Kinetic energy conservation in matter/energy transform

[Ookke]

... C1.... C2->
P->

In the above illustration, P is a particle, C1 and C2 are detectors able to measure the energy/mass of a light pulse. C1 is at rest, P and C2 move horizontally to right with the same velocity.

Let's say that P is actually a matter-antimatter pair that annihilates to two light pulses, one of which heads horizontally towards the detectors and other heads in the opposite direction. The rest mass of P is 2 mass units and the produced pulses have 1 mass unit each.

From C1's frame, P is moving towards it and has kinetic energy, so the produced light pulse is expected to have mass more than 1 unit, when measured at C1. From C2's frame, P is at rest, so the pulse is expected to have mass 1 at C2.

I find it a bit strange that the mass can be measured to be different (isn't it the same pulse after all?), but let's continue.

Let's put a filter in C1. The filter checks the pulse mass (perhaps comparing it to a weight of rest mass 1) and let's the pulse through if its mass is above 1. If mass is 1 or below, the pulse is trapped in filter.

From C1's own frame it seems clear that the pulse must be passed through the filter. From C2's frame, it seems clear that the pulse must be trapped by the filter in C1.

Trapping/passing the pulse sounds like an event that both frames must agree. What have I missed, or does this though experiment make sense at all?. Thanks for help.

 

[Bill_K]

Ookke, Wrong terminology. A light pulse has energy, but certainly not mass. What C1 observes is that the light pulse is Doppler shifted towards the blue. This increase in energy is not "kinetic energy", what it means is that the frequency of the light in C1's frame is greater, and the energy of the pulse is greater. Energy is relative, and there is no reason to expect C1 and C2 to agree.

 

[Ookke]

Bill K: ok, maybe I'm mixing things. But at least that part was correct that C1 gets higher energy pulse. Maybe C1 and C2 could anyway measure the pulse energy level and calculate what mass it corresponds?

Edit: I think it's roughly ok to speak of "mass" of a light pulse meaning the rest mass that corresponds the pulse energy. With that translation, the original terminology should do.

 

[Ookke]

I could illustrate a bit further how the "filter" might work.

Let C1 have an energy reservoir that is able to hold energy that corresponds 1 unit mass by the formula E=mc^2. When the light pulse and C1 meet, the reservoir drains energy from the pulse as much as there is room to: all pulse energy if it fits, 1 mass unit equivalent otherwise.

From C1's frame:
..... C1 ..... C2->
---->
Light pulse has energy that corresponds to more than 1 mass unit. When light pulse and C1 meet, the reservoir is not able to drain all energy, and therefore some of the energy (a weakened pulse) is passed from C1 to the direction of C2. For momentum conservation, some of the light pulse momentum (the trapped part) is transferred to C1, reducing the relative velocity between C1 and C2.

From C2's frame:
... <-C1 ..... C2
-->
Light pulse has energy that corresponds exactly 1 mass unit. The reservoir is moving along with C1, so it's capacity is actually larger than 1 mass unit, because of mass/energy increase by velocity (though 1 mass unit would be enough). When pulse and C1 meet, there is more than enough room in the reservoir, so all pulse energy is drained and nothing is left to pass from C1 to direction of C2. All light pulse momentum is transferred to C1 in this case, also reducing the relative speed.

The frames of C1 and C2 seem to disagree whether or not light is passed from C1 to C2, which is an absolute event that all observers should agree.

 

[jcsd]

Ookke is the kinetic energy conserved under a Gallilean transformation? (no it isn't)

Just considering Newtonian physics, let's say I erected a barrier that would stop all bullets with K.E. less than A (let's assume all bullets have the same mass so K.E. is just a function of momentum) and then fired a bullet at it with K.E. of less than A in the barrier's frame. Now let's say I choose to view the situation a frame where the K.E. of the bullet is greater than A, does it change the outcome?

 

[Ookke]

jcsd: It wouldn't change the outcome, it's all about relative velocity. But if you choose a frame where your bullets move faster, the barrier will move too and its stopping capability kind of increases, doesn't it?

I have tried to be careful with different frames, in particular when comparing pulse energy to reservoir capacity, because that's pretty much the whole point of this thought experiment. I can certainly try harder, but I don't think it's all about transformations either. Higher pulse energy and blueshift in C1's frame compared to C2's frame is basically a measurable quantity, not just something that can be calculated. The energy really _is_ higher there, and that fact can be used.

Thanks for the reply. Hopefully I will get clear understanding to this some day.

 

[Nugatory]

Thanks for the reply. Hopefully I will get clear understanding to this some day.

