I'm not clear about this. I've always assumed (and I could be very mistaken) that "time reversibility" is just a quirk arising when describing a physical process using mathematics. The two are not the same.

"And they have shown that the problem is not with the simulations after all."

Well, they're doing computer simulations in an environment of exceptional chaotic behavior. So I don't know what to think about reversing the actions.

Time for a real physicist to chime in with their opinions. Beyond me. :chin:

This user has been deleted and all their posts removed.

Time for a real physicist to chime in with their opinions — jgill

(y) I was hoping

Just Three Orbiting Black Holes Can Break Time-Reversal Symmetry, Physicists Find
Michelle Starr
ScienceAlert
Mar 2020

Nifty work.
At some point "micro chaos" "bubbles up" to give temporal irreversibility, albeit in a round-about way here. — jorndoe

Thats really cool. We've all seen the common grid 3d example of how general relativity works and i've always wondered how that example would work with more objects coming from different directions.

I'm an amatuer arm chair physicist like most people here, but i feel that going back in time is impossible without some wacky and "religious" solutions.

Thanks for the post.

Awesome.

Thanks, this is interesting!

Here is the full paper: Gargantuan chaotic gravitational three-body systems and their irreversibility to the Planck length

I'm not clear about this. I've always assumed (and I could be very mistaken) that "time reversibility" is just a quirk arising when describing a physical process using mathematics. The two are not the same.

"And they have shown that the problem is not with the simulations after all."

Well, they're doing computer simulations in an environment of exceptional chaotic behavior. So I don't know what to think about reversing the actions. — jgill

Physics enters the picture when they show that in some fraction of initial configurations the sensitivity to initial positions is so high that a displacement of a magnitude less than the Planck length can result in divergent solutions. They interpret this result as the system being "fundamentally unpredictable" when it starts from one of those configurations.

Mathematically, if we don't take into account the Planck length limitation, the system is still only chaotic at most, and therefore fully time-reversible.

By the way, the choice of supermassive black holes is only for astronomical verisimilitude, because in their solution they still use the Newtonian approximation, as in the classic n-body problem.

As a concrete application of our result, we consider three black holes, each of a million solar masses, and initially separated from each other by roughly one parsec. Such a configuration is not uncommon among supermassive black holes in the concordance model of cosmology and hierarchical galaxy formation... [W]e estimate that the closest approach between any two black holes is on average between 10^-2.5 and 10^-2 parsec, during which the Newtonian approximation still holds. A parsec equals 10⁵¹ Planck lengths. Hence... we estimate that up to 5 percent of triples with zero angular momentum are irreversible up to the Planck length, thus rendering them fundamentally unpredictable. — Boekholt et al.

I haven't read the article yet, but given that according to Noether's theorem time reversibility is equivalent to conservation of energy, does this result not imply a violation of conservation of energy? If it does, is energy being created, or destroyed? (NB that it is already known that conservation of energy does not hold universally; dark energy violates it already).

Classically, the three-body problem is time-reversible, and this result doesn't prove otherwise. Indeed, qualitatively this result doesn't prove anything new: the three body problem is already known to be chaotic in the technical sense, i.e. for a certain class of initial initial conditions, any disturbance, no matter how small, results in an unbounded divergence over time.

The direct result of this work pertains to numerical simulations of a class of three-body problems. The claimed physical relevance comes from making numerical errors (which act as perturbations) smaller than the Planck length.

From: https://royalsocietypublishing.org/doi/full/10.1098/rsta.2015.0161

"This fact leads to a paradox if one ponders the reversibility and predictability properties of quantum and classical mechanics. They behave very differently relative to each other, with classical dynamics being essentially irreversible/unpredictable, whereas quantum dynamics is reversible/stable."

"Although, the dynamics are predictable and reversible with an exact representation of the state of the system and an exact implementation with a perfectly precise set of equations of motion, any deviation of either leads to exponential instability in the predictability of the reversed dynamics."

Chaotic systems can arise from SDIC: sensitive dependence on initial conditions. Reversibility seems far-fetched to me even though it can be theoretically possible, having worked with deterministic systems in the complex plane. But I lack the credentials to speculate about the physical world.

