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!

## Noble Prize in Gravitational-Wave Physics

Congratulations to Rainer Weiss, Barry Barish and Kip Thorne who have been award the 2017 Noble Prize in physics “for decisive contributions to the LIGO detector and the observation of gravitational waves“.

More information can be found on the Nobel Prize website.

Also a nice piece about Rai Weiss at MIT. The Caltech announcement for Barry and Kip is here.

## 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 http://www.ligo.org/detections/GW170814.php. Mark Hannam also has a nice blog post about the event.

## 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!

## Royal Society – Science Foundation Ireland University Research Fellowship

I have been very fortunate to have been awarded a Royal Society – Science Foundation Ireland University Research Fellowship. The Fellowship will see me continuing to work at University College Dublin for 5 years after my Marie Curie Fellowship finishes at the end of September.

During the Fellowship I will continue my work on black hole perturbation theory. As well as computing new perturbative results a particular focus of the Fellowship will be incorporating those results into practical waveform template generation schemes ready for when LISA launches.

Further information about the Fellowship can be found at the Royal Society webpage along with the full list of awardees.

## Kerr orbit visualizer

In March I wrote a code to compute the frequencies and constants of motion associated with generic, bound, timelike Kerr geodesics. Back then I said I wanted to write an online tool for visualizing the associated orbits. A few weeks back I did just that and I finally have time to share it and write a little about it now. Rather than trying to embed it in the WordPress layout the Kerr timelike orbit visualizer can be found here.

The tool is basic but it will plot most orbits. Note it wont tell you if you enter parameters that do not correspond to Kerr a bound geodesic, you’ll just get a blank plot. Also, special cases like \(a=0, e=0\) and \(\theta_\text{inc}=0\) are not implemented (but you can set a very close to zero value to get it to work).

You can plot the orbit in Boyer-Lindquist coordinates and also in co-rotating coordinates [as defined in Eq. (3.2) of arXiv:0904.3810] where the orbit usually looks much simpler. A nice little feature, which works in most modern browsers, is you can `play’ the orbital frequencies*. I like really like this feature as it allows you to hear the character of each orbit. If you find a set of parameters near a resonance you can hear the beating between the fundamental frequencies.

This visualizer is pretty basic but I’ve had discussions with Leo Stein and Scott Hughes about improving it. In particular, it would be nice to make it more user friendly, provide more information as the orbits are plotted, give a nice set of example orbits, plot the black hole, allow animation of the orbit, etc. It might also be possible to speed up the calculating using fast Fourier transform methods.

The current code uses plotly.js to visualize the orbit. The orbit is computed by numerically integrating the geodesic equations with a fixed-step RK4 integrator found in JSXgraph. Much of the inspiration to create this online visualizer came from Leo Stein’s visualizer for bound, spherical null geodesics in Kerr. Documenting the equations the code solves, as Leo does so nicely, is something that also needs to be done.

* the frequencies do not correspond to any astrophysical extreme mass-ratio binary as such systems have frequencies in the milli-Hertz regime which cannot be heard by humans. Instead I just increase all the frequencies by a constant multiple to make them audible.

## Invited lecturer at Spring School in 北京 (Beijing)

I have been invited to lecture on numerical approaches to black hole perturbation theory and the self-force problem at a spring summer school in Beijing. The two-week long school is focusing on numerical relativity and gravitational waves. The school will take place from 15th-25th of May at the Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing and is aimed at PhD students and postdocs. More information about the school can be found here: http://ssnr2017.csp.escience.cn/.

Other lectures include: David Hilditch (Friedrich Schiller University of Jena), Koutarou Kyutoku (KEK, IPNS), Matthias Hanauske (Goethe University Frankfurt), Andrea Taracchini (Max Planck Institute for Gravitational Physics), and David Weir (University of Helsinki). They will cover topics such as post-Newtonian (PN) theory, effective-one-body theory (EOB), numerical relativity (NR), and gravitational wave sources such as neutron star (NS) binaries, black hole (BH)-NS binaries, and exotics sources.

My own lectures will concentrate on black hole perturbation as applied to small-mass ratio binaries. I’m still sketching the content of the individual lectures but once that is done I will update this page.

