Time-reverse symmetry of the principle of relativity

[Chrisc]

Please look at the attached diagram and let me know
if there is a reason for the asymmetric dynamics due
to the relative position of rest or, if I have incorrectly
interpreted the mechanics.
Kev, I haven't forgotten.

 

[Chrisc]

Not a single response tells me either it is too confusing in the
graphic form I posted, or I've pointed out something no one can
rationalize with relativity.
In case it's the former, I've included below a less detailed
text version of the question.

The time-reverse symmetry of the dynamics in a simple two body collision seems
to hold only when the observer is initially at rest with the lessor of the two massive bodies.

If this is as straight forward as I think it is, it has significant implications for
the principle of relativity.
Have I misinterpreted the mechanics, or is this stumping everyone?

 

[Vanadium 50]

Not a single response tells me either it is too confusing in the
graphic form I posted, or I've pointed out something no one can
rationalize with relativity.

I think it's more likely that people don't want to spend the time hunting down your mistake. Relativistic mechanics is time symmetric.

 

[DrGreg]

Please look at the attached diagram and let me know
if there is a reason for the asymmetric dynamics due
to the relative position of rest or, if I have incorrectly
interpreted the mechanics.

I can't work out what you think the problem is. As far as I can see, everything in your diagrams is time-symmetric and you haven't explained why you think there is asymmetry.

However, your diagrams are correct only in Newtonian mechanics, not in relativistic mechanics.

In relativity, momentum is

$$p = \frac {mv} {\sqrt{1 - v^2/c^2}}$$​

and velocities are transformed from one frame to another using

$$w = \frac {u - v} {1 - uv/c^2}$$​

Rest mass $m$ is not conserved in collisions but energy

$$E = \frac {mc^2} {\sqrt{1 - v^2/c^2}}$$​

is conserved in elastic collisions.

 

[Chrisc]

I can't work out what you think the problem is. As far as I can see, everything in your diagrams is time-symmetric and you haven't explained why you think there is asymmetry.

When the observer is initially at rest with B1(#5), the time-reverse symmetry of the kinematics shown in #8 violate the laws of dynamics.
B2 will not bring B1 to rest. The kinematics all appear time-reverse symmetric as kinematics are the quantitative expressions of the mechanics with T set to negative.
When the observer is initially at rest with B1, the correct time-reverse dynamics of #5 would present the same as #2.
The time-reverse symmetry of this event is only upheld when the observer is initially at rest with B2,
a situation that implies there is a problem with the symmetry of relativistic mechanics through time-reversal.

 

[MeJennifer]

B2 will not bring B1 to rest.

What do you mean by bringing something to rest. Rest is a relational not an absolute concept in relativity. Something is only at rest in relation to something else.

 

[Chrisc]

What do you mean by bringing something to rest. Rest is a relational not an absolute concept in relativity. Something is only at rest in relation to something else.

I mean with respect to the observer.
When the observer is initially at rest with respect to one of the two bodies (B1 and B2)
they observe differing but valid mechanics after the collision.
The problem arises when the same event is considered through time-reversal.
In the first case where the observer is initially at rest with respect to B2, the
time-reverse mechanics are valid.
When the observer is initially at rest with respect to B1, the time-reverse
mechanics (when held to the symmetry of the forward-time kinematics)
require dynamics that violate or contradict the laws (Newton's).

 

[MeJennifer]

IWhen the observer is initially at rest with respect to B1, the time-reverse mechanics (when held to the symmetry of the forward-time kinematics)
require dynamics that violate or contradict the laws (Newton's).

Not true.

 

[Dale]

f = dp/dt is obviously time symmetric. If you have some scenario which is not time symmetric then it cannot follow the laws of mechanics.

 

[Chrisc]

Not true.

Not helpful.

 

[Chrisc]

f = dp/dt is obviously time symmetric.

A body B1 with mass M and velocity v wrt to the observer collides with a body B2 with mass 1/2M at rest with respect to the observer.
The mass B1 comes to rest wrt the observer and the body B2 moves away with velocity 2v wrt the observer.
This is the (invalid) time-reverse description of the (valid) time forward event. It does not happen in nature as it conserves momentum via increased kinetic energy.
So it appears as a statement of kinematics, f = dp/dt is time-reverse symmetric, but it is not necessarily always time-reverse symmetric as a statement of dynamics.

If you have some scenario which is not time symmetric then it cannot follow the laws of mechanics.

I am not pointing out a scenario that is not time-reverse symmetric and upholds the laws, I am pointing out a scenario that appears time-reverse symmetric in its kinematics but as such its dynamics must violate the laws.
If you take that to mean that it is just not time reverse symmetric in the first place, then you are recognizing what I am saying as it is the same event that is time-reverse symmetric when the observer is at rest wrt B2.

