I thought of this question while studying inelastic collisions in special relativity, where kinetic energy is converted into mass-energy.

I was wondering if it's possible to formalize a version of special relativity where mass conservation still holds. Basically I'm imagining that, in this hypothetical theory, the loss of kinetic energy in an inelastic collision would be accounted for in the same way as in classical kinematics: By assuming that the kinetic energy associated with center-of-mass-motion is converted into kinetic energy associated with disorganized, relative motion about the center of mass - that is, thermal energy.

Of course, this theory would be empirically wrong. But here is my question: Would it be wrong simply because Nature "decided" not to do things that way, or is there a compelling theoretical reason why we ought to doubt it? For example, would such a theory contradict Einstein's postulates of special relativity in some way? Or, perhaps, would it violate certain symmetries?


Edit: To make my question more concrete, here's an example of some calculations in this hypothetical theory, per Ismasou's request:

Consider a collision between two objects, of masses $m_1$ and $m_2$. Suppose these two masses "stick together" upon collision.

Given mass conservation, the total mass of the resulting object is just $m_1 + m_2$. Now, let's restrict our attention to the inertial frame where $m_2$ is at rest, and $m_1$ is moving at some initial speed $v_i$ (assume this is a 1-dimensional problem). What is the final speed $v_f$ of the composite mass?

Well, supposing that relativistic 3-momentum conservation still holds, we know that the initial and final relativistic momentum $p$ is given by:

$$p=\frac{1}{\sqrt{1-v_i^2/c^2}} m_1 v_i=\frac{1}{\sqrt{1-v_f^2/c^2}} (m_1+m_2) v_f$$

If my calculations are right, solving this equation for $v_f$ yields:

$$v_f = \frac{1}{\sqrt{1-\left[1-\left(\frac{m_1}{m_1+m_2}\right)^2 \right]\frac{v_i^2}{c^2}}} \frac{m_1}{m_1+m_2} v_i$$

Notice that in the limit of low speeds, the square root factor approaches 1, and we recover the result obtained in a classical inelastic collision problem.

Now, here, it is not obvious that relativistic energy $E=\gamma m c^2$ is conserved - it seems like we've had to give up energy conservation to maintain mass conservation. However, in this theory, relativistic energy might be conserved in the same way that kinetic energy is conserved in a classical inelastic collision: By postulating that the kinetic energy which moved $m_1$ forward has simply been dispersed into the many disorganized motions of the composite mass about its center of mass (thermal energy).

  • $\begingroup$ I don't exactly understand your question. But you should know that a classical system is just a "lower" limit of special relativity, in some sense it's the first order of perturbation, if you consider your small parameter to be the velocity. So, what happens in a classical system technically shouldn't contradict special relativity (the other way can be false though). Why would your hypothesis be wrong ? $\endgroup$ – Ismasou Jan 23 '17 at 17:38
  • $\begingroup$ @Ismasou I agree - but in this hypothetical version of special relativity, mass conservation would be something that holds not only in the "lower limit," but at all speeds. You could imagine it as an alternate extension of classical mechanics to the relativistic regime. My question is: Is there a strong theoretical reason why this is the "wrong" way to generalize classical mechanics, or are standard special relativity and this hypothetical theory both equally reasonable candidate theories (prior to empirical evidence of mass-non-conservation)? $\endgroup$ – Wade Hodson Jan 23 '17 at 17:50
  • $\begingroup$ I hope I understand what you mean now, let me answer you this way. So the difference between classical and relativity, is the magnitude of velocity, so at low velocity which means low kinetic energy the collision will only transfer some low energy which will easily be seen in the macro world as heat, while for a high energy momentum, the energy is so high that it has to move your objects somewhere, and the boost will be solely in the direction of your momentum. And also because the kinetic energy is pretty high it can explode your object, then you start thinking about mass conservation. $\endgroup$ – Ismasou Jan 23 '17 at 18:03
  • $\begingroup$ @JohnRennie Could you expand on that comment? It runs counter to my understanding of inelastic collisions in special relativity, as illustrated in, for example, this solution: feynmanlectures.info/solutions/… $\endgroup$ – Wade Hodson Jan 23 '17 at 18:08
  • $\begingroup$ @Ismasou sorry, we may be talking past each other a bit. what about the case of an inelastic collision in the center of mass frame? in that case, in both classical and relativity, we have no macroscopic movement at the end of the collision. regardless, my question isn't really about the particular mechanisms by which mass is or isn't conserved. it's about if my hypothetical theory is a good candidate theory to begin with. for example, my theory would be arguably bad if, somehow, mass conservation implied a non-constant speed of light, in contradiction with Einstein's relativity postulate. $\endgroup$ – Wade Hodson Jan 23 '17 at 18:31

Conservation of momentum plus conservation of energy (AKA conservation of four-momentum) and the notion of mass as the magnitude of the four-momentum fundamentally break conservation of mass.

This is just the Minkowski version of the triangle inequality $$ |x + y| \ge |x| + |y| \;,$$ where $|v|$ should be understood to be the magnitude of the four-vector $v$.

So you can't restore the conservation of mass without either losing a cherished conservation rule or redefining mass.

Aside: I think this is one of the most overlooked differences between Einstein's world and Newton's, and I find that emphasizing to students that the mass of a system is generally different from the sum of the masses of it's components often help them start seeing how to work within special relativity rather than trying to cram it into a Newtonian mold.

  • $\begingroup$ this is interesting! any chance you could elaborate a bit? it's not obvious to me how mass non-conservation follows from that inequality. $\endgroup$ – Wade Hodson Jan 24 '17 at 0:03
  • $\begingroup$ The invariant mass (AKA the 'rest mass' to those who insist on using the dated, unhelpful, and unnecessarily confusing notion of relativistic mass) is (to within some boring factors of $c$ that depend on exactly how you construct your four-vector) the norm of the energy-momentum four-vector: $m \propto |p|$. So the mass of a system is the norm of the sum of the four-momenta of parts: $M \propto |p_1 + p_2| \ge |p_1| + |p_2|$ and is not in general equal to the sum of the masses of the parts: $M \ge m_1 + m_2$. $\endgroup$ – dmckee --- ex-moderator kitten Jan 24 '17 at 0:14
  • $\begingroup$ Like everything else that is `wierd' in relativity this can be viewed as a direct consequence of the geometry of Minkowski space, but that is the space that supports the symmetries of Maxwell's equations (and agrees with more direct observations). $\endgroup$ – dmckee --- ex-moderator kitten Jan 24 '17 at 0:17

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