Preprint
(17.01.2003)
Date: Fri, 17 Jan 2003 21:08:18 GMT
From: redshift0@narod.ru (Alexander Chepick)
Organization:
NePaRTsgroups: sci.physics, sci.astro,
alt.sci.physics.new-theories
Subject: The Doppler Effect Formula in
NePaRT
Key
words:
Doppler effect - Renewed Particular
(Special) Relativity Theory (NePaRT)
- - - - - - --
Inference of the Doppler Effect Formula in NePaRT
Alexander M. Chepick, Nizhni
Novgorod
17.01.2003
Abstract
The Doppler effect formula in New Particular
(Special) Relativity Theory (NePaRT) is inferenced and compared with formula of
the relativistic Doppler effect in Einstein′s RT.
1.
Introduction
In the Einsteins Particular (Special) Relativity Theory (PRT), formulated
for inertial systems of reference (ISR), the formula of the Doppler effect
which expressing dependence of accepted frequency of light f from an emitted frequency of light f ',
relative speed V of a source, speed of light c, and from a corner
α between
direction of movement of the source and a direction of movement of those
photons that will get in the receiver is well-known:
f = f ' / {γ [1-(V/c) cos a]} (1)
where γ = (1−(V/c)2)-1/2.
It is necessary to add, that this formula is deduced for vacuum in the flat
Euclidian static universe for own system of reference of the receiver.
In PRT the formula of Doppler effect for own
system of reference of a source has been deduced also:
f = f ' γ [1 + (V/c) cos δ] (2)
where γ = (1−(V/c)2)-1/2,
V is relative speed of the receiver, c is speed of light, δ is a corner between a direction of
movement of the receiver and a direction, opposite to a direction of movement
of those photons which will get in the receiver. (Fig. 1) In particular, for
δ =0 the receiver strictly approaches to the source.
Check of this formula has served as
confirmation of the Einstein's relativity theory, in particular,
cross-sectional Doppler effect has been got. In [3, p.501] it is told, that in
1986 the formula (1) was checked up with accuracy 310-4. But why
accuracy of performance of the formula (1) is so insignificant? In fact values
included in it are measured with much more accuracy: frequency - 10-16,
speed - 410-9, a corner - 510-10.
Is there the physical reasons, which
not allow to increase accuracy of check of performance of this formula? In fact
opportunities for this purpose are available, moreover, the formula is applied
as to calculations of speeds of streams of gases in galaxies, so and other
appendices where required accuracy of calculations should be not less than 10-5.
There are also other reasons for doubt
in accuracy of this formula, for example, non symmetric distance and redshift
from jets, removing from us, for some quasars in comparison with redshifts of
these quasars and their jets approaching
to us.
As the inference of this formula is based on
PRT, maybe, the point is in this theory?
Basis of the particular relativity theory
are two postulates (Einstein
[5,p.147]): the Principle of an independence of lights speed : " Speed
of light in vacuum in all ISR is constant and does not depend on speeds of a
source and the receiver of light "; and the Principle of a relativity:
" All physical phenomena at identical entry conditions are equally resulted in all ISR ". With their help it is
possible to deduce Lorentz's transformations - the ratio, which tied values of
all coordinates in various inertial systems of reference.
At the same time Logunov [2] has shown, that
for a conclusion of Lorentzs transformations in the flat Euclidian static
universe the Principle of an independence of lights speed is not required,
enough one Principle of a relativity. Moreover, from the Principle of a
relativity existence some constant (we shall designate it cL
a Logunovs-Lorentz's constant) follows in all ISR. It has dimension of
a speed, and on sense of these transformations it is the maximal allowable
speed. But as this value at measurements appeared with the big accuracy is
equal to speed of light Einstein has made the decision to postulate the
Principle of a constancy of lights speed. As to the theoretical proof of that
the constant "" in
the equations of transformation of time and metric coordinates from one system
of reference in another also is speed of light, so without the Principle of a
constancy of lights speed such proof simply is not exist!
Therefore we offered a hypothesis:
The constant a Logunovs-Lorentz's
constant cL differs from speed of light.
Concerning the term "speed of light"
in the formulation of a hypothesis is necessary for noting, that here it is
speed of photons in a system of reference of a motionless source. Photons
radiated by a source, have some speed c, identical in all directions, by
virtue of an isotropy of a conditions of their spreading.
