Dark matter and energy explained by negative mass

[PeterDonis]

The word "repel" suggests the assumption that masses tend to accelerate in a way that tends to reduce potential energy

Yes, it does, and if inertial masses are all positive that is in fact what will happen. But, as you note, if particles with negative gravitational mass also have negative inertial mass, they will move to increase potential energy, not reduce it. Which seems to me to be yet another serious problem for Farnes' model.

 

[PeterDonis]

The paper claims that acceleration up to light speed is possible because the overall mass of the system is 0.

I don't buy this either; I think it's just unjustified hand-waving (which quite a few of the claims in the paper seem to me to be). The individual particles are either massless or they're not. If they are, then they can't have negative mass and the whole model breaks down. If they aren't, then they can't move at the speed of light.

 

[Auto-Didact]

(2) The paper does not actually derive the dynamics from a field equation; Farnes just puts in by hand the dynamics the way he thinks they should be. But this means the model might not be consistent; and in fact it does not appear to be.

After having read all the comments there, I see Hossenfelder saying "You cannot start from a Lagrangian and then just postulate what you want to happen in the Newtonian limit", honestly stating afterwards that she might be wrong but will not accept Farnes idea without a full derivation. That is understandable of course, since doing this has become somewhat standard methodology in theoretical physics in the last century.

Having said that, it is very important to realize that settling for nothing less than a Lagrangian formulation or other type of first principles derivation is actually possibly a too strong focus on a very particular methodology, since it is not the only possible methodology for a theorist to construct a novel theory; phenomenological modelling, i e. putting things in by hand and then simply comparing the results to experiments, is another valid methodology of theorization.

It is therefore largely unjustified to say that a theorists' theorization is unscientific or 'not proper physics' simply because the theorist uses phenomenological modelling instead of giving a first principles derivation, especially if this other methodology has proven to be successful for theorization as phenomenological modelling of course has been in countless cases; for theories in some large scale limit (e.g. hydrodynamics) it actually isn't even directly clear whether a first principles derivation such as a Lagrangian formulation is necessary or even wholly appropriate.

The fact remains that historically most of physics was initially modeled phenomenologically with a first principles derivation only following later (e.g. Newton followed by Lagrange/Hamilton, Faraday followed by Maxwell, Planck followed by Dirac et al., etc). Moreover, both in fluid dynamics and in nonlinear dynamics, i.e. the proper context which Farnes' equations are actually from, phenomenological modelling is still the standard theorization methodology.

Farnes does this pretty well by directly picking up a historically abandoned line of research by Einstein - a different interpretation of GR - which was subsequently made respectable by Bondi and procedes to logically build the new case naturally rederiving known results (Eq. 15). Farnes procedes to not only give simulations which qualitatively match observations, but also a brief review reinterpreting known empirical data reformulated based on new Bayesian priors.

Moreover, Farnes directly gives a host of falsifiable predictions. The correct next step in research based on phenomenological modelling is for others to try reproducing his simulations and then doing a statistical comparison with observational data. If the theory is consistent with the data, then others will naturally start to chew much more on his equations, which is about when I'd advise him to start worrying a bit more about actually trying to give a full derivation from first principles.

 

[PeterDonis]

phenomenological modelling, i e. putting things in by hand and then simply comparing the results to experiments, is another valid methodology of theorization.

Proposing a phenomenological model that does not have any known basis as an approximation to some underlying theory derived from first principles is one thing; yes, that's often done in science.

Proposing a phenomenological model that appears to violate properties that are believed to be essential to even have a basis as an approximation to some underlying theory derived from first principles is something else. Doing that, I think, is much more risky and much harder to defend on the grounds that you're only trying to construct a phenomenological model and the details of the underlying theory can catch up later.

Farnes does this pretty well by directly picking up a historically abandoned line of research by Einstein - a different interpretation of GR

Where are you getting this from?

Farnes directly gives a host of falsifiable predictions.

Yes, and at least one--the "runaway" solutions--is arguably already falsified.

 

[Auto-Didact]

Proposing a phenomenological model that appears to violate properties that are believed to be essential to even have a basis as an approximation to some underlying theory derived from first principles is something else. Doing that, I think, is much more risky and much harder to defend on the grounds that you're only trying to construct a phenomenological model and the details of the underlying theory can catch up later.