 

[George Jones]

A body B1 with mass M and velocity v wrt to the observer collides with a body B2 with mass 1/2M at rest with respect to the observer.
The mass B1 comes to rest wrt the observer and the body B2 moves away with velocity 2v wrt the observer.

This is the (invalid) time-reverse description of the (valid) time forward event. It does not happen in nature as it conserves momentum via increased kinetic energy.

Chrisc, it seems to that you're asking

"Why is there a thermodynamic arrow of time?"

 

[JesseM]

A body B1 with mass M and velocity v wrt to the observer collides with a body B2 with mass 1/2M at rest with respect to the observer.
The mass B1 comes to rest wrt the observer and the body B2 moves away with velocity 2v wrt the observer.

This would be true in Newtonian mechanics but it doesn't work in SR.
Total momentum before collision: $$\frac{M*v}{\sqrt{1 - v^2/c^2}}$$
Total momentum after: $$\frac{(M/2)*(2v)}{\sqrt{1 - (2v)^2/c^2}}$$
These are not equal, so this can't be correct (momentum should be conserved in collisions in SR just like it is in Newtonian mechanics).

 

[George Jones]

This would be true in Newtonian mechanics but it doesn't work in SR.

I don't think that this affects Chrisc's argument. Chrisc's argument is that inelastic collsions occur only one way in time.

Roughly, heat (internal energy) is more disordered then translational kinetic energy of an entire object, hence, by thermodynamics, inelastic collisions only happen one way in time.

 

[JesseM]

I don't think that this affects Chrisc's argument. Chrisc's argument is that inelastic collsions occur only one way in time.

Ah, he didn't specifically refer to the collision as inelastic, but now I see that he mentions the kinetic energy changes (in Newtonian terms as well as relativistic ones). And yes, inelastic collisions involve a change in entropy (kinetic energy of the center of mass being transformed into heat, which is random kinetic energy of many molecules in different directions) which is why they are extremely unlikely to happen in reverse, although in terms of the fundamental non-thermodynamic laws of physics there is nothing physically impossible about the reversed scenario.

 

[Dale]

A body B1 with mass M and velocity v wrt to the observer collides with a body B2 with mass 1/2M at rest with respect to the observer.
The mass B1 comes to rest wrt the observer and the body B2 moves away with velocity 2v wrt the observer.
This is the (invalid) time-reverse description of the (valid) time forward event. It does not happen in nature as it conserves momentum via increased kinetic energy.
So it appears as a statement of kinematics, f = dp/dt is time-reverse symmetric, but it is not necessarily always time-reverse symmetric as a statement of dynamics.

I'm sorry, but what you are saying here doesn't make any sense. Newton's laws are the laws of dynamics. Newton's laws are time symmetric. Therefore dynamics are time symmetric. Kinematics are just dynamics w/o the forces, so if the dynamics are time symmetric then the kinematics are also time symmetric. This is obvious and clear from the laws themselves, you don't need to worry about specific cases because the laws are symmetric in general.

In your example, the explanation is simple, in the forward case momentum is conserved through a decrease in KE (KE->thermal energy), in the reverse case momentum is conserved through an increase in KE (thermal energy->KE). The fact that the reverse case doesn't happen in nature is due to the non-time symmetry of thermodynamics, not any asymmetry in dynamics.

A minor point is that your analysis is non-relativistic.

 

[yuiop]

Please look at the attached diagram and let me know
if there is a reason for the asymmetric dynamics due
to the relative position of rest or, if I have incorrectly
interpreted the mechanics...

Your diagrams are obviously non relativistic. There seems to be an error in the calculations in your diagrams when the calculations are done using the Newtonian equations.

The equation for a head on elastic collision is given here: http://hyperphysics.phy-astr.gsu.edu/Hbase/elacol2.html#c1

Using the notation given in that link, you have initial conditions:

Ball B1: $$m_1=2m, v_1=2v$$
Ball B2: $$m_2=1m, v_2=0v$$

The final velocity of mass m1 is:

$$v_1' = v1 \frac{m1-m2}{m1+m2} = 2v\frac{2m-1m}{2m+1m} =2/3v$$

The final velocity of mass m2 is:

$$v_2' = v1 \frac{2m_1}{m_1+m_2} = v\frac{4m}{2m+1m} =8/3v$$

The initial total momentum of the system is $$(m_1 v_1 )+(m_2 v_2) = 4mv$$

The final total momentum of the system is $$(m_1 v_1')+(m_2 v_2') = 4/3mv+8/3mv = 12/3mv = 4mv$$

Total momentum before and after the collision is conserved.