I think it's more that if you want to predict the position of something based on whatever variables you have, and you perturb the position variables by the plank length, you can no longer tell which trajectory - what path the bodies trace over time - you're on.

They ran the system of planets for a while, until t say. Then they perturbed the planet positions at t by the plank length, then they tried to run the system back in time, and got a different end point (or more generally evaluated trajectory) 5% of the time.

The climate system is similar. People perturb the weather variables by 10^-64 or 10^-32 (either the plank length ^2 or 100 times the plank length), and they don't get the same results within days or weeks. Even if the underlying laws are time reversible, the system iterations might amplify any perturbation so much what trajectory you're on changes beyond an error thresh-hold.

Yeah I gather that, but if the perturbations are in principle unobservable because they're below the Planck length, it still seems like you could end up with something that effectively violates conservation of energy.

It's like if you're doing some accounting and you keep rounding off to the nearest cent at every step, you might end up with a net excess or deficit of money at the end. (Think Superman 3, or Office Space). The Planck limit is like rounding off of decimal representations of the otherwise continuous laws. If you can change the system by "zero" (less than half a penny / less than a Planck length) and then get a noticeable difference later, it seems like you got or lost something along the way.

From Wiki:

"In mathematics, a dynamical system is time-reversible if the forward evolution is one-to-one"

(most of the functions I've used fail to be univalent, so irreversible)

"[In physics] T-symmetry implies the conservation of entropy. Since the second law of thermodynamics means that entropy increases as time flows toward the future, the macroscopic universe does not in general show symmetry under time reversal."

There could be a thread on the concept of time-reversibility. There seems to be a slight conflation here between forward and backward dynamics.

You know about time-energy uncertainty, right? It is less straightforward than the other Heisenberg uncertainties, but it is a feature of quantum mechanics. However, the uncertainty only manifests on the quantum scale; on the classical scale it averages out.

Yeah I'm familiar with that (I think it's actually easier to understand in terms of the underlying wave mathematics than position-momentum uncertainty is, honestly). I'm not sure what that has to do with this though?

It has to do with your worry about energy conservation due to Heisenberg uncertainty. Not much with "this" if by "this" you mean the OP.

There could be a thread on the concept of time-reversibility. There seems to be a slight conflation here between forward and backward dynamics. — jgill

A while ago we had a thread on Norton Dome - a simple Newtonian setup that (arguably) gives rise to indeterministic (and therefore irreversible) behavior. Classical mechanics allows for some edge cases where such things can happen. This is distinct from chaotic behavior (which is, in a technical sense, reversible) and also from the second law of law of thermodynamics, which is decidedly irreversible.

I wasn't worried about energy conservation due to Heisenberg uncertainty.

This is distinct from chaotic behavior (which is, in a technical sense, reversible) — SophistiCat

Explain what you mean by "which is, in a technical sense, reversible". Please provide a reference.

"In mathematics, a dynamical system is time-reversible if the forward evolution is one-to-one"

QM effects are already non-reversible...

I suspect that this is a precision related problem that manifests between quantum granularity and the continuous spectra Newtonian mechanics (and relativity) approximates the system as...

Perhaps one way to state this is that the magnitudes of forces (and the differences in them which are caused by smaller and smaller deviations in the evolving positions of the black holes) become extreme enough in this case that quantum states of underlying particles can have lasting effects, where otherwise they seem to cohere or converge toward linear behavior at great time and scale.

QM effects are already non-reversible... — VagabondSpectre

From Physics StackExchange: Reversibility in Physics

" The point is that you can't focus on the particle alone and have reversibility. If you focus on the particle alone in a measure process, then the process is irreversible. On the contrary, if you consider the whole system "particle + measure instrument", the dynamics is reversible. If after the measurement the whole system is described by the product of a state for the particle and one for the instrument (unentangled), using the reversible evolution of the whole system backwards you get the original complete state before measurement. Of course such state is usually entangled."

