SciPost Submission Page
Pseudospectral implementation of the Einstein-Maxwell system
by Jorge Expósito Patiño, Hannes Robert Rüter, David Hilditch
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Jorge Expósito Patiño · Hannes Rüter |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202505_00035v1 (pdf) |
| Date submitted: | May 15, 2025, 4:58 p.m. |
| Submitted by: | Jorge Expósito Patiño |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Electromagnetism plays an important role in a variety of applications in gravity that we wish to investigate. To that end, in this work, we present an implementation of the Maxwell equations within the adaptive-mesh pseudospectral numerical relativity code BAMPS. We perform a thorough analysis of the evolution equations as a first order symmetric hyperbolic system of PDEs. This includes both the construction of the characteristic variables for use in our penalty boundary communication scheme, as well as radiation controlling, constraint preserving outer boundary conditions which, in the language of Kreiss-Agranovich-Métivier, are shown to be boundary-stable. After choosing a formulation of the Maxwell constraints that we may solve for initial data, we move on to show a suite of numerical tests. Our simulations, both within the Cowling approximation, and in full non-linear evolution, demonstrate rapid convergence of error with resolution, as well as consistency with known quasinormal decay rates on the Kerr background. Finally we evolve the electrovacuum equations of motion with strong data, a good representation of typical critical collapse runs.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report #2 by Nils Deppe (Referee 2) on 2025-7-7 (Invited Report)
- Cite as: Nils Deppe, Report on arXiv:scipost_202505_00035v1, delivered 2025-07-07, doi: 10.21468/SciPost.Report.11524
Report
The paper presents a new formulation, implementation, and verification of the Einstein-Maxwell equations. The authors demonstrate stability, convergence, and applicability of the approach to studying critical behavior and quasinormal modes of black holes. The paper is very well written and provides important contributions to the field of numerical relativity. I have a few suggestions for the authors to consider before publishing, but I do not need to review the paper again before it is published.
- Line 64: It would be good to be explicit about units. Presumably G=c=1, but what about the EM units? With matter there's now a 3rd scale involved and so more units need to be chosen.
- Line 71: The authors should clearly define the convention for Levi-Civita tensor and symbol that they are using, since there are several different ones used in the literature.
- Line 79: It would be good to explicitly state and define the Lie derivative, mostly for completeness.
- Line 81: it would be good to explicitly write out how
D_iis defined in terms of partial derivatives, etc. Again, just for completeness and making the paper easier to reproduce! - Fig. 2: since the solution here ranges from positive and negative, naively I would've expected a divergent colormap centered at 0. I encourage the authors to do so or to clearly state why they went with the colormap they are using.
- Line 447: I suggest stating the approximate time the pulse reaches the outer boundary to help guide reader's eyes.
- Line 445: In case it's helpful, SpEC and SpECTRE also have a very hard time being limited by time stepper accuracy. In BBH simulations this only seems to kick in once spatial errors are driven down to about 1e-13 while the time integrator is using a tolerance of 1e-8. However, this seems to be an artifact of the adaptive stepper not really staying consistently below the CFL, and then rapidly decreasing the time step. spectre even has a scalar wave solution that satisfies the discrete DG equations just to test time stepper convergence: https://spectre-code.org/classScalarWave_1_1Solutions_1_1SemidiscretizedDg.html
- Fig. 8: I encourage the authors to clearly label the quantities in the figure itself, not just the caption, and to use a different colormap for the Kretschmann scalar to distinguish it further. (And it changes sign, though not clear that's significant)
- Line 538: possible typo "electovacuum" -> "electrovacuum" (missing "r")?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report #1 by Anonymous (Referee 1) on 2025-6-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202505_00035v1, delivered 2025-06-23, doi: 10.21468/SciPost.Report.11444
Strengths
- Very clear and detailed analysis of the Einstein-Maxwell system, including: 1a. well-posedness analysis; 1b. characteristic analysis; 1c. careful treatment of constraint-preserving boundary conditions.
- Mostly self-contained.
- Very clearly written and well-organised paper.
- Very careful analysis of the presented results.
Weaknesses
- The work itself and the presented results are not very original.
- As far as I could tell, the code that was developed and presented in this paper is not available to the wider community.
