Doppler effect and Hubble effect in different models of space-time in the case of auto-parallel motion of the observer

0.12 MB·English

by  Sawa Manoff

Doppler effect and Hubble effect in different
models of space-time in the case of auto-parallel
motion of the observer
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Sawa Manoff
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2 Bulgarian Academy of Sciences

n Institute for Nuclear Research and Nuclear Energy
a
Department of Theoretical Physics
J
Laboratory of Solitons, Coherency, and Geometry
4
1784 Sofia - Bulgaria

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v E-mail address: [email protected]
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1 Abstract
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Doppler effect and Hubble effect in different models of space-time
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0 in the case of auto-parallel motion of the observer are considered. The
/ Doppler effect and shift frequency parameter are specialized for the case
c
of auto-parallel motion of the observer. The Hubble effect and shift fre-
q
quency parameter are considered for the same case. It is shown that by
-
r the use of the variation of the shift frequency parameter during a time
g
period,considered locally intheproperframeofreference ofanobserver,
:
v one can directly determine the centrifugal (centripetal) relative velocity
i and acceleration as well as the Coriolis relative velocity and acceleration
X
of an astronomical object moving relatively to the observer. All results
r are obtained on purely kinematic basis without taking into account the
a
dynamic reasons for theconsidered effect.
PACS numbers: 98.80.Jk; 98.62.Py; 04.90.+e; 04.80.Cc;
Contents
1 Introduction 2
1.1 Abbreviation and symbols . . . . . . . . . . . . . . . . . . . . . . 5
2 Doppler effect and shift frequency parameter in the case of an
auto-parallel motion of the observer 6
2.0.1 Longitudinal (standard) Doppler effect and the shift fre-
quency parameter . . . . . . . . . . . . . . . . . . . . . . 6
2.0.2 TransversalDopplereffectandtheshiftfrequencyparameter 9
1
2.1 Hubble effect and shift frequency parameter in the case of an
auto-parallel motion of the observer . . . . . . . . . . . . . . . . 10
2.1.1 Longitudinal (standard) Hubble effect and the shift fre-
quency parameter . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 TransversalHubbleeffectandtheshiftfrequencyparameter 10
3 Conclusion 10
1 Introduction
1. Modern problems of relativistic astrophysics as well as of relativistic physics
(dark matter, dark energy, evolution of the universe, measurement of veloci-
ties of moving objects etc.) are related to the propagation of signals in space
or in space-time [1], [2]. The basis of experimental data received as results of
observations of the Doppler effect or of the Hubble effect gives rise to theoret-
ical considerations about the theoretical status of effects related to detecting
signals from emitters moving relatively to observers carrying detectors in their
laboratories.
2. In the classical (non-quantum) field theories different models of space-
time havebeenusedfordescribingthephysicalphenomenaandtheirevolution.
The 3-dimensional Euclidean space E3 is the physical space used as the space
basis of classical mechanics [3]. The 4-dimensional (flat) Minkowskian space
M4 is used as the model of space-time in special relativity [4]. The (pseudo)
Riemannian spaces V4 without torsion are considered as models of space-time
in general relativity [5], [6]. In theoretical gravitational physics (pseudo) Rie-
mannianspaceswithouttorsionaswellas(pseudo)RiemannianspacesU4 with
torsion are proposed as space-time grounds for new gravitational theories. To
the most sophisticated models of space-time belong the spaces with one affine
connection and metrics [(L ,g)-spaces] and the spaces with affine connections
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and metrics [(L ,g)-spaces].
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Thespaceswithoneaffineconnectionandmetrics[(L ,g)-spaces]haveaffine
n
connections whose components differ only by sign for contravariantand covari-
ant tensor fields over a differentiable manifold M with dim M =n.
Thespaceswithaffineconnectionsandmetricshaveaffineconnectionswhose
components differ not only by sign for contravariantand covarianttensor fields
over a differentiable manifold M with dim M =n.
