What is the smallest possible unit of time?
Is time infinitely divisible, or is there a smallest possible unit of time measurement? Is space
infinitely divisible, or is there a smallest possible unit of spatial measurement? The following thought
experiment suggests that neither space or time are infinitely divisible.
Suppose I want to “demonstrate” the “value” of any very large number by counting down one
number at a time to zero and that I want to do this so that the count will not take longer than a
typical working day, say 10 hours/600 minutes/36,000 seconds/3.6 x 104 seconds.
Lets start by taking a number that is around 3.6 x 1010. To count down from this number in the time
set we need to count 106 numbers each second. This means that each number in the final column of
our counting device needs to be displayed for 10-6 second. During this period light will travel around
300m.
What if our number is a bit larger, say around 3.6 x 1020. To count down from this number in the
time set we need to count 1016 numbers each second. This means that each number in the final
column of our counting device needs to be displayed for 10-16 second. During this period light will
travel around 300nm. But 300nm is less than half the wavelength of visible light – humans could
never see that the number had changed from looking at the lowest digit.
Theoretically there is no limit to the size of number we could want to count down in our experiment.
What if our number is a bit larger, say around 3.6 x 1030. To count down from this number in the
time set we need to count 1016 numbers each second. This means that each number in the final
column of our counting device needs to be displayed for 10-26 second. During this period light will
travel around 300 x 10-10nm. But 300 x 10-10nm is so small that it is probably not detectable by
anything a human could make. Just how small is the smallest measurable distance? Unfortunately
when we get down to the size of things like photons we are into quantum mechanics, with its
consequential uncertainty of position and hence size. The Proton Compton wavelength, however, is
10-6nm, so we are thinking here of something orders of magnitudes smaller than a proton.
If we consider the Proton Compton wavelength (or any of the other Compton wavelengths, such as
those for neutron and the marginally smaller tau) as the smallest measurable thing there is, and
compare this with the speed of light, 3 x 108m/s, can we come up with a smallest possible
measurable unit of time? There would seem to be no more than 3 x1023 measurable units within the
space covered by light in a second. Doesn’t this imply that the shortest measurable time is 3 x 10-24s?