Table Of ContentThim’s Experiment and Exact
Rotational Space-Time Transformations
4 Leonardo Hsu∗
1
Department of Postsecondary Teaching and Learning,
0
2 University of Minnesota, Minneapolis, Minnesota 55414,
n and
a Jong-Ping Hsu†
J
Department of Physics,
0
3 University of Massachusetts Dartmouth,
North Dartmouth, MA 02747
]
c
q February 3, 2014
-
r
g
[
Abstract— Thim measured the transverse Doppler shift using a system con-
1 sistingofastationary antennaandpickup,inadditiontoanumberofintermedi-
v ate antennasmountedon therim of a rotating disk. Nosuch shift was detected,
2 although the experiment should have had enough sensitivity to measure it, as
8
predicted by the Lorentz transformations. However, using the Lorentz transfor-
2
mations to analyze the results of experiments involving circular motion, while
8
commonly done, is inappropriate because such an analysis involves non-inertial
.
1 frames,whichareoutsidetherangeofvalidityofspecialrelativity. Inthispaper,
0 we re-analyze Thim’s experiment using exact rotational space-time transforma-
4
tions, findingthat hisnull result is consistent with theoretical predictions.
1
:
v
i
X
r 1 Introduction
a
In the paper ‘Absence of the relativistic transverse Doppler shift at microwave
frequencies,’[1] which was followed by a comment and rebuttal [2, 3], Thim de-
scribes an experiment in which microwave signals from a stationary source are
received andretransmitted bytwosets of antennasmounted oncounter-rotating
∗E-mail: [email protected]
†E-mail:[email protected]
1
disks, and finally received by a stationary antenna. Although the Lorentz trans-
formations of special relativity predict that the final signal received by the sta-
tionarysourceshouldexhibitatransverseDopplershiftcomparedtotheoriginal
signalemittedbythesource,nonewasmeasured,eventhoughsuchashiftshould
havebeen easily within thedetection sensitivity of the apparatus. Thedifficulty
here, however, is that two reference frames that are rotating with respect to one
anothercannotbothbeinertialframesandthusspecialrelativityandtheLorentz
transformation are not applicable tothis situation.
While it is true that the Lorentz transformation has been used to analyze
experiments involving rotational or orbital motion and that in some cases, the
theoreticalpredictionsareconsistentwithexperimentalresults(forexampleinthe
case of calculating thelifetime dilation of unstableparticles moving in a circular
storage ring[4]) the fact remains that, rigorously speaking, such applications are
inappropriate. If the relative motion between two reference frames is rotational,
then objects movingwith constant velocity in one frame havenon-zeroaccelera-
tions as viewed from the otherframe. Thus, at least one of thetwo frames must
be non-inertial, a situation out of the purview of special relativity. In Pellegrini
andSwift’sanalysisoftheWilsonexperiment[5],theydemonstratethattheexact
rotationaltransformationscannotlocally bereplacedbytheLorentztransforma-
tions. Onemajordifficultyisthattheredoesnotyetexistasetofexactlycorrect
transformations for rotating reference frames that is widely accepted and used.
Nevertheless, the preceding arguments suffice to demonstrate that the results of
Thim’s experiment cannot imply any contradiction in special relativity, which
has been supported by a nearly uncountable number of experiments, including
the transverse Doppler shifts predicted by Lorentz transformations for radiation
sources moving with constant velocities.
Although therearenowidely accepted exact rotational transformations, this
does not mean that no exact rotational transformations exist. Indeed there are
several in the literature, some of which are consistent with existing experimen-
tal tests.[6, 7] The lack of a consensus choice is more a reflection of the fact
thatestablishingsucharotational transformation isnot viewedascritical toad-
vancing our understanding of physics. In this comment, we use one such exact
transformation[8] tore-analyze Thim’s experiment,findingthat hisnull result is
fully consistent with the predictions of this transformation. Thus, rather than
serving as a test of special relativity, Thim’s experiment actually gives us clear
clues about the nature of non-inertial frames in our universe and can help serve
as a test of proposed space-time transformations between rotating frames.
