Why does the equivalence principle not violate special relativity?

Question

If the equivalence principle asserts that there is no way to determine whether our reference frame is accelerating or is being acted upon by gravity (i.e. the laws of physics are the same in both situations), then what is wrong with the following (thought) experiment: suppose I set-up a laboratory in a closed environment in which I cannot see whether there is any large mass nearby. Now I set-up my acceleration measuring equipment and I measure a constant acceleration of (for example) $\frac{c}{60} \text{ s}^{-1}$ (in SR units). Now when I monitor the acceleration over time it is expected to remain the same when by lab is being acted upon by gravity, but in the case that we are dealing with an accelerating reference frame, this acceleration cannot continue indefinitely as this would raise the speed of our reference frame (a lab in some rocket or similar) beyond the speed of light at some finite time as measured from the moment when the acceleration measurements began (in our example $61 \text{ s}$ if we assume that $v_{start}=0$). Equivalence of both situations would therefore violate special relativity.

Why can't this experiment be used to determine in which of the two situations our lab is (gravitational acceleration, reference frame acceleration)?

Answer 1

The acceleration you'd measure in a lab with an accelerometer would be the proper acceleration, which in relativity (unlike Newtonian physics) is distinct from the coordinate acceleration in some inertial frame (though at any given moment, the proper acceleration is equal to the coordinate acceleration in the inertial frame in which the object has an instantaneous velocity of 0 at that moment, often referred to as the 'comoving frame' or some variant like the 'momentarily comoving inertial frame' referred to here). It is possible for an object to have a constant proper acceleration indefinitely, in this case a given inertial frame will see its coordinate acceleration continuously decreasing as its velocity relative to that frame approaches the speed of light, for the detailed equations see the relativistic rocket page.

Answer 2

If you think of a right triangle, the base of the triangle is the mass, the height of the triangle is the momentum, and the energy or relativistic mass is the hypotenuse.

A felt acceleration in the lab corresponds to a change in velocity ($\frac{momentum}{mass}$). This is the $tangent$ of the base angle of the triangle. It appears to the lab occupants like they are getting faster all the time.

The velocity from the perspective of an observer is proportional to $\frac{momentum}{energy}$ or $\frac{momentum}{relativistic \ mass}$. This is the $sine$ of the base angle of the triangle. It is clear that no matter how much momentum is accumulated, an observer will never see them going faster than the speed of light as sine will never reach 1. What the observer sees is that time is actually slowing down for the occupants of the lab, so that to the occupants, it appears as if the are forever getting faster.

As $tangent$ can increase without bound, so the velocity of someone travelling through space is unbounded from the perspective of the traveller. As $sine$ is bound to less than 1 (for an object with mass), so the velocity of the traveller from the perspective of any observer will always be less than $1c$.

I am not a professional in this area, so please correct me if I have made a mistake.