The total initial kinetic energy of the system is:

$$1/2m_1 (v_1)^2 + 1/2m_2 (v_2)^2 = 4mv^2$$

The total final kinetic energy of the system is:

$$1/2m_1 (v_1')^2 + 1/2m_2 (v_2')^2 = m(2/3v)^2 + 1/2m(8/3)^2 = 4mv^2$$

Total KE before and after the collision is conserved.


The final conditions in your diagram of:

Ball B1: $$m_1=2m, v_1'=v$$

Ball B2: $$m_2=1m, v_2'=2v$$

do not satisfy the conservation of momentum AND conservation of KE laws.

Another condition that is normally satisfied in a head on elastic collision, is that the velocity of approach equals the velocity of separation. In the example I gave the velocity of approach is 2v and the velocity of separation is also 2v (8/3v - 2/3v). The velocity of approach in your diagram is 2v while the velocity of separation is 1v.

I am pretty sure that when you do the forward calculations correctly, they will time reverse correctly.

 

[George Jones]

Your diagrams are obviously non relativistic. There seems to be an error in the calculations in your diagrams when the calculations are done using the Newtonian equations.

The equation for a head on elastic collision is given here: http://hyperphysics.phy-astr.gsu.edu/Hbase/elacol2.html#c1

Have you read posts #12 through #16? The equations in the link don't apply, since they are for elastic collisions. Chrisc analyzes inelastic collisions that are physically realistic, and that cannot be excluded from consideration.

Consider a more extreme example.

Two equal mass objects collide and stick together. Before the collision, the objects move with equal speeds in opposite directions with respect to a particular frame. By conservation of momentum, the combined object does not move after the collision.

This is a completely plausible physical scenario, i.e., think putty.

The time reverse of the collision is not plausible at all. A blob of putty does not separate into two smaller blobs spontaneously, with each of the two smaller blobs moving in different directions.

This is in accord with the laws of thermodynamics and statistical mechanics.

I am pretty sure that when you do the forward calculations correctly, they will time reverse correctly.

I think Chrisc has done the forward calculations correctly for plausible inelastic collisions. Even though Chrisc didn't make an error by choosing to analyze inelastic collisions, as they happen all the time in the real world, I think Chrisc would agree that elastic collisions are time reversible.

 

[yuiop]

Have you read posts #12 through #16? The equations in the link don't apply, since they are for elastic collisions. Chrisc analyzes inelastic collisions that are physically realistic, and that cannot be excluded from consideration.

In the diagram he attached to post#1 he shows the case for the elastic collision in frames 1 to 4 and the inelastic case in frames 5 to 8. I took the time to check his calculations in frames 1 to 4 (the elastic case) and when I found them to be in error and I did not really pursue the thread further. Ignoring the fact that there is an error in his elastic case I can see now that he is making the case that elastic collisions appear to be reversible while inelastic collisions do not appear to be reversible.

Consider a more extreme example.

Two equal mass objects collide and stick together. Before the collision, the objects move with equal speeds in opposite directions with respect to a particular frame. By conservation of momentum, the combined object does not move after the collision.

This is a completely plausible physical scenario, i.e., think putty.

The time reverse of the collision is not plausible at all. A blob of putty does not separate into two smaller blobs spontaneously, with each of the two smaller blobs moving in different directions.

This is in accord with the laws of thermodynamics and statistical mechanics.I think Chrisc has done the forward calculations correctly for plausible inelastic collisions. Even though Chrisc didn't make an error by choosing to analyze inelastic collisions, as they happen all the time in the real world, I think Chrisc would agree that elastic collision are time reversible.

Looking at the equation for an inelastic collision http://hyperphysics.phy-astr.gsu.edu/Hbase/inecol.html#c1 the final velocity of the combined mass of B1 and B2 should be 4/3v and not the 1v shown in frames 4 to 8. Despite the fact Chrisc has made an error in both the elastic and inelastic cases it should not distract us from the case he is making that inelastic collisons appear to be non-reversible. As I understand it, classical dynamics does not forbid a blob of putty separating into two smaller blobs spontaneously, with each of the two smaller blobs moving in different directions when analysed at the molecular scale. All it says is that it is statistically unlikely. A freek set of unlikely collisons at the molecular level producing that sort of reverse reaction is unlikely but not imposssible. It is basically an example of the arrow of time and increasing entropy. Another example is a glass falling off a table and breaking into a million pieces. The reverse situation of the glass reassembling itself and ending back up on top of the table is not impossible, just statistically extremely unlikely in classical dynamics.