This topic is becoming increasingly complicated. And there are various definitions of chaotic behavior. Plus there are differences between physics and mathematics. :roll:

(I know you know what floating points are, but many readers will not)

They took a classical system; the three body problem but with black holes; evolved it forward in time from t = 0 to t = t. They then took the position of the objects in the system at t and perturbed them by the Plank length. They then ran the system backwards in time, using the same equations, from the perturbed position at t. They looked at how often the forward time trajectory wasn't just replicated, but backwards, and found it to be 5% of the time.

What this says is. quoting the article:

The movement of the three black holes can be so enormously chaotic that something as small as the Planck length will influence the movements," Boekholt said. "The disturbances the size of the Planck length have an exponential effect and break the time symmetry.

.

The original paper describes its methodology as:

The main idea of our experiment is the following. Each triple system has a certain escape time, which is the time it takes for the triple to break up into a permanent and unbound binary-single configuration. Given a numerical accuracy, , there is also a tracking time, which is the time that the numerical solution is still close to the physical trajectory that is connected to the initial condition. If the tracking time is shorter than the escape time, then the numerical solution has diverged from the physical solution, and as a consequence, it has become time irreversible. Only the systems with the smallest amplifications factors will pass the reversibility test. However, by systematically increasing the numerical accuracy (decreasing epsilon), we aim to increase the tracking time of each system. An increasing fraction of systems will obtain a tracking time exceeding its escape time, thus gradually decreasing the fraction of irreversible solutions

Key points:

(1) There's a numerical accuracy parameter; the numerical accuracy parameter arises because computers don't just store numbers like pi exactly like we can write down symbols for on paper, they store floating point approximations of them, and there's always some error. For example, if you can only represent 3 decimal places, the number 0.00099999 will be 0, despite it being close to 0.001. The computers only have so many bits to represent the number with.

(2) The numerical accuracy parameter can be varied with the simulation. The numerical accuracy parameter encodes how precisely all the variables in the system can be represented. If the numerical accuracy parameter for giving directions to your neighbour was 100km, you could give them "accurate" directions within that error-threshold by telling them that you live in the city you live in.

(3) A "solution" of these equations defining the three body system is a numerical solution; all the calculus and numbers are represented by computer approximations using these error prone floating points and functions taking floating points and spitting out floating points. (The scarequotes are not to say this is an illegitimate procedure, the scarequotes are to distinguish a numerical solution from an analytic - pen and paper - one, the analytic one is exact).

They highlight this distinction in the discussion:

In the limit of infinite accuracy (epsilon → 0) we retrieve the microscopic time-reversibility of Newton’s equations of motion. In the presence of perturbations of size epsilon, whether numerical or physical, a fraction of systems becomes irreversible...

Super take home message: the laws as you'd write them down on paper are time reversible.

(4) (1-3) are just maths, the next bit is physics. Something physically interesting would be if the accuracy of the numerical approximations broke down badly sometimes at a physical length scale; like say the diameter of an atom, or the Planck length.

(5) How to establish this? When they run the equations backwards from the end point - using the end point as a new start point - with perturbation of the order of 10^-74 for the system's motion relevant variables (position, velocity, acceleration at a time point it looks like); they receive a result which is statistically indistinguishable from the unperturbed one. But there are negligible differences, "micro differences" as the paper puts it, what this means is that for every forward run, there are a collection of backward runs which are negligibly different from it.

(5) What if we put in an error thresh-hold of the Planck length, run the system forward, then perturb its state by the plank length (approx 10^-32) and counted what proportion of times the system outputs a backward trajectory which is statistically distinguishable from the one we get from retracing the steps from the forward iterations exactly? Turns out this is 5%.

The paper does not discuss any QM effects, and whether there is anything physically meaningful about this result; the Planck-length result is merely suggestive of something significant without specifying what it is. As some speculation about what it is, the paper shows that even if you represent a very chaotic system's dynamics exactly, if there is some underlying "uncertainty" or "fluctuation" in the exact state variables, it will pull trajectories apart - it turns out that this is true 5% of the time for the three body system and the Planck length, which every material thing is presumably bigger than.

The paper's methodological take home message seems to be more about measuring how chaotic systems are numerically using this % of irreversible trajectories as a function of the error thresh-hold.

But fewer people would care about the paper if it didn't suggest (with plausible deniability in that typical academic way) that it has something to say about time irreversibility of physical/natural trajectories as opposed to time irreversibility of numerical algorithms representing them.

