# Relativistic Two-Way Momentum

1. Mar 18, 2012

### 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...

2. Mar 18, 2012

### Staff: Mentor

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

3. Mar 18, 2012

### John232

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?

4. Mar 18, 2012

### 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, px, py, pz) and that vector is "covariant" i.e. its components change in the same way as (t, x, y, z) changes when you change coordinates.

5. Mar 18, 2012

### Staff: Mentor

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.

Correct.

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

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

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

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.

6. Mar 18, 2012

### 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?

7. Mar 18, 2012

### Staff: Mentor

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.

8. Mar 18, 2012

### elfmotat

Don't you mean contravariant?

9. Mar 18, 2012

### DrGreg

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.)

10. Mar 18, 2012

### John232

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.

Last edited: Mar 18, 2012
11. Mar 18, 2012

### lugita15

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.

12. Mar 18, 2012

### Staff: Mentor

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

13. Mar 18, 2012

### John232

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

14. Mar 18, 2012

### lugita15

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.

15. Mar 18, 2012

### John232

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

16. Mar 18, 2012

### lugita15

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.

17. Mar 18, 2012

### John232

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

18. Mar 18, 2012

### lugita15

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.

19. Mar 18, 2012

### John232

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

20. Mar 18, 2012

### lugita15

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.