Basically it comes down to the fact that converting coherant motion (the parallel motion of the molecules that make up the ball) to incoherant random motion (heat) is more likey than the reverse in nature, (but not impossible). An example of the reverse in nature would be the thermal heat of magma beneath the surface of the Earth being converted into coherant motion of the water and steam being ejected from a geyser.

 

[George Jones]

In the diagram he attached to post#1 he shows the case for the elastic collision in frames 1 to 4 and the inelastic case in frames 5 to 8. I took the time to check his calculations in frames 1 to 4 (the elastic case) and when I found them to be in error and I did not really pursue the thread further. Ignoring the fact that there is an error in his elastic case I can see now that he is making the case that elastic collisions appear to be reversible while inelastic collisions do not appear to be reversible.

Looking at the equation for an inelastic collision http://hyperphysics.phy-astr.gsu.edu/Hbase/inecol.html#c1 the final velocity of the combined mass of B1 and B2 should be 4/3v and not the 1v shown in frames 4 to 8. Despite the fact Chrisc has made an error in both the elastic and inelastic cases it should not distract us from the case he is making that inelastic collisons appear to be non-reversible.

I admit that I only checked Chrisc's calculation in post #11; I didn't have the stamina to examine the whole thread in detail. Post #11 was enough to show me Chrisc's point. In the real world, inelastic collisions are not reversible. I checked to see if anyone made the connection with thermodynamics.

As I understand it, classical dynamics does not forbid a blob of putty separating into two smaller blobs spontaneously, with each of the two smaller blobs moving in different directions when analysed at the molecular scale. All it says is that it is statistically unlikely. A freek set of unlikely collisons at the molecular level producing that sort of reverse reaction is unlikely but not imposssible. It is basically an example of the arrow of time and increasing entropy. Another example is a glass falling off a table and breaking into a million pieces. The reverse situation of the glass reassembling itself and ending back up on top of the table is nto ot impossible, just statistically extremely unlikely in classical dynamics.

This just isn't going to happen in the real world. The difference in phase space volumes is more than enormous.

 

[yuiop]

Hi George and JesseM,

I admit that I only checked Chrisc's calculation in post #11; I didn't have the stamina to examine the whole thread in detail. Post #11 was enough to show me Chrisc's point. In the real world, inelastic collisions are not reversible. I checked to see if anyone made the connection with thermodynamics...

This just isn't going to happen in the real world. The difference in phase space volumes is more than enormous.

OK, I will restate it as, the reverse process is more than enormously statistically improbable.

 

[George Jones]

So, Chrisc, why are inelastic collisions not time reversible? Because of the second law of thermodynamics.

Why is there a second law of thermodynamics? I don't know if there is agreement on this, but some physicists, including Roger Penrose and Sean Carroll, think that the second law has a cosmological origin. In the blog entry

http://cosmicvariance.com/2007/06/11/latest-declamations-about-the-arrow-of-time/

Sean Carroll concludes

But if you want to describe why the Second Law actually works in the real world in which we actually live, cosmology inevitably comes into play.

 

[George Jones]

I have split this thead. The rest appears as the new thread Entropy and Cosmology

https://www.physicsforums.com/showthread.php?t=244736

in the Cosmology forum.

Any comments about the entropy and the second law of thermodynamics with respect to cosmology should be placed in the new thread.

 

[Dale]

In your example, the explanation is simple, in the forward case momentum is conserved through a decrease in KE (KE->thermal energy), in the reverse case momentum is conserved through an increase in KE (thermal energy->KE). The fact that the reverse case doesn't happen in nature is due to the non-time symmetry of thermodynamics, not any asymmetry in dynamics.

I just realized that the reverse case can in fact happen in nature. All that is necessary is that in the forward case the energy is stored, e.g. in a locking spring, instead of dissipated into thermal energy. Then in the forward case momentum is conserved through a decrease in KE (KE->elastic energy), in the reverse case momentum is conserved through an increase in KE (elastic energy->KE).

 

[Chrisc]

I had hoped to get back to this sooner but I'm in the middle of a "money pit" renovation that is
taking all my free time, so I can't address every post right now, but George Jones has made it easier
for me to address the point I'm trying to make.
These are inelastic collisions designed to show the principle of time-reversal is not only or always
a simple matter of mathematical symmetry that conserves momentum (total energy of the system).
There is a difference between the time-reverse symmetry of the laws and the time-reverse symmetry of mechanics.
The laws must, in principle, uphold under time-reversal or they would be expressions of or indications of
faulty axiomatic foundations.
Mechanics on the other hand are not so easily reversed. The kinematics of an event are the measurable dimensions
of the system, which are easily reversed as they are simply "quantitative" expression of dimension.
To reverse the direction of time you simply flip the sign to negative and everything runs (equates) backwards.
The dynamics are the problem, as they define the forces (classically) that give rise to the kinematics.
This means a translation of momentum between differing masses must consider the "empirical" evidence
of the second law of thermodynamics. Just because Newton's laws are "quantitatively" symmetric through
time-reversal, (i.e.: equal and opposite) does not mean we will ever see a fly stop a freight train.