You might want to try setting up this paradox in ordinary classical Newtonian mechanics, see if you can resolve it there. The same resolution will work for the relativistic case.

(How to create the same problem in classical mechanics? Easy... The kinetic energy is frame-dependent in classical mechanics too, so all you need is a filter that passes the kinetic energy as seen in one frame but not the other).

 

[Ookke]

You might want to try setting up this paradox in ordinary classical Newtonian mechanics, see if you can resolve it there. The same resolution will work for the relativistic case.

Ok, let's try that.

..... C1 .... C2->
P->

C1 is at rest, P and C2 move with same velocity (velocities are not close to c). P fires a bullet horizontally to right, the bullet has kinetic energy 1 mass unit equivalent in P's reference frame (just the kinetic energy, not including the bullet rest mass converted to energy).

The filter in C1 is a machinery that is able to convert kinetic energy of the bullet to energy of some other form (perhaps chemical), which it stores in a reservoir it has. Filter has capacity to hold energy at most 1 mass unit equivalent. Let's also put a meter into filter, showing how much of the capacity (0...100%) is in use.

From C1's frame:
P is moving towards C1, so the fired bullet has kinetic energy more than 1 mass unit equivalent in this frame. When bullet and C1 meet, the filter takes as much kinetic energy as possible but is not able to take all, so after the encounter the filter is 100% full and the bullet continues it way past C1 with lower kinetic energy.

From C2's frame:
P is at rest and C1 is moving away from C2. The bullet has kinetic energy exactly 1 mass unit equivalent in this frame. But because C1 is moving, it will take actually slightly more than 1 mass/energy unit to fill the filter reservoir: to fill it completely, you need to put in 1 unit of "rest energy" + some more energy that transfers to kinetic energy. Or look the other way: if you empty an moving reservoir that has nominal capacity 1 filled, you will get out the 1 unit of energy + some kinetic energy. So when you put in all the kinetic energy of the bullet, there will be some capacity still left in the repository.

At the end of the encounter, looked in C2's frame, the bullet has lost its all kinetic energy in respect to C2, i.e. it's at rest. Because C1 is moving left in this frame, it appears like bullet continues its way past C1, which is consistent with the observation in C1's frame. However it's not consistent that the meter shows 100% in C1's frame but less than that in C2's frame. This is still an absolute event, which should be agreed by both frames (but is not).

So it turns out that there is some (apparent) inconsistency also with this setting. Since the C1's frame sounds to be right, I cannot come up with any other resolution that we must somehow "invalidate" the viewpoint of C2 frame and stick into C1 frame, where the things happen. If the things were happening in C2's frame, we would probably need to stick in that. But then we would need to admit that the frames are not equivalent, but the "local" frame is privileged, which sounds bad for the Principle of Relativity. I think this would be an unsatisfactory solution. Luckily, there will still be some mistake in this thought experiment I have become blind to, looking too long at this.

 

[Ookke]

I'm starting to think this is, to my great surprise, a real paradox. And it works in both cases, light and bullet.

It seems absolutely clear that in C1's frame, the filter is able to extract enough energy from the wave/bullet to get the reservoir 100% full.

But in C2's frame, the reservoir will inevitably leave less than 100% full, because the light/bullet cannot release more energy into filter than it has (which is 1 mass unit equivalent, and this simply is not enough, as shown above) and energy cannot be created from nothing either.

So I think that either of these needs to be compromised:
- conservation of energy: we would need to allow that energy can be created from nothing or vanish to nowhere
- principle of relativity: we would need to admit that the local frame of C1 (as the things are happening at C1 location) is the only correct and C2's point of view is simply wrong, although it's an inertial observer

I would rather keep the conservation of energy and let the principle of relativity go. Any comments or ideas?

 

[Austin0]

Ok, let's try that.

..... C1 .... C2->
P->

C1 is at rest, P and C2 move with same velocity (velocities are not close to c). P fires a bullet horizontally to right, the bullet has kinetic energy 1 mass unit equivalent in P's reference frame (just the kinetic energy, not including the bullet rest mass converted to energy).

The filter in C1 is a machinery that is able to convert kinetic energy of the bullet to energy of some other form (perhaps chemical), which it stores in a reservoir it has. Filter has capacity to hold energy at most 1 mass unit equivalent. Let's also put a meter into filter, showing how much of the capacity (0...100%) is in use.

From C1's frame:
P is moving towards C1, so the fired bullet has kinetic energy more than 1 mass unit equivalent in this frame. When bullet and C1 meet, the filter takes as much kinetic energy as possible but is not able to take all, so after the encounter the filter is 100% full and the bullet continues it way past C1 with lower kinetic energy.