3. Recently, it has been shown [7], [8] that every differentiable manifold M
(dimM = n) with affine connections and metrics [(L ,g)-spaces] [9] could be
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used as a model of space-time for the following reasons:
• The equivalence principle (related to the vanishing of the components
of an affine connection at a point or on a curve) holds in (L ,g)-spaces
n
[10]÷[12].
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• (L ,g)-spaces have structures similar to these in (pseudo) Riemannian
n
spaceswithouttorsion[V -spaces]allowingfordescriptionofdynamicsys-
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tems and the gravitational interaction [8].
• Fermi-Walker transports and conformal transports exist in (L ,g)-spaces
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as generalizations of these types of transports in V -spaces [13], [14].
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• A Lorentz basis and a light cone could not be deformed in (L ,g)-spaces
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as it is the case in V -spaces.
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• All kinematic characteristics related to the notions of relative velocity
andof relativeaccelerationcouldbe workedoutin (L ,g)-spaceswithout
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changing their physical interpretations in V -spaces [8], [15]÷[18].
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• (L ,g)-spacesinclude alltypes ofspaceswithaffine connectionsandmet-
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rics used until now as models of space-time.
4. If a (L ,g)-space could be used as a model of space or of space-time
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the question arises how signals could propagate in a space-time described by
a (L ,g)-space. The answer of this question has been given in [18], [19], and
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[20]. Bythatthesignalsaredescribedbymeansofnull(isotropic)vectorfields.
A signal could be defined as a periodical process transferred by an emitter
and received by an observer (detector) [19]. A wave front could be considered
as a signal characterized by its wave vector (null vector) as it is the case in
the geometrical optics in a V -space [21]. All results are obtained on purely
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kinematic basis (s. [18], [22], [23]) without taking into account the dynamic
reasons for the considered effect.
Onthebasisofthegeneralresultsinthepreviouspaperswecandrawarough
scheme of the relations between the kinematic characteristics of the relative
velocity and relative acceleration on the one side, and the Doppler effect and
the Hubble effect on the other. Here the following abbreviations are used:
CM - classical mechanics
SRT - special relativity theory
GRT - general relativity theory
CRT - classical relativity theory [7], [8].
5. The considerations of the Doppler effect and of the Hubble effect show
that the Doppler effect is derived in the physical theories (with exception of
generalrelativity)as a resultofthe relative motionofanobserverandanemit-
ter, sending signals to the observer, from point of view of the proper frame of
referenceoftheobserveranditsrelationstotheproperframeofreferenceofthe
emitter. On the other side, the Hubble effect could be considered as a result of
the Doppler effect and the Hubble law assumed to be valid in the correspond-
ing physical theory. In a rough scheme the relations between Doppler effect,
Hubble effect, and Hubble law could be represented as follows:
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Relativemotion Doppler effect Hubbleeffect Hubblelaw
characterized characterized characterized characterized
by as as
CM constant ⇒ corollary → corollary ⇐ by definition
relative velocity
SRT constant ⇒ corollary → corollary ⇐ by definition
relative velocity
GRT changeof the ⇒ corollary → corollary ⇐ by definition
metrics of space-time of the metrics
CRT relative velocity and ⇒ corollary → corollary ← as corollary
relative acceleration ⇓ ⇑
⇒ −→ → −→ → ⇑
6. Since the Doppler effect and the Hubble effect as kinematic effects could
be described by different theoreticalschemes and models of space-time the rich
mathematicaltoolsofthespaceswithaffineconnectionsandmetrics,considered
asmodelsofspace-time,areusedfordescriptionofboththeeffects. Theaimhas
been to work out a theoretical model of the Doppler effect and of the Hubble
effect as corollaries only of the relative motion between emitter and observer
determined by the kinematic characteristics of the relative velocity and the
relative acceleration between emitter and observer from point of view of the
properframeofreferenceoftheobserver. Forthistaskthe(n−1)+1formalism
hasbeenusedrelatedtotheworldlineofanobserveranditscorrespondingn−1
dimensional sub space interpreted as the observedspace in the proper frame of
the observer [22].
