Relativistic Two-Way Momentum

[John232]

Say you have an observer that is not on planet Earth. He then tries to measure the momentum of an object traveling on the Earth itself. He notices that when the object travels in one direction it has an added velocity of the Earths rotation, then in the other direction the object loses velocity due to the Earths rotation. On Earth an observer, Newton for instance, measures that the inertia of an object is the same when the object is pushed in either direction. So then how could an observer not on the Earth show that the relativistic two-way momentum is the same? Assuming they both measure about the same relativistic mass, the two observers would measure two different velocities, but that does not give the same answer for their momentum...

 

[Dale]

Momentum is a frame-variant quantity. Different observers will generally disagree on how much momentum something has.

 

[John232]

Momentum is a frame-variant quantity. Different observers will generally disagree on how much momentum something has.

So what does that mean? Is it then impossible to show something to have the same momentum from another frame of reference? Does it make it a fictitious force?

The more I think about it, it seems like an object at rest on Earth can be treated the same as an object in constant motion. The type of interactions between things on Earth act as though they don't respond to things that are accelerating like they should be. Like someone could walk a tight rope and not have to worry about compensating for their acceleration due to the Earth, and not have any more trouble walking the tight rope going one way or the other. So then is it true that an object can always determine that it is accelerating when I am right now and I can't even tell the difference of being at rest?

Would momentum have to be described in terms of an aether? And somehow the Earth has managed to have no relative motion to this aether? I am sure momentum has been tested many times before and has worked out, but why should it work any differently from another frame of reference?

 

[DrGreg]

Even in Newtonian physics, momentum is mv, and v depends on the observer who measures it. So momentum has always been observer-dependent.

In special relativity we can form a 4-dimensional vector (E, p_x, p_y, p_z) and that vector is "covariant" i.e. its components change in the same way as (t, x, y, z) changes when you change coordinates.

 

[Dale]

So what does that mean?

A frame-variant quantity is one whose value depends on the reference frame in which it is measured. Note that being invariant is a completely different concept than being conserved.

Is it then impossible to show something to have the same momentum from another frame of reference?

Correct.

Does it make it a fictitious force?

No, it doesn't even have the right units to be a force, fictitious or otherwise.

So then is it true that an object can always determine that it is accelerating when I am right now and I can't even tell the difference of being at rest?

Yes, simply use an accelerometer. An accelerometer would definitely show you that you are accelerating right now.

Would momentum have to be described in terms of an aether? And somehow the Earth has managed to have no relative motion to this aether?

No, I have no idea where that strange idea came from.

I am sure momentum has been tested many times before and has worked out, but why should it work any differently from another frame of reference?

See DrGreg's answer above. Momentum depends on velocity and velocity depends on the frame of reference. This has nothing to do with special relativity and is true in Newtonian mechanics also.

 

[John232]

I brought up aether because it is hard to see how there could not be an absolute frame of reference that governs momentum. From what I have gathered, it seems like there is only one frame of reference you can use that gives the correct values for momentum. This would mean that momentum can't be seen as being relative at all, so how does that keep momentum from throwing out the whole idea of relativity all together? One frame could be seen as being more true than another when it comes to momentum. Then how do we know that relativistic momentum is true when it isn't true at any other velocity, than just assuming that you are at rest? Or can that not always be assumed when dealing with momentum?

 

[Dale]

From what I have gathered, it seems like there is only one frame of reference you can use that gives the correct values for momentum.

You have gathered incorrectly. Every frames value is correct in that frame. It is conserved and works exactly as momentum should. Simply because two frames disagree about a frame variant quantity doesn't make one right and one wrong.

 

[elfmotat]

In special relativity we can form a 4-dimensional vector (E, p_x, p_y, p_z) and that vector is "covariant" i.e. its components change in the same way as (t, x, y, z) changes when you change coordinates.

Don't you mean contravariant?

 

[DrGreg]

Don't you mean contravariant?

Yes, you could say that. I was using "covariant" in the wider sense that it is sometimes applied to any tensor quantity, without making a distinction between "vector" and "covector". (And there are arguments for defining 4-momentum as a covector $(E, -p_x, -p_y, -p_z)$; but I don't think John232 would appreciate this level of technicality.)

 

[John232]

Yes, you could say that. I was using "covariant" in the wider sense that it is sometimes applied to any tensor quantity, without making a distinction between "vector" and "covector". (And there are arguments for defining 4-momentum as a covector $(E, -p_x, -p_y, -p_z)$; but I don't think John232 would appreciate this level of technicality.)

You should speak for yourself, I really don't mind. I don't think there is a level of technicality that you could provide for me on this subject that would satisfy my question.
No one showed me how relativity could be used in the case of momentum, so then I would be forced to use classical Newtonian mechanics to describe it.

 

[lugita15]

You should speak for yourself, I really don't mind. I don't think there is a level of technicality that you could provid for me on this subject that would satisfy my question.