Landau and Lifshitz in [1] have shown independence from
ISR of a the supremum speed of interaction of a matter (that gives equivalence
of this value and the Logunovs-Lorentz's constant, as in one space two
constants can not be simultaneous by maxima). And for speed of light they
esteemed only two hypotheses: the Einsteins principle of independence and
"ballistic" hypothesis ", which one states: c ' =c+V,
where c ' - speed of photons in the second reference system moved by
speed V relatively the first reference system, in which one the speed of
photons equals "c". Having produced arguments against the
second hypothesis, they have considered, that have demonstrated necessity first
one, though there was not demonstrated alternation of these hypotheses.
Actually in conditions of Einsteins Principle of relativity, alternative to an Einsteins Principle of
independence is the hypothesis tendered to consideration in this article, but
it is not a ballistic hypothesis.
As a consequent of a hypothesis, offered to
consideration, receives, that many conclusions of PRT are executed with small
modification, namely:
- The value of relative velocity of one body in
own system of reference of the second body is equal to value of relative
velocity of the second body in an own system of reference of the first;
- There is a slowness factor of pace of time
γL, which one looks like: γL=
[1- (V/cL)2 ]-½;
- The Lorentz transformation laws linking
values of coordinates in two ISR are executed;
- There are formulas of the relativistic sum of
speeds and formula of the sum of their projections; in particular, for the
motionless receiver the speed of a photon from a source, is strict moving away
from the receiver with speed V, looks like:
c2= (c-V)/(1 c V / cL2)
But now speed of photons is constant only in
ISR of a source, in other ISR, driving relatively a source, the speed of a
photon can have miscellaneous value, it can be both more c, and less c, in
particular(personally), it is equal to zero relatively of parallel flying
photon. Thereby photons exclusiveness taken off, they were in equivalence with
customary particles.
Therefore the theory, which has been grown out
of this hypothesis, has been named the new particular relativity theory, or
NePaRT.
In article [4] on the
basis of the suggested hypothesis about discrepancy of lights speed in vacuum and the supremum speed of
interaction of a matter are given explanations of dependence on distance for
the w-factor of curve absolute luminosity Supernovae Ia type; and to not
explained till now s-factor.
In this article consideration of this
hypothesis proceeds by the example of the Doppler effect formula.
2.
Restrictions
of applicability of physical theories, principles and postulates; and Supremum
of the interaction speed of a matter
1. Any
physical theory has the limited scope.
First, it is applied to the limited class of
the phenomena.
Second, it has the limited range of definition
that is carried out with some given accuracy for the parameters limited in the
values in aggregate and-or separately. For unlimited basically parameters such
restriction always happens both from above and from below, for limited - near
to limiting values.
2. The formula of any theory treating the physical law corresponding to it, gives approached (and sometimes probable) value .
First, because of the limited accuracy of
definition of parameters and constants included in the formula.
Second, because of the approached formula
representing this law. Frequently in formulas insignificant members because of
their small value are rejected. There are formulas for which there is no exact
expression (such as not taken integral).
Third, because of the approached, idealized
form of the description of this law.
For example, Coulomb formula F=KQq/r2 expressing dependence of
force of an attraction (pushing away) between electric charges. The formula is carried
out for motionless charges which sizes are considered as dot, that is
infinitesimal in comparison with distance between them. The formula is not
carried out for small distances, smaller the size of atom, but not for the
reason, that we do not know the size of a charge in the charged particles (the
size of an electron with sufficient accuracy is dot in comparison with the size
of atom) but that on such distances there is a phenomenon of shielding. The
formula cannot be checked up for very big distances and for very big charges.
Feasibility of the formula for charges, smaller an electron charge as we have
no those is unknown also.
3. As
principles and the postulates concerning to physical values, inherently are
physical theories they, as well as theories, can be applied only with the
certain restrictions.
For example, Einstein's postulate on a
constancy of lights speed in any
inertial system of reference. It concerns to real physical parameter - speeds
of photons. It is possible to check up a postulate only for a limited speed of
a source. And for the speed of a source close to limiting value, the postulate
obviously cannot be applied by virtue of the previous reasons. So it is quite
natural, that in the theories constructed on its basis the results can turn
out, which are not compatible with physical sense
4.