7. Our task in the present paper is to investigate the Doppler effect and
the Hubble effect in the case of an auto-parallel motion of the observer and its
influence on the frequency shift parameter. In section 2 the Doppler effect and
shift frequency parameter are considered in the case of an auto-parallelmotion
oftheobserver. InSection3theHubbleeffectandshiftfrequencyparameterare
considered for the same case. It is shown that by the use of the shift frequency
parameter, considered locally in the proper frame of reference of an observer
[22],wecandirectlydeterminethecentrifugal(centripetal)relativevelocity and
acceleration as well as the Coriolis relative velocity and acceleration. Section 4
comprisessomeconcludingremarks. Mostofthe detailsandderivationomitted
in this paper could be found in [19] and in [20].
1.1 Abbreviation and symbols
• The vector field u is the velocity vector field of an observer: u ∈ T(M),
dim M =n, n=4.
• The contravariant vector field v = ∓l ·n = H ·l ·n = H ·ξ is
z vz ⊥ ξ⊥ ⊥ ⊥
orthogonal to u and collinear to ξ . It is called centrifugal (centripetal)
⊥
relative velocity.
• The function H =H(τ) is called Hubble function.
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• The contravariant vector field a = ∓l ·n = q ·l ·n = q ·ξ is
z az ⊥ ξ⊥ ⊥ ⊥
orthogonal to u and collinear to ξ . It is called centrifugal (centripetal)
⊥
relative acceleration.
• The function q =q(τ) is called accelerationfunction (parameter).
• The contravariantvector field v =∓l ·m =H ·l ·m =H ·η
ηc vηc ⊥ c ξ⊥ ⊥ c ⊥
is orthogonalto u and to ξ . It is called Coriolis relative velocity.
⊥
• The function H =H (τ) is called Coriolis Hubble function.
c c
• The contravariantvector field a =∓l ·m =q ·l ·m =q ·η
ηc aηc ⊥ ηc ξ⊥ ⊥ ηc ⊥
is orthogonalto u and to ξ . It is called Coriolis relative acceleration.
⊥
• The fuction q =q (τ) is called Coriolis acceleration function (parame-
ηc ηc
ter).
• ω is the frequency of a signal emitted by an emitter at a time τ −dτ of
the proper time of the observer.
• ω is the frequency of a signal detected by the observer at a time τ of the
porper time of the observer.
• dτ is the time intervalin the proper frame ofreference ofthe observerfor
which the signal propagates from the emitter to the observer (detector)
at a space distance dl in the proper frame of reference of the observer.
2 Doppler effect and shift frequency parameter
in the case of an auto-parallel motion of the
observer
It has been shown [19], [20] that in a (L ,g)-space longitudinal (standard) and
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transversal Doppler effects could appear when signals are propagating from an
emitter to an observer (detector) moving relatively to each other.
2.0.1 Longitudinal(standard)Dopplereffectandtheshiftfrequency
parameter
1. Letusnowconsidertheshiftfrequencyparameterwhentheobserver’sworld
lineisanauto-paralleltrajectory,i.e. whenthevelocityvectoruoftheobserver
fulfills the equation
∇ u=f ·.u , f ∈C∞(M) .
u
Then, because of g[h (u)]=0,
u
a =g[h (a)]=f ·g[h (u)]=0 ,
⊥ u u
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(∇ a) =g[h (∇ (f ·u))]=g[h ((uf)·u+f ·a)]=
u ⊥ u u u
=(uf)·g[h (u)]+f ·g[h (a)]=0 .
u u
Letasignalwithfrequencyω isemittedbyanemitter[19]atthetimeτ−dτ
and is received by an observer (detector) at the time τ.
Since ω =ω(τ −dτ) and ω =ω(τ), we can expand ω in a Teylor row up to
the second order of dτ
dω 1 d2ω
ω =ω(τ −dτ)≈ω(τ)− ·dτ + · ·dτ2+O(dτ) .
dτ|τ 2 dτ2|τ
Then
dω 1 d2ω
dω =ω−ω ≈− ·dτ + · ·dτ2 ,
dτ 2 dτ2
dω ω−ω 1 dω 1 1 d2ω
2
= ≈− · ·dτ + · · ·dτ .
