# Another paradox from the force transformation laws?

1. Dec 28, 2009

### jeffwu78

Hi all,

I've been recently digging into SR in my attempts to better understand electromagnetics. One thing that has been giving me trouble is the force transformation laws, which seem to give rise to some pretty gnarly paradoxes, one of which I've listed here:

Consider two objects, where object 2 is moving at a velocity v relative to object 1. There is a moment when both objects just touch, at which time object 1 exerts a force F (perhaps via an expertly-timed spring release) perpendicular to the direction of motion of object 2. Assuming Newton's 3rd law holds, then object 2 exerts a force -F on object 1.

Now consider the same scenario from the inertial frame of object 2, so that object 2 is now at rest, and object 1 moves with velocity -v relative to object 2. According to the force transformation laws, the force exerted by object 1 on object 2 is F/sqrt(1 - v^2/c^2), but the force exerted by object 2 on object 1 is F*sqrt(1 - v^2/c^2). The paradox is that Newton's 3rd law does not seem to be preserved by changes in the inertial frame.

Besides the lack of symmetry, the thing that most troubles me about this paradox is how can we have the conservation of momentum hold between inertial frames and yet Newton's 3rd law fail to hold?

Any answers would be greatly appreciated. But please answer only if you know what you're talking about.

Thanks,

Jeff

2. Dec 28, 2009

### bcrowell

Staff Emeritus
I haven't taken the time to go through this in detail, but I suspect that the resolution of the paradox has to do with the relativistic transformation of angles. The perpendicular and parallel components of the force transform differently. A force that in one frame of reference is perpendicular to the line of motion will not be perpendicular to the line of motion in another frame.

Another thing to consider is that you really have three motions: the motion of object 1's center of mass, the motion of object 2's center of mass, and the motion of the point of contact.

3. Dec 28, 2009

### Al68

The v in those equations refer to the components of velocity in the same direction of the force applied, not in a perpendicular direction.

4. Dec 28, 2009

### vin300

No, it's perp.
The force is y^3 times longitudinal acc and y times transverse acc, whole multiplied by rest mass

5. Dec 28, 2009

### vin300

Fact is:In both cases, it is what you said in the second case.Difference is:The first case uses momentum is mv, the second uses ymv

6. Dec 29, 2009

### jeffwu78

>> Fact is:In both cases, it is what you said in the second case.Difference is:The first case uses momentum is mv, the second uses ymv

Thanks for all the replies. But what do you mean by the first or second case? The two scenarios are describing the same events, but viewed from different inertial frames.

Jeff

7. Dec 31, 2009

### jeffwu78

After going crazy about this paradox for a couple of days, I think I have a good hunch of how to resolve the paradox. The key is that object 1 must use some of its internal energy (for example, in a compressed spring) in order to exert the force on object 2 for any nonzero period of time, and thus loses some of its "rest mass" in the process, due to mass-energy equivalence.

If one defines force to be the rate of change of momentum, and the conservation of momentum always holds, then Newton's third law also must hold. The issue is with the force transformation laws, which implicitly assume that the rest mass of the object in question is constant. But here the "rest mass" of object 1 changes with time, so the force transformation laws for object 1 no longer hold.

If/when I can give a mathematical proof for the resolution of this paradox, I will post it on this thread.

Jeff