A Dark matter and energy explained by negative mass

[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.

 

[Elroch]

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.

Good job! I have now corrected the typo. This effectively made the gravitational constant time-dependent without changing the sign of any interactions. I have verified that the main conclusion about the way the relative velocity fails to change over time stands.

 

[PeterDonis]

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).

Negative inertial masses accelerate in the direction of increasing potential energy, yes. And the equivalence principle requires particles with negative gravitational mass to also have negative inertial mass. If both of those things are true, then yes, masses of opposite sign do not accelerate apart; their accelerations are equal in magnitude and in the same direction, not equal in magnitude and opposite in direction.

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

Then in your frame, the centre of the particles does not accelerate, by definition: it is at rest. Which is what I said before.

You can't have it both ways: if you want to say the centre of the particles accelerates, you have to use an ordinary inertial frame, in which the centre is not at rest (except possibly at a single instant at the start). If you want to use a frame centred on the centre of the particles, then you can't say the centre of the particles accelerates, because in that frame, it doesn't.

And actually, as I said before, you can't consistently define a frame "centred on the centre of the particles", at least not if you want to use the rest of the mathematical machinery normally associated with such a frame (as in your example of the hydrogen atom), since at least one key part of that machinery, the reduced mass, is undefined in the case we are discussing.

 

[Elroch]

Negative inertial masses accelerate in the direction of increasing potential energy, yes. And the equivalence principle requires particles with negative gravitational mass to also have negative inertial mass. If both of those things are true, then yes, masses of opposite sign do not accelerate apart; their accelerations are equal in magnitude and in the same direction, not equal in magnitude and opposite in direction.

Glad we can finally agree on that!

Then in your frame, the centre of the particles does not accelerate, by definition: it is at rest. Which is what I said before.

You can't have it both ways: if you want to say the centre of the particles accelerates, you have to use an ordinary inertial frame, in which the centre is not at rest (except possibly at a single instant at the start). If you want to use a frame centred on the centre of the particles, then you can't say the centre of the particles accelerates, because in that frame, it doesn't.

And actually, as I said before, you can't consistently define a frame "centred on the centre of the particles", at least not if you want to use the rest of the mathematical machinery normally associated with such a frame (as in your example of the hydrogen atom), since at least one key part of that machinery, the reduced mass, is undefined in the case we are discussing.

We are not obliged to use the same frame all the time: different ones are useful for different things. I don't see any flaw in my reasoning that is due to switching frames. The centre of the two particles accelerates with respect to an observer in a Galilean/inertial frame. We can use an accelerating frame centred on the centre of the two particles, as long as we are careful not to assume it is Galilean or inertial.
Well done for spotting the special problem with the centre of mass frame when the total mass is zero which stops us using that transformation. It is useful otherwise. I had actually spotted that the reduced mass blew up, but was looking for a way round this. The answer is probably to take masses that almost perfectly balance, say mass m and -m(1-δ) The reduced mass is near enough (-m / δ) and the total mass of the system is mδ, so the product of the two is near enough -m², and this must determine the dynamics of the transformed system, taking the (well-behaved) limit, you can derive the behaviour of the system with total mass exactly zero.

 

[PeterDonis]

We are not obliged to use the same frame all the time

Of course not. I never said we were. I only said that the particular "frame" you were trying to construct is problematic for the particular case we are discussing.

taking the (well-behaved) limit

No, the limit is not well-behaved, because the reduced mass diverges. The dynamics of the system in the transformed frame is determined by the reduced mass, not the product of the reduced mass and the total mass.

 

[PeterDonis]

I had actually spotted that the reduced mass blew up, but was looking for a way round this.

The simplest solution is to compute the dynamics in an ordinary inertial frame, and then compute the relative position and relative velocity between the two particles (by simply subtracting vectors). This tells you the relative motion of the two without having to do any problematic frame transformations.

 

[Auto-Didact]

No, the limit is not well-behaved, because the reduced mass diverges.

None of this is actually physically problematic if the negative masses aren't fundamental particles with the same absolute mass as their positive mass counterparts; Farnes doesn't claim that they are either. In other words, if the creation tensor generates effective negative masses through some dynamical mechanism wherein the masses are proportional to the creation process, then this problem disappears.