It is risky, but because this approximation is based on Newtonian field theory - our old and sometimes forgotten friend - his interpretation makes intuitive sense even if it won't end up working; I'd prefer if he directly derived it on the basis of the Newton-Cartan formalism.

Where are you getting this from?

Farnes' paper (Einstein 1918)

Yes, and at least one--the "runaway" solutions--is arguably already falsified.

That's good, I'm all for quick falsification, as long as the necessary care is taken. i.e. in this case that the model is treated as a preliminary theory in nonlinear dynamics. Since it seems to address the two most important separate core issues simultaneously (nonlinear dynamics & open system non-equilibrium statistical mechanics) it is extremely interesting mathematically, regardless of its physical correctness.

In any case, runaway doesn't seem to be typical in any of his simulations, nor does it increase when varying the dimensionless group which might affect it; this is to be expected given that the geometrodynamics is highly nonlinear.

Moreover, this is to be expected intuitively as well since runaway only occurs in the highly idealized situation of identical dislikes colliding symmetrically, i.e. in the naive 'particle physics setting'.

Given the above, there actually isn't that much reason to believe that runaway should be too big of a problem for the model, since the negative masses might also just be effective negative masses, as Farnes himself also says.

What I'd actually be more worried about is that creation tensor; I've seen similar constructions though, it would be very interesting to see if those were really mathematically consistent with Farnes' proposal.

 

[LURCH]

and at least one--the "runaway" solutions--is arguably already falsified.

If we then discard the idea of runaway pairs, does the proposed theory fall apart? I would contend that the creation of runaway pairs would be an oddity, rather than a necessary prediction from this model, and that the rest of the concept could continue to exist, even if (as has been proposed) this particular phenomenon is not possible. Perhaps the conditions required to generate these pairs really are unattainable, and so they cannot form in any real-world situation. What does that leave us with?

I have been doing some more thinking about galactic collisions, and have expanded that thinking to include cluster and supercluster formation. I have not yet run the simulator program that Farnes created, but I’m having trouble understanding how, under the proposed conditions, any two galaxies could be gravitationally drawn toward each other at all. If all galaxies are more than 80% negative mass, then they can fairly be described as negative mass structures, with traces of impurities. It seems that they should all be repelling one another.

EDIT: Auto-Didact posted his while I was still typing.

 

[Auto-Didact]

I have been doing some more thinking about galactic collisions, and have expanded that thinking to include cluster and supercluster formation. I have not yet run the simulator program that Farnes created, but I’m having trouble understanding how, under the proposed conditions, any two galaxies could be gravitationally drawn toward each other at all. If all galaxies are more than 80% negative mass, then they can fairly be described as negative mass structures, with traces of impurities. It seems that they should all be repelling one another.

Intuitively, I would say by Landau damping with galaxies as the analogues of electrons.

 

[LURCH]

Intuitively, I would say by Landau damping with galaxies as the analogues of electrons.

An interesting notion. Will need to ponder it a while.

Meanwhile, the more I think about it, the more problems are eliminated by discarding the runaway pairs. Reactionless propulsion, perpetual motion, light speed travel; they all go away. Now I only need to cope with spontaneous creation ex nihilo, and I’m practically on board with this!

Except for the galactic collisions, of course.

 

[Elroch]

If this is true, then it would be very badly misstated.

Yes, the phrasing I quoted exactly should not be in the paper.

Aren’t these the exact conditions necessary for runaway pairs? If so, are you pointing out that there can’t be any such phenomena?

It is not something that will happen in the real world at all because it require the two particles to have exactly opposite masses and exactly identical velocities, exactly aligned with their relative position. This happens with probability zero. So what matters are the less extreme interactions where masses don't agree and velocities are distributed widely.

Yes, and it is another difficulty I am having with this concept. Farnes says that these pairs can get up to light speed because they are massless. But if that is true, then they cannot move at sub-light speeds, so they can’t “accelerate to” light speed. However, if they do exist and do accelerate, then it would seem that their rate of acceleration must be infinite. When mass is zero, acceleration is infinite, isn’t it? I suppose this could mean that the acceleration is instantaneous; meaning that the speed is c as soon as the pair is spawned. This would avoid the problem of massless particles moving at sub-light speed.
.