Supremum speed of interaction of a matter is the constant
How then to act with the statement, that the
supremum speed of interaction of a matter in the universe are the
constant?
First, it not a postulate, only the theorem.
Really, if to assume, that this value depends on speed of a source or the
receiver it will not be the top side.
Second, this value does not correspond with any
physical object. It is not speed of any object, it is the top limit of such
speeds.
Therefore the destiny of the theory or a
postulate does not threaten this statement, it will exist in a constant kind
until the universe exists and there is substance in it.
3.
Geometrical
inference of the Doppler effect
formula
When
the source is motionless, photons radiated by it have some speed c,
identical in all directions, because of
isotropic conditions of their moving.
When the source moves with speed V, the
photons radiated by it, get additional speed depending on a direction of
movement of a photon, agrees NePaRT. Their resulting speed c2
is determined under the formula of the relativistic sum of projections of
speeds c and V.
In PRT two formulas of Doppler effect - for
system of reference of the receiver and system of reference of a source have
been deduced.
For a conclusion of similar formulas in NePaRT
also we shall consider these two systems of reference.
It is necessary to understand precisely, that
in the formulas (3) and (4) deduced below, as well as in formulas (2) and (1)
in standard PRT, completely different corners are there. In the formula (3) it
is a corner in own system of reference of source (OSRS) between a direction of
speed of the receiver and a direction opposite to movement of photons at the
moment of the beginning of reception of a lights signal. (Fig. 1). It is
natural, that at the moment of the beginning of radiation of a lights signal
the corner between these directions was same, but it did not coincide with a
corner between speed and a real direction on the receiver at that moment. And
in the formula (4) it is a corner in own system of reference of receiver (OSRR)
between a direction of speed of a source and a direction of movement of photons
at the moment of the beginning of radiation of a lights signal. (Fig. 2).
It is natural, that at the moment of the beginning of reception of a lights
signal the corner between these directions is same, but it does not coincide
with a corner between speed and a real direction on a source at this moment.
4.
The
source is motionless, the receiver moves
Let's designate in own system of reference of
source (OSRS) the moment t'2 the beginning of reception of light, the moment t'3 -
the termination of reception of light3. Accordingly, total
time of lights reception we shall designate t': t'=t'3 - t'2.
Let at the moment t'2 receiver is on distance R from a
source, has speed V and moves under a corner δ to a
direction of movement of photons from a source. Let L is distance between
a source and the receiver at the moment t'3. In flat
Euclidian static universe L2= R2+V2t' 2-2Vt'
cos δ. (Fig. 1). We shall designate the radiated frequency as
f '. Photons radiated by a motionless source, have some speed c,
identical in all directions, and, accordingly, length of a wave λ' = c
/ f '.
Let's designate in own system of reference of
receiver (OSRR) t2 is the moment of the beginning
of reception of monochrome light, the moment of the termination of lights
reception t3. Let t - total time of reception of light
in OSRR, t=t3 - t2. On NePaRT: t'= γL t . We shall designate accepted frequency through
f.
Fig. 1.
Relative movement of the receiver
Let's look, how many waves will pass through
the receiver.
During time t' in OSRS the receiver passes a layer of width |R-L
| of sphere with the center in a source. In a layer of such width at any moment
is |R-L | / λ' waves.
For same time t' in OSRS through anyone motionless in OSRS point
will pass n' =f 't' waves.
Therefore, in a case when the receiver only
came nearer to a source, that is, cos δ > 0, V t'<Rcos
δ, number of waves n, past through the receiver, will make on a
clock of OSRS a number of the waves incoming in a layer in time t ',
plus number of the waves that are taking place in the specified layer by
the initial moment of movement of the receiver through this layer:
n=f ' t' +(R - L ) / λ'
In a case cos δ <= 0, cos
δ <= 0 , that is the receiver only left, number of waves n,
past through the receiver, will make on a clock of OSRS including in a layer in
time t ', minus number of the waves which are taking place in a
layer by the final moment of movement of the receiver through this layer (it is
natural, it is supposed, that the receiver does not overtake photons, that
is | V cos δ |
<c):
n=f 't' - (L-R) / λ '
where
λ' f ' =c, that on a clock of OSRR will give average accepted
frequency in both cases:
f = n / t= {f ' t' +[R - (R2 +(Vt')2 -
2RV t' cos δ)1/2
]/ λ′ }/ t=
={ f ' t' +( f ' / c )[2RV t' cos
δ - (Vt')2]/ [ R + (R2 +(Vt')2- 2RV t' cos
δ)1/2 ] }/ t=
= f ' γL {1 +(V/)[cos δ - (Vt'/2R)]/[1/2+(1/4
+(Vt'/2R)2 - (V
t'/2R) cos δ)1/2] }
that in a limit at t-> 0 describes
all directions of movement (as one of these assumptions is necessarily carried
out at small enough t), and the formula similar to the formula (2) turns
out.
