Preprint (26.04.2009)
Date: Sun,26 Apr 2009 23:08:18 GMT
From:redshift0@narod.ru (Alexander Chepick)
Organization:
Newsgroups: sci.physics, sci.astro, alt.sci.physics.new-theories
Subject: СЭТ
Key words: Absolute - LAST - inertial frame of reference -
Tangherlini transformation - length - time.
PACS: 01.55.+b, 02.10.-v, 03.30.+p, 98.80
The Time and Length in the LAST
Alexander M. Chepick
Nizhni Novgorod
e-mail: redshift0@narod.ru
Abstract
In the LAST there are no paradox of twins and paradox of length, longitudinal lengths of a rod and periods between two events in different IFRs are ordered on speed of these IFRs in the Absolute system of reference (AFR).
1. Introduction
"The size of a figure decreases in a direction of its movement". Fitzgerald and Lorentz have deduced such conclusion from a condition of performance of the Maxwell equations in all inertial system of reference (IFRs), and then Einstein has deduced it from Lorentz's transformations in the special theory relativities (SRT).It turn out that not only in Lorentz's theory and in the SRT this conclusion is true, but also in the theory of a stationary ether (LAST). In this theory the transformation of coordinates from Absolute system of reference (AFR) into any inertial system of reference (IFR), moving in AFR, differs from Lorentz's Transformations.
2. Tangherlini transformation
F.R.Tangerlini [1] was the first who has suggested to consider this transformation (and in honour of which it is named Tangerlini Transformation (TT) ), after that - S.Marinov [2], and R. De Witte [3]. In Russia N.Kupryaev [4] was the first who has declared about such transformations. He at present is the best expert on TT. All of them invented this transformation independently from each other, as prior to the beginning of 21 centuries the LAST-theory had not enough wide propagation.
In an article "Main principles of the Absolute" is shown [5,5] (from the postulates: the existence of AFR motionless in aether; from zero result of Michelson-Morley experiment, and also from a method of natural synchronization [5,3]) a conclusion of Tangherlini Transformation of coordinates (t, x, y, z) of any point in AFR into coordinates (t', x', y', z') of the same point in IFR (the IFR moves in the AFR with a speed V, parallel to axis X):
A(V): t'=t/γ; x'=γ(x-Vt); y'=y; z'=z , где γ=γ(V)=[1-(V/c)2]- . (1)
For recalculation of coordinates from IFR1 into IFR2 it is necessary first to transform them from IFR1 into AFR, and then from AFR into IFR2, so we receive two-parametrical transformations B(V1,V2):
B(V1,V2)=A-1(V1)A(V2) (2)
If a direction of vector V (V is velocity of IFR J' in AFR) is any, then there is such a turn P(V), after of execution of which the direction of axis X of AFR will coincide with a direction of vector V. We shall determine P(0)=E. We shall turn 3 spatial axes of AFR with the same P(V), and we shall obtain a new AFR'. Now we shall choose such IFR J' that at the initial moment t=0 its corresponding axes of coordinates would coincide with axes of AFR'. Then transformations A(V) of coordinates (t,x,y,z) into (t',x',y',z') between AFR and IFR looks like:
A(V)=P-1(V)A(V) (3)
And for recalculation of coordinates from IFR1 into IFR2 the formula has been gained:
B(V1,V2)=A-1(V1)A(V2)=A-1(V1)P(V1)P-1(V2)A(V2) (4)
Formulas (3) and (4) are generalization of formulas (1) and (2) for a case of such IFRs, whose velocities are not necessarily coincide with axis X of AFR.
3. Properties of time in LAST and the SRT
During comparison of properties of transformations of coordinates in LAST and the SRT we shall consider that AFR in the SRT is such IFR in which the observer is motionless. In such a case it turns out that if in LAST an object moves in AFR with speed V, then in the SRT this object will move in AFR with same speed V.
In transformations A() and B() there is no dependence of time from x-coordinate. Note, that in Lorentz's Transformations (LT) such dependence there is. We note also, that formulas for recalculation of intervals of time from AFR into IFR for SRT and LAST coincide. But in SRT this property is only for events linked to one clock:
t'2-t'1=(t2-t1)/γ (5)
In LAST a property of an absolute simultaneity follows from this formula : if in some IFR two events are simultaneous, they are simultaneous in AFR and any other IFR.