If the negative mass particles are however equal in magnitude to the positive mass, as it would be in the case of elementary particles with their mass quantized in the same manner, then the essential question is whether this divergence is inherent or removable; there are of course multiple ways to answer this question.

Perhaps one of the simplest ways to do this would be if mathematically the limit from the left and the limit from the right just corresponds physically to whether the negative mass is the active or passive gravitational mass within the duo; I have a feeling that such an Ansatz misses something essential but I can't quite put my finger on it.

 

[Elroch]

Of course not. I never said we were. I only said that the particular "frame" you were trying to construct is problematic for the particular case we are discussing.

No, the limit is not well-behaved, because the reduced mass diverges. The dynamics of the system in the transformed frame is determined by the reduced mass, not the product of the reduced mass and the total mass.

You can do the calculation for a small δ. If it is very similar for sufficiently small δ, all is fine. In fact what you find is the finite force determined by the product of the two transformed masses acts on the reduced mass, which grows to infinity as δ shrinks. The conclusion is that the fictitious reduced mass accelerates less and less as δ shrinks, which implies that the two particles have a constant relative velocity in the limit.

 

[Auto-Didact]

You can do the calculation for a small δ. If it is very similar for sufficiently small δ, all is fine. In fact what you find is the finite force determined by the product of the two transformed masses acts on the reduced mass, which grows to infinity as δ shrinks. The conclusion is that the fictitious reduced mass accelerates less and less as δ shrinks, which implies that the two particles have a constant relative velocity in the limit.

Indeed, but assuming identical particles of opposite mass, it is exactly this limit which is problematic. If the positive mass, which starts off at rest, is impinged upon by the force of the negative mass, would the two go off with a constant relative velocity or remain at zero velocity?

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.

Exactly. This is precisely why one would prefer a Newton-Cartan treatment, because the geometrodynamics of the problem - which might turn out to be essential in some manner, perhaps even w.r.t. removing the divergence - is fully retained, in stark contrast to the pure Newtonian case.

 

[PeterDonis]

The simplest solution is to compute the dynamics in an ordinary inertial frame, and then compute the relative position and relative velocity between the two particles (by simply subtracting vectors).

This was simple enough that I went ahead and coded it. Here is the code:

#!/usr/bin/env python3

from math import *G = 6.67e-11def negate(f):
    # Avoids "minus zero" floating point result
    return f if f == 0.0 else - fclass V3(object):
    # Poor man's 3-vector implementation, just the properties we need for this program
   
    def __init__(self, x, y, z):
        self.x = x
        self.y = y
        self.z = z
   
    def __str__(self):
        # For convenient print output
        return "({}, {}, {})".format(self.x, self.y, self.z)
   
    def __neg__(self):
        # For unary - operator
        return V3(negate(self.x), negate(self.y), negate(self.z))
   
    def __add__(self, other):
        # For + operator for two vectors
        return V3(self.x + other.x, self.y + other.y, self.z + other.z)
   
    def __sub__(self, other):
        # For - operator for two vectors
        return V3(self.x - other.x, self.y - other.y, self.z - other.z)
   
    def __mul__(self, other):
        # For * operator for vector and scalar
        if isinstance(other, (int, float)):
            return V3(other * self.x, other * self.y, other * self.z)
        raise TypeError("Cannot multiply V3 by {}".format(type(other)))
   
    __rmul__ = __mul__  # allows * to work regardless of order of operands
   
    def radius(self):
        # For convenience in calculations
        return sqrt(self.x**2 + self.y**2 + self.z**2)
   
    def unit(self):
        # Unit vector pointing in the same direction as this vector
        r = self.radius()
        return V3(self.x / r, self.y / r, self.z / r)
   
    def is_zero(self):
        # For convenience in calculations
        return (self.x == 0.0) and (self.y == 0.0) and (self.z == 0.0)class Run(object):
    # Simulation run with given initial conditions
   
    headings = "r_pos v_pos r_neg v_neg r_diff v_diff".split()
   
