GLHTT
- From: "EL" <hemetis@xxxxxxxxx>
- Date: 13 Jun 2005 06:51:54 -0700
[EL]
The Galilean, Lorentzean, Hemetean Time Transformations {GLHTT}
Let _T_ represent the temporal interval delta-t_0, which we usually
call _Proper Time_.
Thus _T_ is a time interval measured locally using a clock at rest with
the observer but that such coordinate system, which we will call S' is
moving relative to another remote system S and observed by its origin.
The velocity of S' in S is measured relative to S and denoted as _v_.
The interval _T_ observed remotely from S and denoted by _t_ must be
dilated under the relation:
t = T / (1 - (v^2/c^2))^(0.5)
By substituting as many values for _v_ between the limits {0} and {1}
we may conclude that t can be anything from Unity to Infinity.
No provisions are made by this equation for remote _time interval_
contraction.
Since roots of a square may be positive or negative, then it is up to
the mathematician to match the sign including a negative time, which is
not logically consistent with any imminent observation of time
intervals.
On the other hand we have a variant velocity _v_ that is always
observed remotely in S but the time interval is mediated by light
travelling between the moving object and the axis of the coordinate
system.
1- How can the velocity of the object be in meters per seconds where
those seconds are relevant to the observers proper time in S while
applied to the velocity of the remote object the time of which is
deformed as observed by S also?
2- If we do accept two scales of proper time to exist, then say that
light travels between those two scales with a constant velocity _c_,
which is Length / time, then the velocity of light in meters per second
must be moving between two scales in which those seconds have a variant
magnitude from that of the seconds to which light speed is being
referred.
My verdict is that a time interval dilation is applicable to intervals
between the origin and event-points only, because remote events are
evidentially only dilated (under the logic of time required for light
to travel the distance) and can never be contracted unlike remote world
lines (the deformation of whom are direction dependent).
Of course, if we start with the accepted relativistic equations we
would be treating symptoms and not the causes behind the symptoms.
The correct procedure to investigate the apparent contradiction is to
go right to Lorentz transformation and its foundational concepts.
Lorentz began by a formula that must be reducible to the Galilean
transformation at low speeds and must equate it at rest.
_In Search for k_
The idea then was to _assume_ a constant of proportionality to exist,
hence:
x' = k (x - vt)
Using the first postulate of SR
x = k (x' + vt')
Oops, there is something unsettling here so let us rewrite them again
and investigate deeper
x' = k_1 (x - vt)
x = k_2 (x' + vt')
If k_1 = k_2 = 1, _AND_ t = t', then the equations are reduced to the
Galilean Transformation.
If it was assumed that k_1 = k_2 = k, then the derivation should have
substantiated a proof and not outright use the assumption to prove the
assumption; besides, where is the proof of t =/= t' that was also an
assumption?
Either {[t =/= t'] _AND_ [k_1 =/= k_2]} _OR_ {[t = t'] _AND_ [k_1
= k_2 = k]}, which is what the dimensional consistency of assumptions
dictates to arrive at the Galilean Transformation when k = 1.
Neglecting the Equity of t and t' was unjustified and deliberate.
Then:
x = k_2 (k_1 (x - vt) + vt') = k_1.k_2.x - k_1.k_2.vt + k_2.vt'
And
k_2.vt' = x - k_1.k_2.x + k_1.k_2.vt
t' = (x - k_1.k_2.x + k_1.k_2.vt) / k_2.v
Thus, we obtain an irreducible form unless we assumed the Galilean case
or the forced assumptions that we are out to prove OR disprove.
The second assumption is based on the second postulate of SR.
where:
x = ct
And
x' = ct'
Which is only acceptable of [c] = [L/T], where [T] is an invariant
scale as postulated.
Therefore, remote scale variance is absolutely acceptable while local
scale variance is utter nonsense.
So how do we reconcile this messy contradiction!
We have empirical data that confirms remote world-line-deformations and
we need to formulate a solution to estimate values for our variables.