The formula of relation of intervals of time in two IFRs for LAST:
g(V1,V2) = (t''2-t''1)/(t'2-t'1) = γ1/γ2 (6)
differs from the formula for the SRT, and where γm=γ(Vm); m=1,2; Vm - absolute speed of IFRm .
And in LAST the parities (5) and (6) are true for any pair of events, but in the SRT - only for those events, at which x'2=x'1, y'2=y'1, z'2=z'1. Formula (6) is a definition of rate of time IFR2 relatively IFR1.
In LAST a relative rate of time IFR2 in IFR1 is inverse to relative rate of time IFR1 in IFR2. Relative rate of time in LAST can be greater or less than 1, depending on a parity of absolute speeds V1 and V2. Therefore the paradox of time is impossible in LAST. Now in case of a repeated meeting of two twins the twin-traveler will be younger, as his speed in the AFR was varied. Generally, those clocks whose way was longer in AFR, will show smaller time at a repeated meeting.
4. Properties of length and longitudinal length in LAST and the SRT
A difference of coordinates of rod's length on axis X' of IFR is named a longitudinal length of a rod. It is natural that fixations of positions of the ends of a rod moving in IFR are made simultaneously.
In LAST and the SRT for rod moving in AFR the formulas of reduction of longitudinal length l=x2-x1 coincide
l=x2-x1=(x'2-x'1)/γ(V) =l0/γ(V) (7)
where l0=x'2-x'1 - longitudinal length of the rod in its own IFR J' (where this rod is immobile), V - speed of J' (and of the rod) in AFR.
And in any IFR J" with axis X": X"|| X, the longitudinal length l" of the rod due to absolute simultaneity will be equal:
l'' = l0γ(W)/γ(V), (8)
where W - velocity of observer J'' : W ||X in AFR, V - velocity of the rod in AFR.
From here follows that the (longitudinal) length of a rod always will be minimal in AFR irrespective of the rod moving or immobility in AFR, and restriction on the maximal length of a rod does not exist.
This conclusion does not coincide with a conclusion in the SRT where the rod has the maximal length in own IFR, and there is no restriction on its minimal length.
For a case of the nonparallel velocities, if to designate ξ as
a corner between vectors W and V in AFR, the rod's projection
to axis X" (X"||V" - X" parallel to V")
equals l''x''=l0γ(W) cos ξ
/γ(V),
the rod's projection to axis Y" equals l''y''=l0
sin ξ/γ(V),
and the full length of a rod equals
|
l'' = l0 |
γ(W) |
[1 - |
W2 |
sin2ξ ]1/2 |
(9) |
|
γ(V) |
c2 |
In particular, if W= -V then the length of a rod in S'' equals l''=l0 , as well as in own IFR of a rod though in this S'' the rod is not immovable at all!
4. A parity of lengths of unit-vectors in two systems of reference
Above we have considered length of the same rod in two different IFRs. But in "paradox of lengths" in the SRT there are different objects - rods of unit-lengths e1 and e2 , directed along axis X, motionless in IFR1 and IFR2, accordingly. Each of these rods has the length of 1 meter obtained in the own IFR by procedure of the Sevre's standard of length (1983).[6,v.3,p.124]
Really, in the SRT there is consequence of LT (incorrectly named as "paradox of length") according to which in IFR1 the longitudinal length of a rod e2 which is moving in IFR1, is less than length of a rod e1; and in IFR2 the longitudinal length of a rod e1 which is moving in IFR2, is less than length of a rod e2. The reason of occurrence of this "paradox" in the SRT is in that that we measure lengths in different systems of reference at the different moments of time because of property of a relative simultaneity.
Though this consequence is made simply enough from LT, but it appeared what to realize the received parities of lengths of rods much more difficultly. But for this purpose it is enough to understand a difference between phrases "the length of a rod in own IFR" and "the length of a rod in other IFR". First of all we shall pay attention, that the traversal sizes of a body in different IFRs remain constant, only longitudinal sizes of a body vary. Thus it turns out, that if we measured a body at the different moments of time the parity of lengths would be another. Hence, procedure of measurement essentially influences results of measurement of lengths. We have linked measurements to the moments of time, but the moments of time are measured in different places, hence, they should be set by the accepted procedure of synchronization. Another procedure of synchronization would give another length of a rod.
