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$BACKGROUND:$

I was recently watching a youtube video about Unruh radiation:https://youtu.be/7cj6oiFDEXc

In this video, the host said that we need to accelerate at $10^{20}m/s^2$ to raise the temperature by $1 kelvin$.

This got me thinking about relativistic acceleration because if you accelerate at $10^{20}m/s^2$, you will pass the speed barrier in less than a second if we apply the math of Newtonian mechanics in which if you are travelling at v and you start accelerating at a then after time t your velocity will be $v+at$ which cannot be true so I want to know the relativistic formula for acceleration.

(If you are thinking that how on earth can someone talk about Unruh radiation but don't know the formula for relativistic acceleration, then let me point out that I am not a physicist, I am just a high school student who tries to learn as many things as possible from the internet. Sometimes that information can be incomplete but slowly I manage to gain the full information.)

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  • $\begingroup$ This explains it rather well; en.wikipedia.org/wiki/Unruh_effect#Temperature_equation $\endgroup$ – Adrian Howard Jan 2 at 7:18
  • $\begingroup$ The Unruh effect is a prediction, observations are in debate, and there is some doubt about the existence of Unruh radiation. $\endgroup$ – Adrian Howard Jan 2 at 7:25
  • $\begingroup$ @Adrian howard, I think you should read her question again. She used Unruh effect as an example for her question about relativistic acceleration. $\endgroup$ – Ranjit Kumar Sarkar Jan 2 at 8:44
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    $\begingroup$ In SR you can accelerate forever and never go faster than light. See this article. $\endgroup$ – G. Smith Jan 2 at 18:08
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    $\begingroup$ @Jhuma Sarkar. You may find it better to find a suitable book on SR and learn from that. It will be more work than watching video, but that's what it takes to understand a subject. Also, you may find that even reading the first chapter will give you most of the info you want to know. Finally, books, the good ones, tend to be wrong less often than videos (@G. Smith already gave a good reference, so my reply is just rambling :-)) $\endgroup$ – Cryo Jan 2 at 22:19
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The acceleration that is relevant to the Unruh radiation is what is called proper acceleration. It is the acceleration of an object as observed by an inertial frame with respect to which the object is momentarily at rest. As pointed out in the comments, an object performing such motion of constant proper acceleration performs the hyperbolic motion. As you can see in the spacetime diagram in the Wiki article on hyperbolic motion, the (spacetime) trajectory of such a particle is (surprise, surprise) a hyperbola with the light cone as its asymptote. Thus, it doesn't reach the speed of light after any finite amount of time.

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