You don't need to be concerned because even with the perfect conditions for runaway pair, they merely accelerate indefinitely, never reaching the speed of light.

 

[Elroch]

Yes, it does, and if inertial masses are all positive that is in fact what will happen. But, as you note, if particles with negative gravitational mass also have negative inertial mass, they will move to increase potential energy, not reduce it. Which seems to me to be yet another serious problem for Farnes' model.

This was a surprise to me and I am concerned about it too, but not to the extent I would be if, say, it implied that entropy decreased! For it to be a genuine problem for Farnes' ideas we need it to lead to some inconsistency. It would be good to look at this from the point of view of the principle of least action and Lagrangians.

 

[Elroch]

That might be necessary if one actually worked out a quantum field theory that had the phenomenological model in Farnes' paper as a Newtonian approximation, yes. But Farnes has certainly not done anything to work out such a theory. (Nor do I think one could be worked out consistently, but that's probably off topic here.)

In physics we presently deal with gravity as a classical phenomenon acting on particles that are quanta of field theories and the same would be true here. We don't have a quantum gravity theory, so it would be unreasonable to object to Franes not having one for gravity extended to negative masses: this is a separate matter to an analogous theory to the known field theories for normal matter. At least we do know that there are theories analogous to what I describe for the interaction of many or all of the particles in the standard model, all expressible in Feynman diagrams.
We are as much in the dark about the physics of dark matter as about Franes' hypothetical substance.

 

[PeterDonis]

this approximation is based on Newtonian field theory

That doesn't address the issue I raised at all. Neither would deriving it from Newton-Cartan theory (if it could be done).

Farnes' paper (Einstein 1918)

This refers to Einstein's postulating a positive cosmological constant in order to have a static universe. Farnes claims that "negative mass" can have the same effect as a negative cosmological constant. Also, as has already been commented here, this claim does not appear to be correct.

 

[PeterDonis]

If we then discard the idea of runaway pairs, does the proposed theory fall apart?

You can't "discard" the idea. It's a prediction of the model. As Hossenfelder points out in her article, if you want your theory to be consistent with GR, you can't choose the mass and independently choose the dynamics. The mass determines the dynamics.

 

[PeterDonis]

It is not something that will happen in the real world at all because it require the two particles to have exactly opposite masses and exactly identical velocities, exactly aligned with their relative position

That is how "runaway" pairs are described in the paper, in order to make the idea intuitively plausible. But the actual interaction Farnes postulates between negative and positive mass particles will cause their motion to approach that exact "runaway" solution regardless of their initial relative velocity (as long as that velocity is small, which it should be under the postulated conditions in the paper), if they approach each other closely enough for their two-body interaction to dominate their dynamics (which should happen often under the postulated conditions in the paper).

 

[PeterDonis]

We don't have a quantum gravity theory, so it would be unreasonable to object to Franes not having one for gravity extended to negative masses

I'm not talking about a quantum gravity theory. I'm talking about a quantum field theory, on a classical background spacetime, that has a field with negative mass in it. As Hossenfelder points out, any such theory makes the vacuum unstable. This has been a well known property of standard QFTs for decades.

At least we do know that there are theories analogous to what I describe for the interaction of many or all of the particles in the standard model, all expressible in Feynman diagrams.

None of the Standard Model particles fields have negative mass. That's not just a coincidence; it's true for a good reason (see above).

 

[Auto-Didact]

That doesn't address the issue I raised at all. Neither would deriving it from Newton-Cartan theory (if it could be done).

I didn't say it did, I'm only saying that in order to obtain a first principles derivation, it might be easier to proceed backwards towards a modified GR Lagrangian by unlimiting from a modified Newton-Cartan theory than from a modified Newtonian theory alone.

This refers to Einstein's postulating a positive cosmological constant in order to have a static universe. Farnes claims that "negative mass" can have the same effect as a negative cosmological constant. Also, as has already been commented here, this claim does not appear to be correct.

I'll post the entire paper, in order to make the point clearer:

When I wrote my description of the cosmic gravitational field I naturally noticed, as the obvious possibility, the variant Herr Schrödinger had discussed. But I must confess that I did not consider this interpretation worthy of mention.

