Ibix said: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 said: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).
Elroch said:Regarding my frame, it was centred on the centre of the particles and stayed so, regardless of their motion at the start or later.
Glad we can finally agree on that!PeterDonis said: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.
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.PeterDonis said: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 said:We are not obliged to use the same frame all the time
Elroch said:taking the (well-behaved) limit
Elroch said:I had actually spotted that the reduced mass blew up, but was looking for a way round this.
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.PeterDonis said:No, the limit is not well-behaved, because the reduced mass diverges.
PeterDonis said: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.
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?Elroch said: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.
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 said: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.
PeterDonis said: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).
#!/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("")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.Auto-Didact said: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.
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).PeterDonis said:This was simple enough that I went ahead and coded it. Here is the code:
Python:#!/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 said:my program had used the same procedure
Elroch said: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.
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.PeterDonis said: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.
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).PeterDonis said: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.)
rootone said:I think that the concept of a negative mass is silly.
By analogy, a velocity of less than zero is silly.
rootone said:By analogy, a velocity of less than zero is silly.
Elroch said: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).
Elroch said: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.
PeterDonis said: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.
Hi Elroch:Elroch said:For velocities to rise indefinitely, it would be necessary for the masses to be perfectly matched and the particles to have exactly zero initial relative velocity, so it simply would not occur naturally.
Well, both particles accelerate in the direction of the positive mass particle but, if the masses are equal, their relative velocity remains constant. So, given your description, they get further apart. The distance between them is thus O(time) and the interaction forces and the accelerations are O(time-2). So the velocities are asymptotically constant (the integral of an inverse square is bounded), as is also true for two positive masses that are not bound together (For anyone not familiar with the terminology, google "big O notation" and "asymptotic").Buzz Bloom said:Hi Elroch:
I do not understand the above quote. I wonder if you would please explain in some detail what would happen if there were a +mass particle near a -mass particle, and the initial conditions are that relative to the center of mass, the velocity vectors of these two particles have the same magnitude and opposite directions. Note that the initial relative velocities are not zero. I do not have the skills to do the math with any confidence that I will not make mistakes, but intuitively it seems that if the two particles initially are moving towards each other, then the results would be that particles would follow each other at an accelerating speed which eventually would overwhelm the initial velocity.
Regards,
Buzz
Hi Elroch:Elroch said:Well, both particles accelerate in the direction of the positive mass particle but, if the masses are equal, their relative velocity remains constant. So, given your description, they get further apart. The distance between them is thus O(time) and the interaction forces and the accelerations are O(time-2). So the velocities are asymptotically constant
Yes. but mass doesn't have a direction.PeterDonis said:Not if you take direction into account.
If you mean a magnitude of velocity less than zero is silly, that's true.