f = f ' γL (1+ (V/c)cos
δ) (3)
Not considered in OSRS the condition in case of moving away of the receiver
|V cos δ |> =c
It gives invisibility of a source and absence of shift of frequency, or even arrival of photons from a direction opposite to a source.
5.
The
source moves, the receiver is motionless
Let's designate in own system of reference of
the receiver t0 - the moment of the beginning of
radiation of monochrome light, the moment of the termination of radiation of
light t1, the moment of the beginning of reception
of light t2, the moment of the termination of
reception of light t3. Let at the moment t0
the source has speed V and moves under a α to a direction on the receiver, simultaneously being direction of
movement of those photons which will get in the receiver. Let in OSRR R
is distance between the receiver and a source at the moment t0,
L - distance between them at the moment t1.
(Fig. 2).
Fig. 2.
Movement of a source
Let t - total time of radiation of light
in OSRR, t=t1 - t0. Light, moving
with some speed 2 (it is natural, that examined
movement of a source is those, that light from it reaches the receiver),
will achieve the receiver at the moment of time t2
: t2 = t0
+R/c2. Accordingly, reception of light will end at some
moment of time t3: t3 = t1 +L/c2 ; t3 - t2 = t
+ (L - R)/c2 ;
where
in flat Euclidian static universe L2=
R2+V2t2-2Vt cos α .
Let's designate accepted frequency through f.
Let's designate in OSRS t'0 -
moment of time of the source radiations beginning and moment of the termination of radiation t'1.
We shall designate radiated frequency f '. Accordingly, total time of
light\s radiation we shall designate t': t'=t'1 - t'0.
On NePaRT the ratio is carried out: γL t' = t.
Let's look, how many waves will pass through
the receiver in this case.
Let's assume, that the source only came nearer
to the receiver, that is cos a> 0 and Vt <R cos a;
or that the source only left, that is cos a <=0. One of these
assumptions is necessarily carried out at small enough t. In time t
the source has passed a layer of width |R-L | of spher with the center in the
receiver. The number of the radiated waves n' is equal n' = f 't'
in OSRS. And the number of the accepted waves n is equal n = f( t3-t2) in OSRR. As on a condition all radiated
waves have been accepted, so n' = n, that gives average accepted
frequency:
f = n / (t3-t2)= f '
t' / { t + [(R2 +(Vt)2-
2RV t cos a)1/2- R]/c2 }=
= f ' (t'/t)/{1+ (V/2) [V t -2R cos a]/[(R2
+(Vt)2- 2RV t cos a)1/2 +R ] }=
That in a limit at t-> 0 gives the formula
f ' = f γL-1 / [1- (V/c2)cos a] (4)
Not considered condition |V cos a
|> =c gives in OSRR in case if the source is moving away its
invisibility and absence of shift of frequency.
6.
Calculation
of speed of photons
Fig. 3.
Movement of the
receiver in OSRS
Speed of photons c2 in OSRR
it is determined as the relativistic sum of projections of speeds V and
c on a direction of speed V.
In OSRS for the photon has got in the moving
receiver, the direction of speed of a photon should "outstrip" the receiver.
Thus the corner δ between a direction of movement of photons from a
source and vector V does not vary during movement. Then in OSRR the
speed of photons 2 and the corner a
between a direction of their movement from a source and vector V is
determined under formulas of the relativistic sum of speeds projections on a
direction of speed V (Fig. 3):
c2,X = (V + c cos δ)/(1+
c V cos δ /c2L)
c2,Y = c
sin δ (1 -V2 /c2L)½
/ (1+ c V cos
δ/c2L)
c2=( c22,X + c22,Y
) ½
cos a = c2,X / c2
where c2, X
and c2, Y - accordingly x-and
y-projections of speed of photons 2, a -
the same corner that appears in the formula (4), and a corner δ
in the formula (3). Whence we receive speed c2,
expressed through c, V, cL and a.
