Re: Relativity question

[Joe Heafner <heafnerj@VNET.NET>]

On Tuesday, Jun 10, 2003, at 17:10 US/Eastern, 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(!)"

Ratios are DIMENSIONLESS, but not necessarily UNITLESS. The ratio of an
arc length along a circle the radius of the circle is a dimensionless
quantity, but it has units of radians. In studying simple harmonic
motion, the argument of cos(wt) is dimensionless but has units of
radians. I've even seen the eccentricity of a Keplerian orbit given
units of radians too. In relativistic systems of units, speed is
dimensionless.

> 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!

Historically, we have been fooled into thinking that energy, momentum,
and mass were three different quantities, each with its own unit.
However, Einstein showed us that this just isn't necessary and that if
we adopt a system in which c = 1, we see the true connections among
these quantities more easily. Since they are all just different
manifestations of the same thing, it doesn't matter which of the three
units we choose. I always choose units of energy because I try to
stress the fundamental importance of energy to my students. Some
authors use units of distance for both space and time, and therefore
speak of meters of light travel time and meters of spatial separation.
Some authors use units of time for both space and time, and therefore
speak of seconds of light travel time and seconds of spatial
separation. Again, it doesn't matter which convention you adopt as long
as you're consistent with it. After all, Einstein also showed us that
space and time are merely different aspects of the same entity --
spacetime.

I HIGHLY recommend Taylor and Wheeler's _Spacetime Physics_ (2nd ed.)
for extensive application of relativistic units.

Cheers,
Joe Heafner

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