Re: Relativity question

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding the part of John Denker's post where he wrote:

> The number and kind of units is another matter of
> taste.  Logically, stoichiometry should be done
> by counting atoms one, two, three, but that is
> often inconvenient so people use moles instead.

Or sometimes they use dozens, scores or grosses.

> Similarly logically, entropy should be measured
> in bits, but that is often inconvenient, so people
> use Joules per Kelvin instead.

Alas!

> Similarly it would
> be logical to measure temperature in Joules but
> people commonly use Kelvins instead.

I think John means that temperatures would logically be measured in
joules *per bit* (not just joules).  Recall that both joules and bits
measure *extensive* quantities and temperature is an *intensive*
quantity that can be expressed as a partial deriviative of
extensive internal energy w.r.t. extensive entropy (under quasistatic
conditions where no macrowork is done during the variation).  Other
expressions for thermodynamic temperature can be also derived by
playing such games as making various Legendre transformations.  But
all of them have the temperature come out as a limiting quotient of
an energy-denominated quantity divided by an entropy-denominated
quantity.

> It's not a big deal;  the conversion factors are
> well known.
>
> It turns out that for typical classroom measurements,
> measuring distances in terms of seconds of light-travel
> time is inconvenient, so people use metres and such.

Meters might be convenient for classroom measurements of
distance, but astronomical and cosmological distances are
often most conveniently measured in conventional time units.
So what is convenient depends on the particular physical
setting at hand.

> Another way of saying the same thing: it depends
> on whether you want to live in three dimensions
> and treat relativity as a correction that crops
> up when the velocity gets large ... or whether
> you want to embrace four dimensions as the normal
> state of affairs, and treat nonrelativistic physics
> as merely the first-order approxmation, valid in the
> limit of small velocity.
>
> I find the D=4 point of view helpful for intuitive
> qualitative thinking as well as for calculations.
>
> In relativity class, you really ought to try to
> embrace D=4.  You can always go back to D=3 at the
> end of the term if you want to.  You might or might
> not want to.
>
> BTW, choosing units such that c=1 is not a problem
> in practice.  If a calculation tells you that the
> energy is 17 kg and you want to re-express that
> in some other units (e.g. Joules), it's pretty
> obvious what conversion factor must be applied
> (c^2 in this example).

Not only can c be thought of as a simple unit conversion factor
between spacetime interval components denominated in spatial units
and those denominated in temporal units, but many of the other
universal dimensioned constants of nature can be thought of as
actually representing unit conversion factors between quantities that
are ostensibly different, but actually are seen as the same by a
deeper theory that makes the particular constant enter as a
fundmental parameter.

In SR c enters as a unit conversion factor between time-denominated
spacetime component intervals and space-denominated ones.  In
quantum mechanics the universal dimensioned quantity h-bar enters as
a unit conversion factor that relates the amount of action that
represents a radian of quantum amplitude phase difference, and h is
the amount of action that represents a full cycle of phase
difference.  In quantum mechanics the elapsed phase of an amplitude
for some process is measured by the elapsed action for that process.
In statistical mechanics the universal dimensioned constant k
(Boltzmann's constant) enters as a conversion factor that converts
between entropy measured in nats (1 bit = ln(2) nat) and entropy
measured in J/K effectively relating a thermal quantity to an
information-theoretic one.  In GR the quantity G is also related to
a conversion factor.  In GR the expression 8*[pi]*G/c^4 is the unit
conversion factor between the units, (i.e. inverse square length)
of spacetime curvature (as contained in the Einstein curvature
tensor) and the units of the stress-energy-momentum tensor whose
units are usually taken as those of an energy density or a pressure
(or mass density * c^2).

In each case the theory that has the dimensioned quantity (universal
constant of nature) enter in a fundamental way relates two supposedly
different kinds of things that are in a deep way really the same
'stuff' in some deep respects.  In SR time intervals are
interconvertable with distance intervals. In GR a spacetime curvature
is always associated with a corresponding gravitational effect whose
sources are the elements of the stress-energy-momentum tensor.  In QM
the action for a process is associated with the phase of its
amplitude for occuring.  In SM we see that the amount of information
needed for full omniscience about the microscopics is related to the
entropy of the macroscopic system.  In a deep sense we can consider
each of these as a relation between concepts that are fundamentally
equivalent in terms of their underlying 'stuff'.  Temporal intervals
are of the same underlying 'stuff' as spatial intervals are.  The
phase of a quantum amplitude is the 'stuff' of action.  Different
component aspects of spacetime curvature is the 'stuff' of energy
density, pressure, anisostropic stress, & momentum density.
Missing information about the microscopic state is the 'stuff' of
entropy.

None of this ought to be all that disconcerting or unfamiliar in
philosophy.  It is exactly the same sort of thing that happens
whenever a theory unites previously disparate physical phenomena.
For instance, in classical thermodynamics before Joule demonstrated
the equivalence of heat and work these two concepts were considered
as fundamentally different kinds of 'stuff'.  Work was measured as
a mechanical property of a force times a distance yielded along the
force (foot-pounds, newton-meters, etc.), and heat was measured by
its ability to change the temperature of a standard amount of a
standard substance, i.e. water (BTU, calorie, etc.).  Once Joule
demonstrated that work and heat were both of the same underlying kind
of stuff (namely energy transfers) the universal constant of
4.816 J/cal was seen to be just a unit conversion factor between the
energy 'stuff' being measured in work units and it being measured in
heat units.  By choosing to make this conversion factor unity by
denominating all heats in units of work resulted in that conversion
factor no longer appearing in and cluttering up the first law of
thermodynamics (and other expressions derived from it) with an
extranous constant that had no intrinsic conceptual value.  The same
thing happens in the later theories of SR, GR, SM, & QM when we set
the appropriate dimensioned constant of nature equal to unity.

(I know that John already knows all this stuff, but I took the
opportunity of his last comment above as an excuse to expound
further anyway.)

David Bowman