Transformations it is carried out in real
assumptions in our conditions: V <c. Condition V> =c in
this article is not considered.
We shall substitute expression for c2,
X and c2, Y in formulas for c2
and cos a:
c2 = (c2
+V2 +2 V c cos δ - c2 sin2 δ V2
/c2L) ½ / [1+ c V cos δ /c2L]
cos2 a= [V + c cos
δ]2 / (c2
+V2 +2 V c cos δ - c2 sin2 δ V2
/c2L)
From last ratio we shall work out the equation
for cos δ :
(1- V2 /c2Lcos2
a) cos2 δ+2 V/c sin2 a cos
δ + V2/c2 sin2 a -
cos2 a + V2 /c2Lcos2
a= 0
We shall solve it, having chosen that decision
which at aspiration c and cL to infinity
gives a ratio cos δ =cos a:
|
cos δ= |
-V/ c sin2 a+ cos a
(1 - V2 /c2L)
½ ( -V2 /c2sin2 a
+ 1 - V2 /c2L
cos2 a )½ |
|
1- V2 /c2Lcos2
a |
Let's designate β = V /cL ; γ =
(1-(V/c)2)- ½ ; γL =
(1-(V/c L)2)- ½ ;
γK-1= ( -V2 /c2sin2 a +
1 - V2 /c2L
cos2 a )½ = (γ -2sin2 a+ γL-2 cos2 a )½;
The parameter γK
can be named Gamma circular as by the definition it changes from γL
up to γ depending on a corner a.
Let k=1 c/cL= 2,810-11
- factor of difference between speed of light and supremum speed of
interaction of a matter, empirically determined in article [4].
In these designations it is received:
0 <= β < 1-k
c=cL(1-k)
V2 /c2= β2/(1-k)2
γL-2=1- β2
γ-2=1- β2/(1-k)2
|
cos δ= |
cos a γL-1γK-1 (1- cos 2a) β/(1k) |
|
1-β2cos 2a |
|
c2= |
cL [-(1-(1-k)2)(1-
β2)+(1+β(1-k)cos δ )2]½ |
|
1+β(1-k)cos δ |
Let's notice, that a denominator 1 + ß
(1-k) cos δ we shall present as fraction:
|
1+β(1-k)cosδ = γL-1 |
γL-1 +β(1-k) γK-1cos a |
|
1-β2cos 2a |
If we
substitute last expression in expression for c2,
we shall receive
|
c2= |
cL [(γL-1 +β(1-k) γK-1 cos a) 2-(1-(1-k)2)(1-
β2cos 2a) 2]
½ |
|
γL-1 +β(1-k) γK-1cos a |
or
|
c2=
cL[1
|
(1-(1-k)2) (1- β2cos 2a) 2 |
] ½ |
(5) |
|
(γL-1 +β(1-k) γK-1 cos a ) 2 |
This
formula has independent value for the proof NePaRT, in fact it gives value of
speed of a photon depending on a direction and a speed of a source.
In particular, at strict moving away of a
source, that is cos a = -1, it turns out already used formula in
article[4]:
|
c2= cL [1 - |
(1-(1-k)2) γL-2 |
] ½ = |
c-V |
|
(1 - β(1-k)) 2 |
1 cV/cL 2 |
In connection with changeability of speed of photons it is necessary to make distinction in terminology: a speed of photons is a speed of particles in any concrete system of reference, and a speed of light means constant speed of photons in OSRS.
7.
Comparison
of the formula of Doppler effect in PRT and NePaRT
Speed of light is determined [Physical
encyclopedia, t.4, p. 549] for the present moment as c =299792458,0
(+-1,2) [m/c]. Though accuracy here makes 410-9, at
the present stage accuracy of measurement of lights speed 5,010-11 is
achievable.