In terms of the Newtonian theory, the problem to be solved can be phrased more or less as follows. A spatially closed world is only thinkable if the lines of force of gravitation, which end in ponderable bodies (stars), begin in empty space. Therefore, a modification of the theory is required such that “empty space” takes the role of gravitating negative masses which are distributed all over the interstellar space. Herr Schrödinger now assumes the existence of matter with negative mass density and represents it by the scalar $p$. This scalar $p$ has nothing to do with the internal pressure of “really” ponderable masses, i.e., the noticeable pressure within stars of condensed matter of density $\rho$ ; $\rho$ vanishes in the interstellar spaces, $p$ does not.

The author is silent about the law according to which $p$ should be determined as a function of the coordinates. We will consider only two possibilities:
1. $p$ is a universal constant. In this case, Herr Schrödinger’s model completely agrees with mine. In order to see this, one merely needs to exchange the letter $p$ with the letter $\Lambda$ and bring the corresponding term over to the left-hand side of the field equations. Therefore, this is not the case the author could have had in mind.

2. $p$ is a variable. Then a differential equation is required which determines $p$ as a function of $x_1... x_4$ . That means, one not only has to start out from the hypothesis of the existence of a nonobservable negative density in the interstellar spaces but also has to postulate a hypothetical law about the space-time distribution of this mass density.

The course taken by Herr Schrödinger does not appear passable to me, because it leads too deeply into the thicket of hypotheses.

Farnes is taking this pathway, as is clear in Eqs. 12, 13, 14, 15, 23 and 29 among others.

I'm not talking about a quantum gravity theory. I'm talking about a quantum field theory, on a classical background spacetime, that has a field with negative mass in it. As Hossenfelder points out, any such theory makes the vacuum unstable. This has been a well known property of standard QFTs for decades.

When speaking about relativistic gravity, QFT is frankly speaking completely irrelevant because it is fundamentally incapable of dealing with the concept of multiple vacua which are necessarily there per the equivalence principle in geometrodynamics; this has already been shown using the Newton-Cartan formalism. If QFT is fundamentally unable to accurately handle positive gravitational masses, why would you expect the situation to change for negative masses?

 

[Elroch]

That is how "runaway" pairs are described in the paper, in order to make the idea intuitively plausible. But the actual interaction Farnes postulates between negative and positive mass particles will cause their motion to approach that exact "runaway" solution regardless of their initial relative velocity (as long as that velocity is small, which it should be under the postulated conditions in the paper), if they approach each other closely enough for their two-body interaction to dominate their dynamics (which should happen often under the postulated conditions in the paper).

Why would you think that? I have just written a little simulation where the velocities do not perfectly match to start off with, and the particles steadily drift apart over time as I would have guessed. Note that the two particles are not bound: intuitively, as they get further apart, the negative mass loses potential energy and the positive one gains it, and the two effects balance perfectly.
I summarised somewhere that when the net mass of a two particle system is positive, it can be bound and when it is negative it never is. When the total mass is zero only a neutral equilibrium is possible, not a stable one.

 

[PeterDonis]

I have just written a little simulation where the velocities do not perfectly match to start off with, and the particles steadily drift apart over time as I would have guessed

I'm not sure why you would guess that. See below.

intuitively, as they get further apart, the negative mass loses potential energy and the positive one gains it

Potential energy is not a property of either particle in isolation; it's a property of the two-particle system. As the two particles get further apart, the potential energy of the two-particle system decreases for this case (whereas for two masses of the same sign, it increases).

If we assume that the negative gravitational mass particle also has negative inertial mass, then it will accelerate in the direction of increasing potential energy, i.e., towards the positive mass particle. The positive mass particle will accelerate in the direction of decreasing potential energy, i.e., away from the negative mass particle. Since the potential depends on the distance between them, the two particles will have identical accelerations: same magnitude, same direction. If they have a nonzero relative velocity to start off with, that will not affect this fact. So I would not expect the particles to steadily drift apart, since if they get further apart, the acceleration of both decreases.

I can't comment on your simulation since I haven't seen the source code. The two-body problem should be solvable analytically, but it might take a little time before I can try that.

 

[andrew s 1905]

PeterDoins and Elorch , I don't think you will resolve your difference with words. Why not write down the equations and assumptions and or constraints you are implicitly using in your verbal description of what will happen then it should be possible to clear up the issue one way or the other.

