Mass & Velocity Paradox: Solving a Relativity Conundrum

[H8wm4m]

Imagine for a moment a closed system. All that exists in this system is two like-charged bodies which would have the tendency to repel each other. These bodies of mass are connected by a short string. Say this string is cut or suddenly snaps, and the bodies begin to fly apart. This means the tension energy in the string is changed into kinetic energy, and no energy is created or destroyed. According to Relativity, however, an increase in velocity creates an increase in mass. This means the total mass of the system is increased, which is potential energy. Is this not in violation in the Law of Conservation?

I assume I am missing some factor or just don't fully understand the theory, so maybe a more knowledgeable person can solve this parodox or point out some aspect of relativity I failed to consider.

 

[jostpuur]

Increase in the relativistic mass is the same thing as the increase in kinetic energy.

EDIT:
H8wm4m, I answered in the same language as you asked the question. What "mass" is supposed to mean is a different matter then. It is not difficult to predict that this thread will attract some preaching about it. I could in advance throw my own opinion on the matter as well: Merely knowing what concepts are recommended, and what are not, (or what are said to be correct and what incorrect,) does not help you to understand anything, or help you to solve paradoxes.

 

[RandallB]

Mass does not change with speed.
At best early interpretations of E=mc² might say “think of it as if mass was increasing” as speed approaches “c”. Modern views of SR use intrinsic mass m₀ alone, but you still apply “gamma” to the momentum “p”.
Just do not apply "gamma" to mass and expect to get anything “real”, just because “p” increases by “gamma” does not require that real mass increase to cause it.

relativistic mass is an abstraction not a real thing

 

[jostpuur]

In fact the mass of a charged particle that moves in an electric field is more complicated than one might first think. I'll move onto a role of the one who is asking questions now.

If we think that the potential energy of the particle is located precisly in the particle, then the rest mass of the particle changes as it moves in the potential field. Suppose the particle is then accelerating as it moves away from some other point charge. Now the particle is moving to a lower potential, so its rest mass decreases. And doesn't it decrease in precisly such manner, that the relativistic mass is remaining constant, as the speed increases??

One might say that the potential energy is not in the particle, but in the field, but if the energy density is still localized close to the particle, then what's the difference?

 

[pmb_phy]

Imagine for a moment a closed system. All that exists in this system is two like-charged bodies which would have the tendency to repel each other.

This system has energy due to the charged particles and potential energy related to the repulsive force.

Note: I have a web page dedicated to physics. This page

http://www.geocities.com/physics_world/sr/sr.htm

Has several links which are concerned with the concept of mass as does an article I wrote which is located at

http://www.geocities.com/physics_world/mass_paper.pdf

These bodies of mass are connected by a short string. Say this string is cut or suddenly snaps, and the bodies begin to fly apart. This means the tension energy in the string is changed into kinetic energy, and no energy is created or destroyed.

Sounds okay so far but I'm unclear about this tension energy thing. Perhaps others can explain this to me?

According to Relativity, however, an increase in velocity creates an increase in mass. This means the total mass of the system is increased, which is potential energy. Is this not in violation in the Law of Conservation?

The energy in the system consists of electrostatic energy, mass-energy and potential energy. An increase in the mass comes at the cost of the decrease of the mass associated with the potential energy - thus the mass of the system is conserved. What you've outlined could be a very simple model of a nuclear fission. Take a look at this page

http://www.geocities.com/physics_world/sr/nuclear_energy.htm

I think it may help you.


Note: When RandallB says "Mass does not change with speed." he is speaking about a different definition of the term "mass" that you are speaking. You were speaking of what is also called "relativitistic mass" aka "inertial mass". His comment "relativistic mass is an abstraction not a real thing" can be strongly argued against. The only "real" things are things like position and speed. All else are defined in terms of these two quantities. Calling something which is defined in terms of position and speed "real" or not is ambiguous at best.

Pete

 

[H8wm4m]

So it seems I have merely fallen prey to the common confusion between relativistic mass and intrinsic mass.
There is probably some sort of lesson to be learned here.
Many thanks

 

[robphy]

So it seems I have merely fallen prey to the common confusion between relativistic mass and intrinsic mass.
There is probably some sort of lesson to be learned here.
Many thanks

One lesson is certainly:
clearly [and as unambiguously as possible] define ones terms,
especially terms used in Galilean and Newtonian physics that have refined meanings in Special and General relativity.

 

[pmb_phy]

So it seems I have merely fallen prey to the common confusion between relativistic mass and intrinsic mass.

Not at all. You knew what you were talking about. The only mistake you made was to exclude the mass of the electrostatic potential energy.

Pete