Let's consider, that in formulas (1) and (2)
speed of light c coincides with value cL
=299792458,000 [m/c] in formulas (3) and (4), in which in turn c = (1-k)
cL = 299792457,992 [m/c]. Here k=1- c/cL=2,810-11
- factor of difference between speed of light and supremum speed of interaction
of a matter, empirically determined in article [4]. This assumption will
not affect on value of the fraction of
frequencies f / f ', as it is within the limits of achievable accuracy.
Therefore for formulas (1) and (2) multiplier
γ is designated γL.
From comparison c and cL
we see, that distinction between them is inside accuracy of definition of
lights speed accessible now. Probably,
the reason of this insufficient accuracy of definition is made in inconstancy
of lights speed .
For the analysis of formulas we shall divide
the fraction of frequencies f / f ' from the formula (4) on similar
value f / f ' from the standard formula (1). The value u(ß,a)
determined thus, at u(ß,a)>1 will speak that at the data
ß, a and f ' the formula (4) should show the bigger
accepted frequency, than the standard formula (1), and at u(ß,a)<1 is smaller. As before, β=V/cL.
u(β,a)={ γL-1 / [1- (V/c2)cos a] } / {1 / γL [1- (V/cL)cos a]}
Let's substitute value 2
from the formula (5):
|
u(β,a)= |
1 β cos
a |
|
(6) |
|
1 β cos
a |
[1- (1-(1-k)2) (1- β2cos 2a) 2 /(γL-1 +β(1-k) γK-1 cos a ) 2]
-½ |
The
analysis of the formula (6) shows, that at anyone a at ß <0.6
it turns out | u(ß,a)-1 | <510-11, and at ß
<0.95 it turns out | u (ß,a)-1 | <5.310-10, that
is distinction of values of formulas in this range of definition is outside
detection at the present stage. The greatest rejection from value of the
standard formula at ß <=1 - 10-9 is marked in area at a
= π.
We receive at ß <=0.972 accuracy of
concurrence to standard formula of Doppler u(0.972, π)= 1 - 10-9
At the bigger speeds the grater distinctions
turn out: u(0.99999972, π)= 1 - 10-4
and
it is only at a = π. At other corners the distinction is less.
8.
Conclusions
1.
The
value f / f ' in the formula of Doppler effect deduced for NePaRT,
differs less, than on 10-N, for relative speed of source V
<(1-2,810-11+N)cL at N = (1÷10), from value f / f ' in the Standard
relativistic Doppler effect formula . At V>=(1 - 2,8 10-11)cL
difference of these values can make more than 10 %.
2.
Accuracy
of concurrence 10-4 of value f / f ' in formulas of Doppler
effect for PRT and NePaRT is achieved for all corners a and almost all range
of speeds of a light source, namely at V<(1-2,8 10-7)cL
.
3.
At
cross-section Doppler effect of the formula (1) and (4) coincide, in PRT and
NePaRT accordingly.
4.
Orbital
speed of the Earth practically does not influence on speed of photons from
stars, only on a corner of an aberration. Speed of stars and galaxies
relatively the Earth does not influence almost on speed of photons from them.
5.
The
above-stated calculations show, that all experiences executed till now on
definition of lights speed and on check
of the formula of Doppler effect do not contradict NePaRT.
6.
At
V cos a <=-c Doppler effect is not observed for NePaRT (a cone of
invisibility). In Standard PRT in this case also Doppler effect is not
observed, as in PRT such condition is not feasible.
7.
The
formula (5) can be used for check of speed of photons from a moving source.
8.
On the
basis of the formula (6) it is possible to carry out the specified check of the
formula (1) in the field of the high speeds of a source and the corners close
to π, with the
purpose of a rejection or confirmation of an offered hypothesis.
9.
For
check of NePaRT supervision of the secondary photons flying in an almost
opposite direction from a downpour of ions is of interest, formed by a space
particle with ultrahigh energy.
Reference:
[1] L.D.Landau,
E.M.Lifshitz. The field theory. (., Nauka, 1988.)
[2]
A.A.Logunov. Bases of the theory of a relativity, (., Nauka, 1982.)
[3]
The physical encyclopedia, t.3., (., Soviet encyclopedia, 1992.)
[4]
A.M.Chepick, Supremum
speed of interaction of a matter,
Spacetime and Substance, No. 3-2002, p.122)
[5] A.
Einstein, L. Infeld. The Evolution of Physics, (M., Nauka, 1965.)
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Last correction 15.03.2003 15:18