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Given the definition of the Planck length

$$ \ell_\mathrm{P} =\sqrt\frac{\hbar G}{c^3} \ $$

and the Planck time

$$ T_{P}\equiv \sqrt{\frac{\hbar G}{c^5}}, $$

if we take into account an observer moving with respect to us with speed $v$, then due to Lorentz contraction Planck length and time would be different for different observers. How to explain this with special relativity?

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These values are simply defined values that qualitatively express the limits of the theory.

They're not the real lengths of physical objects so contraction does not affect these values.

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  • $\begingroup$ The second sentence is correct, the first only debatably so. While it is conjectured that some of the Planck units represent such limits, in no currently unequivocally accepted theory could we definitely deduce that the Planck units have any real physical significance. E.g. the Planck mass is around 21 µg and has certainly no special significance within current QFT or GR. $\endgroup$ – ACuriousMind Jan 5 '17 at 13:55
  • $\begingroup$ My understanding is that it is generally accepted that current mainstream theories are not reliable below the Plank length, hence my use of the word "qualitatively". $\endgroup$ – StephenG Jan 5 '17 at 14:17

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