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The cosmic speed limit is 299792458 m/s $(c)$.

According to our current physics, if anything with positive mass will start to accelerate then the object will get arbitrarily closer to the speed of light.

But there's a problem with the notion of getting arbitrarily getting closer to some speed.

According to our current physics, the smallest distance is the plack length $(l_p)$.

So the closest speed possible to c is $c-l_p/s$.

So if an massive object is travelling at $c-l_p/s$ will need to accelerate at $l_p/s^2$ in order to get to $c$. But since we need $infinite$ amount of energy to get to $c$. So this means that our object has to spend $infinite$ infinite energy in order to accelerate.

So, my question is:

Is my claim true or my math is wrong?

Dale
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Rounak Sarkar
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2 Answers2

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More than your math, your assumptions are wrong.

First, the Planck length is not “the smallest distance”. The Planck length is simply the scale at which we expect quantum gravitational effects to become large. At that scale it is expected that our current theories will break down, and the physical theory that works at that scale has not yet been developed. We do not know if that theory will involve a smallest distance nor do we know how large that unknown smallest distance of the unknown future theory would be compared to the Planck length.

Second, there is nothing fundamental about the length of time defined as one second by the BIPM. One Planck length per second is not an important quantity. In particular, it is not a minimum speed. If you could go one Planck length in one second then there is no reason you couldn’t go one Planck length in two seconds, which is a slower speed.

Dale
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I think you are making a few mistakes in your reasoning, which I will try to straighten out for you.

It might help if you imagine you are on a platform and I am on a train which passes you at the speed you mentioned, namely one Planck-length per second less than the speed of light. In my train, I can get out of my seat and go for a walk. I can accelerate in my frame, and the energy I need to do it, as measured in my frame, is entirely independent of the speed at which I am moving past you. The point is not that I need infinite energy to accelerate in my frame, but that the result of my acceleration seems to be a negligible increase in my speed when viewed from yours.

Professor Sushing
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