Regards Andrew

 

[Davephaelon]

Per my comment on Sabine Hossenfelder's Backreaction blog at 10:40 AM, December 9, 2018, using the analogy of Pac-man icons eating each other should negative-matter in a galactic halo meet the positive matter in the galactic disc, this would seem to afford a potential way to test Jamie Farnes model. Assuming the negative-energy matter in his model is constituted of the same baryonic matter that makes up our positive-energy matter, with the energy sign reversed, and the negative-energy electrons and protons generally don't coalesce into atoms due to mutual gravitational repulsion, then numerous free, negative-energy electrons and protons might be passing through the Milky-Way's galactic disc gobbling up their positive-energy counterparts. Further assuming that on a time averaged basis that the disappearance of electrons and protons in a chunk of positive-energy matter is not perfectly balanced, then for some period of time that chunk of positive-energy matter would not be electrically neutral. By conducting an experiment similar to the search for fractional charges in bulk matter, it might be possible to check for occasional, tiny charge imbalances in an aggregate of ordinary matter in a carefully controlled laboratory setting.

 

[Elroch]

I'm not sure why you would guess that. See below.

Potential energy is not a property of either particle in isolation; it's a property of the two-particle system. As the two particles get further apart, the potential energy of the two-particle system decreases for this case (whereas for two masses of the same sign, it increases).

If we assume that the negative gravitational mass particle also has negative inertial mass, then it will accelerate in the direction of increasing potential energy, i.e., towards the positive mass particle. The positive mass particle will accelerate in the direction of decreasing potential energy, i.e., away from the negative mass particle. Since the potential depends on the distance between them, the two particles will have identical accelerations: same magnitude, same direction. If they have a nonzero relative velocity to start off with, that will not affect this fact. So I would not expect the particles to steadily drift apart, since if they get further apart, the acceleration of both decreases.

I can't comment on your simulation since I haven't seen the source code. The two-body problem should be solvable analytically, but it might take a little time before I can try that.

It doesn't really matter why I would guess the particles continue to drift apart over time if their initial relative velocities are not equal: it's a fact, as my elementary finite difference simulation, based on the definitions of the forces and Newton's laws (with negative masses) confirms. This is important, as my guesses cannot be relied upon! The line in your reasoning which is misleading is "So I would not expect the particles to steadily drift apart, since if they get further apart, the acceleration of both decreases." The confusion here is between the acceleration of the centre of the particles (which does decrease as they get further apart) and their relative velocity, which is unaffected by that. The drifting apart is solely due to the initial relative velocity persisting in this special case where the sum of the masses is zero.

One valid way to reason about potential energy is to work in a frame centred on the midpoint between the two particles and to observe that the net mass is like a mass of zero at that point, with the distance to either of the two masses providing the other co-ordinate. This is a co-ordinate transformation similar to that used for modelling two body systems like the hydrogen atom. Given that the radial forces on the two particles are equal and opposite (instantaneous Newton's third law or conservation of momentum), the rate of change of potential energy with their common distance from their centre (dV/dr, say) is zero (for two separate reasons!)

Anyhow, working in this midpoint centred frame, if the particles have an initial relative velocity, they are either getting closer or further away along the line between them. While this is happening the potential energy is not changing, so the relative velocity is just maintained. This is the rather dull result of my simulation too.

 

[Lindsayforbes]

Sabine Hossenfelder has posted about this paper:

http://backreaction.blogspot.com/2018/12/no-negative-masses-have-not.html

Peter, I read the press stuff and the paper, but Sabine's blog is highly critical. She is pretty well informed.

 

[kent davidge]

@PeterDonis, who is Sabine Hossenfelder?

 

[Elroch]

google is your friend, kent. ;) Sabine Hossenfelder is a German researcher in quantum gravity.

Note that Sabine made what I identified as an slip in her blog post. It was about whether negative masses repel each other, where she claimed that Franes had got it wrong (but in fact he had got it right because of the odd fact that the acceleration of a negative mass is in the opposite direction to the force on it). She has accepted my correction in the comments to the blog post but has not explicitly responded to it.

