Re: Question on Spacetime dilation.
- From: "Ken S. Tucker" <dynamics@xxxxxxxxxxxx>
- Date: Mon, 22 Dec 2008 01:09:13 -0800 (PST)
Hi John.
On Dec 21, 8:09 pm, John Polasek <jpola...@xxxxxxxxxx> wrote:
On Thu, 18 Dec 2008 14:12:06 -0800 (PST), "Ken S. Tucker"
<dynam...@xxxxxxxxxxxx> wrote:
Hi XXein and guys....
...
xxein: Welcome aboard to both you and Allen. I hope you enjoy theI posted this to SPF...
ride.
To Theo and all theoreticians who embrace the signature (+---),
I'm unable to understand it so far, but I'm trying!
On Dec 18, 12:16 pm, Theo Wollenleben <alpha0...@xxxxxxxx> wrote:
Ken S. Tucker schrieb:
On Dec 16, 2:50 pm, Theo Wollenleben <alpha0...@xxxxxxxx> wrote:
Ken S. Tucker schrieb:
ds^2 =g_uv dx^u dx^v , {u,v=0,1,2,3},
= dx_u dx^u ,
= dx_0 dx^0 + dx_i dx^i , {i=1,2,3} , Eq.(1).
dt' = ds = (0.6) dt, Eq.(2).
What differential coefficients should be subbed
into Eq.(1) to yield Eq.(2)?
g_uv = diag(1,-1,-1,-1)
dx_0 = dx^0, dx_i = -dx^i
ds^2 = (dx^0)^2 - (dx^i)^2
ds = sqrt(1 - (dx^i/dx^0)^2)*dx^0 = sqrt(1-(v/c)^2)*cdt
If "dr" is a spatial displacement *vector*, and
e^i and e_i are the 3D spatial basis vectors, the
text book suggests that,
dr = e_i dx^i = e^i dx_i .
and also
dr^2 = dr.dr , (that's a dot/scalar product)
= e_i.e_j dx^i dx^j = g_ij dx^i dx^j
= e^i.e^j dx_i dx_j = g^ij dx_i dx_j
does that seem reasonable to you Theo and all?
Where the scalar product is given by x.y=g(x,y). That is correct, though
"being spatial" in Minkowski space is not an invariant concept, of
course. If we choose another basis, the same vector could have a nonzero
time component.
Ok, I've studied metrics such as g_i0, however they are not popular,
and I have been advised to understand the signature (1,-1,-1,-1)
you (Theo) introduced above, that are conventionally used by most
relativists and what they mean by that.
A minor problem I'm stuck on is this, if
dr = e_i dx^i = e^i dx_i , and dx^i = - dx_i then e_i = -e^i .
The tensor textbook requires that the Kronecker Delta " delta^u_v "
has values of 1,0 such as
delta^u_v = {1,0}when {u=v , u=/=v}, defined by
delta^u_v = e^u.e_v , but
I doubt it.
Well I have no doubt, one of us knows what I'm
doing.
The LHS is second rank, and the RHS 0 rank. The equation
has no value.
Well that's easy, dismiss the foundation of
tensor analysis because you don't understand it!
What YOU need to prove, when adopting the
signature (+---) is that it will NOT generate
3 values for the Kronecker delta, ( +1 0 -1),
and if you cannot do that, prove a 3 valued
Kronecker in any CS is logically valid, and if
you cannot do that, then that specialized CS
is not valid.
What do you think mathematical physics is,
a circus ??
Back in the 60's and 70's, in consultation with
a few of the best GR theoreticians this planet
has seen, privately informed me that the (+---)
was awkward at best and likely wrong.
And furthermore, tasked me to straighten out
the "mess". At that time NO ONE could maintain
a consistency from basis vectors (unitary base
vectors) directly to the GR metric boundary.
In the absence of that solution, that I've
requested in this group and also, SPF and
spr, that no one knows now (2008), and back
then 1970's, we decided by 1983 to solve the
"mess" by interdefining the meter and second,
for a brief see,
http://physics.trak4.com/modern-spacetime.pdf
and a mathematically and physically proven
signature becomes (++++) in accord with the
ISU, following my tasked results.
I did the task in the late 70's, published
the results, the ISU accepted them and then
redefined the meter, in 1983, and no one paid
much attention to that, which at the time was
reasonably quite unimportant to most people.
It would have remained unimportant until
a test of the Kerr metric was conceived and
brought to fruitation by satellite experiment,
the GP-b, (something NASA argued against, but
were politically pressured to support).
So now Tucker is back in main-stream, (I'm ok
with that), explaining the impact of the 1983
decision on the Kerr metric and the resulting
null "frame dragging" here,
http://physics.trak4.com/MST_Kerr.pdf
It appears to me in hind-sight, by adopting
signature (+---) will suggest a Kerr metric
with antisymmetric rotational artifacts from
the "g_01" metric fed-back into ds^2, for ref
see Weinberg's "Grav&Cosmo" pg 240, that was
developed from the (+---) signature.
OTOH, dating back to 1983, my ref above
excludes the Kerr metric using the (++++),
signature, funny how things turn out...
The GP-b becomes an experiment to determine
which signature is correct, like a battle
between (++++) and (+---), waged in outer
space by two opposing groups of scientists,
by a satellite with Tucker at the fulcrum,
but I digress...
e^1.e_1 = -1 , e^0.e_0 = +1 , e^1.e_0 = 0 , etc.
Yes John, I look forward to your improvements.
Regards
Ken S. Tucker
.
- References:
- Question on Spacetime dilation.
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