Does increase in relativistic mass with speed violate principle of relativity?

Question

From special relativity we know that,
$$ m = \frac{m_0}{\sqrt {1-\frac {v^2}{c^2}}} $$

where, $m$, $m_{0}$ and $v$ are the relativistic mass, rest mass of the object and speed relative to the observer respectively.

Here's my question:
Postulate of relativity: Laws of physics are the same in all inertial frames of reference.

Let's imagine 3 frames, one of them being a stationary frame in space, let's call it frame-A, a second frame, frame-B is a spaceship that is currently at rest with respect to frame-A. Let's imagine another frame, frame-'C' which is traveling at a constant velocity of 'v m/s' with respect to A and B. Let both spaceships 'B' & 'C' be of identical construction and have the exact same rest mass. Both B and C fire on their engines begin accelerating. Now, since 'B' was stationary its mass(m) as shown in (i) will be lower than that of 'C'. Because of this although engines exert exactly the same force, 'C' will experience less acceleration than 'B', thus by measuring acceleration with an onboard accelerometer and later comparing his acceleration with B's, C finds out that he was moving originally. Doesn't this count as an experiment that I can use to find whether it is me who is moving or not. It's not about finding out whether C is accelerating but C finds out whether it is him who was moving or not when he was an inertial frame. While performing the experiment as he accelerates, his frame was no longer inertial, yes. But with that experiment he was able to find out whether or not he was in motion when he was an inertial frame.

NOTE: I may have gotten several things wrong in the question. I have not yet learned special relativity formally but have learned it from a few books, little Internet research, and youtube videos. I think maybe the effects of time dilation compensate for this but I don't yet know G.R. and thus can't calculate time dilation in an accelerating frame. The question may be totally stupid but I am curious about what I may have gotten wrong. Thank you for any answers.

OP, increase of relativistic mass with speed is a prediction of Special Relativity which is based on the principle of relativity. Of course it doesn’t violate it. Also, nowadays no one uses the outdated concept of relativistic mass anymore; it turns out to not be useful — Prof. Legolasov, Oct 10 '21 at 08:00
Relativistic mass is an utterly confusing and outdated concept. Good teachers do not teach it. Please get rid of any books you have that use it, NOW! — m4r35n357, Oct 10 '21 at 08:45
Actually, it's not any physics textbooks. It's from books of Brian greene, Richard feynman(Feynman Lectures), Stephen Hawking and Einstein himself. Like I said I haven't yet learned special relativity formally as I am not in undergraduate level yet. I think that's why I am confused. Thanks for the correction. — Samip Gyawali, Oct 10 '21 at 10:30
FWIW Feynman and Einstein are not modern physicists and Greene/Hawking's presentation to the public is targeted at laypeople and necessarily includes flashy half-truths for the purpose of interest. — Charlie, Oct 10 '21 at 11:28
If you encounter a treatment of relativity that discusses variation in mass with velocity, then it's not wrong in the sense of making wrong predictions, but it's 50 years out of date. — PM 2Ring, Oct 10 '21 at 12:18
@m4r35n357 I'm with OP here. It might be an obsolete concept (like I said in my comment), but it doesn't mean it isn't worth studying, if anything, because the history of 20th century discoveries in fundamental physics is fascinating. — Prof. Legolasov, Oct 10 '21 at 13:22
@Prof.Legolasov I agree it is interesting from a historical perspective, but it is severely misleading as a teaching concept. — m4r35n357, Oct 10 '21 at 14:53

Marco Ocram · Accepted Answer · 2021-10-10T09:00:20.957

The point of relativity is that all inertial frames are equivalent, which means that the mass of an object does not increase in its own rest frame. Suppose C was moving at almost the speed of light compared with A. When C fires its engines, it hardly seems to accelerate at all from A's perspective, in spite of the huge thrust of the engines, which suggests to A that C's mass has grown enormously. However, from C's perspective, the rocket is at rest, and firing the engines causes it to accelerate as usual, so an accelerometer with show the full acceleration C would normally expect.

If you are new to SR, a couple of key principles to bear in mind are that

In your own rest frame, nothing changes- your clocks never run slow, your length is never contracted.
The effects are entirely reciprocal between reference frames. For example, you might have heard of muon decay times as an example of time dilation. Where a muon is moving relative to us, the time it takes to decay increases, the increase being related to its speed just as SR predicts, suggesting that it is experiencing time dilation. In the frame of the muon, however, it decays at the usual rate, and it is us, whizzing past it at great speed, who are time dilated.

John Hunter · Answer 2 · 2021-10-10T10:20:36.143

The resolution to your apparent contradiction may be along these lines, the mathematical details are likely to be quite complicated.

when viewed from $A$s reference frame, the distance between the spaceships increases. Without any effects from relativity it would be $vt$.

With relativity theory spaceship $C$ accelerates with a lower acceleration than $B$ due to its higher mass. However $B$ doesn't catch up $C$ (to do so it would have to reach the same speed and hence also have its acceleration reduced) - but the distance between the ships is lower than expected.

That's explained by relativity theory as there is a length contraction - so the $vt$ distance between the moving spaceships, as viewed from $A$, would also be reduced.

Does increase in relativistic mass with speed violate principle of relativity?

2 Answers2