Correction: she has responded and appears open to being convinced that the reasoning about what has to be true in a Newtonian approximation is indicative of what has to be true in general relativity, which I am now rather certain is the case (supported by Hermann Bondi and those who followed him).

 

[Elroch]

A short light discussion of some more anti-intuitive (but consistent) behaviour of negative masses. The observation that if you were able to push a negative mass like a normal mass, this would cause acceleration of the mass towards you is bizarre. However, the notion of pushing involves contact of two surfaces and a force which derives from the electromagnetic force. We cannot assume that anything like this would be possible with a negative mass, if such a thing existed.
Negative mass can be positively amusing - Richard Price, University of Utah

 

[PeterDonis]

it's a fact, as my elementary finite difference simulation, based on the definitions of the forces and Newton's laws (with negative masses) confirms. This is important, as my guesses cannot be relied upon!

Neither can your simulation if I can't see the source code. Is it available somewhere?

 

[PeterDonis]

who is Sabine Hossenfelder?

Aside from the excellent advice @Elroch gave you, there is also the fact that I posted a link to an article on her blog, which has an "About" page that directly answers your question.

 

[PeterDonis]

The confusion here is between the acceleration of the centre of the particles (which does decrease as they get further apart)

No, that's not correct. The acceleration of each particle gets smaller as the distance between them increases--because the "slope" of the potential "hill" between them gets smaller as the distance between them increases. There is no "acceleration of the centre of the particles" if we are considering an isolated two-body system; the system as a whole has no external forces on it and moves in a straight line at a constant speed.

This is a co-ordinate transformation similar to that used for modelling two body systems like the hydrogen atom.

And it doesn't work for the case under discussion, because the reduced mass $\mu = m_1 m_2 / (m_1 + m_2)$ is undefined.

if the particles have an initial relative velocity, they are either getting closer or further away along the line between them. While this is happening the potential energy is not changing

Yes, it is; if the distance between the particles is changing, the potential energy is changing, since it depends on the distance between the particles.

This is the rather dull result of my simulation too.

Do you see why I don't trust the results you are claiming from your simulation?

 

[Elroch]

I thought I should make it a little more user friendly first. Python with standard numpy and matplotlib libraries only. It was written in a jupyter notebook, so should work nicely if pasted into one. The graphs show the behaviour quite nicely when there is an initial relative velocity. The maximally simple implementation is in theory more vulnerable to numerical errors than say Runge-Kutta, but it confirms the expected behaviour perfectly here.

import numpy as np
import matplotlib.pyplot as plt

n_step = 100

# x's are positions in 2D, v's are velocities, "neg" means negative mass, "pos" positive mass
# g determines the strength of gravity

x_pos = np.zeros((n_step, 2))
x_neg = np.zeros((n_step, 2))
v_pos = np.zeros((n_step, 2))
v_neg = np.zeros((n_step, 2))
x_rel = np.zeros((n_step-1, 2))

x_neg[0]  = np.array([0, 0])
x_pos[0]  = np.array([1, 1])
v_pos[0]  = np.array([0, 0.1])
v_neg[0]  = np.array([0, 0])

g = 0.1

for i in range(1, n_step):
    x_rel[i-1] = x_pos[i-1] - x_neg[i-1]
    v_pos[i, :] = v_pos[i-1] + g * x_rel[i-1] / (np.sum(x_rel[i-1]**2) ** (3/2))
    v_neg[i, :] = v_neg[i-1] + g * x_rel[i-1] / (np.sum(x_rel[i-1]**2) ** (3/2))
    x_pos[i, :] = x_pos[i-1] + v_pos[i]
    x_neg[i, :] = x_neg[i-1] + v_neg[i]
 
plt.plot(x_pos[:,0], x_pos[:,1])
plt.plot(x_neg[:,0], x_neg[:,1])
plt.title("The paths of the two particles")
plt.show()
plt.plot((v_pos[:,0] + v_neg[:,0])/2)
plt.title("First component of velocity of particles")
plt.show()
plt.plot((v_pos[:,1]+v_neg[:,1])/2)
plt.title("Second component of velocity of particles")
plt.show()
plt.plot(x_rel[:,0])
plt.title("First component of relative position")
plt.show()
plt.plot(x_rel[:,1])
plt.title("Second component of relative position (the one with an initial non-zero relative velocity at the start)")
plt.show()

 

[Elroch]

No, that's not correct. The acceleration of each particle gets smaller as the distance between them increases--because the "slope" of the potential "hill" between them gets smaller as the distance between them increases. There is no "acceleration of the centre of the particles" if we are considering an isolated two-body system; the system as a whole has no external forces on it and moves in a straight line at a constant speed.