John232, there's no need for any technicality. As DrGreg said, just consider Newtonian mechanics. In that case, different observers think the same object has different velocities, right? And since p=mv, they disagree about what momentum the object has, right? So do you agree that in Newtonian mechanics, the fact that momentum is frame-dependent does not at all contradict the principle of relativity? The relativistic case is just a generalization of that.

 

[Dale]

No one showed me how relativity could be used in the case of momentum, so then I would be forced to use classical Newtonian mechanics to describe it.

As DrGreg, lugita15, and I all mentioned, the same thing happens in Newtonian mechanics. So that probably is the best place to start anyway.

 

[John232]

John232, there's no need for any technicality. As DrGreg said, just consider Newtonian mechanics. In that case, different observers think the same object has different velocities, right? And since p=mv, they disagree about what momentum the object has, right? So do you agree that in Newtonian mechanics, the fact that momentum is frame-dependent does not at all contradict the principle of relativity? The relativistic case is just a generalization of that.

So how do you know that you have chosen the correct frame to use?

 

[lugita15]

So how do you know that you have chosen the correct frame to use?

That's the whole point of the principle of relativity, that there is no "correct" frame in some sense; all inertial reference frames are considered equally valid because the laws of physics (as opposed to the quantities of physics) are the same in all of them.

 

[John232]

That's the whole point of the principle of relativity, that there is no "correct" frame in some sense; all inertial reference frames are considered equally valid because the laws of physics (as opposed to the quantities of physics) are the same in all of them.

But everyone else just said that momentum wasn't equally valid in all frames...

 

[lugita15]

But everyone else just said that momentum wasn't equally valid in all frames...

No they didn't! They said momentum was frame-dependent, meaning it has different values in different frames, just like velocity. But like all physical concepts, according to the principle of relativity momentum is an equally valid concept in ALL frames.

 

[John232]

No they didn't! They said momentum was frame-dependent, meaning it has different values in different frames, just like velocity. But like all physical concepts, according to the principle of relativity momentum is an equally valid concept in ALL frames.

So then you know how to show how that momentum is the same in two different frames?

 

[lugita15]

So then you know how to show how that momentum is the same in two different frames?

It's not the same, it's different! I told you, momentum is frame-dependent, so its value varies from frame to frame, but it is an equally valid concept in all frames. Just like velocity. You think your velocity is zero, but someone riding on a train thinks you are moving. The concept of velocity makes equally good sense to both of you, it's just that you disagree on what the value of the velocity is. And by the principle of relativity, you can both be said to be equally right, or in other words you can say that there is no "correct" frame to measure the velocity in.

 

[John232]

It's not the same, it's different! I told you, momentum is frame-dependent, so its value varies from frame to frame, but it is an equally valid concept in all frames. Just like velocity. You think your velocity is zero, but someone riding on a train thinks you are moving. The concept of velocity makes equally good sense to both of you, it's just that you disagree on what the value of the velocity is. And by the principle of relativity, you can both be said to be equally right, or in other words you can say that there is no "correct" frame to measure the velocity in.

But then wouldn't a different value for momentum give a different answer for a collision? No wonder why Nasa still uses Classical Physics.

 

[lugita15]

But then wouldn't a different value for momentum give a different answer for a collision? No wonder why Nasa still uses Classical Physics.

Classical physics and special relativity do not differ on this point. They both say the same thing: if you measure velocity and momentum in different frames, you'll get different answers. This includes the case of collisions. Different frames will calculate different values for the initial velocities and the final velocities (and of course the initial momenta and the final momenta). But all frames, even though they may disagree on the value of the total momentum of the system, will agree that the momentum is conserved.

 

[John232]

Okay, I think I got it. You could only measure momentum by its true relative velocity to the object it is traveling relative to in the collision.

 

[lugita15]

Okay, I think I got it. You could only measure momentum by its true relative velocity to you.

Yes, exactly.

 

[John232]

Yes, exactly.

Darn, I just changed it to correct it. lol

 

[lugita15]

Okay, I think I got it. You could only measure momentum by its true relative velocity to the object it is traveling relative to in the collision.

Your correction is wrong. You can measure the momentum of any object relative to any other object using its relative velocity to that object. It's not true that you can only measure the momentum relative to the object it's colliding with. But keep in mind, as I said if you measure the momentum in different frames you will get different answers. But all frames agree that whatever the total momentum is equal to, its value is conserved.

 

[John232]

Your correction is wrong. You can measure the momentum of any object relative to any other object using its relative velocity to that object. It's not true that you can only measure the momentum relative to the object it's colliding with. But keep in mind, as I said if you measure the momentum in different frames you will get different answers. But all frames agree that whatever the total momentum is equal to, its value is conserved.

