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I've frequently seen two different explanations for why, in SR, it's impossible for an massive object to reach $c$:

  1. As a massive object approaches $c$, its kinetic energy starts being converted to mass and so if it keeps getting more and more energy from somewhere, it will also just keep gaining more and more inertia (aka relativistic mass) and so never reach $c$.

  2. It's impossible for objects with mass to move through space at the speed of light, because all objects are already "moving" through spacetime, which is hyperbolic, at the speed of light. For objects with mass, some of that speed must be through time and so it can never all be through space. Most of this explanation is from a video by physicist Don Lincoln of Fermilab. To paraphrase him, "trying to move faster than light is like trying to go further north than the north pole -- there's just no more speed to gain."

I know both of these explanations are oversimplifications, the first more so than the second, and my impression is that most physicists think the idea of relativistic mass (as related to the first explanation) is outdated and not particularly useful. It also seems like most of them also think the geometric explanation has more explanatory power and is more useful and I agree.

However, if I understand correctly, the first explanation isn't actually wrong per se and is actually mathematically equivalent to the geometric explanation, if both arguments are made more rigorous -- it's easy enough to show algebraically that the relativistic mass of an object does approach infinity as its speed approaches $c$ afterall.

My question is, how do we make both these arguments more rigorous and show that they're mathematically equivalent (assuming I'm not mistaken about that being the case)? I've seen the algebra for the first argument and it's simple and easy to follow. I've also seen spacetime diagrams used in demonstrating the second argument, both via analogy to Euclidean space, as Dr. Lincoln does in his video, and more accurately using hyperbolas, but the geometry focused explanations were sort of hand-wavy in terms of the actual math, so I'm unsure how to directly relate them to the first argument or demonstrate that they follow from the postulates of SR.

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    $\begingroup$ Just a note: kinetic energy isn't "converted" into mass. It's probably best to leave "mass" out of the conversation entirely. Energy has inertia (that's in fact what Einstein proved in the paper that produced his famous equation) and mass (aka "rest mass") is just the minimum amount of energy a body can have. $\endgroup$
    – Eric Smith
    Commented Apr 25, 2023 at 23:51
  • $\begingroup$ @Ericsmith, I mean, we talk about potential energy being converted to kinetic energy in classical mechanics, electrical energy being converted to heat in thermodynamics and electromagnetics, etc., so why not talk about kinetic energy being converted to mass in the context of SR? I do agree that the "energy converted to relativistic mass" explanation isn't the best way to think about what happens when a massive object approaches $c$ though. $\endgroup$ Commented Apr 27, 2023 at 5:01
  • $\begingroup$ Just say no! to wibbly mass. $\endgroup$
    – m4r35n357
    Commented Apr 27, 2023 at 8:18
  • $\begingroup$ @MikaylaEckelCifrese : usually when we talk about electrical energy being "converted" to heat we mean that the electrical energy goes away and heat energy appears in its place. But kinetic energy doesn't go away to be replaced by mass -- a moving body always has kinetic energy. In fact the kinetic energy is just the difference between its total energy and its rest energy (i.e. rest mass). Speaking in terms of total energy is usually clearer than risking confusion about the meaning of "mass". $\endgroup$
    – Eric Smith
    Commented Apr 27, 2023 at 11:37

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There is nothing mathematically problematic with either argument, and they are both rigorous and "correct". Yes, you can show that they are mathematically equivalent. The problem is not there. The problem is that the concept of relativistic mass is in itself a source of confusion and does not need to be there in the first place. It is pedagogically a bad idea and so we are hoping that everybody can stop using bad ideas like that.

I will only have rest mass. Everybody is much less confused when we only talk about the one true mass$^\text{TM}$.

The Einstein energy-momentum relation reads as $$ \left ( \frac E c \right )^2 - p^2 = (m c)^2 \qquad \implies \qquad E = m c^2 \text{ only if not moving!}$$ It is actually rather bad to be considering the energy and momentum as velocity functions multiplied by rest mass, because photons have energy and momentum without rest mass, and this really shows that it is energy and momentum that are the correct independent variables, subject to the above constraint, and that defines the velocity. But we can at least display their relations $$E = \gamma m c^2 = \frac{mc^2}{\sqrt{1-\left (\frac v c \right )^2}} = m c^2 \cosh \chi \qquad \bigwedge \qquad \vec p = \gamma m \vec v = \frac{m \vec v}{\sqrt{1-\left(\frac v c\right)^2}} = m c \hat{\vec v} \sinh \chi$$ You might have seen the relations in front, but maybe not the hyperbolic functions. They are very nice because they naturally satisfy $$\cosh^2 \chi - \sinh^2\chi = 1$$ which thus makes it very obvious that Einstein's energy-momentum relation is obeyed. It also makes it clear that Lorentz boosts are somewhat of a rotation in Minkowski spacetime.

We can write the energy-momentum as a 4-vector $$ \begin{pmatrix} E / c \\ \vec p \end {pmatrix} = m c \begin{pmatrix} \cosh \chi \\ \hat{\vec v} \sinh \chi \end {pmatrix} = m ( \gamma c , \gamma \vec v )^T$$ where, if you just divide out the rest mass, then the 4-vector is the 4-velocity, that has the ``always moves in spacetime at velocity $c$" that you saw from Don Lincoln. At normal speeds, every particle is mostly moving in time at velocity $c$, but if you keep speeding up, the particle seems to lengthen its 4-velocity (if you plot it in Minkowski spacetime) and aims more and more to move at 45 degree angle, and there is just no way to get exactly there. You can speak of it as a hyperboloid; it is also obvious from the hyperbolic angle view, which goes on and on without bound.

When you speak of the relativistic mass, that corresponds to $\gamma m$. You can heuristically argue that this thing will grow without bound as you move closer to speed of light. You can go higher in mathematical sophistication by talking about the 4-acceleration, and show the same behaviour, that more and more of the energy given to a system is sent to increasing $\gamma m$ than to increasing $v / c$, and show that it is mathematically equivalent to the above.

But if you start going into those directions, you will run into issues like Longitudinal and transverse mass and quickly pull your hair out. The fact of the matter is that these are all bad ways to look at things.

It is easier to realise that $\gamma m$ is better seen as $E/c^2$, that it has never been the mass that is increasing, but rather the total energy that is increasing without bound. If you just do one division, you will see that $$ \frac{\vec v}c = \frac{\vec p c}E $$ and then you can either consider things by momentum or energy $$ \frac v c = \frac{|\vec p|}{\sqrt{(mc)^2 + p^2}} < 1 \\ \frac v c = \frac{\sqrt{E^2 - (mc^2)^2}}{E} < 1 $$ i.e. no matter how much momentum or energy you give to a massive, originally slowly moving object, there is just no way to get them to arrive at the speed of light. You can give it infinite amounts of kinetic energy, and it would just reach the speed of light.

It is just bad habit that we cling on to ideas about mass. Gravitational pull is due to total energy and momentum too, i.e. a hot cup of tea has a greater gravitational pull than the same cup of tea cooled. Inertia, as shown above, is due to total energy. It just so happens that in old physics, we always moved so slowly that the rest mass energy dominated the total energy. For the physics that only cared about energy differences, purely kinetic energy sufficed. Those that used to care about mass, are actually really all caring about total energy. Rest mass only appears in the invariant Einstein's energy-momentum relation, and in the fundamental equations (e.g. Dirac equation). This view is much cleaner, easier for students to understand, and much less confusing, because we have only one thing we call mass, not 4 different things we call mass that we have to remember when to use what.

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