﻿ The relativistic Pythagorean three-body problem - by Tjarda C.N. Boekholt 2021 - Article review

## The relativistic Pythagorean three-body problem - by Tjarda C.N. Boekholt 2021 - Article review

This document contains article review "The relativistic Pythagorean three-body problem " by Tjarda C.N. Boekholt 2021
To order to read the article select:https://arxiv.org/pdf/2109.07013.pdf
• The text in italics is copied from the article.
• Immediate followed by some comments

### Abstract

We study the influence of relativity on the chaotic properties and dynamical outcomes of an unstable triple system; the Pythagorean three-body problem.
Three questions.
• Why mention the word relativity? Do all n-body systems show a relativistic behaviour?
• Why study unstable triple system? Unstable triple systems are very rare in nature and the trajectories are very difficult to measure.
It makes more sense to study stable triple system, because they can be measured and simulated and the results can be compared with the reality.
Even more important is to answer the question: What makes such a system unstable?
• Why using the word chaotic? All systems evolve influenced by all the objects in their neighbourhood.
The degree to which our system is relativistic depends on the scaling of the total mass (the unit size was 1 parsec).
To what extend does the total mass influence the behaviour of the trajectories of the individual objects in a special way? How is that influence explained? What has the concept "relativistic" to do with this. A more detailed physical explanation is required.
When we take into account dissipative effects through gravitational wave emission, we found that the duration of the resonance, and the amount of exponential growth of small perturbations depend on the mass scaling
To what extend is this in line with actual observations?
(1) For a unit mass <= 10M0, the system behavior is indistinguishable from Newton's equations of motion, and the resonance always ends in a binary and one escaping body.
I expect that this means that the total mass is <= 10M0. For example 2M0, 3M0 and 4M0. When you take all the masses more or less identical the whole system behaves rather symmetrical. This means that all the objects have more or less the same speed.
This is completely different if one mass is large and two are small or two are large and one is small.
(2) For a mass scaling up to 10^7M0, relativity gradually becomes more prominent, but the majority of the systems still dissolve in a single body and an isolated binary.
That means no mergers take place.
(3)The first mergers start to appear for a mass of ~10^5M0, and between 10^7M0 and 10^9M0 all systems end prematurely in a merger.
What is the explanation for specific range between 10^7M0 and 10^9M0?
These mergers are preceded by a gravitational wave driven in-spiral
What is the explanation?
For a mass scaling => 10^9M0, all systems result in a gravitational wave merger upon the first close encounter.
I expect that the masses involved are still: 0.2 10^9M0, 0.3 10^9*M0 and 0.4 10^9M0. That means they are almost the same.

### 1. Introduction.

In the subsequent evolution, the bodies fall towards the center of mass of the triple, after which the prolonged, chaotic interaction takes place.
After a first close encounter between the three bodies, they experience multiple close encounters until the least massive body escapes, leaving the other two in a stable binary orbit.
Is this result: leaving the other two in a stable binary orbit, a valid assumption for all kind of masses? Even when Black holes are involved?
The Pythagorean problem exhibits exponential sensitivity to small changes in the initial condition.
There are two issues;
1. To perform this experiment in real life.
2. To simulate the same experiment
However
• To perform the experiment in real life is very difficult.
• To simulate that same experiment as done is also very difficult. Because it requires very accurate measurements.
All experiments that involve close encounters are very difficult to simulate.

### Page 2

In other words: if the velocities of the bodies start to approach the speed of light, general relativistic e ects, such as precession and gravitational wave emission, have to be included in order to recover the correct physical behavior.

### 2. Method

In Fig. 1 we show three solutions to the Pythagorean problem for log10 ζ = 3.45, 4.2 and 4.3. These values correspond to fm = 2.63 × 10^6, 8.32 × 10^4, and 5.25 × 10^4 solar masses, respectively.
That means all the three masses are almost identical.
The left panels demonstrate the orbital chaos in each solution, while the right panels show a zoom-in of the gravitational inspiral and precession (top two rows) and head-on collision for the most relativistic case (bottom row).
Why calling this chaos?
The most important question is why should this system collide? Why is this called relativistic ?

### 3. Time Reversibility IN Chaotic Reversibility Triple Systems

Newton’s laws of motion are symmetric with respect to the arrow of time
The first question is how important is the physical meaning of the concept: "symmetric with respect to the arrow of time"? IMO this has absolute no physical meaning. The concept of the arrow of time does not exist.
But there is a more severe problem. Newton's Law assumes that gravity acts instantaneous. That is wrong. In reality the speed of gravity (Gravity waves) is final, which makes 'symmetrical' a tricky issue to test the accuracy of any simulation.
Time reversibility has therefore been used as a proxy for the accuracy of an N-body simulation.
I expect that the idea behind this is, that at any point in time you can replace t by -t in the equation of motion and then the process will return back to its initial state.
What you can also do is replace any speed v by -v. This means physical that each object, straight, bounces back against a wall ahead. This implies that the object does not loses mass.
In a first experiment, we perform a time-reversibility test for the relativistic Pythagorean problem. The aim is to determine whether we can obtain time-reversible solutions to the relativistic Pythagorean problem, and to measure how relativistic terms affect the numerical accuracy needed to reach a converged solution.

### Reflection 2 - 1 Body system - 2 Body system

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Created: 22 Februari 2022

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