The problem as I see it is the "qualitative" expression of the laws under time-reversal.
My point, or question is not why is there a second law of thermodynamics, it is that the second law
conditions the mechanics according to our frame of reference.
The mechanics of time-reversal measured by an observer are different according to their frame of reference.
This is a trivial observation in most cases, but in the example I've given it makes the difference between
the law of conservation of momentum displaying increased entropy or decreased entropy.
In other words, from one of only two frames, both involved in the event, the second law is upheld
in one and contradicted in the other.
This seems to indicate a "preferred" frame with respect to the laws of mechanics.
More importantly it says something fundamentally significant about the principle of relativity
and time. The (ideal) instantaneous exchange of momentum in the collision of two differing
masses, presents a "qualitative" change in dimension depending on the frame of the observer.

 

[Chrisc]

I just posted when I noticed you split this thread.
I understand that it was moving to a discussion of thermodynamics but to my OP it is
a question of relativity that becomes apparent in thermodynamics. So I would hope
to keep this thread here and steer back to my point which is more "relative" to
this forum.

 

[George Jones]

I just posted when I noticed you split this thread.
I understand that it was moving to a discussion of thermodynamics but to my OP it is
a question of relativity that becomes apparent in thermodynamics. So I would hope
to keep this thread here and steer back to my point which is more "relative" to
this forum.

Any aspects of thermodynamics and relativity that pertain to your scenarios can be posted here. I didn't mean to give the impression that this thread is dead.

Any general posts about the cosmological origin of the second law of thermodynamics should go in the new thread.

 

[JesseM]

There is a difference between the time-reverse symmetry of the laws and the time-reverse symmetry of mechanics.
The laws must, in principle, uphold under time-reversal or they would be expressions of or indications of
faulty axiomatic foundations.

Why do you say that? It is perfectly possible to imagine mathematical laws that are not invariant under time-reversal, which physically just means that if you saw a movie of a physical system played backwards, you would see it was not obeying the same fundamental equations as in the forward version. In fact, in our own universe the laws of physics are not purely T-symmetric but actually exhibit a symmetry called CPT-symmetry; physically this means that if you play a movie of a system backwards, you also have to reverse the labels of positive and negative charges (so matter becomes antimatter and vice versa), and also flip the system to its mirror image on each of the 3 spatial axes (reversing 'parity'), in order for the backwards movie to appear to be obeying precisely the same fundamental laws as the forwards movie in all possible circumstances.

This means a translation of momentum between differing masses must consider the "empirical" evidence
of the second law of thermodynamics.

The second law can be derived theoretically from the underlying laws using statistical mechanics. The basic idea is that when we ignore all the microscopic details of a system and characterize it by macroscopic parameters like pressure and temperature, then there may be many more "microstates" (exact microscopic details of the system's state) for some "macrostates" (a particular set of values for the macroscopic parameters) than for other macrostates, and the ones with more microstates have a higher entropy. The evolution of the system's microstate over time, guided by the underlying physical laws, is statistically more likely to take it to macrostates that have a higher number of microstates.

Just because Newton's laws are "quantitatively" symmetric through
time-reversal, (i.e.: equal and opposite) does not mean we will ever see a fly stop a freight train.

I think you're confused here, the fact that we never see a fly stop a freight train has nothing to do with thermodynamics, even if the collision is perfectly elastic you'd never see a situation where the collision causes a major change in the train's velocity, unless the fly is moving at a significant fraction of light speed. The time-reversed situation isn't seen either--you never see a fly cause a freight train at rest to acquire a significant velocity.

My point, or question is not why is there a second law of thermodynamics, it is that the second law
conditions the mechanics according to our frame of reference.
The mechanics of time-reversal measured by an observer are different according to their frame of reference.

No idea what you mean here. If some behavior before and after a collision is thermodynamically improbable in one frame, then when you translate that to another frame, it's equally improbable. There's no frame-dependence in the likelihood of particular types of collisions.

This is a trivial observation in most cases, but in the example I've given it makes the difference between
the law of conservation of momentum displaying increased entropy or decreased entropy.
In other words, from one of only two frames, both involved in the event, the second law is upheld
in one and contradicted in the other.