And it doesn't work for the case under discussion, because the reduced mass $\mu = m_1 m_2 / (m_1 + m_2)$ is undefined.

Yes, it is; if the distance between the particles is changing, the potential energy is changing, since it depends on the distance between the particles.

Do you see why I don't trust the results you are claiming from your simulation?

Firstly, with two body system with total mass zero, the centre of the system does exhibit the weird acceleration given Newtonian dynamics, because the two forces are in opposite directions, but F=ma makes both accelerations in the same direction. The centre of the particles has the average of these two accelerations.

EDIT: I find I do agree with you on the dependence of potential energy on distance for two masses of opposite sign. When they are close together, they have more potential energy. But this does not mean they tend to move apart, because negative masses accelerate to where they have more potential energy.

 

[Ibix]

Secondly, I'd like to see your argument why the potential energy depends on the distance. When a negative mass falls towards a positive mass it gains potential energy (simply because a zero mass is a sum of a positive and a negative one). When a positive mass falls towards a negative mass it loses potential energy (it's the opposite of falling towards a positive mass). Is is such a surprise that these two equal and opposite effects cancel out?

So if potential energy is not changing, where is the energy coming from to accelerate your particles?

 

[Elroch]

So if potential energy is not changing, where is the energy coming from to accelerate your particles?

When the two particles are moving at velocity v, the kinetic energy of the first is 0.5mv² and the kinetic energy of the second is -0.5mv².
I imagine you see the answer to your question now. ;)
[None of this implies I am convinced that negative masses exist, but this is what an extrapolation of known physics leads to].

 

[PeterDonis]

with two body system with total mass zero, the centre of the system does exhibit the weird acceleration given Newtonian dynamics, because the two forces are in opposite directions, but F=ma makes both accelerations in the same direction. The centre of the particles has the average of these two accelerations.

Ah, I see, you switched frames in midstream. In an inertial frame in which the center of the system is originally at rest, this is true, yes. But I thought you were talking about the "frame" you claimed to construct by a coordinate transformation similar to the one used to model two-body systems like the hydrogen atom. (Which, as I pointed out, doesn't work for this case anyway.)

I'd like to see your argument why the potential energy depends on the distance.

It's just the standard Newtonian formula $U = - G m_1 m_2 / r$. If $m_1$ and $m_2$ are of opposite signs, $U$ is positive, and gets more positive as $r$ decreases and less positive as $r$ increases. In GR, there is a more complicated equation involving the stress-energy tensor that gives the Newtonian formula in the appropriate approximation. As far as I can tell, Farnes accepts all this as given in his paper; he just doesn't fully consider the implications for his model.

When a negative mass falls towards a positive mass it gains potential energy (simply because a zero mass is a sum of a positive and a negative one). When a positive mass falls towards a negative mass it loses potential energy (it's the opposite of falling towards a positive mass).

You're doing it wrong. There aren't two potential energies in a two-body system. There is only one. It's given by the formula I gave above (in the Newtonian approximation, which we are using for this discussion).

 

[Elroch]

I agree with you about the potential energy (I was editing my last post when you were replying). My deleted reasoning had one sign flip too few! Or maybe it was too many.

However, negative masses tend to accelerate to where they have more potential energy according to Newton's laws, so it would be a mistake to think this causes opposite sign masses to repel each other (i.e. accelerate apart).

Regarding my frame, it was centred on the centre of the particles and stayed so, regardless of their motion at the start or later.

 

[Ibix]

When the two particles are moving at velocity v, the kinetic energy of the first is 0.5mv² and the kinetic energy of the second is -0.5mv².
I imagine you see the answer to your question now. ;)
[None of this implies I am convinced that negative masses exist, but this is what an extrapolation of known physics leads to].

Interesting.

Incidentally, there's a bug in your code. x**3/2 is interpreted as (x**3)/2, whereas what you want is x**1.5.