4 ms·

So the stability of the Solar System is highly dependent on general relativity. That seems a bit misleading. If you have a complex system that reaches a measur

by causi 4y ago

So the stability of the Solar System is highly dependent on general relativity. That seems a bit misleading. If you have a complex system that reaches a measure of stability over a large timescale and then change any of the underlying rules, it's going to become unstable. In this scenario there's nothing special about special relativity, it just happens to be the rules they messed with.

infogulch 4y ago
That's a fair criticism. If a dynamical system is stable under one model, it's not that surprising if suddenly swapping out the model causes an instability. Basically, different models have different stability islands, if you would. But the question remains, how dependent on GR is the long term stability of an N-body system in general? Here's an experimental idea: create a "random N-body system" generator, and compare what % of systems are stable under Newtonian vs Einsteinian physics models.
- raattgift 4y ago
  > in general That's too broad to answer. Let's narrow to our solar system, or a close approximation, first. Below, I'll step up to more complicated arrangements, and binary black holes. These are only three or four families of (regions of) general curved spacetime; there is a very large infinity of other families one could reasonably associate with "in general". In a (model) solar system, there is a large central mass and several other objects with extreme mass ratios to the sun. There are interesting secular resonances among Mercury, Venus, and Jupiter, and between Earth and Mars, where that means that from time to time (over many orbits), the precessions of the near-to-the-sun parts of the planets' orbits match. These resonances may increase the eccentricity of the smaller mass's orbit, which can in extremis yeet the small body out of the solar system, or cause a collision between planets (e.g. Mercury's orbit may near aphelion cross Venus's orbit). For the Mercury-[Venus-]Jupiter the closeness of Mercury to the sun tends to break up the resonance enough to significantly stabilize the shape of Mercury's orbit. Any sort of slow-down on the inner body when it and heavier outer bodies are relatively close together with the sun is liable to suffice, and that's what the authors of the link at the top argue what we get with (good approximations of) GR. (The approximations used by the authors are a post-Newtonian expansion <https://en.wikipedia.org/wiki/Post-Newtonian_expansion https://en.wikipedia.org/wiki/Post-Newtonian_expansion> and numerical methods). Very roughly, thanks to the large central mass and the very different distances of the planets involved, the outer body rushes ahead or equivalently the inner body lags behind, which avoids stabilizing the resonance. Many -- possibly even most -- stars are in binaries, rather than singletons like our sun. Planetary orbits in a binary (or triple) star system are likely to be dramatically different. In binaries, the width and mass-ratio between the partners are relevant quantities. Triples can be arranged in all sorts of ways. PSR J0337+17 is one known arrangement <https://www.youtube.com/watch?v=oDgfqq_W_uM https://www.youtube.com/watch?v=oDgfqq_W_uM>. PSR B1620-26, a pulsar-white dwarf binary, has at least one planet. I have only weak intuitions about what Jupiter or Earth mass bodies would do in such systems, or even how to select a reasonably astrophysical set of initial orbital values. (There are known circumbinary planets. See <https://en.wikipedia.org/wiki/Circumbinary_planet https://en.wikipedia.org/wiki/Circumbinary_planet> and <https://sci-hub.ru/10.1007/978-90-481-8687-7 https://sci-hub.ru/10.1007/978-90-481-8687-7> in which Chapter 9 details how these can be simulated). It is not too fanciful however to suppose that periastrons could be close enough that post-Newtonian effects are non-negligible, even if the stellar masses are non-compact. That doesn't say much about whether secular resonances (in the sense above) are likely in such systems. I have no idea, but I'd guess they aren't forbidden. > compare what % of systems are stable under Newtonian vs Einsteinian [gravitation] That's what the authors do in their paper <https://arxiv.org/abs/2303.05567 https://arxiv.org/abs/2303.05567>, although "systems" here is a simplified model of the solar system (they discuss several simplifications and their justifications: the oblateness of the sun and the presence of moons are probably interesting "future work" but are most likely not going to result in large corrections see their §3.2). For non-isolated systems, interactions with the environment can magnify whether these properties drive dynamical stabilization (merger) or instability (fly apart). The authors in their other fresh paper even more freshly summarized at <https://astrobites.org/2023/03/14/chaos-planets-future-of-the-solar-system/ https://astrobites.org/2023/03/14/chaos-planets-future-of-th...> discuss how "environmental events" (nearby stars and their potential to approach our solar system) might lead to e.g. collisions between Earh and Mars, or an ejection of one or more planets into deep space. Astrobites has also freshly summarized another "environment can change orbits" (for 2-body + dusty environment) at <https://astrobites.org/2023/03/13/will-it-merge-investigating-how-gas-disks-can-determine-the-future-of-a-binary-black-hole-system/ https://astrobites.org/2023/03/13/will-it-merge-investigatin...>. ("Dust" in the GR sense of <https://en.wikipedia.org/wiki/Dust_solution https://en.wikipedia.org/wiki/Dust_solution>, rather than in the sense of fine particles of e.g. chalk that allegedly accumulate on sedentary GR theorists). > Here's an experimental idea Been done (and improving) for many decades. See for example Fig. 1 in <https://arxiv.org/abs/1105.1082v1 https://arxiv.org/abs/1105.1082v1>.