What two frames are you talking about? Different frames in SR don't disagree on which direction of time is the forward direction--if event B happens after event A on the worldline of some object in one frame, then B happens after A in every frame. So if you are thinking of viewing a given collision both forwards and backwards, this cannot be a valid example of the same situation viewed from the perspective of two different SR frames. And as long as you stick to SR frames related by the Lorentz transformation, you'll always find that if the second law is upheld in one frame, then the second law is also upheld when you view the same events in another frame.

 

[yuiop]

First, my appologies for sending this thread on a large detour into thermodynamics in the context of cosmology. George was right to split the thread and I am sorry about the extra work I created for George. My only excuse is that I spend a lot of time in both the Relativity forum and the Cosmology forum and I probably forget which forum I am in most of the time. (I know... poor excuse :tongue:)

...
There is a difference between the time-reverse symmetry of the laws and the time-reverse symmetry of mechanics.
The laws must, in principle, uphold under time-reversal or they would be expressions of or indications of
faulty axiomatic foundations.
Mechanics on the other hand are not so easily reversed. The kinematics of an event are the measurable dimensions
of the system, which are easily reversed as they are simply "quantitative" expression of dimension.
To reverse the direction of time you simply flip the sign to negative and everything runs (equates) backwards.
The dynamics are the problem, as they define the forces (classically) that give rise to the kinematics.
This means a translation of momentum between differing masses must consider the "empirical" evidence
of the second law of thermodynamics. Just because Newton's laws are "quantitatively" symmetric through
time-reversal, (i.e.: equal and opposite) does not mean we will ever see a fly stop a freight train.

The problem as I see it is the "qualitative" expression of the laws under time-reversal.
My point, or question is not why is there a second law of thermodynamics, it is that the second law
conditions the mechanics according to our frame of reference.
The mechanics of time-reversal measured by an observer are different according to their frame of reference.
This is a trivial observation in most cases, but in the example I've given it makes the difference between
the law of conservation of momentum displaying increased entropy or decreased entropy.
In other words, from one of only two frames, both involved in the event, the second law is upheld
in one and contradicted in the other.
This seems to indicate a "preferred" frame with respect to the laws of mechanics.
More importantly it says something fundamentally significant about the principle of relativity
and time. The (ideal) instantaneous exchange of momentum in the collision of two differing
masses, presents a "qualitative" change in dimension depending on the frame of the observer.

There are examples of the "arrow of time" and increasing entropy even when perfectly elastic collisions are considered. A classis example is a cluster of gas molecules in one corner of a box. Random (elastic) collisions between the gas molecules, disperses the gas throughout the box. The time reverse, where random motion of the gas molecules initially spread out in the box, results in all the gas molecules clustered in one corner is not impossible, but statistically it is enormously improbable.

In the inelastic case, when the part of the kinetic energy of the ball of putty is converted into thermal energy, a close examination of the interactions during the collision at the molecular level reveals that a lot of the molecular interactions are also elastic collisions. The difference is that the linear kinetic energy of the ball becomes randomly directed kinetic motion of the individual molecules which is one form of what we call heat. The situation gets more complicated if the individual molecules get sufficiently excited that they radiate photons or if chemical bonds are formed when the molecules are in close proximity. That would be a bit like the "locking spring" Dalespam refers to.

So whether elastic or inelastic collisions are considered, the principle of increasing entropy is basically about what is statistically more probable. While classical Newtonian physics does not rule out time reversal of kinetic interactions, thermodymanics suggests that events occurring in one direction are much more probable that events happening in the reverse sequence.

However, time does not require a "thermodynamic arrow of time" in order to advance. Take this example of throwing a ball up in the air and taking a photograph of the ball on its way up and another photograph of the ball at the same height but on its way down. With a very fast camera, so that any blur due to motion is undetectable, you would not be able to tell which is the picture of the ball on its way up and which is the picture of the ball on its way down. Thermodynamic considerations would not predict what is statistically more likely to happen next to the ball in any given picture. The arrow of time, in this case, is determined by momentum, or put another way, what will happen in the next instant is determined by the current instant and what happened in the previous instant.

 

[Dale]

I had hoped to get back to this sooner but I'm in the middle of a "money pit" renovation that is taking all my free time

Good luck on your projects!

There is a difference between the time-reverse symmetry of the laws and the time-reverse symmetry of mechanics.

You are mistaken here. If the laws that describe a process exhibit some symmetry then that process also exhibits that symmetry. There is no difference between the time-reverse symmetry of the laws and the time-reverse symmetry of mechanics. There is likewise no difference between the asymmetry of the laws of thermodynamics and the asymmetry of thermodynamic processes.

