Generalized Doppler effect in spaces with a transport along paths

0.1 MB·English

by  Bozhidar Z. Iliev

Generalized Doppler Effect in Spaces
with a Transport along Paths
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Bozhidar Zakhariev Iliev 1
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Published: Communication JINR, E5-95-160, Dubna, 1995
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a http://www.arXiv.org e-Print archive No. math-ph/0401033
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2000 MSC numbers: 70B05,83C99, 83E99

2003 PACS numbers: 04.90.+e, 45.90.+t
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3 The LATEX2ε source file of this paper was produced by converting a ChiWriter 3.16
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may exist.
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p
- Abstract
h
t AnanalogoftheclassicalDopplereffectisinvestigatedinspaces(man-
a
m ifolds) whose tangent bundle is endowed with a transport along paths,
which, in particular, can be parallel one. The obtained results are valid
:
v irrespectively to the particles mass, i.e. they hold for massless particles
i (e.g. photons) as well as for massive ones.
X
r
a 1. INTRODUCTION
The present paper continues the began in [1,2] applications of the theory
of transports along paths (in fibre bundles) [3] to the mechanics of material
point particles. Here is studied a phenomenon consisting in the comparison of
the (relative) energies of a material (massive or massless) point particle with
respect to two other arbitrary moving point particles (observers). An evident
special case of this problem is the well known Doppler effect [4,5]. When the
1 Permanent address: Laboratory of Mathematical Modeling in Physics, Institute for
Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Boul. Tzarigradsko
chauss´ee72,1784Sofia,Bulgaria
E-mailaddress: [email protected]
URL:http://theo.inrne.bas.bg/∼bozho/
1
mentioned transport along paths is linear and, in fact, only paths without self-
intersections are taken into account the above problem was investigated in [6].
Here we closely follow [6] without supposing these restrictions.
Sect. 2 contains the strict formulation of the problem of the present work
and the derivation of the main results, which in Sect. 3 are applied to the
general and special relativity. Sect. 4 closes the paper with some comments.
Belowtheneededforthefollowingmathematicalbackgroundissummarized.
All considerations in the present work are made in a (real) differentiable
manifoldM[7]whosetangentbundle(T(M),π,M)isendowedwithatransport
along paths I[3]. Here T(M) := ∪ T (M), T (M) being the tangent to M
x∈M x x
space at x ∈ M and π : T(M) → M is such that π(V) := x for V ∈ T (M).
x
Besides,the tangentbundle (T(M),π,M)issupposedtobe equippedalsowith
a real bundle metric g, i.e. [8]a map g : x → g ,x ∈ M, where the maps
x
g : T (M) ⊗ T (M) → R are bilinear, nondegenerate and symmetric. For
x x x
brevitythedefinedbygscalarproductsofX,Y ∈T (M),y ∈M willbedenoted
y
by a dot (·), i.e. X ·Y := g (X,Y). The scalar square of X will be written as
y
(X)2for it has to be distinguished from the second component X2of X in some
local basis (in a case when dim(M)>1). As g is not supposed to be positively
defined, (X)2can take any real values.
By J and γ : J → M are denoted, respectively, an arbitrary real interval
and a path in M. If γ is of class C1, its tangent vector is written as γ˙.
The transport along paths I and the bundle metric are supposed to be con-
sistent, i.e. I preserves the defined by g scalar products of the vectors (see
[3],eq.(2.9)):
A·B =(Iγ A)·(Iγ B),A,B ∈T (M), s,t∈J,
s→t s→t γ(s)
where Iγ is the transport along γ from s to t[3].
s→t
Fordetailsconcerningtransportsalongpathsthereaderisreferredto[3]and
for the ones about relative mechanical quantities (such as velocity, momentum
and energy) - to [1].
2. STATEMENT OF THE PROBLEM AND
GENERAL RESULTS
Let a material object (a point particle which may be massless as well as
massive) be moving in M along the path γ : J → M (its world line) which
is parameterized with r ∈J. Let γ intersects the paths x : J → M,a = 1,2,
a a
representingtheworldlinesoftheparticles1and2,whichwecallobservers;i.e.
forsomer ∈J ands0 ∈J ,wehaveγ(r )=x (s0),a=1,2. Ifitis necessary,
a a a a a a
the parameters r ∈ J and s ∈ J ,a = 1,2 will be considered as proper times
a a
of the corresponding particles. As a special case of this construction we can
pointthecasewhenthematerialobjectisemittedfromthefirstparticleand/or
is detected from the second one, or vice versa. In particular, if the considered
materialobjectisaphoton,thenthelastsituationrealizestheclassicalDoppler
effect [4,5].