    def __init__(self, r_pos, v_pos, r_neg, v_neg):
        # Initialize time step arrays with initial conditions
        self.r_pos = [r_pos]
        self.v_pos = [v_pos]
        self.r_neg = [r_neg]
        self.v_neg = [v_neg]
   
    def increment(self, vec, dvec, dt):
        # Convenience operation to add new value to time step array
        vec.append(vec[-1] + dt * dvec)
   
    def output_line(self, *fields):
        # Convenience for line output
        return " ".join(str(f) for f in fields)
   
    def execute(self, steps=10000, dt=0.001, show_output=True, full_output=False):
        # Run simulation for given number of time steps
        for _ in range(steps):
            r_vec = self.r_pos[-1] - self.r_neg[-1]
            if r_vec.is_zero():
                break  # stop iterating if particles collide
            u_vec = r_vec.unit()
            r = r_vec.radius()
            a = (G / r**2) * u_vec
           
            self.increment(self.r_pos, self.v_pos[-1], dt)
            self.increment(self.r_neg, self.v_neg[-1], dt)
           
            self.increment(self.v_pos, a, dt)
            self.increment(self.v_neg, a, dt)
       
        if not show_output:
            return
       
        lines = []
        lines.append(self.output_line(*self.headings))
        lines.append("")
        itemcount = min(steps + 1, len(self.r_pos), len(self.r_neg), len(self.v_pos), len(self.v_neg))
        indexes = range(itemcount) if full_output else (0, -1)
        for i in indexes:
            r_pos = self.r_pos[i]
            r_neg = self.r_neg[i]
            v_pos = self.v_pos[i]
            v_neg = self.v_neg[i]
            r_diff = r_neg - r_pos
            v_diff = v_neg - v_pos
            lines.append(self.output_line(r_pos, v_pos, r_neg, v_neg, r_diff, v_diff))
        return linesZERO = V3(0.0, 0.0, 0.0)
INCX = V3(0.000001, 0.0, 0.0)
INCY = V3(0.0, 0.000001, 0.0)initial_datasets = [
    # r_pos, v_pos, r_neg, v_neg
    (ZERO, ZERO, INCX, ZERO),  # base case, no relative velocity
    (ZERO, ZERO, INCX, INCX),  # moving apart
    (ZERO, ZERO, INCX, - INCX),  # moving together
    (ZERO, ZERO, INCX, INCY),  # moving transverse
]if __name__ == '__main__':
    for dataset in initial_datasets:
        R = Run(*dataset)
        for line in R.execute():
            print(line)
        print("")

And here is a quick summary of the output for the four cases. Each case runs for 10000 time steps, which equals 10 seconds of simulated time. The masses are plus and minus 1 kg, and they start out separated by 1 micrometer in the x direction. The positive mass is at the origin and is motionless. The negative mass is at x = plus 1 micrometer and has the given initial velocity for each case.

Base case: no relative velocity. At the end of the simulation, the masses are at about x = - 3334 meters and moving at about - 667 meters per second in the x direction. Their separation is still 1 micrometer and they still have zero relative velocity.

Moving apart: the negative mass starts out moving at + 1 micrometer per second in the x direction. At the end of the simulation, the masses are at about x = - 507 meters and moving at about - 60 meters per second in the x direction. Their separation in the x direction has increased to about 10 micrometers, and their relative velocity is unchanged.

Moving together: the negative mass starts out moving at - 1 micrometer per second in the x direction. Here the simulation ends before the full 10,000 steps because the particles collide. Their x velocity is well in excess of the speed of light at this point so the simulation is clearly unphysical anyway for this case, it would need to be redone to properly capture relativistic motion.

Moving transverse: the negative mass starts out moving at +1 micrometer per second in the y direction. At the end of the simulation, the masses are at about x = - 604 meters, y = - 467 meters, and are moving at about - 66 meters per second in the x direction and about - 60 meters per second in the y direction. Their separation is about 1 micrometer in the x direction and about 10 micrometers in the y direction, and their relative velocity is unchanged.

The quick and dirty summary of all this is that the runaway solution is present even with (small) nonzero relative velocity. The particles do slowly increase their separation if they have a nonzero relative velocity away from each other, but that does not prevent them from both drastically running away from their original position in an inertial frame. (Note that this latter fact points out a key limitation of using a frame in which the center of the particles is always at rest: it prevents you from seeing the runaway motion.)