I don't think conservation is an issue. But, I think I am right because momentum in classical mechanics only determines the relative velocity of two objects. It wouldn't have been discovered if objects in space traveling in relative motion could change the results of that experiment.

 

[lugita15]

But, I think I am right because momentum in classical mechanics only determines the relative velocity of two objects. It wouldn't have been discovered if objects in space traveling in relative motion could change the results of that experiment.

Remember, in classical mechanics momentum is just mass times velocity, so if you think that classically momentum is the same in all frames, do you also think that classically velocity is the same in all frames? That would mean all frames would agree on what objects are at rest and what objects are moving. Don't you see that that's absurd?

 

[John232]

Remember, in classical mechanics momentum is just mass times velocity, so if you think that classically momentum is the same in all frames, do you also think that classically velocity is the same in all frames? That would mean all frames would agree on what objects are at rest and what objects are moving. Don't you see that that's absurd?

I think both objects would have to agree on their relative velocity. If you say one is at rest and the other is in motion then their relative velocity to each other would be how you would determine how much momentum the object had in the collision.

 

[lugita15]

I think both objects would have to agree on their relative velocity. If you say one is at rest and the other is in motion then their relative velocity to each other would be how you would determine how much momentum the object had in the collision.

That's a completely different issue. Classically, the relative velocity between two objects will be agreed upon by all frames. But in the relativistic case, things are more complicated. See the bottom of this page: http://math.ucr.edu/home/baez/physics/Relativity/SR/velocity.html

And just so you know, these relativistic formulas are confirmed on a daily basis by particle accelerator experiments.

 

[John232]

That's a completely different issue. Classically, the relative velocity between two objects will be agreed upon by all frames. But in the relativistic case, things are more complicated. See the bottom of this page: http://math.ucr.edu/home/baez/physics/Relativity/SR/velocity.html

And just so you know, these relativistic formulas are confirmed on a daily basis by particle accelerator experiments.

Special Relativity isn't used in quantum mechanics for time dialation. They use the Lorentz Transformation. Addition of velocity in Special Relativity also doesn't show the same movement for galaxies. IDK what addition of velocities even represents, or why you would add them together this way to find a result. You would have to know the momentum of the objects in order to find out that a velocity would even be fully transferred in a collision. Then to find the momentum you would have to find the difference in velocity of the two objects. Then to find the momentum involved in the collision you could just subtract them from each other.

 

[lugita15]

The Lorentz transformation is part of special reativity, and the formulas used in special relativity for velocity addition and relative velocity can be derived from the Lorentz transformation. And as I said, there is voluminous experimental confirmation of these formulas.

 

[Dale]

You could only measure momentum by its true relative velocity to the object it is traveling relative to in the collision.

No, you can measure momentum relative to any frame you like. That is the whole point of the principle of relativity (whether Galilean relativity in Newtonian mechanics or special relativity in Einstein's mechanics).

In each frame the momentum will be different, but in each frame using the momentum for that frame will give you the correct answers in that frame for the collision.

 

[John232]

No, you can measure momentum relative to any frame you like. That is the whole point of the principle of relativity (whether Galilean relativity in Newtonian mechanics or special relativity in Einstein's mechanics).

In each frame the momentum will be different, but in each frame using the momentum for that frame will give you the correct answers in that frame for the collision.

Really? So there is a parallel universe that pops up that shows different results for a collision? I still don't see how you could use a different frame to calculate momentum correctly, the whole reason for this discussion. Could you show an example where the collision is the same in two separate frames of references?

 

[Dale]

Really? So there is a parallel universe that pops up that shows different results for a collision?

Nonsense.

I still don't see how you could use a different frame to calculate momentum correctly, the whole reason for this discussion. Could you show an example where the collision is the same in two separate frames of references?

Sure, work any collision problem you like in any frame you like and post the results, and I will transform it to another frame and post the results.

 

[John232]

Sure, work any collision problem you like in any frame you like and post the results, and I will transform it to another frame and post the results.

Newton hits a bowling ball that weights 1kg with another bowling ball that weights 1kg. One is at rest on a table, and the other is moveing at 1 m/s. The ball at rest then gets hit and then moves 1m/s in the opposite direction.

 

[lugita15]

Newton hits a bowling ball that weights 1kg with another bowling ball that weights 1kg. One is at rest on a table, and the other is moveing at 1 m/s. The ball at rest then gets hit and then moves 1m/s in the opposite direction.

I think you mean the same direction; in frame S1 initially ball A is moving to the right at 1m/s, and it collides elastically with the stationary ball B, so then ball A comes to a stop and ball B moves to the right at 1m/s. Now consider frame S2, which is moving at 3m/s to the left with respect to frame S1. In frame S2 initially ball A is moving at 4m/s to the right and ball B is moving at 3m/s to the right, and they collide elastically, so then ball A is moving at 3m/s to the right and ball B is moving at 4m/s to the right.