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!

PhD studentship

I have funding for a PhD student to work on modeling gravitational wave emission from extreme mass-ratio inspirals with me at University College Dublin, Ireland. For more information, and to contact me if you are interested, see the advert. If you have any questions about the project please do not hesitate to get in touch.


Gravity and Light

The gravitational wave news just keeps coming. This time it is really big news: the first gravitational wave event with an electromagnetic counterpart. Roughly 130 million years ago in the galaxy NGC 4993 two neutron stars ended their billions of years long dance with a violent collision. In their final throes they disturbed space and time just enough for the outgoing gravitational waves to be detectable after their 130 million light year journey to Earth. Shortly after the gravitational waves were detected a gamma-ray burst was detected by the Fermi space telescope and so began the era of multi-messenger astronomy. A great many follow up observations were made and a lot of exciting science was done by thousands of scientists around the world.

With all the information released today it will take a lot of time to digest it all. Better get reading!

Screenshot 2017-09-27 18.20.52

Three-detector observation of gravitational waves from a binary black hole coalescence

Congratulations to the LIGO-Virgo collaboration on the 4th detection of gravitational waves! The source of this detection is again merging black holes. This one is special as it is the first time the Virgo collaboration has joined the detection. Having a third detector results in substantially better sky localization (down from ~1000 deg^2 to ~100 deg^2) and the ability to test the polarization states of the incoming waves.

Information on the detection and the detection announcement paper can be found at Mark Hannam also has a nice blog post about the event.

Screenshot 2017-09-14 19.43.37

Happy 2nd birthday GW150914!

It’s hard to believe that it has already been two years since the first detection of gravitational waves*. So much has happened since then: two more confirmed detections of merging black holes, the success of the LISA Pathfinder mission, and the funding by the European Space Agency of the LISA mission. With rumors swirling again about a possibly detection of merging neutron stars, it remains a very exciting time to be gravitational wave scientist.

There have been a few things released today to celebrate GW150914’s birthday. I enjoyed this introduction to gravitational waves on TEDEd:

* After 1.5 years, the detection paper has over 1700 citations already!