2
Weputthefollowingproblem. Onthebasisoftheintroducedin[1]concepts
we wish to compare the relative energies of the material object with respect
to the observers 1 and 2 at the points γ(r ) = x (s0) and γ(r ) = x (s0)
1 1 1 2 2 2
respectively.
Let p(r) be the momentum of the materialobject (the observedparticle) at
γ(r) and V (s ) = (s )a = 1,2 be the velocities of the observers. For brevity
a a a a
we let
p :=p(r ),V :=V (s0), a=1,2. (1)
a a a a a
In accordance with the definition of a relative energy (see [1], sect. 4) we
have to compare the relative energies
E :=ǫ((V )2)p ·V E :=ǫ((V )2)p ·V , (2)
1 1 1 1 2 2 2 2
of the observed particle with respect to the particles 1 and 2 at the points
γ(r )= x (s0) and γ(r )=x (s0) respectively. Here ǫ(λ):= +1 for λ>0 and
1 1 1 2 2 2
ǫ(λ):=−1 for λ≤0.
We introduce the quantities:
(V ) :=Iγ V , (3)
2 1 r2→r1 2
∆p(r ,r ;γ):=p(r )−Iγ p(r )=p −Iγ p . (4)
1 2 2 r1→r2 1 2 r1→r2 1
The quantity (4) is defined analogously to the relative momentum (cf.[1],
sect. 3), but it defines along γ with the help of the transport along paths I the
change of the momentum of the observed particle when it moves from γ(r ) to
1
γ(r ).
2
Now we shall prove that the quantity
∆E :=ǫ((V )2)∆p(r ,r ;γ)·V (5)
21 2 1 2 2
is closely connected with the change of the energy of the material object along
γ with respect to the point γ(r ) = x (s0), at which its world line intersects
2 2 2
theworldlineofthesecondparticle,whentheobjectmovesfromγ(r )toγ(r )
1 2
along γ. In fact, using (1),(4) and (5), we get
∆E =ǫ((V )2)(p −Iγ p )·V =E(r ,r ;γ)−E(r ,r ;γ). (6)
21 2 2 r1→r2 1 2 2 2 1 2
Here
E(r,r ;γ)=ǫ((V )2)(Iγ p(r))·V , (7)
2 2 r→r2 2
as a consequence of the considerations of [1], is the defined along γ by means
of the transport along paths relative energy of the observed particle when it
is situated at γ(r),r ∈ [r′,r′′] with respect to the second particle when it is
situated at γ(r )=x (s0).
2 2 2
Fromonehand,puttingr =r in(7)and(duetotheconsistencybetweenthe
1
metric and the transport along paths) applying to the both multiplies Iγ
r2→r1
and, from the other hand, letting r = r in (7) and using (2), we, respectively,
2
get
3
E(r ,r ;γ)=ǫ((V )2)p ·(V ) , E(r ,r ;γ)=E . (8)
1 2 2 1 2 1 2 2 2
So, from (6), we find
E =∆E +ǫ((V )2)p ·(V ) . (9)
2 21 2 1 2 1
Further,supposing(V )2 6=0,whichisinterpretedasamovementofthefirst
1
observer with a velocity different from (less than) the one of light in vacuum,
we shall express p ·(V ) through E . Representing (V ) in the form (V ) =
1 2 1 1 2 1 2 1
(V )k+(V )⊥,wherethelongitudinal(V )kandthetransversal(V )⊥components
2 1 2 1 2 1 2 1
with respect to V are given by
1
(V )k :=V (V ·(V ) )/(V )2 ((V )k·V =(V ) ·V ), (10a)
2 1 1 1 2 1 1 2 1 1 2 1 1
(V )⊥ :=(V ) −(V )k ((V )⊥·V =(V )⊥·(V )k =0), (10b)
2 1 2 1 2 1 2 1 1 2 1 2 1
we obtain:
p ·(V ) =p ·(V )⊥+p ·(V )k =p ·(V )⊥
1 2 1 1 2 1 1 2 1 1 2 1
+(p ·V )(V ·(V ) )/(V )2 =p ·(V )⊥
1 1 1 2 1 1 1 2 1
+ǫ((V )2)E [(((V ) )2−((V )⊥)2)/(V )2]1/2, (11)
1 1 2 1 2 1 1
wherewehaveused(2)andtheequality(V ·(V ) )/(V )2 =[((V )k)2/(V )2]1/2 =
1 2 1 1 2 1 1
[(((V ) )2−((V )⊥)2)/(V )2]1/2 (thesquarerootsignisuniquelydefinedby(V ·
2 1 2 1 1 1
(V ) )/(V )2| = +1), which follows from ((V )k)2 = (V ·(V ) )2/(V )2
2 1 1 (V2)1=V1 2 1 1 2 1 1
and ((V )⊥)2 =((V ) )2−((V )k)2).
