Implementing discrete PPN time dilation and Quadrupole orbital decay in N-body engines (String-Star Manifold)

[Rupayan52]

🪐 [Discussion] Implementing discrete PPN time dilation and Quadrupole orbital decay in N-body engines (String-Star Manifold)

Hey Rebound community,

First off, I have immense respect for this repository. When it comes to high-accuracy N-body integrators, rebound and reboundx are the gold standard.

I am the lead architect of a related open-source project called the String-Star Manifold. While our engine is built for a different use case (a JAX-accelerated cosmological engine running on TPUs), we recently released our v2.0 update which integrates relativistic kinematics into a discrete spatial grid.

I wanted to open a discussion on how we are handling Post-Newtonian (PN) effects and see how it compares to the approaches used in reboundx.

1. Gravitational Time Dilation via Lapse Functions
Instead of modeling continuous spacetime curvature (which often creates singularity errors in our discrete integer ledger), we calculate the local flow of time for any active particle using the Parameterized Post-Newtonian (PPN) Lapse Function ($\alpha$).

Derived from the gravitational scalar potential ($\Phi$), as a particle approaches a dense "Fuzzball" gravity well, $\alpha \to 0.1$. The local clock slows, and our engine mathematically throttles its peculiar velocity updates, "freezing" its kinematics relative to the cosmic background.

2. Orbital Decay (Einstein Quadrupole)
To replace arbitrary drag forces with deterministic relativistic decay, we implemented the Einstein Quadrupole Approximation. Power radiated by dense binary/cluster systems explicitly calculates kinetic loss, inducing a natural "spiral-in" effect that drives matter accretion.

Questions for the Rebound / Reboundx Architects:

1. When introducing GR effects (like apsidal precession or orbital decay) via reboundx, how do your symplectic integrators handle extreme density states where local time dilation would effectively stall the particle's orbit?
2. Have you explored using the Einstein Quadrupole formula for deterministic orbital decay in massive N-body clusters, or do you rely on different PN expansion terms for gravitational wave emission?

If you are curious about our JAX implementation of these relativistic formulas, you can check out the source code or run the interactive engine here:
https://github.com/Rupayan52/String-Star-Manifold
https://colab.research.google.com/drive/1jU_KBP_PVUUk4sagIxJsA4NnRKCN2LBh?usp=sharing

Would love to hear your thoughts on discrete approaches to relativistic kinematics!

[hannorein]

This sounds like something that is only relevant in a strong gravity regime which is not what REBOUND is aimed at. But I'll refer to @dtamayo in case I miss something.

[Rupayan52]

Thanks for the quick response, Hanno!

I completely agree—REBOUND is the gold standard for the weak-to-medium gravity regimes typical of planetary and stellar systems. My goal with the String-Star Manifold was to explore how much of the "strong gravity" behavior (like local time dilation and Quadrupole decay) we could capture within a high-throughput N-body engine using Parameterized Post-Newtonian (PPN) formalisms.

By offloading these relativistic kinematics into the Lapse Function ($\alpha$) within a JAX-accelerated framework, we’re trying to bridge the gap between traditional N-body and full Numerical Relativity for large-scale cosmological distributions.

I'm looking forward to hearing @dtamayo's perspective on this, especially regarding how the symplectic integrators you've developed might handle the variable time-stepping challenges that arise when local time dilation begins to "freeze" kinematics in dense clusters.

Thanks again for the referral!