2 Exact rotational space-time transformation
The space-time transformations involving accelerations and rotations that we
employherehavebeendiscussedandderivedindetailelsewhereintheliterature[6,
8]. Wesimplygivetheresulthere. SupposeF ,withcoordinates(w ,x ,y ,z ),is
I I I I I
an inertial frame and F(Ω), with coordinates (w,x,y,z), is a frame that rotates
with a constant angular velocity Ω (to be defined more precisely below) with
2
respect to F . The origins of both frames coincide at all times and, for reasons
I
tobeseen later, we useaCartesian coordinate system in both frames. Thetime
variable is expressed in units of length (where 1 meter is the amount of time it
takes for a light signal in a vacuum to travel a distance of 1 meter in an inertial
frame) to avoid complications associated with including the quantity c in the
transformation equations for non-inertial frames, in which the speed of light is
not necessarily universal or constant[6, 9, 10]. The exact rotational space-time
transformation equations are[6, 8]
w =γw, x =γ[x cos(Ωw)−y sin(Ωw)],
I I
y =γ[x sin(Ωw)+y cos(Ωw)], z =z; (1)
I I
γ =1/p1−ρ2Ω2, ρ2=x2+y2;
and theinverse transformations are
w 1
w= I, x= [x cos(Ω w )+y sin(Ω w )],
γ γ I I I I I I
1
y= [−x sin(Ω w )+y cos(Ω w )], z=z ; (2)
γ I I I I I I I
1 dφ dφ dw Ω
γ = , Ω ≡ I = I = ,
p1−Ω2I(x2I+yI2) I dwI dw dwI γ
where φ = φ+Ωw. The quantity Ω is the constant angular velocity of F(Ω)
I
with respect to F as measured by observers in the F(Ω) frame and Ω is the
I I
angular velocity of F(Ω) with respect to F as measured by observers in the F
I I
frame. The relationships between these angular velocities are
2 2 2
w Ω =wΩ, ρ Ω =ρΩ, ρ =x +y . (3)
I I I I I I I
The validity of these transformations has also been discussed elsewhere in
detail[6, 8]. Here we make only two brief notes:
(A)Transformations(1)and(2)reducetotheLorentztransformationsinthe
appropriatelimit. Thetransformationequations(1)areactuallyaspecialcaseof
a more general set of transformation equations between an inertial frame and a
non-inertialframe whose origin orbits theorigin of theinertial frame at aradius
R. The transformations between those two frames are[6, 8]
w =γ(w+ρ·β), x =γ[x cos(Ωw)−(y−R) sin(Ωw)],
I I
y =γ[x sin(Ωw)+(y−R) cos(Ωw)], z =z; (4)
I I
β =|Ω×S|=Ωpx2+(y−R)2=ΩS <1, (5)
ρ·β=xRΩ, γ =(1−β2)−1/2. (6)
In the limit of zero acceleration, i.e., when R → ∞ and Ω → 0 such that the
productRΩ=β isafinitenon-zeroconstantvelocity,transformation(4)reduces
o
to theLorentz transformations
w =γ [w+xβ ], x =γ [x+wβ ], y =−∞, z =z, (7)
I o o I o o I I
3
where one may shift the y-axis so that y = y. Thus, we see that Cartesian
I
coordinatesallow theexactrotationalspace-timetransformation (4)tohavethis
property, known as limiting Lorentz-Poincar´e invariance[8, 11]. Furthermore,
transformation (4) reduces to the classical rotational transformation in the case
where β is small.
(B)Transformations(1)and(2)areconsistentwithallwell-knownexperimen-
taltests. Forexample,considerthelifetimedilationofunstableparticlestraveling
inacircularstoragering[4]. Furthermore,basedontherotationaltransformations
of the covariant momentum vector, the expression for the energy of a particle,
with rest mass m, traveling in a circle is
pI0 =γm, (8)
in agreement with with the well-established results of high energy experiments
performed in an inertial laboratory frame F .[8]
I
3 Thim’s experiment
Thim’sexperiment[1]canbeanalyzedusingtherotationaltransformationsofthe
covariant wave vector kµ = (k0,k1,k2,k3) which, like the covariant momentum
p ,hasthesametransformationpropertiesasthecovariantcoordinatedifferential
µ
vectorsdx anddx . Therotationaltransformationsforthecovariantcoordinate
µ Iµ
differential vectors can be obtained from (1) with dxµ = ηµνdx for inertial
I Iν
frames and dxµ = Pµνdx for rotational frames[8, 11].1 The transformation
ν
equations between the wave vector k measured in F and the wave vector k
Iµ I µ
measured in the rotating frame F(Ω) are[6, 8]
kI0 =γ−1(k0+Ωyk1−Ωxk2),
kI1 =(cid:2)−γ−2Ω2wxI(cid:3)k0+γ−2(cid:2)γcos(Ωw)−Ω2xIx−Ω3wxIy(cid:3)k1
+γ−2(cid:2)−γsin(Ωw)−Ω2xIy+Ω3wxIx(cid:3)k2, (9)
kI2 =(cid:2)−γ−2Ω2wyI(cid:3)k0+γ−2(cid:2)γsin(Ωw)−Ω2yIx−Ω3wyIy(cid:3)k1
+γ−2(cid:2)γcos(Ωw)−Ω2yIy+Ω3wyIx(cid:3)k2,
1
kI3 =k3, γ = p1−ρ2Ω2.