2 1 2 1 2 1
Substituting (11) into (9), we get:
E =∆E +ǫ((V )2)ǫ((V )2)E [(((V ) )2−((V )⊥)2)/(V )2]1/2
2 21 1 2 1 2 1 2 1 1
+ǫ((V )2)p ·(V )⊥. (12)
2 1 2 1
This formula is the answerof the stated aboveproblem and it expresses the
”generalized” Doppler’s effect for the considered process.
Now we will put the last term of (12) into a slightly different form (cf.[4]).
Let the vector N ∈ T (M) be defined in the following way. If p and
1 γ(r1 1
V arenotcollinear,thenN iscoplanarwiththem,i.e. N =aV +bp ,forsome
1 1 1 1 1
a,b∈R, and satisfies the conditions
N ·V =0, (13a)
1 1
(N )2 =N ·N =ǫ((p )2−(E )2/(V )2), (13b)
1 1 1 1 1 1
N ·p <0. (13c)
1 1
In this case N is uniquely defined and its connection with p and V may be
1 1 1
written, for example, as
p =V (V ·p )/(V )2
1 1 1 1 1
4
−N ǫ((p )2−(E )2/(V )2)|(p )2−(E )2/(V )2 |1/2, (14a)
1 1 1 1 1 1 1
where|λ|:=λǫ(λ)istheabsolutevalueofλ∈R.(Notethat(p )2−(E )2/(V )2 =
1 1 1
0 if and only if p and V are collinear; see (1).) If p and V are collinear,
1 1 1 1
i.e. p = λV for some λ ∈ R(which, because of (V )2 6= 0, is given by
1 1 1
λ=p ·V /(V )2), then we put N =0:
1 1 1 1
N =0 for p =λV . (14b)
1 1 1
(Note that due to (14a) from N = 0 follows p = λV , which is equivalent to
1 1 1
(p )2−(E )2/(V )2 =0.)
1 1 1
Hencethedefinedby(14)vectorN ∈T (M)isorthogonaltoapathx at
1 γ(r1 1
the point γ(r ) = x (s0)(N ·V = 0) and if p and V are not collinear, then it
1 1 1 1 1 1 1
is a unit vector ”pointing from x to x ”((N )2 =ǫ((p )2−(E )2/(V )2)).
1 2 1 1 1 1
By definition the recession speed ω of the particle 2 from the particle 1 is
21
the projection of (V ) on N (see [4]), i.e.
2 1 1
ω =(V ) ·N =(V )⊥·N . (15)
21 2 1 1 2 1 1
If p and V are collinear, then ω =0.