Edit: something this post didn't cover, and maybe suggests wrongly, is that the weird irreversibility can be blamed on the researchers' implementation/code; and that's wrong. The time irreversibility is framed as a feature of the system (under the numerical algorithms), not a bug. The whole graph representing "what proportion of simulations turn out to be time irreversible at this error threshold" is a system property, the system (or the equations defining it) imbues the numerical approximations with properties like that.

Explain what you mean by "which is, in a technical sense, reversible". Please provide a reference. — jgill

I won't hunt for a reference, but as I understand it, a reversible system would pass a reversibility test: Allow the system to evolve for some time T, then reverse the time direction of all dynamical properties (flip the direction of all velocities, moments, etc.) and allow the system to evolve further for the same amount of time T. A reversible system would end up in the same state from which it started, but with all of its dynamical properties in reverse.

"In mathematics, a dynamical system is time-reversible if the forward evolution is one-to-one" — jgill

Well, your wiki reference gives rather more succinct definitions, though they may require some unpacking.

Helpful post, thanks. Possible typos:

t = 0 to t = t. They they — fdrake

t = 0 to t = T?

They then? Obvs.

If you can only represent 3 decimal places, — fdrake

Fixed not floating, then? (And fixed by Planck length, if I understand you.)

Fixed not floating, then? (And fixed by Planck length, if I understand you.) — bongo fury

It was an example. I edited it with "for example". Thanks.

The internet told me floating was sig figs not dp?

The internet told me floating was sig figs not dp? — bongo fury

It is. I changed to DP because it seemed easier to talk about rounding errors in that context.

Yeah fine. They both cause rounding errors, but Planck length would be a dp thing rather than sf?

The idea insofar as they concern rounding errors is similar in both cases.

For floating points, you have a given number of bits that represent significant digits. Then a given number of bits that shift the significant digits up and down (like powers of ten shift digits up and down their tens/hundreds etc columns). If you can only have 3 numbers to represent significant digits, and in reality you get the number 1199, you'll lose the last 9 if the rounding occurs by truncation, and it'll be 1190. If it occurs by some other method, you might get 1200.

Planck length would be through fixed rather floating? — bongo fury

I have no idea how they have implemented the Planck length. I guess that they represent it with a floating point number. If your intuition is that the Planck length is represented as fixed because it is a physical constant, I don't think that holds, because numbers like pi are often given floating point representations. I don't think the Planck length is known exactly anyway.

But fewer people would care about the paper if it didn't suggest (with plausible deniability in that typical academic way) that it has something to say about time irreversibility of physical/natural trajectories as opposed to time irreversibility of numerical algorithms representing them. — fdrake

In the conclusion the authors also try to present their work as being relevant to astronomy, but it should be noted that the problem that they actually consider is a very, very special case of the three-body problem, which is notoriously difficult to treat in any general way. They consider three equal masses in free fall with no initial velocities (which also makes this a planar problem, unlike the more general case, which is 3D).

One fact is here taken for granted, and I wonder whether this is a necessary outcome in this setup, or whether this is an additional assumption: after some time the system ejects one body that flies off into the infinite distance, leaving behind a binary system. This is a dramatic transition in the system's dynamics, which helps understand the criterion of "irreversibility" that they use:

The main idea of our experiment is the following. Each triple system has a certain escape time, which is the time it takes for the triple to break up into a permanent and unbound binary-single configuration. Given a numerical accuracy, , there is also a tracking time, which is the time that the numerical solution is still close to the physical trajectory that is connected to the initial condition. If the tracking time is shorter than the escape time, then the numerical solution has diverged from the physical solution, and as a consequence, it has become time irreversible.

Since such an escape happens more-or-less stochastically, if your simulation doesn't track its onset closely, then from that point on it will quickly diverge from reality, and the error will only increase over time.

If your intuition is that the Planck length is represented as fixed because it is a physical constant, — fdrake

More like, that some error threshold imposed by the Planck length is represented as fixed.

What if we put in an error threshold of the Planck length, — fdrake

That would be a dp thing rather than sf?