ω ω ω dτ 2 ω dτ2
On the other side, the general results in the case of an auto-parallelmotion
of the observer read [19], [20]
ω−ω dω l 1 dl
= =ω·(1− vz + · ·l )=
ω ω l 2 l2 az
ξ⊥ u
dl l 1 dl2 l
=ω·(1− · vz + · · az) .
l l 2 l2 l
u ξ⊥ u ξ⊥
Since
dl dl2
2
=dτ , =dτ ,
l l2
u u
we obtain the relations
l 1 l
ω =ω·(1− vz ·dτ + · az ·dτ2) .
l 2 l
ξ⊥ ξ⊥
2. If we consider only infinitesimal changes of the frequency ω for the time
intervaldτ,i.e. ifdω =ω−ω,wecanexpresstheshiftparameterz =(ω−ω)/ω
as an infinitesimal quantity
dω
z = =d(log ω)=
ω
1 dω 1 1 d2ω
=dz =− · ·dτ + · · ·dτ2 =
ω dτ 2 ω dτ2
l 1 l
=− vz ·dτ + · az ·dτ2 .
l 2 l
ξ⊥ ξ⊥
On the other side, dz could be considered as a differential of the function z
depending on the proper time τ of the observer, i.e. z = z(τ). It is assumed
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that z(τ) has the necessary differentiability properties. The function z at the
point z(τ −dτ) could be represented in Teylor row as
dz 1 d2z
z(τ −dτ)=z(τ)− ·dτ + · ·dτ2+O(dτ) .
dτ 2 dτ2
Then z(τ −dτ) and dz = z(τ −dτ)−z(τ) could be written up to the second
order of dτ respectively as
dz 1 d2z
z(τ −dτ)=z(τ)− ·dτ + · ·dτ2 ,
dτ 2 dτ2
dz =z(τ −dτ)−z(τ)=
dz 1 d2z
=− ·dτ + · ·dτ2 =
dτ 2 dτ2
1 dω 1 1 d2ω
=− · ·dτ + · · ·dτ2 =
ω dτ 2 ω dτ2
l 1 l
=− vz ·dτ + · az ·dτ2 .
l 2 l
ξ⊥ ξ⊥
2
The comparisonofthe coefficientsbefore dτ anddτ inthe last(above)two
expressions leads to the relations
dz 1 dω d2z 1 d2ω
= · , = · ,
dτ ω dτ dτ2 ω dτ2
dz l d2z l
= vz , = az .
dτ l dτ2 l
ξ⊥ ξ⊥
The vector ξ could be chosen as a unit vector, i.e. l = 1, equal to
⊥ ξ⊥
the vector n showing the direction to the emitter from point of view of the
⊥
observer. Then
dz d2z
=l , =l .
dτ vz dτ2 az
Therefore, if we can measure the change (variation) of the shift frequency
parameter dz in a time interval dτ we can find the centrifugal (centripetal)
relativevelocityandcentrifugal(centripetal)relativeaccelerationoftheemitter
with respect to the observer. The above relations appear as direct way for a
check-up of the considered theoretical scheme of the propagation of signals in
spaces with affine connections and metrics. On the other side, the explicit
form of l and l as functions of the kinematic characteristics of the relative
vz az
velocity and relative acceleration could lead to conclusions of the properties of
thespace-timemodelusedfordescriptionofthephysicalphenomena. Thesame
relations could lead to more precise assessment of the Hubble function H and
the acceleration function q at a given time.
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2.0.2 Transversal Doppler effect and the shift frequency parameter
In analogous way we can find the relations between the absolute values of the
Coriolis relative velocity and the Coriolis relative acceleration and the shift
frequency parameter in the case of auto-parallel world line of the observer.