1 1 21
From (14),(15) and (10b), we obtain
p ·(V )⊥ =− ǫ((p )2−(E )2/(V )2)|(p )2−(E )2/(V )2 |1/2 . (16)
1 2 1 21 1 1 1 1 1 1
Substituting this expression into (12), we find the looked for connection
between the relative energies E and E in the form
1 2
E =∆E +ǫ((V )2)ǫ((V )2)E [(((V ) )2−((V )⊥)2)/(V )2]1/2
2 21 1 2 1 2 1 2 1 1
−ǫ((V )2)ǫ((p )2−(E )2/(V )2) |(p )2−(E )2/(V )2 |1/2 . (17)
2 1 1 1 21 1 1 1
NamelythisformulagivestheDoppler’seffectintermsofparticlesandtheir
energies in the considered process, which can be called a ”generalized Doppler
effect”. The corresponding to it ”red shift” [4] is given by the equality
(E −E )/E =1− ∆E /E +ǫ((V )2)ǫ((V )2)[(((V ) )2
2 1 2 21 1 1 2 2 1
(cid:8)
−((V )⊥)2)/(V )2]1/2−ǫ((V )2)ǫ((p )2−(E )2/(V )2)
2 1 1 2 1 1 1
×ǫ(E ) |(p )2/(E )2−1/(V )2 |1/2 −1. (18)
1 21 1 1 1
(cid:9)
3. EXAMPLES: GENERAL AND SPECIAL RELATIVITY
In this section are consider the applications of the obtained in Sect. 2 gen-
eral results to the cases of the Doppler’s effect in general relativity and the
”generalizedDoppler effect” (for constant velocities) in special relativity.
The Doppler effect in general relativity is investigated, e.g. in [4],ch. III,
section7, where the condition(13c) is notstated explicitly but it is used in the
calculation, consist in the following. At the point x (s0) is emitted a photon
2 2
5
whichmovesalongthe isotropicgeodesicpath γ to the point x (s0), where itis
1 1
detected. The problem is to be compared the detection and emission energies
E and E respectively.
1 2
Choosing the manifold M to be the space-time of general relativity with
signature(−+++)andIγtobeaparalleltransportalongγ,wefind(seesection
2 and [1]) : (V )2 = (V )2 = −c2(c is the velocity of light in vacuum),(p )2 =
1 2 1
(p )2 =0,E >0,∆p(r ,r ;γ)≡0,∆E =0 and N =−[p +V (V ·p )]/E .
2 1 1 2 21 1 1 1 1 1 1
So, (17) and (18) take respectively the form:
E =E ω +(1+((V )⊥)2/c2)1/2], (19)
2 1 21 2 1
(E −E )/E =1−[ω +(1+((V )⊥)2/c2)1/2]−1. (20)
2 1 2 21 2 1
Byusingalocalorthonormalbasisthisresultisderivedin[4],ch. III,section
7,whereE ,E ,ω and(V )⊥ aredenotedasE,E′,ω andω2respectivelyand,
1 2 21 2 1 R
besides, there are used units in which c=1.
In the case of special relativity M is the Minkowski’s space-time and Iγis
also a parallel transport along γ. Let the considered particles be moving with
constant 3-velocities v,v and v with respect to a given frame of reference, i.e.
1 2
wehaveγ(r)=y+(ct,tv)andx (s )=y +(ct,tv ),a=1,2,wherey,y ,y ∈
a a a a 1 2
M are fixed, t is the time in that frame, r = t for | v |= c and r = t(1−
v2/c2)1/2for | v |< c, and s = t(1−v2/c2)1/2,a = 1,2 are the corresponding
a a
proper times (cf. [2,4,5]). (Of course, we suppose that γ intersects x and x .)
1 2
From here we find the momenta of the observed particle as p(r) == p =
1
p = µc(1,v/c), where µ := E/c for | v |= c(E is the particle’s energy in
2
the given frame) and µ := m(1 −v2/c2)−1/2for | v |< c(m is the particle’s
rest mass),v =c(1,v/c)(1−v2/c2)−1/2,a=1,2,(V ) =V ,∆p(r ,r ;γ)≡0
a a 2 1 2 1 2
and ∆E = 0. The computation of the remaining quantities concerning the
21
phenomenonunderconsiderationissimplebutlong;thatiswhywepresenthere
only the final result:
1−v2/c2 1/2 1−v ·v/c2
E =E 1 · 2 , (21)
2 1(cid:16)1−v2/c2(cid:17) 1−v ·v/c2
2 1
which also may easily be obtain from the expressions E == ǫ((V )2)p ·V =
a a a a
µc2(1−v ·v/c2)(1−v2/c2)−1/2,a=1,2.
a a
Inthe caseofthe ”usual”Doppler effect[5]we havea photon(v=cn,n2 =
1) emitted with energy E = E and detected with energy E = E, which
2 0 1
according to (21) is
1−v2/c2 1/2 1−v ·n/c2
E =E 2 · 2 , (22)
0(cid:16)1−v2/c2(cid:17) 1−v ·n/c2
1 1
Letusnotethatwhenv =0,theformulae(21)and(22)maybederivedalso
1
asacorollaryfromthedefinitionofrelativeenergy(see[1],sect. 4). Thisresult
is in agreementwith the consideredin[5]Doppler effect in specialrelativity (in
terms of frequencies; cf. the quantum relation E = hν,h being the Planck’s
constant).