S' is a system moving within S, where S harbours a Divine Robot acting
as the observer of the details within S.
Our DR is relentlessly tracking the spacetime coordinates of the origin
of S'.
We shall re-enact the scenario of the sneezing astronaut with whom an
open communication channel was open.
At spacetime events (x1,t1)' and (x2,t2)' the astronaut sneezes twice.
Electromagnetic waves move at _c_ and takes time to return from the
spatial coordinates, such that:
t_1 = t'_1 + (x'_1 / c)
t_2 = t'_2 + (x'_2 / c)
delta t = t_2 - t_1 = (t'_2 + (x'_2 / c)) - (t'_1 + (x'_1 / c))
delta t = ((t'_2.c + x'_2 )/c) - ((t'_1.c + x'_1 )/c)
delta t = (t'_2.c + x'_2 - t'_1.c - x'_1 ) / c
But x'_2 - x'_1 = delta x'
and t'_2 - t'_1 = delta t'
Then
delta t = ((delta t'.c) + (delta x'))/c
Notice that the term (dt'.c) is compliant with Minkowski's world
formalism.
In that way we do add spatial dimensions properly and by relating it
back to the speed of light we get the relative time after deformation.
My formalism preserves the direction identity in the term (dx') because
while (dt'.c) is an absolute quantity we ensure that (dx') is a vector
quantity to reflect the correct deformation whether contractive or
expansive.
My formalism is neither Galilean Nor Lorentzean, it is Hemetean. ;-)
By testing for a body at rest in S' with S' at rest in S, we find that
dt = dt'.
The sign between the two terms decides whether the deformation was
contractive or expansive.
_Time Transformations_
Galilean
t = t' .................. (Euclidean Space Compliant) (Newtonian Laws
Compliant)
Lorentzean
t = t' / (1 - (v^2/c^2))^(0.5) .................. (?) (SR Compliant)
Hemetean
delta t = (( delta t'.c) + ( delta x'))/c .................. (Minkowski
Space Compliant) (Relativistic)
_Discussion_
The Galilean transformations stands on the Newtonean concept of a
Universal time scale, and the simple idea behind it is that you cannot
determine scale changes of time unless you compare them to a universal
scale to make such a decision, hence the scale must be invariant.
Galileo, in his transformation did not consider "remote observation"
and that is why only spatial changes can be reflected in relative
velocities.
Lorentz, was motivated by Michelson and Morley experiment and by
Einstein's SR as much as by the then new Minkowski space. Under the
pressure of the moment a misconception of time and a dimensional
confusion took place, from which science is suffering till today. The
artefacts of that confusion was mainly Time Travel and the Twin-Paradox
along with strong confusion between real and apparent time dilations.
Hemet (who happens to be me) is offering a new relativistic time
transformation in which strong distinction between real and apparent
time-interval-deformations, the later of which can be calculated by the
simple formulas:
Hemetean time-interval-deformations
delta t = (( delta t'.c) + ( delta x'))/c
delta t' = (( delta t.c) - ( delta x'))/c
Where delta x' = 0 when events are spatially symmetrical, and is always
equal to the difference between the two events' spatial coordinates.
And _c_ is the constant speed of light in the homogenous and isotropic
medium in which it mediates the remote information.
This formula shows the relation between a remote time interval (delta
t) as observed locally (delta t') when the remote information is
mediated by light.
Consequently:
f_o = Total-Observer-Cycles-Count / delta t
f_s = Total-Source-Cycles-Count / delta t'
But
Total-Observer-Cycles-Count = Total-Source-Cycles-Count
Then
f_o = f_s . delta t'/delta t ........... Hemetean frequency shift.
Where:
f_s is the frequency of the wave in the frame of its source.
f_o is the frequency of the wave in the frame of the remote observer.
>>From here we can extrapolate to find unknown remote velocities if
enough local variables were measured and at least the remote frequency
should be known.
EL
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