 

[Elroch]

Indeed, but assuming identical particles of opposite mass, it is exactly this limit which is problematic. If the positive mass, which starts off at rest, is impinged upon by the force of the negative mass, would the two go off with a constant relative velocity or remain at zero velocity?
Exactly. This is precisely why one would prefer a Newton-Cartan treatment, because the geometrodynamics of the problem - which might turn out to be essential in some manner, perhaps even w.r.t. removing the divergence - is fully retained, in stark contrast to the pure Newtonian case.

There is no doubt about the answer to your question which may have been motivated by a little confusion between the relative velocity of the particles and the velocity of one of the particles relative to a Galilean/inertial frame.
Whatever the initial relative velocity of two identical particles of opposite mass, it never changes afterwards.

 

[Elroch]

This was simple enough that I went ahead and coded it. Here is the code:

#!/usr/bin/env python3

from math import *G = 6.67e-11def negate(f):
    # Avoids "minus zero" floating point result
    return f if f == 0.0 else - fclass V3(object):
    # Poor man's 3-vector implementation, just the properties we need for this program
 
    def __init__(self, x, y, z):
        self.x = x
        self.y = y
        self.z = z
 
    def __str__(self):
        # For convenient print output
        return "({}, {}, {})".format(self.x, self.y, self.z)
 
    def __neg__(self):
        # For unary - operator
        return V3(negate(self.x), negate(self.y), negate(self.z))
 
    def __add__(self, other):
        # For + operator for two vectors
        return V3(self.x + other.x, self.y + other.y, self.z + other.z)
 
    def __sub__(self, other):
        # For - operator for two vectors
        return V3(self.x - other.x, self.y - other.y, self.z - other.z)
 
    def __mul__(self, other):
        # For * operator for vector and scalar
        if isinstance(other, (int, float)):
            return V3(other * self.x, other * self.y, other * self.z)
        raise TypeError("Cannot multiply V3 by {}".format(type(other)))
 
    __rmul__ = __mul__  # allows * to work regardless of order of operands
 
    def radius(self):
        # For convenience in calculations
        return sqrt(self.x**2 + self.y**2 + self.z**2)
 
    def unit(self):
        # Unit vector pointing in the same direction as this vector
        r = self.radius()
        return V3(self.x / r, self.y / r, self.z / r)
 
    def is_zero(self):
        # For convenience in calculations
        return (self.x == 0.0) and (self.y == 0.0) and (self.z == 0.0)class Run(object):
    # Simulation run with given initial conditions
 
    headings = "r_pos v_pos r_neg v_neg r_diff v_diff".split()
 
    def __init__(self, r_pos, v_pos, r_neg, v_neg):
        # Initialize time step arrays with initial conditions
        self.r_pos = [r_pos]
        self.v_pos = [v_pos]
        self.r_neg = [r_neg]
        self.v_neg = [v_neg]
 
    def increment(self, vec, dvec, dt):
        # Convenience operation to add new value to time step array
        vec.append(vec[-1] + dt * dvec)
 
    def output_line(self, *fields):
        # Convenience for line output
        return " ".join(str(f) for f in fields)
 
    def execute(self, steps=10000, dt=0.001, show_output=True, full_output=False):
        # Run simulation for given number of time steps
        for _ in range(steps):
            r_vec = self.r_pos[-1] - self.r_neg[-1]
            if r_vec.is_zero():
                break  # stop iterating if particles collide
            u_vec = r_vec.unit()
            r = r_vec.radius()
            a = (G / r**2) * u_vec
   
            self.increment(self.r_pos, self.v_pos[-1], dt)
            self.increment(self.r_neg, self.v_neg[-1], dt)
   
            self.increment(self.v_pos, a, dt)
            self.increment(self.v_neg, a, dt)
 