antognini 4y ago
No, I don't think that follows. It is plausible that a system could remain stable under perturbations to the dynamical laws. In their case they find that there seems to be no critical threshold for GR on the stability of the system. Presumably if you strengthened GR somewhat rather than weakening it you would find increased stability. (Though I don't know that they ran that experiment.) I should note that you could make these changes two ways: you could just suddenly change the dynamical rules or you could gradually change them (a so-called "adiabatic" change). Making a sudden change is more likely to result in an instability. In this paper they were looking at adiabatic changes.
- raattgift 4y ago
  Please forgive what I suspect is something of a rant. I'm not comfortable with the wording "strengthened GR" because there's an implicit assumption that one can straightforwardly map the authors' time-dependent parameter on the perihelion procession (their eqs (1), (2), and (3) in <https://arxiv.org/abs/2303.05567 https://arxiv.org/abs/2303.05567>) into a relativistic theory at all. The experience of various alternatives to GR (and in particular MONDian alternatives) suggests that assumption may be a bad one. What they are doing is essentially exploring a family of theories of gravitation (which may or may not solve the Einstein Field Equations, or even a comparable set of equations) where each theory in the family produces a different evolution of the orbit of a planet. General Relativity is one of the theories in that family. But how would you even get from eqns 1 & 2 with the parameter fixed at unity (and not varying in time) to the EFEs? What do the Einstein and stress-energy tensors look like? I do sympathize with the implied (and even partially-stated in §3.1) idea that each theory in the family of theories maps to a different set of initial (stress-energy) values for the planet (ignoring backreaction onto the metric) in standard GR, but I wouldn't bet much on that being right. The authors write about the (and in particular "some critical") general relativistic precession rate (or frequency), and I don't mind that wording.
raattgift 4y ago
> nothing special about special relativity What's special about special relativity is that the spacetime is flat, i.e., there is no gravitation at all. This work is about the sun's gravity, i.e. the (weakly) curved spacetime sourced by the central mass of our solar system, and so is about general relativity and not special relativity. > change any of the underlying rules, it's going to become stable The relevant thing here is chaos. [italicizations below are mine] §3.1 in the authors' preprint at <https://arxiv.org/abs/2303.05567 https://arxiv.org/abs/2303.05567> makes the point: "because the systems are chaotic and we use slightly different general relativistic corrections in each simulation, the simulations diverge quickly. This is effectively the same as varying the initial conditions of one planet by a tiny amount". Quoting Wikipedia's Chaos Theory article: "Chaos theory is [...] focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions". (The initial conditions in question are the orbital parameters--or more broadly the momentum--of Mercury. The equivalence is fairly obvious, but I think I prefer to look it at similar to your wording, "... change any of the underlying rules ...", although not just any rule, they change a specific rule (or closely related family of rules, more on that below).) In the paper's §3.2 they identify the most prominent post-Newtonian contribution ("... the precession from GR (the gravito electric effect) .. follows closest behind Newtonian planet-planet interactions and is more than three orders of magnitude more influential than any additional effects"), and the paper in general focuses on the question of whether simulated planet-planet interactions' tendencies to eject Mercury from the solar system are dampened by GR's influence on the perihelion precession of Mercury. (Their answer: yes - Mercury sticks around for many billions of years even if the gravito-electric effect is small, but if it's zero, simulated Mercury is typically yeeted right out of simulated solar system in relatively few billions of years). More specifically they look at the evolution of Mercury's orbital eccentricity and use it as a proxy for Mercury's ejection <https://www.urbandictionary.com/define.php?term=Yeet https://www.urbandictionary.com/define.php?term=Yeet>. Mercury's orbit becomes more elliptical if over long periods of time its perihelion coincides with the perihelion of Jupiter. With practically any perihelion or ascending node precession of Mercury or Jupiter, Mercury's orbit remains roughly circular. Even more specifically in their eqn. (2) they parameterize the general relativity corrections to the planets' perihelion precessions, and vary the parameter (eqn (3)). Eqns. (1) & (2), with the parameter \alpha set to unity, are straightforwardly extracted from an approximate solution to the Einstein Field Equations (of General Relativity; the EFEs), but going the other way from a modification of the parameter \alpha to a solution of the EFEs is something reviewers or a paper in reply may question, since it's not as straightforward. (One might compare how (Milgrom's) MOND can be seen as a parametrization in the same spirit eqns (1) & (2), "simples", but nobody has completely reverse-engineered the parametrized equation back into a set of relativistic field equations. The authors touch on this at the bottom of the first page's second column.)

