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I’m currently in the process of learning about special relativity, of which one fundamental concept is that nothing can move faster than the speed of light.

There appear to be numerous sources for this “cosmic speed limit” suggested. The one that my textbook gives is that based off the principal of relativistic mass and $E = mc^2$, saying that as you approach $c$, relativistic mass increases and thus as relativistic mass approaches infinity, so does the energy needed to move the object.

But from my understanding, this change in mass would not be felt from the frame of reference of the moving object, so from that FOR there is nothing capping the speed. I’m sure that somewhere in this explanation there’s a fatal flaw in my reasoning, so if anybody would be able to explain why nothing can reach the speed of light that would be greatly appreciated.

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  • $\begingroup$ Just for your information: The mass of an object is a relativistic invariant. There is no such a thing as relativistic mass. $\endgroup$ – Gonenc Mogol Oct 24 '17 at 20:59
  • $\begingroup$ @gonenc It is perhaps not the standard treatment, but some sources define a "relativistic mass" given by $\gamma m_0$ $\endgroup$ – Chris Oct 24 '17 at 21:00
  • $\begingroup$ @Chris Yes I know! But I think that the misconception of relativistic mass should be cleared, even though the high school and also some undergrad textbooks keep perpetuating it. $\endgroup$ – Gonenc Mogol Oct 24 '17 at 21:01
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    $\begingroup$ @gonenc It is helpful as an introduction to the topic, as it fits both the definitions of mass as "resistance to acceleration" and "source of gravity" in a far more obvious manner than the rest mass. It's not uncommon usage even at higher levels in some fields. $\endgroup$ – Chris Oct 24 '17 at 21:09
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    $\begingroup$ Definitely the worst pedagogical thing ever. For those conscious and insisting on pedagogy. Moreover , I am afraid that most teacher just say it with no much thinking. I am not a very intelligent one. But it took me years to overcome the confusion generated by this. Can someone please explain how a textbook can claim what OP "quoted". $\endgroup$ – Alchimista Oct 24 '17 at 21:49
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But from my understanding, this change in mass would not be felt from the frame of reference of the moving object, so from that FOR there is nothing capping the speed.

At the center of mass of the system a uniquely, not velocity dependent mass, is called the invariant mass of the system and is the "length" of the four momentum vector describing the system in any Lorentz covariant frame. There is no flaw to your argument.

The limit of the speed of light to c comes from Lorentz transformations .

One has to have a clear understanding of what physics and its theories are. Physics uses mathematics as tools in order to describe observations and measurements, and be able to predict for new boundary conditions. After the time of Newton and the extensive use of differential and integral equations, it became necessary to impose physics "axioms" on the set of possible mathematical solutions, in order to pick up the subset of those solutions that were descriptive and predictive of physical observables. These physics axioms are called laws, or postulates, or principles, as in Newton's laws, or the postulates of quantum mechanics. and the "heisenberg uncertainty principle".

The limit of the speed of light originally came up when Maxwell combined the electric and magnetic laws of the time into an elegant electromagnetic theory, which predicted the existence of electromagnetic waves that transferred energy with a specific velocity in a specific medium. It was Lorentz who noticed that under Lorentz transformations the systems remained covariant, and that is why the transformations have his name, though they are inherent in the Maxwell system.

In classical physics, light is described as a type of electromagnetic wave. The classical behaviour of the electromagnetic field is described by Maxwell's equations, which predict that the speed c with which electromagnetic waves (such as light) propagate through the vacuum is related to the electric constant ε0 and the magnetic constant μ0 by the equation

c=1/(ε0.μ0)

So, in a mathematically convoluted way the limit of c for electromagnetic radiation is based on the laws of Ampere and Faraday ... , because Maxwell's equations describe the observations based on these fundamental laws.

When one hits on fundamental laws, the answer to "why" is "because it describes and predicts observations and measurements perfectly up to now" .

c became a universal speed limit both in classical and quantum mechanics because of the thinking out of the box of Einstein. trying to reconcile moving frame transformations and Maxwell's elegant and successful theory. The link give an idea of the thought processes.

Thus the general speed limit of c comes as an axiom in the theory of special relativity, which has been validated innumerable times in particle experiments, (for example).

The only answer then to the "why nothing is moving faster than c" is "because the axiom iss necessary to describe all existing data and the theory based on it predicts future measurements perfectly".

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To illustrate @annav's conclusion (her last paragraph), let's have a look at the Large Hadron Collider (LHC). This gigantic machine is able to give to a single proton the kinetic energy of a flying mosquito [*]. Now that kinetic energy is put whole into a particle which is about $10^{-21}$ times lighter than the mosquito. But still, those proton do not fly faster than the speed of light. It is amusing to compute how fast they would be if Newtonian mechanics was correct: the speed of that proton shall then be 118 times that of light in vacuum.

This is this experimental fact and the many thousands of its kind that justify the postulate that the speed of light is an upper bound.

[*] mosquito: 2 milligrams, flying at 3.7 kilometers/hour (yes, I know, a mosquito is about half slower than that); LHC: 6.5 TeV (energy of the current runs, starting from 2015)

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In that frame of reference, the speed is always zero! Which is a much harder cap than the speed of light. So there's no need for the mass to increase to keep it under the cosmic speed limit.

Two observers needn't agree on either the speed of an object or its coordinate acceleration. An observer who sees an object nearing the speed of light will observe that object to be accelerating more slowly than an observer who sees it starting at rest.

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  • $\begingroup$ It would be so nice if teachers could point to the fact that SR is kinematics. ... $\endgroup$ – Alchimista Oct 24 '17 at 21:53
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That the speed limit is $c$ is a consequence of the shape of Minkowski space, $M$. $c$ is the scaling between the time dimension and the space dimension. Everything in $M$ has a four velocity:

$u_{\mu} = \gamma(c, {\bf v})$

with magnitude $c$---for everything in all reference frames.

Moreover, 4-acceleration is always orthogonal to the 4-velocity: everything moves through $M$ at $c$ no matter what.

With that, the magnitude of the 3-velocity is bounded by $c$. Massless particles, like the photon, travel at $c$--which is conveniently named, "the speed of light".

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... which one fundamental concept is that nothing can move faster than the speed of light.

That's not entirely true. Sure, entities with speed greater than $c$ 'cause' problems with causality but this isn't the essential problem.

The essential problem is this: an entity with speed $c$ in an inertial reference frame necessarily has speed $c$ in all inertial reference frames, i.e., a entity with speed $c$ has no inertial reference frame.

Since an object with speed less than $c$ has an inertial reference frame (it is at rest with respect to itself), it cannot have speed $c$ in any other inertial reference frame (thought it can have speed arbitrarily close to $c$).

From another perspective, there are an infinity of (locally) inertial reference frames in which you have speed arbitrarily close to $c$. Think about that. Do you feel like you're traveling arbitrarily close to $c$ right now?

But there is no (locally) inertial reference frame in which you have speed $c$ because, if that were the case, you would have speed $c$ in all of them!

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