From C2's frame:
P is at rest and C1 is moving away from C2. The bullet has kinetic energy exactly 1 mass unit equivalent in this frame. But because C1 is moving, it will take actually slightly more than 1 mass/energy unit to fill the filter reservoir: to fill it completely, you need to put in 1 unit of "rest energy" + some more energy that transfers to kinetic energy. Or look the other way: if you empty an moving reservoir that has nominal capacity 1 filled, you will get out the 1 unit of energy + some kinetic energy. So when you put in all the kinetic energy of the bullet, there will be some capacity still left in the repository.

At the end of the encounter, looked in C2's frame, the bullet has lost its all kinetic energy in respect to C2, i.e. it's at rest. Because C1 is moving left in this frame, it appears like bullet continues its way past C1, which is consistent with the observation in C1's frame. However it's not consistent that the meter shows 100% in C1's frame but less than that in C2's frame. This is still an absolute event, which should be agreed by both frames (but is not).

So it turns out that there is some (apparent) inconsistency also with this setting. Since the C1's frame sounds to be right, I cannot come up with any other resolution that we must somehow "invalidate" the viewpoint of C2 frame and stick into C1 frame, where the things happen. If the things were happening in C2's frame, we would probably need to stick in that. But then we would need to admit that the frames are not equivalent, but the "local" frame is privileged, which sounds bad for the Principle of Relativity. I think this would be an unsatisfactory solution. Luckily, there will still be some mistake in this thought experiment I have become blind to, looking too long at this.

Why do you think that in C2 frame the bullet has only 1 mass KE relative to the meter in C1 that is moving towards it? It has that KE relative to a meter at C2 ,no?

 

[Ookke]

Why do you think that in C2 frame the bullet has only 1 mass KE relative to the meter in C1 that is moving towards it? It has that KE relative to a meter at C2 ,no?

Yes, the bullet has more than 1 mass unit energy, if you look from C1.

But if you look from C2, the bullet has only 1 mass unit energy. That's really all it has. The bullet cannot give C1 filter/meter more energy than it has. The energy conservation law must hold in C2 frame too.

 

[Dale]

You should work it out mathematically, and don't forget conservation of momentum also. Usually in these problems failure to account for the conservation of momentum is the problem.

 

[Ookke]

You should work it out mathematically, and don't forget conservation of momentum also. Usually in these problems failure to account for the conservation of momentum is the problem.

Fair enough, I will try that. I have started studying some of the formulas and need practice, especially the momentum part seems tricky. I'm not going to spam this thread if there is no real progress, but anyway this is how I have started working on this.

....... C1 .... C2->
P-> B--->

C1 is at rest, P and C2 move with 0.8c. P fires a bullet B that goes 0.8c in P's rest frame. C1 has very little rest mass of 1 mass unit. B has much larger rest mass of 100 mass units. The masses of P and C2 don't seem to matter here, so let's leave them undefined.

In C1's frame:
By addition of velocities, bullet velocity is about 0.9756c. Lorentz factor (gamma) for that is 4.55, bullet total energy gamma*m = 455, bullet momentum gamma*m*v = 444.

Let C1 have a dynamo system able to extract kinetic energy from the bullet and a battery where to store that energy. Let battery capacity be 200 mass unit equivalent of energy, initially the battery is empty.

Since the bullet's rest mass is 100 and most of the total energy is kinetic energy, it sounds fine that the dynamo and battery are able to get 200 mass unit equivalent of kinetic energy from the bullet during B and C1 encounter. I don't know yet how the momentum would be preserved, but maybe somehow, so that this would still be allowed (or, maybe not). I would expect also that C1 will experience some acceleration during the encounter with much more massive bullet, but that shouldn't invalidate the C1's observation of getting battery full, because that's a local event in C1 and should be absolute (even if the frame is not inertial all the time).

In C2's frame:
The bullet velocity is 0.8c with Lorentz factor 1.67, bullet total energy 167 and momentum 133. C1 is also moving 0.8c in this frame (to left), with total energy 1.67 and momentum 1.33.

Even if the bullet and C1 combine their kinetic or total energy, it doesn't sound possible that the battery gets itself full of energy with capacity 200 mass unit equivalent, which nevertheless seems to be happening in C1's frame.

In any case, it won't hurt me to study some formulas, so I will continue with that.

 

[Dale]

Fair enough, I will try that. I have started studying some of the formulas and need practice, especially the momentum part seems tricky. I'm not going to spam this thread if there is no real progress, but anyway this is how I have started working on this.