InThim’sexperiment,thesourceislocatedontheaxisofrotationoftheF(Ω)
frame. Since all points on this axis of rotation are at rest in both the rotating
andinertialframes,thespace-timepropertiesofbothframesatallpointsonthat
axisarethesame. Thus,thefrequencyoftheradiation from thesourcefI ≡kI0
1Thenon-vanishingcomponentsofPµν aregivenbyP00=1,P11=−γ2[1+2γ2Ω2x2
−γ4Ω4x2(w2−x2−y2)],etc. ThecontravariantmetrictensorsPµν areP00=γ−2[1
−Ω4w2(x2+y2)], P11=−γ−2[γ−2(1−Ω2x2)−2γ−2Ω3wxy+Ω6w2y2(x2+y2)], etc.
4
measured from F is the same as the frequency of the radiation of the source at
I
rest relative to F(Ω) f(rest)≡k0(rest) measured from F(Ω),
f =f (rest)=f(rest). (10)
I I
Consider one specific detector located at the rim of the rotating disk located at,
say,ρia =(xa,ya,0). Radiation with wavevectorki=(k1,k2,0)propagates from
the center of the disk to this detector along the radius vector ρi = (x,y,0), so
that k1/k2 =x/y.Thus, the first equation in (9) leads to
fI =γ−1f, xk2=yk1, γ = p1−1ρ2Ω2. (11)
a
Combining (10) and (11), we obtain
f =γf =γf(rest). (12)
I
Thisresult, which holdsforeach of theeight detectors, implies thatobserversat
rest with respect to the detectors (i.e., observers at rest in the rotating frame)
will measure a shift by a factor of γ in the frequency of theradiation as a result
of the orbiting motion of the detectors. Another way to think about this result
istoseethataccordingtotherotationaltransformations(1),clocksinarotating
frame located at a radius ρ are slowed by a factor of γ, resulting in an increase
in the detected frequency f bya factor of γ.
However, result (12) actually does not contradict Thim’s null result because
in the experiment,[1] the frequency of the radiation f received by the orbiting
detectors is not measured by observers in the rotating frame F(Ω). Instead, the
signal received by the detectors on the rotating disk is transferred through a
second rotating disk to a stationary detector, where the frequency is measured
by apparatus situated in the inertial laboratory F . The process of transferring
I
thesignal back to theinertial laboratory frame is simply thereverse of the first,
inwhichthefrequencyoftheradiation was increased byafactorofγ. Thus,the
frequencyoftheradiationf (detector)receivedbythefinaldetector,asmeasured
I
in the laboratory, will be smaller than the frequency measured by the orbiting
detectors by a factor of γ because the clocks in the inertial frame run faster by
the factor of γ, f (detector)=f/γ. Combining these two processes leads to the
I
result
f (detector)=f(rest)=f . (13)
I I
Thus, the rotational transformations (1) imply that when the frequency of the
radiationf (detector)receivedbythedetectorismeasuredbystandardmixerand
I
interferometertechniquesintheinertiallaboratoryframeandcomparedwiththe
frequencyf(rest)ofthesource,noshiftshouldbemeasured,consistentwiththe
results obtained byThim.
4 Discussion and conclusion
The previous conclusion is independent of the angular velocity Ω′ of the second
disk. This can be seen by regarding the second disk as a second rotating frame
5
F(Ω′). The rotational transformations for F , F(Ω) and F(Ω′) are
I
w =γw=γ′w′, etc. (14)
I
f =γ−1f =γ′−1f′, etc. (15)
I
1 1
γ = , γ′ = ;
p1−ρ2Ω2 p1−ρ′2Ω′2
a
where ρ and ρ′ are constant. If the frequency of the radiation were to be mea-
a
suredbyapparatusatrestineithertheF(Ω)(disk1)ortheF(Ω′)(disk2)frame,
itwoulddifferfromitsoriginalvalue(measuredintheinertiallaboratoryframe).