by causi 4y ago

infogulch 4y ago

That's a fair criticism. If a dynamical system is stable under one model, it's not that surprising if suddenly swapping out the model causes an instability. Basically, different models have different stability islands, if you would. But the question remains, how dependent on GR is the long term stability of an N-body system in general? Here's an experimental idea: create a "random N-body system" generator, and compare what % of systems are stable under Newtonian vs Einsteinian physics models.

raattgift 4y ago
> in general That's too broad to answer. Let's narrow to our solar system, or a close approximation, first. Below, I'll step up to more complicated arrangements, and binary black holes. These are only three or four families of (regions of) general curved spacetime; there is a very large infinity of other families one could reasonably associate with "in general". In a (model) solar system, there is a large central mass and several other objects with extreme mass ratios to the sun. There are interesting secular resonances among Mercury, Venus, and Jupiter, and between Earth and Mars, where that means that from time to time (over many orbits), the precessions of the near-to-the-sun parts of the planets' orbits match. These resonances may increase the eccentricity of the smaller mass's orbit, which can in extremis yeet the small body out of the solar system, or cause a collision between planets (e.g. Mercury's orbit may near aphelion cross Venus's orbit). For the Mercury-[Venus-]Jupiter the closeness of Mercury to the sun tends to break up the resonance enough to significantly stabilize the shape of Mercury's orbit. Any sort of slow-down on the inner body when it and heavier outer bodies are relatively close together with the sun is liable to suffice, and that's what the authors of the link at the top argue what we get with (good approximations of) GR. (The approximations used by the authors are a post-Newtonian expansion <https://en.wikipedia.org/wiki/Post-Newtonian_expansion https://en.wikipedia.org/wiki/Post-Newtonian_expansion> and numerical methods). Very roughly, thanks to the large central mass and the very different distances of the planets involved, the outer body rushes ahead or equivalently the inner body lags behind, which avoids stabilizing the resonance. Many -- possibly even most -- stars are in binaries, rather than singletons like our sun. Planetary orbits in a binary (or triple) star system are likely to be dramatically different. In binaries, the width and mass-ratio between the partners are relevant quantities. Triples can be arranged in all sorts of ways. PSR J0337+17 is one known arrangement <https://www.youtube.com/watch?v=oDgfqq_W_uM https://www.youtube.com/watch?v=oDgfqq_W_uM>. PSR B1620-26, a pulsar-white dwarf binary, has at least one planet. I have only weak intuitions about what Jupiter or Earth mass bodies would do in such systems, or even how to select a reasonably astrophysical set of initial orbital values. (There are known circumbinary planets. See <https://en.wikipedia.org/wiki/Circumbinary_planet https://en.wikipedia.org/wiki/Circumbinary_planet> and <https://sci-hub.ru/10.1007/978-90-481-8687-7 https://sci-hub.ru/10.1007/978-90-481-8687-7> in which Chapter 9 details how these can be simulated). It is not too fanciful however to suppose that periastrons could be close enough that post-Newtonian effects are non-negligible, even if the stellar masses are non-compact. That doesn't say much about whether secular resonances (in the sense above) are likely in such systems. I have no idea, but I'd guess they aren't forbidden. > compare what % of systems are stable under Newtonian vs Einsteinian [gravitation] That's what the authors do in their paper <https://arxiv.org/abs/2303.05567 https://arxiv.org/abs/2303.05567>, although "systems" here is a simplified model of the solar system (they discuss several simplifications and their justifications: the oblateness of the sun and the presence of moons are probably interesting "future work" but are most likely not going to result in large corrections see their §3.2). For non-isolated systems, interactions with the environment can magnify whether these properties drive dynamical stabilization (merger) or instability (fly apart). The authors in their other fresh paper even more freshly summarized at <https://astrobites.org/2023/03/14/chaos-planets-future-of-the-solar-system/ https://astrobites.org/2023/03/14/chaos-planets-future-of-th...> discuss how "environmental events" (nearby stars and their potential to approach our solar system) might lead to e.g. collisions between Earh and Mars, or an ejection of one or more planets into deep space. Astrobites has also freshly summarized another "environment can change orbits" (for 2-body + dusty environment) at <https://astrobites.org/2023/03/13/will-it-merge-investigating-how-gas-disks-can-determine-the-future-of-a-binary-black-hole-system/ https://astrobites.org/2023/03/13/will-it-merge-investigatin...>. ("Dust" in the GR sense of <https://en.wikipedia.org/wiki/Dust_solution https://en.wikipedia.org/wiki/Dust_solution>, rather than in the sense of fine particles of e.g. chalk that allegedly accumulate on sedentary GR theorists). > Here's an experimental idea Been done (and improving) for many decades. See for example Fig. 1 in <https://arxiv.org/abs/1105.1082v1 https://arxiv.org/abs/1105.1082v1>.