....... C1 .... C2->
P-> B--->

C1 is at rest, P and C2 move with 0.8c. P fires a bullet B that goes 0.8c in P's rest frame. C1 has very little rest mass of 1 mass unit. B has much larger rest mass of 100 mass units. The masses of P and C2 don't seem to matter here, so let's leave them undefined.

In C1's frame:
By addition of velocities, bullet velocity is about 0.9756c. Lorentz factor (gamma) for that is 4.55, bullet total energy gamma*m = 455, bullet momentum gamma*m*v = 444.

Let C1 have a dynamo system able to extract kinetic energy from the bullet and a battery where to store that energy. Let battery capacity be 200 mass unit equivalent of energy, initially the battery is empty.

This is all good up to here.

Since the bullet's rest mass is 100 and most of the total energy is kinetic energy, it sounds fine that the dynamo and battery are able to get 200 mass unit equivalent of kinetic energy from the bullet during B and C1 encounter. I don't know yet how the momentum would be preserved, but maybe somehow, so that this would still be allowed (or, maybe not).

It may "sound fine" to you, but you need to work through the math. There is no way that one object can simply stop another and not be accelerated itself without violating conservation of momentum. C1 is very light compared to B, so it will be accelerated a lot. The more acceleration it experiences the less of B's energy will go into the battery and the more will go into C1's KE. So, in all likelyhood C1 will not be able to collect anywhere near the 200 mass units of energy. In any case, "sounds fine" is not a calculation.

However, may I suggest one thing that may make your work a bit easier. It is to specify the type of collision (elastic, inelastic, plastic). Since there is no energy transferred in an elastic collision, and since the goal here is to transfer as much energy as possible, the natural choice is a plastic collision. I suggest one further simplification. In a plastic collision the rest mass of at least one of the objects must change, so let's assume that the bullet's rest mass is unchanged, and that all of the change in rest mass belongs to C1 and consists entirely of the additional energy stored in the battery. I.e. there is no mechanical deformation or heating of either B or C1.

PS when I worked it out just now, I got 3.46 mass units of energy stored in the battery.

 

[Austin0]

Ok, let's try that.

..... C1 .... C2->
P->

C1 is at rest, P and C2 move with same velocity (velocities are not close to c). P fires a bullet horizontally to right, the bullet has kinetic energy 1 mass unit equivalent in P's reference frame (just the kinetic energy, not including the bullet rest mass converted to energy).

The filter in C1 is a machinery that is able to convert kinetic energy of the bullet to energy of some other form (perhaps chemical), which it stores in a reservoir it has. Filter has capacity to hold energy at most 1 mass unit equivalent. Let's also put a meter into filter, showing how much of the capacity (0...100%) is in use.

From C1's frame:
P is moving towards C1, so the fired bullet has kinetic energy more than 1 mass unit equivalent in this frame. When bullet and C1 meet, the filter takes as much kinetic energy as possible but is not able to take all, so after the encounter the filter is 100% full and the bullet continues it way past C1 with lower kinetic energy.

From C2's frame:
P is at rest and C1 is moving away from C2. The bullet has kinetic energy exactly 1 mass unit equivalent in this frame. But because C1 is moving, it will take actually slightly more than 1 mass/energy unit to fill the filter reservoir: to fill it completely, you need to put in 1 unit of "rest energy" + some more energy that transfers to kinetic energy. Or look the other way: if you empty an moving reservoir that has nominal capacity 1 filled, you will get out the 1 unit of energy + some kinetic energy. So when you put in all the kinetic energy of the bullet, there will be some capacity still left in the repository.

At the end of the encounter, looked in C2's frame, the bullet has lost its all kinetic energy in respect to C2, i.e. it's at rest. Because C1 is moving left in this frame, it appears like bullet continues its way past C1, which is consistent with the observation in C1's frame. However it's not consistent that the meter shows 100% in C1's frame but less than that in C2's frame. This is still an absolute event, which should be agreed by both frames (but is not).

So it turns out that there is some (apparent) inconsistency also with this setting. Since the C1's frame sounds to be right, I cannot come up with any other resolution that we must somehow "invalidate" the viewpoint of C2 frame and stick into C1 frame, where the things happen. If the things were happening in C2's frame, we would probably need to stick in that. But then we would need to admit that the frames are not equivalent, but the "local" frame is privileged, which sounds bad for the Principle of Relativity. I think this would be an unsatisfactory solution. Luckily, there will still be some mistake in this thought experiment I have become blind to, looking too long at this.

Originally Posted by Austin0

Why do you think that in C2 frame the bullet has only 1 mass KE relative to the meter in C1 that is moving towards it? It has that KE relative to a meter at C2 ,no?

Yes, the bullet has more than 1 mass unit energy, if you look from C1.