Report
I have listed some relatively minor points that the authors should clarify in the next section. However, my main criticism of this work is that the code developed, as far as I understand, is not open. In my opinion, this is not a minor detail. Open science has been a significant talking point recently, especially in the context of works funded by the European Union. It also aligns well with SciPost's philosophy of Open Access. And the fact remains that, as it stands, the results from this work are not reproducible since the underlying code is not available to the wider community.
I will leave it to the discretion of the Editors to decide whether this issue constitutes an impediment to publication.
Requested changes
- Minor typo in Eq. 4: index $\nu$ on the rhs should be lowered.
- The overall infrastructure in which this code is implemented (BAMPS) is mentioned very briefly and only in passing (eg, page 5). While I understand that a detailed analysis is available in Ref [14], the present work would be more self-contained if a brief summary was presented, in particular regarding the (multipatch) coordinates used, excision techniques employed, parallelization, etc.
- On page 6, in Eqs. 28-29, it was not clear to me how $E^A$ and $B^A$ are defined.
- On page 8, line 211, I was confused with the phrase "incoming constraint radiation". $\phi_0$, as I understood it from Eq. 57, is the actual physical (incoming) electromagnetic radiation, no? What is the "constraint radiation" in this context?
- On page 13, line 334, is $\psi$ also freely specifiable data?
- General comment on figures 3, 4, 6 and 9: it is not straightforward to see from these plots that the convergence is indeed exponential (as mentioned in the text). While a clear improvement is seen when increasing the number of collocation points, it is not obvious at all from the plots if the convergence is exponential or (high-order) polynomial since there are no expected lines to guide the eye. Also, in figure 3 (left plot): what happens after t>20? Is convergence lost?
- On page 19, line 479, excision surface is mentioned but it was never mentioned how this is treated numerically in practice (related with item 2 above).
- There is no comment on code runtime, number of CPUs used, efficiency and/or scaling properties of the code, or on how the present PSC implementation compares with other (finite-difference) implementations of this system.
Recommendation
Ask for minor revision
We appreciate the referees comments, and we believe the requested changes
clarify both the presentation and results. The changes we have implemented
are as follows, by comment index. We stress the originality of the proof of boundary-stability that we include,
since, to the best of our knowledge, for the Einstein-Maxwell system no proof
of boundary stability existed for constraint preserving boundary conditions
that were employed in the context of numerical relativity. We now make this
point clearer in the manuscript. 2 , 3 and 7. To improve clarity we add a new paragraph to explain the infrastructure,
including a description of the excision method.
We now also give a definition for \(E^A\) and \(B^A\). 4. Regarding the "constraint radiation", the word "constraint" is indeed wrong.
\(\phi_0\) is the physical radiation. 5. We now add extra information to clarify the conformal
factor \(\psi\) is not freely specifiable, but is to be solved for
in the initial data. 6. A new figure is added to show clearly the convergence.
In the text we explain that convergence to the analytic solution
is lost after \(t/\sigma > 20\) due to the causal contact with the
outer boundary. We now added that
explanation also to the caption of figure 3, not just in the text. 8. We have added the cost in core-hours of each simulation. Efficiency and
scaling of the underlying infrastructure are present in the references. We
agree that a code comparison with a different code base would be really
illuminating, but we feel it falls outside of the scope of the current paper,
as it is something that concerns the entire infrastructure. We will return
to this point in future studies. Lastly, there is the question of the closed source code. The authors
of this paper are actively working on the process of preparing the
underlying infrastructure (bamps) to be open source in the near
future. Given that this is a infrastructure used by many researches,
of which the authors don't posses the copyright, it isn't a trivial
process, so it most likely won't be finished by the time this
paper continues the review process.
I thank the authors for their careful reply. I believe all my comments were addressed.
I have noticed just one minor issue which I had not noticed on my first review. On section 4.1, when the authors present the Flat background test, the initial data is provided in equations 123 and 124. This initial data leads to a dynamical evolution, which is seen in Figure 2. This evolution is then used for convergence testing and in Figure 3 (left) it is shown the convergence of the numerical data to the analytical solution as a function of time. However, this analytical solution is never explicitly written down. For completeness, I believe this would be valuable to have.
Other than the minor issue above, I am happy with the current version of the manuscript and I thank the authors once again for implementing my suggestions.
Author: Jorge Expósito Patiño on 2025-09-22 [id 5841]
(in reply to Report 2 by Nils Deppe on 2025-07-07)We are grateful to the referee for their comments, which have pushed us to be more explicit in several places. We have addressed the comments in the revised manuscripts.