 

[Chrisc]

There's a lot of unexpected confusion or ambiguity in the premise of my question.
Obviously I didn't pose it as concise as I should have.

My question comes down to this: the "measured" exchange of momentum
between differing masses in collisions of uniform motion, is(appears to be) a relative
measure that upholds or violates the laws through time reversal depending on the
frame of the observer.

The inertial energy (rest mass) of a small mass will not overcome (is less than) the
energy of momentum of a larger mass, therefore the small mass at rest will not bring
the larger mass in motion to rest.
When collisions of this type (ideal, inelastic collisions of rigid, non-composite bodies -
hypothetical events designed for the purpose of considering the principles of the laws,
not the real mechanics of the event) are observed from a position of rest with respect
to each of the two masses involved, Newton's laws are upheld and Einstein's SR principle
of relativity accounts for the differing but valid observations between each observer.
To test the extent of the principle of these classical laws, I considered the same events reversed in time.
One would assume the second law of thermodynamics predicts the exchange of
momentum is both conserved and leads to observable "decrease" in entropy under
time reversal. Which is to say the law of inertia would simply reverse such that a
small mass would bring a larger mass to rest - a clear indication of the reversal
of the second law.
This is in fact the case when the observer is initially at rest with the larger mass.
It is not the case when the observer is initially at rest with the smaller mass.
The statistical nature of the second law is irrelevant to this asymmetry as this
inconsistency is directly correlated to the masses involved.
My question is whether there is any known reason this symmetry holds from
one frame (larger mass) and fails from the other (smaller).
At first glance it appears this asymmetry suggests time-reversal reveals a preferred
or privileged frame of reference. It seems to suggest the dynamical laws are upheld
only when one of the masses (either) is defined as the mass in motion.
In other words the relativity of momentum is not as clear cut as the relativity of motion.

 

[JesseM]

My question comes down to this: the "measured" exchange of momentum
between differing masses in collisions of uniform motion, is(appears to be) a relative
measure that upholds or violates the laws through time reversal depending on the
frame of the observer.

I still don't understand where the "frame of the observer" comes into it. Do you understand that different frames in SR don't disagree on the direction of future vs. past for causally related events? If so, would you agree that if a particular inelastic collision is thermodynamically unlikely in one frame, it's thermodynamically unlikely in every frame? So what is it that "depends on the frame of the observer" here, specifically?

When collisions of this type (ideal, inelastic collisions of rigid, non-composite bodies -
hypothetical events designed for the purpose of considering the principles of the laws,
not the real mechanics of the event) are observed from a position of rest with respect
to each of the two masses involved, Newton's laws are upheld and Einstein's SR principle
of relativity accounts for the differing but valid observations between each observer.
To test the extent of the principle of these classical laws, I considered the same events reversed in time.
One would assume the second law of thermodynamics predicts the exchange of
momentum is both conserved and leads to observable "decrease" in entropy under
time reversal. Which is to say the law of inertia would simply reverse such that a
small mass would bring a larger mass to rest - a clear indication of the reversal
of the second law.
This is in fact the case when the observer is initially at rest with the larger mass.

Huh? An observer at rest with respect to the larger mass is not going to see a reversal of the second law! He'll see the total kinetic energy of the centers of mass decrease rather than increase, as should be true for any inelastic collision that obeys the second law (because some of the kinetic energy of the centers of mass is converted to heat). If you think otherwise, please explain your reasoning.

 

[Chrisc]

I do and I would until I thought about this example and the time-reverse dynamics required to explain it.
1.
A large mass M is moving with constant velocity v toward a small mass m as measured by an observer R
initially at rest with respect to m.
After a collision through the center of their masses, R measures the velocity of M to be 1/2v and the velocity of m to be v.

Now take the time reverse this event.
2.
R observes the large mass M moving toward him at 1/2v and the smaller mass m moving toward him at v.
M and m collide and m comes to rest with respect to R and M continues at v.

The forward and time-reverse of this event holds to the laws mechanics.
If you could view the time reverse version it would appear as any natural collision of the same proportions forward in time.

Now place the observer initially at rest with respect to M and they observe the following.
3.
The smaller mass m moves toward R and M with velocity v.
After the collision, m is at rest with R and M moves away at 1/2v.

Now take the time reverse of this event.
4.
R observes M moving toward m at 1/2v.
After the collision M is at rest with R and m moves away at v.
Under these mechanics, all large masses should come to rest upon colliding with smaller masses.

So it would seem (4.) is correct in that a reversal of time should result in the reversal of the second law and the converse of the law of inertia.
The problem is with (2) as it presents mechanics that uphold the second law under time-reversal and the law of inertia.
That they differ is my point. They differ when they are the "same" event viewed from different inertial frames.

