Several physical quantities are thought of as having "obvious" upper
or lower bounds in their magnitudes. Temperature, for example, cannot
go below absolute zero, the magnitude of any velocity is at most c,
the speed of light, and time cannot go back past the Big Bang.
In life outside the laboratory, we never get very close to any of these
bounds, so they pose no problem. However, researchers in cryogenics, cosmology,
and high-energy physics work at values that are perceived to be close
to these limits. We read of temperatures one one-hundredth of a degree
above absolute zero (that is, 0.01 K); of velocities at 99.95% of the
speed of light; and of the state of the cosmos at 10-8 seconds
after the Big Bang. In these cases, the standard system of units is inconvenient.
Getting three orders of magnitude closer to 0 K, or to the time of the
Big Bang, or to the speed of light, should not be obscured by the appearance
that the change is merely an infinitesimal improvement.
A change of units that maps zero to minus infinity and, in the case
of velocity, maps c to plus infinity makes it easier to appreciate
improvements that get closer to these bounds by orders of magnitude. This
principle has been in use for a long time by engineers who measure changes
in intensity in decibels (dB), where an increase or decrease by n
orders of magnitude is a change of +10n dB or -10n dB, respectively.
For physical quantities like time and temperature, the new units may
simply be taken as the common logarithms of the old units. Using T
to represent the new unit of temperature, we set 1 K = 0 T with
(10n) K = n T for negative as well as
positive values for n. Thus, 0.01 K = -2 T. More generally,
x K = (log10x) T, for any positive real
number x. A similar approach applies to time since the Big Bang,
where "x seconds after the Big Bang" becomes (log10x)
U, where U is our new, logarithmic measure of time.
For velocity magnitudes v, with 0 ≤ v
≤ c, we can rescale to V= tan(πv/2c),
with 0 ≤ V ≤
+∞. At v = ½c, this gives
V= tan(π/4) =
1.
For nanotechnology researchers, the lower limit of zero for weight,
length, and so forth can be moved to minus infinity by the same logarithmic
technique used above for time and temperature. Names and symbols for these
new units should be recommended by appropriate standards committees for
each of the research areas involved.
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