However, because this radiation is eventually transferred back and measured by
adetectorthatisatrestinthesameinertiallaboratory frameasthesourcefrom
which it was emitted, therewill be no overall frequency shift.
A new experimental test of the exact rotational transformations (1) is to
measuredirectlythefrequency(12)ofthesignalreceivedbytheorbitingantenna
on the rotating disk. Assuming that the centrifugal effects on the apparatus are
negligible, one can test the transverse frequency shift predicted in (12). In fact,
one could test for centrifugal effects by repeating the experiment with different
angular velocities.
In conclusion, the Lorentz transformations cannot be used to analyze the
results of Thim’s experiment, involving orbiting detectors. Using the correct
transformationsforrotatingnon-inertialframespredictsthat,aslongasthesignal
received by the detectors is analyzed by apparatus at rest with respect to the
inertiallaboratoryframefromwhichtheradiationisinitiallyemitted,noDoppler
shiftwillbedetected,consistentwiththeexperimentalobservations. Toobtaina
broaderandmorecompleteunderstandingofphysics,itisdesirabletogeneralize
the physical framework to include non-inertial frames. Inertial frames represent
onlylimitingandidealisticcasesandmoreover,thereisnowstrongevidencethat
the observable universe is expanding with a non-zero acceleration. Therefore,
rotational experimentsof Davies-Jennison[8]and Thim[1]are important because
they can reveal new principles in physics and increase our understanding of the
physicsof non-inertial frames.
Note added in proof. To avoid misunderstanding, we stress that it is incor-
rect to use the usual rotational coordinate transformations (i.e., X = r cos(θ−
ωt), Y =r sin(θ−ωt), Z =z, cT =ct;orequivalently,ds2 =gµνdxµdxν,g00 =
1−ω2r2/c2, etc. See, for example, C Moller, The Theory of Relativity, p. 240)
to discuss the precision experiments of Thim and Davis-Jennison mentioned in
thepresentpaper.[1,8]Thereasonisthatthisusualrotationalcoordinatetrans-
formation holds only approximately in the classical domain with small velocity
rω << c). Furthermore, this approximate rotational transformation is incon-
sistent with the lifetime dilatation of muon decay in circular-orbital motion.[4]
Therefore,onecannotusetheresultofthisapproximationrotationaltransforma-
tion,e.g., arotating radiusr=px2+y2 doesnotcontract, x2+y2=X2+Y2,
to rule out the exact rotational transformations (1) or (4) in the present paper
(which is consistent with the muon lifetime dilatation in circular motion and
6
Thim’s experiment) and to reject its logical consequence that a rotating radius
contract.
7
References
[1] H. W. Thim, IEEE Trans. Instrumentation and Measurement, 52, 1660
(2003).
[2] A.Sfarti, IEEE Trans. Instrumentation and Measurement, 59, 494 (2010).
[3] H. W. Thim, IEEE Trans. Instrumentation and Measurement, 59, 495
(2010).
[4] See, for example, F. J. M Farley, J. Bailey and E. Picasso, Nature 217, 17
(1968).
[5] G.N.PellegriniandA.R.Swift,Am.J.Phys.,63,694(1995)andreference
therein.
[6] J.P.HsuandL.Hsu,inA Broader View of Relativity, General Implications
of Lorentz-Poincar´e Invariance, (Singapore, New Jersey, World Scientific
(2nd Ed.), 2006), pp. 402-415 and pp. 267-318.
[7] M.Nouri-Zonoz,H.Ramazani-AvalandR.Gharachahi,arXiv1208.1913v1,
(2012), and reference therein.
[8] L. Hsu and J. P. Hsu, Euro. Phys. J. Plus, 128: 74 (2013). (DOI
10.1140/epjp/i2013-13074-4).
[9] J. P. Hsu and L. Hsu, Phys.Letts. A, 196, 1 (1994).
[10] L. Hsu and J. P. Hsu, Eur. Phys. J. Plus, 127, 11 (2012). (DOI 10.1140/
epjp/i2012-12011-5).
[11] J.P.HsuandL.Hsu,Space-TimeSymmetryandQuantumYang-MillsGrav-
ity, (Singapore, NewJersey, World Scientific, 2013), Chapter 5.
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