Thus, in different IFRs we have parities between own length of a body and the length of a body measured in other IFR. That we measure length of a body in several different IFRs, it will not change own length of a body. The term "the length of a body" at consideration in different systems of reference loses an unambiguity inherent in it in own system of reference, and becomes ambiguous, dependent in SRT from the term "speed of a body in IFR".
So, in this "paradox" we speak not about opposite parities of own lengths of meter rods, but about parities of measurement's results of these lengths in different IFRs.
In LAST such a relative simultaneity is not present, the length of each rod is measured in all IFRs simultaneously, therefore occurrence of the specified paradox of length is impossible. According to the formula (8) in IFR1 the length of a rod e2 equals l21=γ(V1)/γ(V2), and in IFR2 the length of a rod e1 equals l12=γ(V2)/γ(V1). We receive strictly reverse parity of lengths.
5. A parity of lengths of a rod in three systems of reference
Let's consider in the SRT three systems of reference IFR0, IFR1 and IFR2. Let IFR0 is motionless, IFR1 is moving in IFR0 with constant velocity V1, and IFR2 is moving with the constant velocity V2 : V2 ||V1, and V1 <V2. We shall determine a relative speed V12 of IFR2 in IFR1 by the formula shown in [6,v.3,p.498] :
(1-V122/c2)1/2 = (1-V12/c2)1/2 (1-V22/c2)1/2 / (1-V1V2/c2) (4)
Let a rod length in systems of reference IFR0, IFR1 and IFR2 are accordingly l, l' and l". We shall assume, that the formula (3) is fair in each pair of IFRs, that is, it is possible to fix respective alteration of rod length. We obtain a system of equations:
l = l'(1-V12/c2)1/2 ; l = l''(1-V22/c2)1/2 ; l' = l''(1-V122/c2)1/2 ;
from which with taking into account the formula (4) we obtain:
l' = l''(1-V22/c2)1/2 / (1-V12/c2)1/2
l' = l''(1-V12/c2)1/2 (1-V22/c2)1/2 / (1-V1V2/c2)
Whence V1 = V2. We received the contradiction to condition V1<V2. That is, in three IFRs we would not manage to carry out physical experiment - to measure change of length of the rod corresponding to transformations of Lorentz, as it is impossible for three IFRs to find the necessary moments of time for comparison of lengths of a rod in these IFRs.
Thus, for length of a rod in different IFRs there is no property of transitivity.
Similarly it is possible to show, that transitivity in three IFRs is not carried out and for time interval between two events, as the necessary condition of an immobility of clocks (see formulas (5) and (6)) can be executed only in one of three IFRs.
But in LAST and the rod length in three IFRs possesses transitivity by virtue of the formula (8), and the time interval in three IFRs possesses transitivity by virtue of the formula (6).
6. Conclusions
1. In LAST there are no paradoxes of length and time.
2. In LAST the periods between pair of events and longitudinal lengths of a rod in different IFRs strictly correspond with speed of these IFRs.
References:
[1]. Tangherlini F.R. "The velocity of light in
uniformly moving frame",
PhD Thesis (Stanford: Stanford Univ., 1958) and Tangherlini F.R. Suppl.
Nuovo Cimento 20 1 (1961)
[2]. Marinov S. Eppur si muove (East-West, Graz, 1987,
Engl.), first ed. 1977 (http://www.ptep-online.com/index_files/books_files/marinov1987.pdf)
[3]. R. De Witte, Website http://www.teslaphysics.com/DeWitte/general.htm
R.Cahill, The Roland De Witte 1991 Experiment (to the Memory of Roland De
Witte), http://www.ptep-online.com/index_files/2006/PP-06-11.PDF
[4]. Kupryaev N.V. Analysis of the expanded
representation of transformations of Lorentz, Izv. VUZov. Physics №7, 8
(1999). (http://sciteclibrary.ru/rus/catalog/pages/7521.html)
[5].A. Chepick "Absolute. Main principles", j.
"Modern problems of statistical physics", 2007, v.6, p.111-134
( http://www.mptalam.org/a200709.html
, http://www.mptalam.org/200709.pdf
) (http://redshift0.narod.ru/Eng/Stationary/Absolute/Absolute_Principles_3_En.htm)
[6].The physical encyclopedia, (М., the
Soviet encyclopedia, 1988-1992.)
Last correction 14.05.09 07:18:15