The determination of mass dilation

[swampwiz]

Mass dilation would of course be the relativistic mass (a term which really shouldn't be used.)

It seems none of the books or websites I have researched have explained this to my satisfaction, so I began thinking about this, and here is my derivation.

NOTE: IRF = Inertial Reference Frame

The law of motion of force-mass-acceleration is such that for any object, the time rate of change of the momentum of that object is equal to the net force applied, with the time being used - both for the time rate of change of the momentum and the velocity term of the momentum itself - being that as measured in the IRF in which that object is stationary – i.e., the IRF in which that object is stationary – and thus being proper time. However, the observed acceleration is proportional to the observed length, and this would be observed to be contracted for the object that is traveling relative to the observer, as compared to the same object being stationary with respect to the observer. Thus, the difference in the observation of an object being imparted on by a force is that the length component of the acceleration is observed to be contracted for an object that is in relative motion to the observer (i.e., contracted length), as compared to that same object being at rest with respect to the observer (i.e., proper length.)

So an observer would observe that the force on an object which is stationary with respect to his IRF as being equal to the mass that object (i.e., as observed by that observer) times the 2nd derivative with respect to proper time of the proper length - while observing that the force on an object with the same mass (i.e., if observed in the IRF in which that object is stationary) traveling at some relative velocity as being equal to the mass of that object (i.e., as observed by the same observer) times the 2nd derivative with respect to proper time of the contracted length - and since the force is the same no matter what IRF it is observed in, the length contraction on a proper length is essentially inverted to produce a mass dilation on a proper mass – or a relativistic mass in terms of a rest or proper mass.

What do you all think? The key points are that the time used for the derivative is the proper time of the object, but the length is the observed length (which would be contracted) of the object - and that any force being applied to an object would be observe to be the same from any IRF - i.e., force is invariant.

Alternatively, if anyone knows of a very clear discussion somewhere, I'd be all ears & eyes!

 

[PeterDonis]

It seems none of the books or websites I have researched have explained this to my satisfaction

What exactly do you need explained? "Relativistic mass" is just another word for "total energy". You appear to be asking why an object gains energy when it has work done on it by a force. Isn't that obvious?

If you are really asking what is the correct relativistic version of F = ma, see here:

http://en.wikipedia.org/wiki/List_of_relativistic_equations#Force

 

[swampwiz]

No I am asking how the term for relativistic mass (or relativistic momentum, from which the relativistic mass can be derived) gets derived. The fact that energy is mass derives from the fact that there is this relativistic mass.

Yes, I know F = ma. It's the derivation of m that I am asking about.

 

[Muphrid]

The use of "relativistic mass" has really fallen out of vogue of late. Nowadays, people tend to only use the invariant (rest) mass and use relativistic four-acceleration and four-momentum, and so on.

 

[PeterDonis]

No I am asking how the term for relativistic mass (or relativistic momentum, from which the relativistic mass can be derived) gets derived. The fact that energy is mass derives from the fact that there is this relativistic mass.

No, you have it backwards. Energy is the more fundamental concept; actually, 4-momentum is even more fundamental than that (since energy is just the time component of 4-momentum). The fact that "energy is mass" derives from the fact that energy has inertia; some people insist on viewing total energy as "relativistic mass" because of this, but that's a matter of interpretation, not "derivation". There's no "derivation" of relativistic mass beyond recognizing that it's just another name for the total energy.

Yes, I know F = ma.

But my point was that in relativity, F = ma is not correct, even if you interpret "m" as relativistic mass. The correct relativistic version of that equation is on the page I linked to, and it makes no mention of relativistic mass; the only "mass" involved is the rest mass.