*Now* we know that heat and work are just different methods of changing the
internal energy of a thermodynamic system. Heat was shown to be equivalent
to macroscopic work by Joule. Before this heat and work were not necessarily
thought of as being so interchangable. Heat had been thought of in terms of
a caloric substance. Before the work of Joule if you told someone that
the calorimetric property of heat was "the same stuff" as a force times a
distance (work) that person would probably have thought you were crazy. Heat
and work were thought to have their own separate (engineering) dimensions.
> On the other hand, the
>constant c is the speed of light in vacuum, a measurable quantity.
Not any more. Now c is a *defined* quantity. An experiment done before
1983 that proported to carefully measure c in terms of two independent
definitions for time and length *now* actually would carefully measure
the length of the prior length standard (wavelength of an orange Kr-86 line)
in terms of the modern meter.
>defining the value of c to have an exact number, 299792458, one is only
>defining the meter in terms of the unit of time, the second. But note that
>we are dealing with two different quantities---space and time, albeit,
>components of a 4-vector---which have different dimensions.
One of the main lessons of relativity is that time and space are *not*
such different things that they deserve their own separate dimensions and
units. Time and space are very intimately tied together by relativity
theory into a *single* geometric manifold called spacetime. This is not so
different from the marriage of work and heat by the first law of
thermodynamics into the single energy change concept.
Why would you think giving different components of a (4-)vector different
dimensions is a sensible thing to do?
> If somewhere in
>the universe, the velocity of light is different, then by using our
>definition of the second and the meter, they would get a different length
>for the meter.
Or for the duration of the second, or both. How could you possibly tell if
the speed limit of causation c was different in one place to another given
that time and space are defined in such a way keep c constant? We attribute
differences in local dilations between locally measured proper lengths of
measuring rods and locally measured proper times of clocks in different
regions of spacetime as being due to inhomogeneous local curvature effects in
the spacetime manifold caused by gravitation. If some region of spacetime
effectively had a different speed limit of causation relative to some other
part of spacetime and this effect was not due to different local
concentrations in the stress-energy-momentum tensor of the matter there then
there would be some prior geometry not determined by Einstein's equations and
General relativity would be incorrect. But even in this case the effect
would be interpreted as a geometric distortion of spacetime and not as an
actual change in the speed limit of causation which is the standard by which
the geometry is measured.
> The numbers 2.54 and 4.184 are the same anywhere in the
True. But suppose that in some far away place it actually took 6756 J of
work (via a Joule-type paddle wheel device) to raise the temperature of a
kg of water by 1 deg C, while here it takes only 4184 J of work to
raise the temperature of a kg of water by 1 deg. C. Then your 4.184 J/cal
number would not be as universal as you had thought. Of course we could
always define the calorie so that the conversion factor stayed the same. In
this case we would interpret the effect as there being a weird spatially
dependent specific heat for water. This is analogous to discovering local
differences in a prior geometric curvature of spacetime not generated by the
matter/energy/stress/momentum in it.
Similarly suppose that we had defined the inch via a definition which was
independent of the definition of the centimeter. Say we defined the inch as
1/12 of the length of the average adult human's foot length. Using this
definition we can measure the the length of people's feet in centimeters
and conclude that experimentally there are 2.54 cm/in. Now suppose that in
some other place some local effects take place that stretch or shrink the
foot length of the people who live there. Also assume that travelers who
go there have their foot length similarly modified by the weird effect as
soon as they enter this twilight zone. If we measured the average adult
human foot length in the this twilight zone then we would discover that the
length of the inch (being 1/12 of the average foot length) is different
in this place being something other than 2.54 cm. Your energy and length
examples are not as different from the defined constant c case as you think.
>p.s. Perhaps we do need a referee to settle the issue.
Do you really think so? Where is Don Page anyhow? :-)
P.S. I discovered that the name of the paper in a reference I gave in an
earlier post about a possible spatial directionality or arrow in the universe
(birefringence of the universe) was in error. The correct citation is: Borge
Nodland & John P. Ralston, "Indication of Anisotropy in Electromagnetic
Propagation over Cosmological Distances", Phys. Rev. Lett., 78, 3043, Apr. 21,