I need to make a presentation on natural units. My professor asked me to visualize a world where $c$ and $\hbar$ are actually equal to unity. Like, what are the consequences? I also want to know the philosophical meaning behind natural units.
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1$\begingroup$ Hi! Check out wikipedia first :) en.wikipedia.org/wiki/Natural_units. Having these two equal 1, you then should check en.wikipedia.org/wiki/Planck_units. $\endgroup$– Radek DCommented May 13 at 13:27
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$\begingroup$ Sydney Coleman described a spinning basketball as having a tiny speed but an enormous angular momentum, in natural units. $\endgroup$– AndrewCommented May 13 at 14:19
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$\begingroup$ The philosophical meaning of natural units is that units are a human choice and thus have no physical significance. $\endgroup$– GhosterCommented May 13 at 17:11
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1$\begingroup$ Maybe your professor means a world where, for exampe, $c$ and $\hbar$ have values such that their effects can be seen and experienced by humans? SI units are based around units that are natural to humans: 1 meter, 1 second, 1 kilogram etc. So to make these effects measurable, you would to have to put them to 1 in SI units. For example $c=1m/s$ or $\hbar=j\cdot s$. Natural units are quite different from this. They are just one kind of measuring system and they don't modify the actual values of physical constants. Like metric or imperial units. $\endgroup$– AccidentalTaylorExpansionCommented May 13 at 21:29
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1$\begingroup$ @AccidentalTaylorExpansion yes .....a world where c and h are actually equal to 1 m/s and 1 Js... that's what I'm asking $\endgroup$– ShrishtiCommented May 15 at 1:11
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You already live in a world where natural units apply. The speed of light is exactly $1$ light-year/year, or equivalently $1$ light-second/second.
Or approximately $1$ foot/nanosecond. This is useful for designing computer circuits. It helps you figure out how far a signal can propagate in one clock cycle.
You can choose units where $\hbar$ is also exactly $1$.
This allows you to simplify equations. For example, $E=m$. Or for a beam of light, $d = t$.
This helps clarify some concepts. In space-time, space and time are on equal footing. $d = t$ makes this clearer than $d = ct$.
We live in a world dominated by gravity. Horizontal distance is an entirely different thing than altitude. They are unrelated concepts. They have different names and units. We use kilometers for horizontal distances, $x$, and meters for altitude, $h$.
If you want to calculate a slope, s, you need a special constant, $K$, that I just made up. $K = 0.001$ km/m. $K$ is a fundamental property of the universe.
$$s = Kh/x$$
But if you use natural units and measure everything in meters, equations get simpler. You start to see relationships that were obscured.
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$\begingroup$ But you cannot make everything be unity, because look at the fine structure constant, which always evaluates to about 1/137 no matter what unit system you use. $\endgroup$– CPlusCommented May 13 at 16:18
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1$\begingroup$ Is second really a natural unit? In contrast to a year I know nothing in nature which would lasts exactly one second. $\endgroup$ Commented May 13 at 21:46
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$\begingroup$ I meant that units exist where c and $\hbar$ are exactly one. I didn't mean to imply that those units had to be based on seconds or years. $\endgroup$ Commented May 14 at 13:41