1. The relation between the frequency ω of the emitted signals and the
frequency ω of the detected signals reads [19], [20]
ω =ω·(1− dl · lvηc + 1 · dl2 · laηc)=
l l 2 l2 l
u ξ⊥ u ξ⊥
=ω·(1− lvηc ·dτ + 1 · laηc ·dτ2) .
l 2 l
ξ⊥ ξ⊥
The shift frequency parameter has the form
z = ω−ω =−lvηc ·dτ + 1 · laηc ·dτ2 =dz =
c c
ω l 2 l
ξ⊥ ξ⊥
dz 1 d2z
=− c ·dτ + · c ·dτ2+O(dτ) ≈
dτ 2 dτ2
dz 1 d2z
≈− c ·dτ + · c ·dτ2 .
dτ 2 dτ2
2
The comparisonof the coefficients before dτ and dτ in the two expressions
z = ω−ω =−lvηc ·dτ + 1 · laηc ·dτ2 =dz =
c c
ω l 2 l
ξ⊥ ξ⊥
dz 1 d2z
≈− c ·dτ + · c ·dτ2
dτ 2 dτ2
leads to the relations
dzc = lvηc , d2zc = laηc .
dτ l dτ2 l
ξ⊥ ξ⊥
The vector ξ could be chosen as a unit vector, i.e. l = 1, equal to
⊥ ξ⊥
the vector n showing the direction to the emitter from point of view of the
⊥
observer. Then
dz d2z
c c
=l , =l .
dτ vηc dτ2 aηc
Therefore, if we can measure the change (variation) of the shift frequency
parameter dz in a time interval dτ we can find the Coriolis relative velocity
c
and Coriolis relative acceleration of the emitter with respect to the observer.
The above relations appear as direct way for a check-up of the considered the-
oretical scheme of the propagation of signals in spaces with affine connections
and metrics. On the other side, the explicit form of l and l as functions
vηc aηc
ofthe kinematic characteristicsofthe relativevelocity andrelativeacceleration
could lead to conclusions of the properties of the space-time model used for
description of the physical phenomena. The same relations could lead to more
precise assessment of the Hubble function H and the accelerationfunction q
c ηc
at a given time.
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2.1 Hubble effect and shift frequency parameter in the
case of an auto-parallel motion of the observer
It has been shown [19], [20] that in a (L ,g)-space longitudinal (standard) and
n
transversal Hubble effects could appear when signals are propagating from an
emitter to an observer (detector) moving relatively to each other.
2.1.1 Longitudinal (standard) Hubble effect and the shift frequency
parameter
By the use of the relations [19], [20]
l =∓H·l , l =∓q·l ,
vz ξ⊥ az ξ⊥
dz l d2z l
= vz , = az ,
dτ l dτ2 l
ξ⊥ ξ⊥
we canfind the Hubble function H andthe accelerationfunction (parameter)q
respectively as
dz d2z
=∓H , =∓q .
dτ dτ2
2.1.2 Transversal Hubble effect and the shift frequency parameter
By the use of the relations [19], [20]
l =∓H ·l , l =∓q ·l ,
vηc c ξ⊥ aηc ηc ξ⊥
dzc = lvηc , d2zc = laηc ,
dτ l dτ2 l
ξ⊥ ξ⊥
wecanfindthetransversalHubblefunctionH andthetransversalacceleration
c
function (parameter) q respectively as
ηc
dz d2z
c =∓H , c =∓q .
dτ c dτ2 ηc
3 Conclusion
InthepresentpaperwehaveconsideredtheDopplereffectandtheHubbleeffect
for the case of auto-parallelmotion of the observer in a (L ,g)-space and their
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relations to the shift frequency parameters corresponding to the longitudinal
and transversaleffects. It is shown that these effects lead to direct check-up of
the theoreticalscheme andcouldbe usedfor finding outthe relativecentrifugal
(centripetal)velocitiesandaccelerationsaswellastherelativeCoriolisvelocities
and accelerations of moving astronomical objects from point of view of the
proper frame of reference of an observer (detector).
The Doppler effects andthe Hubble effects areconsideredonthe groundsof
purelykinematicconsiderations. ItshouldbestressedthattheHubblefunctions
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