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I am working on a personal hard science-fiction project but when I analysed the following situation, in the context of special relativity my conclusion seems counter intuitive and perhaps I am misunderstanding the underlying physical principles of SR which is not my strong point.

A particle is permanently moving in a circle at some constant relativistic speed. I wanted to find the radius of this circular trajectory for a given acceleration, and how long it would take for the particle to complete one full revolution in proper time.

I calculated the relativistic period (in proper time) should be $T = \frac{2\pi r}{\gamma v}$ where $r$ is the radius, $v$ is the constant speed, and $\gamma$ is the Lorentz factor which is easy to get from $v$.

For the radius, as there is never space contraction in direction to the center of the circular trajectory (the radius is always perpendicular to the motion) it would be logical to me that the radius has the same value $r = v^2/a$ as for Newtonian mechanichs. However this would imply that for a given acceleration $a$ the radius would never exceed $c^2/a$ for any given speed $v$, which I find counter intuitive. Have I come to the wrong conclusion because I am suffering from some misconception about the fundamental principles of SR?

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I calculated the relativistic period (in proper time) should be T=2πrγv

This part is straightforward and correct.

it would be logical to me that the radius has the same value $r=v^2/a$ as for Newtonian mechanics. However this would imply that for a given acceleration a the radius would never exceed c2/a for any given speed v, which I find counter intuitive.

In the limit as v goes to c, r goes to $c^2/a$ which does indeed imply a maximum allowable radius for a given centripetal acceleration. This is not too mysterious, as by rearranging the equation to $a = v^2/r$, it can be seen that that increasing the radius reduces the centripetal acceleration and this is why the radius is limited for a given acceleration.

The radius is also limited by angular velocity. The equation for angular velocity is given by $\omega = r/v$, which implies that in the limit as v goes to c, the maximum allowable radius is limited to $r<\omega c$, which is simply a consequence of nothing being able to move faster than the speed of light.

KDP
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