        if not show_output:
            return
 
        lines = []
        lines.append(self.output_line(*self.headings))
        lines.append("")
        itemcount = min(steps + 1, len(self.r_pos), len(self.r_neg), len(self.v_pos), len(self.v_neg))
        indexes = range(itemcount) if full_output else (0, -1)
        for i in indexes:
            r_pos = self.r_pos[i]
            r_neg = self.r_neg[i]
            v_pos = self.v_pos[i]
            v_neg = self.v_neg[i]
            r_diff = r_neg - r_pos
            v_diff = v_neg - v_pos
            lines.append(self.output_line(r_pos, v_pos, r_neg, v_neg, r_diff, v_diff))
        return linesZERO = V3(0.0, 0.0, 0.0)
INCX = V3(0.000001, 0.0, 0.0)
INCY = V3(0.0, 0.000001, 0.0)initial_datasets = [
    # r_pos, v_pos, r_neg, v_neg
    (ZERO, ZERO, INCX, ZERO),  # base case, no relative velocity
    (ZERO, ZERO, INCX, INCX),  # moving apart
    (ZERO, ZERO, INCX, - INCX),  # moving together
    (ZERO, ZERO, INCX, INCY),  # moving transverse
]if __name__ == '__main__':
    for dataset in initial_datasets:
        R = Run(*dataset)
        for line in R.execute():
            print(line)
        print("")

And here is a quick summary of the output for the four cases. Each case runs for 10000 time steps, which equals 10 seconds of simulated time. The masses are plus and minus 1 kg, and they start out separated by 1 micrometer in the x direction. The positive mass is at the origin and is motionless. The negative mass is at x = plus 1 micrometer and has the given initial velocity for each case.

Base case: no relative velocity. At the end of the simulation, the masses are at about x = - 3334 meters and moving at about - 667 meters per second in the x direction. Their separation is still 1 micrometer and they still have zero relative velocity.

Moving apart: the negative mass starts out moving at + 1 micrometer per second in the x direction. At the end of the simulation, the masses are at about x = - 507 meters and moving at about - 60 meters per second in the x direction. Their separation in the x direction has increased to about 10 micrometers, and their relative velocity is unchanged.

Moving together: the negative mass starts out moving at - 1 micrometer per second in the x direction. Here the simulation ends before the full 10,000 steps because the particles collide. Their x velocity is well in excess of the speed of light at this point so the simulation is clearly unphysical anyway for this case, it would need to be redone to properly capture relativistic motion.

Moving transverse: the negative mass starts out moving at +1 micrometer per second in the y direction. At the end of the simulation, the masses are at about x = - 604 meters, y = - 467 meters, and are moving at about - 66 meters per second in the x direction and about - 60 meters per second in the y direction. Their separation is about 1 micrometer in the x direction and about 10 micrometers in the y direction, and their relative velocity is unchanged.

The quick and dirty summary of all this is that the runaway solution is present even with (small) nonzero relative velocity. The particles do slowly increase their separation if they have a nonzero relative velocity away from each other, but that does not prevent them from both drastically running away from their original position in an inertial frame. (Note that this latter fact points out a key limitation of using a frame in which the center of the particles is always at rest: it prevents you from seeing the runaway motion.)

Note that my program had used the same procedure and come to the same conclusion except it is inaccurate to say there is a "runaway" solution when there is any initial relative velocity. By contrast, the velocity (in any inertial frame) of each of the particles is asymptotic to a constant value in all such cases, as it is when both masses are positive and not gravitationally bound. The "runaway" situation is when the velocity of the particles tends to the speed of light (in a Galilean theory, they would tend to infinity).

The reason this is so is that the distance between the particles increases at least linearly with time (once they are heading apart), and the two accelerations are determined by the square of this distance. The integral from t=0 to infinity of an inverse square is finite. This bounds the magnitude of the absolute value of the velocity of each particle and ensures the limit exists by absolute convergence.

The sole case in which this is not so is when the initial relative velocity is zero, so it never increases.(I am excluding the case where the particles collide for obvious reasons).

 

[PeterDonis]

my program had used the same procedure

I couldn't understand what your program was doing so it was easier to just implement my own.

 

[Elroch]

Fair enough. I have no doubt we have both gained insight.

 

[PeterDonis]

The reason this is so is that the distance between the particles increases at least linearly with time (once they are heading apart), and the two accelerations are determined by the square of this distance. The integral from t=0 to infinity of an inverse square is finite.