 

[yuiop]

There's a lot of unexpected confusion or ambiguity in the premise of my question.
Obviously I didn't pose it as concise as I should have.

Hi Chrisc,

Your right. I was a bit confused about what you were getting at and after checking it all out again I think I have a better handle on it now and it turns out the calculations in your diagram are correct for a inelastic collision.

The solution to the apparent paradox is this.

First case (observer is initailly at rest with the small ball)
Total momentum of the system before and after collison is 4mv.
Total KE before collision is 4 mv^2
Total KE after collision is 3 mv^2

There is a loss of 1 mv^2 as heat during the collision. When the process is time reversed the system starts with 3 mv^2 of energy and finishes with 4 mv^2 of energy so an input of heat is required and this is shown as the freak lightning bolt that hits the balls as they collide in the attached diagram.


Second case (observer is initially at rest with the large ball)
Total momentum of the system before and after collison is 2mv.
Total KE before collision is 2 mv^2
Total KE after collision is 1 mv^2

There is a still a loss of 1 mv^2 as heat during the collision. When the process is time reversed the system starts with 1 mv^2 of energy and finishes with 2 mv^2 of energy and this is again shown as the freak lightning bolt that hits the balls as they collide in the attached diagram. It is the additional energy supplied by the lightning bolt that brings the large ball to a complete stop.

The thermodynamic arrow of time is clear here. It is statistically unlikely that heat (or lightning) from the surrounding environment concentrates and imparts coherent momentum to the large ball exactly as required at the moment of collision in the reverse time scenario. Or as George would say, "it's not going to happen" :P

Perhaps I should add that the results are consistent from which ever inertial reference frame you look at it from. All the velocities in the second case are simply the velocities in the first case minus 2v. Also it should be clear that inelastic collisions are not time reversible in general.

 

[JesseM]

I do and I would until I thought about this example and the time-reverse dynamics required to explain it.
1.
A large mass M is moving with constant velocity v toward a small mass m as measured by an observer R
initially at rest with respect to m.
After a collision through the center of their masses, R measures the velocity of M to be 1/2v and the velocity of m to be v.

OK, for simplicity let's just use Newtonian formulas, which will be approximately correct if v is small compared to the speed of light. Before the collision the total momentum is Mv, afterwards it's Mv/2 + mv, and since momentum is conserved in inelastic collisions this means m = M/2. So, the kinetic energy before the collision is (1/2)*Mv^2, and afterwards it's (1/2)*M*(v/2)^2 + (1/2)*(M/2)*v^2 = (1/8)*Mv^2 + (1/4)*Mv^2 = (3/8)*Mv^2. So, the kinetic energy decreases as would be expected in an inelastic collision.

Now take the time reverse this event.
2.
R observes the large mass M moving toward him at 1/2v and the smaller mass m moving toward him at v.
M and m collide and m comes to rest with respect to R and M continues at v.

The forward and time-reverse of this event holds to the laws mechanics.
If you could view the time reverse version it would appear as any natural collision of the same proportions forward in time.

The time-reversed version would be mechanically possible, but it would also be a "thermodynamic miracle" since it would require a huge number of random vibrations in the molecules of the objects (heat) to all coincidentally synchronize at the moment of the collision and increase their combined kinetic energy in this frame, from (3/8)*Mv^2 to (4/8)*Mv^2.

Now place the observer initially at rest with respect to M and they observe the following.
3.
The smaller mass m moves toward R and M with velocity v.
After the collision, m is at rest with R and M moves away at 1/2v.

Sure. And still we see that kinetic energy has decreased, as would be expected from thermodynamics. Before the collision, total kinetic energy in this frame is (1/2)*(M/2)*v^2 = (1/4)*Mv^2. After the collision, total kinetic energy in this frame is (1/2)*(M)*(v/2)^2 = (1/8)*Mv^2.

Now take the time reverse of this event.
4.
R observes M moving toward m at 1/2v.
After the collision M is at rest with R and m moves away at v.
Under these mechanics, all large masses should come to rest upon colliding with smaller masses.

But again, the time-reversed version requires a thermodynamic miracle where random heat vibrations suddenly synchronize and give a kick to the masses, doubling their combined kinetic energy in this frame.

So it would seem (4.) is correct in that a reversal of time should result in the reversal of the second law and the converse of the law of inertia.
The problem is with (2) as it presents mechanics that uphold the second law under time-reversal and the law of inertia.

I don't follow. The time-reversed scenario (2) violates the second law just like the time-reversed scenario (4); each one involves heat being spontaneously converted into extra kinetic energy for the center of masses.