Dr Chris Stevens



BSc. Hons First Class (Otago), PhD (Otago)

c.stevens at ru.ac.za



Room 10 Maths Building (Drostdy Lodge)

(046) 603 7547


My research lies in the field of mathematical and numerical relativity. I am particularly interested in

  • Black holes and gravitational waves
  • Initial boundary value problems for Einstein’s field equations
  • The conformal field equations and global properties of space-times
  • Cauchy-characteristic matching and characteristic extraction
  • Conformal Cyclic Cosmology

I am currently working on two major projects; an implementation of Cauchy-characteristic matching and using the conformal field equations to investigate open problems in relativity.

Cauchy characteristic matching

With the recent first direct detections of gravitational waves emitted from binary black hole systems by LIGO, efficient and accurate methods for both the numerical simulations and the subsequent extraction of gravitational waveforms are becoming increasingly important. Gravitational waveforms are calculated by solving the Einstein equations numerically and subsequently templates are made for the purpose of matching to the observed data. From this matching, parameters such as the mass an

d spin of the black holes can be found. 


Our research aims to investigate a novel method for gravitational wave extraction, Cauchy Characteristic Matching (CCM), whose successful implementation has been an outstanding problem in numerical relativity for over 20 years. We have designed a new cutting-edge algorithm to take the problem, which incorporates the use of multiple coordinate domains. This has proven to work successfully in other numerical relativity codes. Our implementation generalises this idea as it also includes the time domain, but previous successful utilisations endows confidence that this will also work.


Not only would the successful completion of this project solve a long outstanding issue in numerical relativity, the successful implementation of CCM, but it is also expected to greatly increase code efficiency for future gravitational wave extractions. As we are now in the era of gravitational wave astronomy, increased code efficiency is desperately needed in order to calculate templates for future detections in a reasonable time frame.


The conformal field equations and applications

The conformal field equations are an extension of the Einstein field equations to a conformally rescaled metric. This new metric acts as a “compactified” physical space-time, whose infinity can subsequently be represented finitely.


We have developed a (numerically) wellposed Initial Boundary Value Problem (IBVP) framework for the conformal field equations that ensures there is no constraint violation through the boundaries. This has been used to investigate nonlinear gravitational perturbations of the Schwarzschild space-time and calculate global properties directly at future null infinity.


An open problem tractable with this framework is the stability of the Kerr space-time under nonlinear gravitational perturbations. It is still not clear whether this space-time is stable under general perturbations, and the IBVP framework could be used to investigate this from a global perspective. Exploring this problem does not require any fundamental changes to the current setup for Schwarzschild space-time, only a new initial data set is needed.


There is also the possibility of investigating the conditions required on an asymptotically flat time symmetric initial data set so that the resulting vacuum solution has a regular null infinity. Friedrich has restricted the problem to how initial data is chosen on the blowup of space-like infinity to a 2-sphere. He has conjectured that the necessary condition for a regular null infinity is that the initial data near null infinity are those induced by asymptotically conformally stationary space-times. This still remains as just a conjecture and hence it would be intriguing to probe this question numerically by evolving sets of initial data that do and do not satisfy the necessary conditions of the conjecture.



  • Frauendiener, J. & Stevens, C., Nonlinear gravitational perturbations of the Schwarzschild space-time. (2019). In preparation, to be submitted to Classical and Quantum Gravity
  • Frauendiener, J. & Stevens, C., The Bondi energy-momentum in numerical space-times (2019). In preparation, to be submitted to Classical and Quantum Gravity
  • John, Anslyn J., and Chris Z. Stevens. "Accretion onto deformed black holes via pseudo-Newtonian potentials." arXiv preprint arXiv:1904.10377 (2019). Accepted conference proceedings for HEASA 2018.
  • Doulis G., Frauendiener, J., Stevens, C.,& Whale, B., "COFFEE - An MPI-parallelised Python package for the numerical evolution of differential equations", SoftwareX (2019) 100283
  • John, A. & Stevens, C., Bondi accretion in the spherically symmetric Johannsen-Psaltis space-time, European Physical Journal C (2019) 79:962
  • Barack, Leor, et al. "Black holes, gravitational waves and fundamental physics: a roadmap." Classical and quantum gravity 36.14 (2019): 143001.
  • Beyer, Florian, et al. “The numerical initial boundary value problem for the generalized conformal field equations.” Physical Review D 96 (2017): 084020.
  • Frauendiener, Jörg, Chris Stevens, and Ben Whale. “Numerical evolution of plane gravitational waves in the Friedrich-Nagy gauge.” Physical Review D 89.10 (2014): 104026.
  • Barker, Richard J., et al. “Closed?population capture–recapture modeling of samples drawn one at a time.” Biometrics 70.4 (2014): 775-782.



Courses 2019

Semester 1:

  • MAT1S coordinator
  • MAT1C lecturer
  • MAP 314 (PDEs) lecturer

Semester 2:

  • MAT1C lecturer
  • Analytical Mechanics honours course lecturer


I am happy to supervise honours, masters or PhD students who have interests in General Relativity or other areas of mathematical or theoretical physics. Feel free to drop by my office to discuss.

Current students:

Jonathan Hakata (MSc): Numerical evolution of gravitational plane waves in the Friedrich-Nagy gauge

Sinakho Gobozi (Hons)

Previous students:

Sebenele Thwala (AIMS MSc): Colliding gravitational plane waves (Distinction)

Last Modified: Tue, 26 Nov 2019 10:27:03 SAST