Is the metric dimensionful?
- From: "Ken S. Tucker" <dynamics@xxxxxxxxxxxx>
- Date: 28 Feb 2007 10:40:22 -0800
Hi Dr. Francis, studied your post.
On Feb 27, 11:55 am, Oh No <N...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
Thus spake Ken S. Tucker <dynam...@xxxxxxxxxxxx>
Hi Dr. Most and all.
On Feb 27, 1:44 am, Andreas Most <Andreas.M...@xxxxxxxxx> wrote:
Ken S. Tucker wrote:
Oh No wrote:
Ken>>Well, g_00 is 1/sec^2,
ds^2 = g_mu_nu dx^mu dx^nu
I would be inclined to think ds is measured in secs, as is dx, which
makes g_00 dimensionless.
We should carefully discuss that.
"ds" is invariant so it can't have relativistic units,
which seconds (or minutes etc...) is.
Invariance has nothing to do with the units of ds.
By definition "ds" is a scalar, and therefore invariant,
(a scalar, is unitless, it's a pure number).
Andreas is right. Scalar quantities, or invariant quantities are not
unitless. cf mass in classical mechanics.
A kilogram mass is a rest relatively to Paris, per ISU standards.
Relatively to a moving FoR or one in a different gravitational
potential that ISU standard Kilogram will have a differing
inertia. The use of the word "mass" scientifically refers
to the ISU standard reference kilogram in France, and
that mass is certainly relativisitic (not scalar, not invariant).
Invariance simply means that ds^2 is independent from the choice
of coordinates in which you calculate the distance.
The coordinate transformation takes care of the unit conversion
if necessary. You just have to agree on what units to use for ds.
I do understand what you are imagining, however
you'll need to prove, or reference to a proof that the
Kronecker Delta is a dimensionless {1,0}, in the
scheme you select.
Kronecker delta is defined in mathematics without any reference to
units. Look it up with google.
Certainly, see pgs 21-24 (html version)...
http://72.14.253.104/search?q=cache:TM3H8ZIcLPQJ:www.grc.nasa.gov/WWW/K-12//Numbers/Math/documents/Tensors_TM2002211716.pdf+tensors+%2B+nasa&hl=en&ct=clnk&cd=1
Please note on pg 24, the Kronecker delta is
given by a scalar product of "unit" vectors.
As I understand it, those unit vectors may be expressed
as inches, centimeters, seconds, years, and General
Covariance permits the free choice of units, as well
as the choice of the CS.
I previously posted a proof,
as to why g_00 = 1/time^2.
and I already corrected it. Had your post found me moderating it would
not have gone through.
Ohno, I'm being sent back to "Shut-up and Calculate" :-).
Seriously though, It is foundational (per SPF charter ) to
consider alternative interpretations to the meaning of an
arbituary "norm" like AB = g_uv A^u B^v, especially applied
to ds^2=g_uv dx^u dx^v.
It is my impression mathematicians and phycists differ
on that interpretation. I think our discussion may bring
that difference into relief.
Regards
Ken S. Tucker
.
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