of Special Relativity

Instead of radar pulses which are essentially delta functions, as employed by Einstein, we use more realistic pulses $\,\mbox{sinc}(2\pi t / \tau)$ , where $\,\mbox{sinc}(x) = \sin(x) / x$ . These pulses are modulated with the signal $\,\exp(i\omega t)\,$ of an atomic clock, while synchronizing the clocks of laboratory O and laboratory O' :

Where it is noticed that the Fourier spectrum of a $\mbox{sinc}(2\pi t / \tau)$
function is a Rectangular function, giving an (idealized) bandwidth $2\pi/\tau$ for the radar pulses.

After recoil, the wavelength of the carrier wave has become different
by Compton scattering,
because light waves interact with *electrons* - rest mass $m_0$ - in a reflector:
$$
\lambda' = 2\pi c/\omega + (\lambda' - \lambda) =
2\pi c/\omega + h / m_0 c (1-\cos \pi ) = 2\pi c/\omega + 2 h / m_0 c
$$
by Compton scattering .
But, in order for the pulse to be modulated, the following condition must be fulfilled:
$$
\omega' \gg 2\pi/\tau \quad \Longrightarrow \quad \tau/2 \gg \pi/\omega'
= \pi/(2\pi c/\lambda') = (\lambda'/c)/2 = \pi/\omega + h / m_0 c^2 > h/m_0 c^2
$$
Conclusion: the spread of the radar-pulses in time must be much greater
than the *Compton time* $= h / m_0 c^2$ , in order to
be able to synchronize clock O with clock O' of a reference frame with rest
mass $m_0$ . **It is impossible to sychronize clocks within intervals
smaller than the Compton time.**

In addition to the above, the following thought-experiment can be performed, which affects the space-like part of the Lorentz transformations:

In somewhat more detail:
$$
\lambda' = \lambda + (\lambda' - \lambda) =
\lambda + h / m_0 c (1-\cos \pi/2 ) = \lambda + h / m_0 c
$$
Conclusion: the spread of the radar-pulses in space must be much greater
than the *Compton (wave)length* $= h / m_0 c$ , in order to be
able to compare lengths in O with those in O' of a reference frame with rest
mass $m_0$ . **It is impossible to compare lengths within intervals
smaller than the Compton (wave)length.**

Therefore, according to Wikipedia as well as according to the above,
the Compton wavelength *expresses a fundamental limitation on measuring for
the position of a particle, taking into account quantum mechanics and special relativity*.
Leaving aside the restrictions on relativistic time, we thus have:

- (common length) - (relativistic length) ≫ (Compton wavelength of electron)

program short; const c : double = 299792458; L : double = 2.4263102389E-12; begin Writeln(c*sqrt(2*L/1-sqr(L/1))); end.Output:

6.60402739611352E+0002