On 06/10/2003 05:10 PM, cliff parker wrote:
>
> "In SR unites, the light-second is considered to be equivalent to the
> second of time, and both units are simply referred to as seconds.
> This means that these units can be canceled if one appears in the
> numerator and the other in the denominator. For example, in SI
> units, velocity has units of meters per second; but in SR units it
> has units of seconds per second = unitless(!)"
>
> The consequence of this is that the SR units for energy, momentum and
> mass all have units of Kg. Now that is interesting. However
> something doesn't feel right here and I am looking for a few of you
> to guide my thinking. I understand that a light-second can be used
> as a unit of distance. And I guess you can call it a second (for
> short) as the author says elsewhere if you want to. But calling it a
> second and having it be equivalent to a second are two different
> things. How can a light-second in the denominator cancel a second in
> the numerator? They are not the same thing!
This raises a number of issues, some trivial and
some profound.
First we must ascertain what we mean by "equivalence".
-- Is a red Volkswagen equivalent to a blue Volkswagen?
-- Is a red ball-point equivalent to a red felt-tip?
Answer: it depends. If all you care about is transportation,
you can define an _equivalence relation_ which does not
take color into account. All Volkswagens would then be
in the same equivalence class. Similarly, if all you
care about is signing something in red, you could define
another equivalence relation that takes color into
account but doesn't take other properties into account.
An author is free to define a new equivalence relation
and use it.
I suggest you give the benefit of the doubt to this
space=time equivalence, because a lot of people find
it very useful.
To some extent it is a matter of taste and a matter
of cultural bias. According to my taste, and according
to the way I was brought up, physics is intrinsically
four-dimensional. The timelike dimension behaves a
bit differently from the spacelike dimensions, but
not completely differently.
In particular, a rotation might mix up the X and Y
coordinates, while a boost mixes up the X and T
coordinates. The analogy between boosts and rotations
is profound. It would be horrifying if east-west
distances were measured in different units from
north-south distances, because that would introduce
needless complexity into rotations. By the same
token it would be horrifying to measure spacelike
intervals in units different from timelike intervals,
because that would introduce needless complexity
into boosts.
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.
Similarly logically, entropy should be measured
in bits, but that is often inconvenient, so people
use Joules per Kelvin instead. Similarly it would
be logical to measure temperature in Joules but
people commonly use Kelvins instead.
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.
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).