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If I could travel a super small distance (proton) in an even smaller time (Planck time), how long would it take me to cross the observable universe?

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If you don't like math: A fraction of a second.

For math lovers:

One proton is $8.414 \cdot 10^{-16}$ meters wide, and one Planck time is $5.391247 \cdot 10^{-44}$ seconds. This means that we are traveling at $\dfrac{8.414 \cdot 10^{28}}{5.391247} = 1.5606779 \cdot 10^{28}$ meters per second. Dividing by $299792458$ m/s, or $c$, we get $5.205861 \cdot 10^{19}c$. The observable universe is 93 billion light years in size, so it would take $\dfrac{93 \cdot 10^9 \cdot 86400 \cdot 365c}{5.205861 \cdot 10^{19}c} \approx 0.0564$ seconds to traverse the observable universe. This obviously is impossible, as faster-than-light travel (username checks out) violates the rules of general relativity (as we know it).

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    $\begingroup$ Strictly speaking, the problem isn't just general relativity but also special relativity. Going FTL means that there are inertial observers that see your trip take different time, including zero or a negative number of seconds. So the impossibility is not just that moving that fast as a material object isn't possible according to our current understanding of physics, but that the answer itself is somewhat undefined. $\endgroup$ – Anders Sandberg May 19 at 10:32

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