This will be my second visit to Beijing. I had a great time first time I went (as part of China-Japan trip in 2013) and am looking forward to heading out there again in May.

## Bound timelike Kerr geodesics: constants of motion and frequencies

\(a:\) | |

\(p:\) | |

\(e:\) | |

\(\theta_\text{inc}:\) | deg |

\(\mathcal{E}:\) | |

\(\mathcal{L}:\) | |

\(\mathcal{Q}:\) | |

\(\Omega_r:\) | |

\(\Omega_\theta:\) | |

\(\Omega_\varphi:\) | |

\(\Gamma:\) |

## Brief Explanation

Any timelike geodesic about a Kerr black hole has associated with it three constants of motion: energy \(\mathcal{E}\), angular momentum \(\mathcal{L}\) and the Carter constant \(\mathcal{Q}\). In addition, generic bound orbits have three frequencies associated with the orbital motion. With respect to Boyer-Lindquist coordinate time these are the rate of azimuthal angle accumulation \(\Omega_\varphi\), radial libration \(\Omega_r\) and polar libration \(\Omega_\theta\). These frequencies can also be defined with respect to Mino time. The Mino time frequencies, \(\Upsilon_\alpha\), can be calculated via \(\Upsilon_\alpha = \Gamma \Omega_\alpha \). The above interactive Javascript code computes these constants and frequencies given the orbital parameters.

## Method, Code and Limitations

The equations to compute the constants of motion come from W. Schmidt’s Celestial mechanics in Kerr spacetime.

A method to compute the orbital frequencies can also be found in that work but that method requires numerically integrating various equations. Instead I use equations in terms of elliptic integrals from Fujita and Hikida’s Analytical solutions of bound timelike geodesic orbits in Kerr spacetime.

To compute elliptic integrals in Javascript I use Leo Stein‘s library which can be found on GitHub.

Limitations: Not all parameters will lead to bound stable orbits. The code will not warn you of this, in part because calculating when a set of parameters leads to a stable orbit is not straight forward for generic orbits. Instead you will get NaN’s (Not a Number). Also the code gives an error for the Schwarzschild \(a=0\) case. It’s on my todo list to fix that (note you can put in very small values of \(a\)).

## Extra

I also wrote, and am making freely available, code to compute the orbital constants and frequencies in:

- Mathematica
- Python
- Python (arbitrary precision using mpmath)

I also have C and C++ (arbitrary precision using Arb) implementations but they need a little tidying up before public release.

## Future work

Plot the associated orbits in an interactive manner similar to Leo’s nice visualization of photon orbits. These orbits have a very rich structure, e.g.,

This orbit has parameters \((a,p,e,\theta_\text{min}) = (0.998,3,0.7,\pi/4) \). The apparent complexity of the orbit can be made simpler by moving to a co-rotating coordinate system. This resulting orbit is quite pretty:

I already have Mathematica code to make the calculation (that code produced the above two images). The Mathematica code could easily be converted to Python but Javascript would be more difficult due to a lack of good numerical libraries.

## X-ray reverberation

In 2016 I spent a little time exploring x-ray reverberation with colleagues from the University of Southampton. We’re interested in modeling flares in accreting black hole systems. Initially we’re considering a simplified lamppost model where the flare takes place along the symmetry axis of the system. Our goal is to calculate the difference in the time of arrival of the photons that travel directly from the flare to the observer (us on Earth) and the photons that reverberate around the black hole + accretion disk system (this includes photons that are bent around the black hole and photons that ‘reflect’ off the accretion disk).

The project, which is still in its infancy, involves ray-tracing a black hole spacetime with an accretion disk. This involves solving the null geodesic equations, working out where and when the photons interact with the accretion disk and calculating the time of arrival. As well as the time of arrival we’re interested in the intensity of the emission and its spectrum. Currently the first of these (time of arrival) is implemented for a Schwarzschild black hole with a simple accretion disk.

The code is written in Mathematica and uses semi-analytic methods to calculate the path of the light. Using the code I made a few short animations which can be found here: http://nielswarburton.net/ray_tracing/ray_trace_Schwarz.html. The page describes what each animation depicts.