhD and they really brought EMRIs to life for me. I hope we can take our new virtual reality black holes and inspire a new generation of gravitational wave scientists.

Their orbit visualization shows the motion of a test body about a rotating black hole. The banner at the top of this post shows an screenshot of their code in action. Its very impressive in the virtual world as you can control the speed of the orbit and walk around the black hole as it takes place. Going forward we have hoping to develop this into a more complete public engagement activity. Watch this space.

Looking at a virtual non-rotating black hole

Looking at the motion of a test body around a rotating black hole

High-order asymptotics for the Spin-Weighted Spheroidal Equation at large real frequency



We recently put out a paper to compute the high-order, large-frequency expansions for the eigenvalue and the eigenfunction of the spin-weighted spheroidal equation. The spin-weighted spheroidal equation turns up in a number of places in physics, and I most often encounter it using the Teukolsky formalism to model gravitational wave emission. A great strength of our paper is the combination of analytic and numerical results. The topic of high-frequency expansion of spin-weighted harmonics has been addressed before. Evaluating the results of that work relied on the results of an earlier work which had an error in it. Careful comparison with numerical calculations brought this to light and our recent paper corrects and extends the literature.


Our work provides formula to compute a high-frequency expansion of the eigenvalue of the spin-weighted spheroidal equation. Code to compute this series expansion has also been made publicly available in the SpinWeightedSpheroidalHarmonics Mathematica package of the Black Hole Perturbation Toolkit. Details are provided in the paper. We also provide an example notebook which shows how to use the new feature and gives code to compute the coefficients of the expansion of the harmonic.


This work is in collaboration with Marc Casals and Adrian Ottewill.



LISA consortium: waveform working group

Shortly after the LISA consortium reboot it was clear that in addition to the Cosmology, Fundamental Physics and Astrophysics Working Groups under the LISA Science Group there was as a need to have a Waveform Working Group. I happy to say this group has now been created with Deirdre Shoemaker, Helvi Witek, Maarten van de Meent and myself as the co-chairs.

If you are a full or associate member of the consortium and interested in waveform modeling for LISA then join the new Waveform Working Group!