But if you look from C2, the bullet has only 1 mass unit energy. That's really all it has. The bullet cannot give C1 filter/meter more energy than it has. The energy conservation law must hold in C2 frame too.

lets take a simpler version. P is a photon emitted in C2 frame. At C2 is an optical high pass filter such that the frequency of P is right at the top of the absorption range and does not go through
Given an identical filter in C1:
"because C1 is moving, it will take actually slightly more than 1 mass/energy unit to fill the filter reservoir:"
do you think this reasoning would apply and the threshold of the filter in C1 would be increased (increased storage capacity) as calculated in C2 ?
Therefore the photon would not be expected to pass?

Or do you think that the frequency of P would be increased relative to the filter in C1 and would pass?

If this is the case, then you would think that the photon had greater momentum relative to C1 than it did to C2 yes? ( frequency is equivalent to momentum wrt a photon.)

Considering that relative velocity between two frames (objects) is frame invariant do you think that any possible inertial frame would disagree on the calculations of whether or not the photon would pass in either frame?
Is there a fundamental difference between a photon and a bullet?

Regarding the bullet scenario: Do you think there would be any difference in physical outcome or calculations of that expected outcome in C2, between your parameters B v=0.8, C1 v=-0.8 ,with rel v=0.97

and an identical bullet with v=0.97 interacting with your same apparatus at rest in C2 ??

If the filter was simply a thickness of metal such that it completely absorbed the momentum and KE in C2 such that it just barely penetrated to the opposite side before stopping. If this was then transported to C1 would you calculate it to have greater absorption capacity there so that the bullet would be stopped somewhere in the middle?

 

[Ookke]

However, may I suggest one thing that may make your work a bit easier. It is to specify the type of collision (elastic, inelastic, plastic). Since there is no energy transferred in an elastic collision, and since the goal here is to transfer as much energy as possible, the natural choice is a plastic collision. I suggest one further simplification. In a plastic collision the rest mass of at least one of the objects must change, so let's assume that the bullet's rest mass is unchanged, and that all of the change in rest mass belongs to C1 and consists entirely of the additional energy stored in the battery. I.e. there is no mechanical deformation or heating of either B or C1.

PS when I worked it out just now, I got 3.46 mass units of energy stored in the battery.

Thanks for your work and suggestions. I made some calculations and it looks like
$$m = E \sqrt{ 1 - p^2/E^2 }$$
is a formula to get the common rest mass after perfectly inelastic collision, here E is the total energy and p momentum. I also tried to give C1 different rest masses (as the rest mass 1 is very small compared to the bullet mass), but for all cases, the mass increase was smaller or equal to the combined kinetic energy of B and C1, which is just as it's supposed to be for energy conservation. The case is pretty much solved for this part.

Of course if there was some magic way (which probably was something that I had in mind) that C1 could extract the kinetic energy from bullet instantaneously and as much as it wants, then it would get very interesting, but I'm sure that conservation of momentum makes such magic impossible.

lets take a simpler version. P is a photon emitted in C2 frame. At C2 is an optical high pass filter such that the frequency of P is right at the top of the absorption range and does not go through
Given an identical filter in C1:
"because C1 is moving, it will take actually slightly more than 1 mass/energy unit to fill the filter reservoir:"
do you think this reasoning would apply and the threshold of the filter in C1 would be increased (increased storage capacity) as calculated in C2 ?
Therefore the photon would not be expected to pass?

I think that's a good question and the optical high pass filter is worth considering. Doesn't it operate just based on the frequency? Velocity shouldn't change anything. The passing or not passing a photon is an absolute local event that all frames must agree. But if so, then we need to admit that a moving filter may seem to behave strangely, like passing a pulse that looks like (considering its frequency) that it should not be passed.

The metal barrier is a bit different, because the bullet and barrier collision would quite clearly take kinetic energy from both of them, if both are moving. I don't think this is one-to-one with the light wave case.

But for bullet, yet another setup could be to put a balance scale in C1. The scale would have a test weight of some known rest mass in one cup, and in other cup it would put the moving bullet (tricky, but I'm just trying to illustrate the idea). Depending on the frame, the bullet and test weight can have different masses. In C1's own frame, the bullet could be more massive, but in C2's frame, the test weight could be more massive. So in C1's own frame one would expect that the scale turns to show that the bullet is heavier, but in C2's frame, the observer would expect that the scale turns to the other direction.

I suppose that there is no real contradiction. The rest frame of the optical filter or balance scale is easy to imagine and see how it works, and there are some conservation or other laws that make it consistent also when looked from the other frame. I wonder what they might be, need to think about it.