I see what you're saying, but I don't think this invalidates the term "runaway solution". First, the acceleration going like the inverse square of the distance is the non-relativistic approximation; when relativistic effects are taken into account this gets modified, and we would have to look at the relativistically correct math.

Second, even if there is a bound on the velocity (or the kinetic energy in the relativistic version), it can still be the case (and probably will be for at least a significant range of initial conditions) that the kinetic energy of the positive particle is much, much larger than it was initially, and that's the key point. Remember that the prediction from Farnes's paper that we're talking about is that these "runaway" pairs can explain the observation of cosmic rays, and in particular cosmic rays with energies beyond the GZK limit (because we only observe the positive mass particle of the pair, not the negative mass particle), so the question is how rare such pairs are predicted to be. Showing a finite bound on kinetic energy is not enough to assess that; we have to know what the finite bound is for the range of expected initial conditions, and how rare that makes such pairs. If the bound is too low, then Farnes's model predicts that such pairs can never produce a positive mass particle with high enough kinetic energy to account for cosmic rays at all. If the bound is too high, the model predicts that such cosmic rays are more common than we actually observe. The bound would have to be just right for the model prediction to match observations, and that seems like a highly fine-tuned fit to me. (My personal guess, as I've already said, is that the model ends up predicting such cosmic rays to be much more common than we actually observe; but I admit that's just a guess, not based on actually doing the math.)

 

[Elroch]

I see what you're saying, but I don't think this invalidates the term "runaway solution". First, the acceleration going like the inverse square of the distance is the non-relativistic approximation; when relativistic effects are taken into account this gets modified, and we would have to look at the relativistically correct math.

No, I was referring to the inverse square law of gravitation which determines the acceleration of the each of the two particles just like for positive masses, except both accelerations are in the same direction. This law gets very accurate as two objects move apart in general relativity (it's pretty accurate all the time unless they are black holes or neutron stars that are very close). This means that if two objects have a constant relative velocity, the force of one particle on the other is of the order of the inverse square of time once they are heading apart, so the same for acceleration, so the integral of this converges rapidly.
To see whether a pair of objects is going to accelerate by a large fraction of the speed of light, just multiply the usual gravitational acceleration by the time they are going to stay close. Eg An Earth and a negative Earth-mass object so close they are nearly touching for a year will reach relativistic speeds (1g x 1 year is roughly the speed of light).

Second, even if there is a bound on the velocity (or the kinetic energy in the relativistic version), it can still be the case (and probably will be for at least a significant range of initial conditions) that the kinetic energy of the positive particle is much, much larger than it was initially, and that's the key point. Remember that the prediction from Farnes's paper that we're talking about is that these "runaway" pairs can explain the observation of cosmic rays, and in particular cosmic rays with energies beyond the GZK limit (because we only observe the positive mass particle of the pair, not the negative mass particle), so the question is how rare such pairs are predicted to be. Showing a finite bound on kinetic energy is not enough to assess that; we have to know what the finite bound is for the range of expected initial conditions, and how rare that makes such pairs. If the bound is too low, then Farnes's model predicts that such pairs can never produce a positive mass particle with high enough kinetic energy to account for cosmic rays at all. If the bound is too high, the model predicts that such cosmic rays are more common than we actually observe. The bound would have to be just right for the model prediction to match observations, and that seems like a highly fine-tuned fit to me. (My personal guess, as I've already said, is that the model ends up predicting such cosmic rays to be much more common than we actually observe; but I admit that's just a guess, not based on actually doing the math.)

My example above indicates that except in the most extreme cases (eg ridiculously perfectly velocity-matched and mass-matched neutron star and negative mass equivalent), the perfectly balanced masses need to be virtually stationary relative to each other over a very long time to have a chance of acquiring relativistic speeds as a pair. A pair of small mass objects can't get to high speeds even if perfectly balanced in mass and velocity (which is too unlikely to happen).
Needless to say the gravitational acceleration generated by a cosmic ray on a velocity matched partner is so insignificant no-one you can very safely ignore it.
So the artificial, delicately balanced "runaway" situation is completely irrelevant to cosmic rays as stands, and Franes should have realized this if that is what he meant.