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4. CONCLUDING REMARKS
We want to emphasize that the basic result of this work is given by the
equality (17) the main difference of which from the usual Doppler effect (e.g.
in general relativity; see (19) or [3]) is the existence in it of, generally said,
nonvanishing term ∆E , defined by (5).A feature of the equation (17) is its
21
validity for particles with arbitrary, zero or nonzero, masses, i.e. our results do
not depend on the mass of the investigated particle.
If ∆E = 0, then eq.(17) takes a form similar to the classical one (in the
21
case of arbitrary mass (cf.[4])). Due to (5)a sufficient condition for this is
∆p(r ,r ;γ)=0, (23)
1 2
i.e. (see (4))
p(r )=Iγ p(r ) (23′)
2 r1→r2 1
whichmeansthatthemomentump(r )is(I-)transportedbymeansofthetrans-
1
port along paths I from the point γ(r ) to the point γ(r )(cf.[3],eq.(2.4)).
1 2
If a point particle is moving along the path γ : J → M, then it is natural
to call it a free particle (with respect to the transport along paths I) if (23′)
holds for every r ,r ∈J, i.e. if its momentum is I-transported along its world
1 2
line γ(cf.[3], definition 2.2). In particular, if Iγis a parallel transport along
γ(generated by a linear connection), then this definition of a free particle coin-
cides with the one in [4], p. 110, given therein as a special case of the geodesic
hypothesis. In our case, the corresponding generalization of the geodesic hy-
pothesis states that the world line γ : J → M of a tree (with respect to I)
particle is an I-path (see [2], definition 2.2), i.e.
γ˙(t)=Iγ γ˙(s), s,t∈J, (24)
s→t
and besides
˙
p(s)=µ(s;γ)(s) (25)
for some scalar function µ(identified with the particles rest mass if it is not
zero).
Ifthetransportalongpathsislinear(seee.g. [3],eq.(2.8)),thensubstituting
(25)into(23′)andcomparingtheresultwith(24),weget(cf.[4],p. 110,eq.(9))
µ(s;γ)=const. (26)
So, the mass parameter µ(the rest mass if it is not zero) of a free particle is
constant if the used transport along paths is linear.
It should also be noted, that as for free massive particles with a rest mass
meq.(25) holds (by definition) for µ(s;γ)=m, then for linear transports along
paths the (”generalizedgeodesic”) hypothesis (24) is a consequence of the con-
dition (23′).
At the end we want to mention the equality (cf.(4))
∆p(r ,r ;γ)=Iγ ∆p(r ,r ;γ) (27)
2 1 r2→r1 1 2
7
for a linear transport along paths I. Hence, in this case eq.(5) can equivalently
(due to the consistency of I and the metric) be written as
∆E :=ǫ((V )2)∆p(r ,r ;γ)·(V ) . (28)
21 2 2 1 2 1
ACKNOWLEDGEMENT
This researchwaspartially supportedby the Fund for Scientific Researchof
Bulgaria under contract Grant No. F103.
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paths, JINR Communication E2−94−188, Dubna, 1994.
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munication JINR, E5−93−299, Dubna, 1993.
4. Synge J.L., Relativity: The general theory, North-Holland Publ. Co.,
Amsterdam, 1960.
5. Moller C., The Theory of Relativity, 2d ed., Clarendon Press, Oxford,
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6. IlievB.Z.,Ononegeneralizationofthe Dopplereffectinspaceswithgen-
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Moscow, 1979 (In Russian).
8
Iliev B. Z.
Generalized Doppler Effect
in Spaces with a Transport along Paths
AnanalogoftheclassicalDopplereffectisinvestigatedinspaces(manifolds)
whose tangent bundle is endowed with a transport along paths, which, in par-
ticular, can be parallel one. The obtained results are valid irrespectively to the
particles mass, i.e. they hold for massless particles (e.g. photons) as well as for
massive ones.
TheinvestigationhasbeenperformedattheLaboratoryofComputingTech-
niques and Automation, JINR.
9