antognini 4y ago

No, I don't think that follows. It is plausible that a system could remain stable under perturbations to the dynamical laws. In their case they find that there seems to be no critical threshold for GR on the stability of the system. Presumably if you strengthened GR somewhat rather than weakening it you would find increased stability. (Though I don't know that they ran that experiment.) I should note that you could make these changes two ways: you could just suddenly change the dynamical rules or you could gradually change them (a so-called "adiabatic" change). Making a sudden change is more likely to result in an instability. In this paper they were looking at adiabatic changes.

raattgift 4y ago
Please forgive what I suspect is something of a rant. I'm not comfortable with the wording "strengthened GR" because there's an implicit assumption that one can straightforwardly map the authors' time-dependent parameter on the perihelion procession (their eqs (1), (2), and (3) in <https://arxiv.org/abs/2303.05567 https://arxiv.org/abs/2303.05567>) into a relativistic theory at all. The experience of various alternatives to GR (and in particular MONDian alternatives) suggests that assumption may be a bad one. What they are doing is essentially exploring a family of theories of gravitation (which may or may not solve the Einstein Field Equations, or even a comparable set of equations) where each theory in the family produces a different evolution of the orbit of a planet. General Relativity is one of the theories in that family. But how would you even get from eqns 1 & 2 with the parameter fixed at unity (and not varying in time) to the EFEs? What do the Einstein and stress-energy tensors look like? I do sympathize with the implied (and even partially-stated in §3.1) idea that each theory in the family of theories maps to a different set of initial (stress-energy) values for the planet (ignoring backreaction onto the metric) in standard GR, but I wouldn't bet much on that being right. The authors write about the (and in particular "some critical") general relativistic precession rate (or frequency), and I don't mind that wording.

raattgift 4y ago

> nothing special about special relativity What's special about special relativity is that the spacetime is flat, i.e., there is no gravitation at all. This work is about the sun's gravity, i.e. the (weakly) curved spacetime sourced by the central mass of our solar system, and so is about general relativity and not special relativity. > change any of the underlying rules, it's going to become stable The relevant thing here is chaos. [italicizations below are mine] §3.1 in the authors' preprint at <https://arxiv.org/abs/2303.05567 https://arxiv.org/abs/2303.05567> makes the point: "because the systems are chaotic and we use slightly different general relativistic corrections in each simulation, the simulations diverge quickly. This is effectively the same as varying the initial conditions of one planet by a tiny amount". Quoting Wikipedia's Chaos Theory article: "Chaos theory is [...] focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions". (The initial conditions in question are the orbital parameters--or more broadly the momentum--of Mercury. The equivalence is fairly obvious, but I think I prefer to look it at similar to your wording, "... change any of the underlying rules ...", although not just any rule, they change a specific rule (or closely related family of rules, more on that below).) In the paper's §3.2 they identify the most prominent post-Newtonian contribution ("... the precession from GR (the gravito electric effect) .. follows closest behind Newtonian planet-planet interactions and is more than three orders of magnitude more influential than any additional effects"), and the paper in general focuses on the question of whether simulated planet-planet interactions' tendencies to eject Mercury from the solar system are dampened by GR's influence on the perihelion precession of Mercury. (Their answer: yes - Mercury sticks around for many billions of years even if the gravito-electric effect is small, but if it's zero, simulated Mercury is typically yeeted right out of simulated solar system in relatively few billions of years). More specifically they look at the evolution of Mercury's orbital eccentricity and use it as a proxy for Mercury's ejection <https://www.urbandictionary.com/define.php?term=Yeet https://www.urbandictionary.com/define.php?term=Yeet>. Mercury's orbit becomes more elliptical if over long periods of time its perihelion coincides with the perihelion of Jupiter. With practically any perihelion or ascending node precession of Mercury or Jupiter, Mercury's orbit remains roughly circular. Even more specifically in their eqn. (2) they parameterize the general relativity corrections to the planets' perihelion precessions, and vary the parameter (eqn (3)). Eqns. (1) & (2), with the parameter \alpha set to unity, are straightforwardly extracted from an approximate solution to the Einstein Field Equations (of General Relativity; the EFEs), but going the other way from a modification of the parameter \alpha to a solution of the EFEs is something reviewers or a paper in reply may question, since it's not as straightforward. (One might compare how (Milgrom's) MOND can be seen as a parametrization in the same spirit eqns (1) & (2), "simples", but nobody has completely reverse-engineered the parametrized equation back into a set of relativistic field equations. The authors touch on this at the bottom of the first page's second column.)