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What I mean is that suppose we can go to a speed greater than the speed of light, i. e., $c+1$, with the speed of light still being $c$. That said, in my new transformations, two events couldn't influence one another if they were a distance greater than $(c+1)t$ away from each other.

Consequently, in my view, we could go faster than light without breaking causality, just by discovering a new speed limit. Is that true or are there something that avoid this to happen?

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    $\begingroup$ Futurama did it first. $\endgroup$ – AccidentalFourierTransform Jun 12 '18 at 22:48
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    $\begingroup$ In which frame is the speed of light still $c$? $\endgroup$ – probably_someone Jun 12 '18 at 23:00
  • $\begingroup$ It would be less confusing to keep $c$ as the space / time scale factor and write the "new" speed of light in a vacuum as $c-1$. That way you don't need to change all the equations. The speed of photons isn't very important, but the space / time scale factor is. $\endgroup$ – PM 2Ring Jun 12 '18 at 23:43
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    $\begingroup$ The speed of light is the highest not because nothing can achieve a faster speed, but because a faster speed does not exist in the hyperbolic geometry of the universe. Time slows down to zero at the speed of light and nothing can be smaller than zero in the absolute value. Your idea is similar to getting 1 mile to the North of the North Pole. The problem is not that you cannot do this. The problem is that such a place does not exist. Nothing is to the North of the North Pole. $\endgroup$ – safesphere Jun 12 '18 at 23:47
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    $\begingroup$ Alternatively, if you mean that the actual speed of photons is slightly slower, this would imply that photons are massive particles and conceptually could be slowed down to be at rest. This would dramatically change the theory and reality of electromagnetism, but would not necessarily be completely impossible. $\endgroup$ – safesphere Jun 12 '18 at 23:52
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First of all, the concept of a "speed of light" in that universe would be meaningless. To see why, all we have to do is consider the relativistic velocity addition formula, which in ordinary Special Relativity reads

$$u=\frac{v+u'}{1+\frac{vu'}{c^2}}$$

for an object that moves with velocity $u'$ in a frame that is moving with velocity $v$ relative to an observer, who sees the object moving at velocity $u$. It's clear from this equation that, no matter what the frame velocity $v$ is, if you set $u'=c$, you get $u=c$, so light travels at the same speed in every frame in ordinary Special Relativity.

In your higher-speed-limit universe, you can do all of the exact same derivations as in normal Special Relativity, except that by throwing things back and forth that travel at a speed of $c+1$ (let's call them "slayerons"), you could influence things at distance $(c+1)t$ from you and no further (because your assumptions imply that nothing has been discovered traveling faster than $c+1$). As such, the whole of Special Relativity survives with the replacement $c\to c+1$, including the velocity addition formula:

$$u=\frac{v+u'}{1+\frac{vu'}{(c+1)^2}}$$

Note that now, when you plug in $u'=c+1$, you get $u=c+1$ no matter what $v$ is. So, with the higher speed limit, we have that anything traveling at $c+1$ will travel at $c+1$ in every frame, just like before. But if you plug in $u'=c$, you will find that $u$ now depends on $v$. With the higher speed limit, the speed of light is now frame-dependent.

Because of this, photons would have a mass. In normal Special Relativity, we have that $E=pc$ for massless things and $E>pc$ for massive things. With your higher speed limit for causality, this becomes $E=p(c+1)$ for massless things and $E>p(c+1)$ for massive things. A photon, which is observed to travel at $c$ in some particular frame (we'll get back to this later), would have $\beta=\frac{v}{c+1}=\frac{c}{c+1}<1$. Our definitions for energy and momentum become $E=\gamma m(c+1)^2$ and $p=\beta\gamma m(c+1)$, so, since $\beta<1$, we have $E>p(c+1)$. Therefore, the photon now has mass, which has dire consequences (to say the least) for electromagnetism and quantum field theory.

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  • $\begingroup$ Another way that I thought myself to see that this would violate something in physics is that the eletromagnetic wave equation $\dfrac{\partial^2}{\partial x^2} \phi + \dfrac{\partial^2}{\partial y^2} \phi + \dfrac{\partial^2}{\partial z^2} \phi - \dfrac{1}{c^2} \dfrac{\partial^2}{\partial t^2} \phi = 0$ wouldn't be invariant under that new transformations. $\endgroup$ – Slayer147 Jun 13 '18 at 11:33
  • $\begingroup$ One important implication of a massive photon theory is that the electromagnetic force wouldn't have infinite range, it would be limited like the weak force is with its massive bosons, and the residual strong force that binds nucleons. $\endgroup$ – PM 2Ring Jun 14 '18 at 9:05
  • $\begingroup$ @PM2Ring This is true, but there are so many other problems with the massive photon (which mostly boil down to the fact that you either need Polyakov electromagnetism or another Higgs-sector-type symmetry breaking operation) that I decided not to go into it here. $\endgroup$ – probably_someone Jun 14 '18 at 18:58

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