Re: sinh, cosh Lorentz Xformations: illumination or obfuscation?
- From: John C. Polasek <jpolasek@xxxxxxxxxx>
- Date: Wed, 31 May 2006 16:13:07 GMT
On Tue, 30 May 2006 23:18:33 -0400, hetware <massless@xxxxxxxxxxxx>
wrote:
In the 1966 version of Taylor and Wheeler's _Spacetime_ _Physics_, theyThe answer I believe, can be found in Misner Thorne Wheeler's 1972
express the Lorentz transformations using hyperbolic trigonometric
operators on a velocity parameter. This is basically the same approach as
Einstein uses in The Meaning of Relativity by introducing an imaginary
coefficient to the time component of 4-vectors.
I found that form to be both suggestive and compelling. I see that in the
1992 edition Taylor and Wheeler opted to drop that aspect of their
exposition. The give no indication of their rationale in the preface of
the second edition. AAMOF, they give no preface at all. I suspect their
reason was that they found that it confused students, and obscured the
essential aspects of special relativity by introducing too much abstraction
at the outset.
I'm curious to know what others think about the use of hyperbolic
trigonometry to express the Lorentz transformations.
"Gravitation" page 51 "Farewell to "ict"". Up till that time the base
vector for general relativity was representable as (ict x y z) As
their headline shows, they evidently abhorred the imaginary
coefficient, since, obviously, it meant that time was different from
xyz. It would be niftier if they had (ct x y z), so they created the
"metric tensor" with signature (-1 1 1 1). In that way the quadratic
product would be -c2t2 + x2 + y2 + z2, as required. But their iniquity
is concealed in the quadratic form because the vectors enter twice.
Two algebraic sins were committed. It is invalid to take a quantity
ict and arbitrarily change it to ct so you can declare time as just a
4th dimension in addition to the other 3. That's unforgiveable.
The 2d sin is to call the metric tensor a tensor. It cannot be
transformed as a tensor, a simple example of which would be to have
the signature change from -1 1 1 1 to 1 -1 1 1. This "tensor" has to
be viewed as an arrangement at best, on which to hang gravity terms.
A tensor is first of all a transformer, so (ct x y z)xG = (-ct x y z)
clearly an unacceptable consequence.
Without ict, your hyperbolic functions disappear, I think.
The fundamental truth, that time is orthogonal to any vector you can
produce, is intrinsic to ict vs x, but that fact has been buried.
In fact, using ict you can dispose of the MM null experiment in one
sentence. The light is sent both upstream and downstream of v the
orbital velocity. Therefore the vector sum is either ic + v upstream,
or ic - v downstream, and the resultants are identical, so that no
fringe shift could occur.
I keept trying to point this out without effect.
John Polasek
http://www.dualspace.net
.
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