Recent experiments with high-speed ions have investigated potential
deviations from the time-dilation predicted by special relativity (SR). The
main contribution of this article is to show that the SR predictions are
matched by the experimental results only when the transverse Doppler effect in
the observed emissions from the ions are neglected in the analysis. However,
the Doppler effect in the emission cannot be neglected because it is similar to
the time dilation effect. Thus, the article highlights the need to consider
Doppler emission effects when validating SR time dilation using high-speed ion
experiments.Comment: 3 pages, 3 figure

C. Novotny

D. Anselmi

D. Colladay

D.-Z. Wang

G. Saathoff

H.E. Ives

J.J. Snyder

J.L. Chkareuli

K. Nozari

L.P. Colatto

M. Kaivola

M.E. Tobar

R.W. McGowan

Santosh Devasia

English

arXiv

Springer - Publisher Connector

Eur. Phys. J. C (2010) 69: 343–346
DOI 10.1140/epjc/s10052-010-1437-9
Letter
Lorentz violation in high-energy ions
Santosh Devasiaa
U. of Washington, Seattle, WA 98195-2600, USA
Received: 20 June 2010 / Revised: 28 July 2010 / Published online: 15 September 2010
© The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract The theoretical interest in small Lorentz viola-
tions has motivated experiments that investigate it by mea-
suring deviations in the time dilation predicted by special
relativity (SR) using high-energy ions. The main contribu-
tion of this article is to show that including the Doppler ef-
fect in the emission (which is of the same order as the time
dilation effect) in the analysis leads to differences between
experimental and theoretical predictions that indicate poten-
tial Lorentz violation.
1 Introduction
Recent theoretical efforts allow for small Lorentz viola-
tions, see, e.g., [1–6]. For example, Doppler shift experi-
ments such as the Ives–Stillwell experiment are sensitive to
Lorentz violating terms in standard model extension [1, 2].
This renews the interest in experiments on high-speed ions
that check for Lorentz violations by experimentally measur-
ing potential deviations in the time dilation predicted by spe-
cial relativity [7–11].
The time dilation predicted by special relativity (SR) has
been verified at low speeds with a number of experiments
starting with the classical experiment using hydrogen canal
rays by Ives and Stilwell [12] at speed v = 0.005c where c
is the speed of light. More recent experiments [7–11] have
evaluated the time-dilation effect by using Doppler-shifted
lasers to excite transitions in high-energy neon (Ne) and
lithium (Li) ions and then observe the emissions at high ion
speeds—as high as v = 0.338c in [11].
The main contribution of this article is to show that in-
cluding the Doppler effect on the emission from high-energy
ions (which is of the same order as the time dilation effect) in
the analysis leads to differences between experimental and
theoretical predictions that indicate potential Lorentz viola-
tion.
a e-mail: devasia@u.washington.edu
2 High-energy ion spectroscopy
2.1 Doppler effect on laser and emission frequencies
The main idea is to use Doppler-shifted lasers to excite
transitions in high-energy ions and then observe the result-
ing emissions to evaluate time dilation. In particular, an
ion moving with speed v = βc with respect to a laboratory
frame FL can be excited by using parallel (co-propagating)
or anti-parallel (counter-propagating) lasers as in Fig. 1.
The relations between associated laser frequencies νp , νa
(parallel and anti-parallel to the ion velocity with respect
to the laboratory frame FL) and ν∗p , ν∗a (parallel and anti-
parallel to the ion velocity with respect to a frame FI at-
tached to the moving ion) are given by relativistic Doppler
expressions as (see, e.g., [10, 13])
ν∗p = νpγ (1 − β), (1)
ν∗a = νaγ (1 + β), (2)
where the time dilation term is
γ = 1/
√
1 − β2. (3)
Similarly, the emission frequency ν∗e of photons from the
ions (with respect to the ion frame FI ) and the detection
frequency νe perpendicular to the moving ions (with respect
to the laboratory frame FL) are related by (see, e.g., [13])
ν∗e = νeγ, (4)
with angle cos θ = −β in the ion frame FI as shown in
Fig. 1.
2.2 Evaluating potential Lorentz violation
The ratio R, given by
R = νpνa
ν∗pν∗a
= 1
γ 2(1 − β2) = 1, (5)
344 Eur. Phys. J. C (2010) 69: 343–346
Fig. 1 Laser frequencies νp , νa in the laboratory frame FL are
Doppler shifted to frequencies ν∗p , ν∗a in a frame FI that is moving
with the ion (circle) at speed v = βc. Photons emitted at frequency
ν∗e in the ion frame FI are observed at frequency νe in the laboratory
frame FL perpendicular to the moving ions
is independent of the speed of the ions. Potential dependence
of the measured ratio R on speed β is used to evaluate the
time dilation predicted by SR and thereby, to evaluate po-
tential Lorentz violation.
The frequency terms ν∗p , ν∗a needed to evaluate the ex-
pression for R (in (5)) are not directly measurable. This in-
ability to measure ν∗p , ν∗a can be avoided if the lasers excite
a known transition, say at frequency ν∗e in the ion frame FI .
For example, in saturation spectroscopy [10, 11], one of the
laser frequencies is kept constant and the other frequency is
varied to observe the Lamb dip in the fluorescence spectrum,
which indicates that both lasers are acting on ions with the
same speed, i.e.,
ν∗p = ν∗e and ν∗a = ν∗e .
Then, the ratio R (in (5)) can be rewritten as
R = νpνa
(ν∗e )2
= 1
γ 2(1 − β2) = 1. (6)
2.3 Transition frequency shift
Ideally, the transition frequency ν∗e in the moving ion frame
FI should be the same as the transition frequency νo for
ions that are stationary in the laboratory frame FL, and is
therefore known—the transition frequency νo can be deter-
mined with high accuracy using stationary ions in the labo-
ratory frame FL. However, the transition frequency for the
moving ions can get shifted (i.e., ν∗e = νo) due to external
fields and charged particles causing Stark and Zeeman ef-
fects, e.g., [11].
The potential shift in the transition frequency implies that
the transition frequency ν∗e excited in the moving ion can-
not be assumed to be exactly the same as the transition fre-
quency νo measured for stationary ions under different ex-
perimental conditions. Moreover, this shifted frequency ν∗e
is not directly measurable (in the moving ion frame FI )—
although the transition frequency νo of the stationary ions
(in laboratory frame FL) is known. Therefore, the ratio R
(in (6)) cannot be evaluated directly from the measurements,
and an expression in terms of the directly measurable fre-
quencies (νa , νp , νo) is sought.
2.4 Effect of PMT pre-filters
Emissions from the ions are used to identify when the
Doppler-shifted laser frequencies match the transition fre-
quencies in the moving ions, e.g., using the Lamb dip in the
observed fluorescence spectrum and to optimize the experi-
ment [8–10].
In general, measurements of the number of photons emit-
ted by the moving ions (to determine excitation of the ion
transition) using photomultiplier tubes (PMTs) should be in-
dependent of the photon frequency. However, the measure-
ments will not be independent of photon frequency if opti-
cal pre-filters are used before the PMTs. For example, the
emission is observed using an interference filter (before the
PMT) centered at the transition frequency νo (of stationary
ions) with a narrow (10 nm) halfwidth in [14] to precisely
detect the laser frequency where the Lamb dip occurs. In this
case, the observed Lamb dip corresponds to emitted photons
of a specific frequency,
νe = νo, (7)
with respect to the laboratory frame FL, which is related
to the emitted photon frequency ν∗e in the ion frame FI by
(from (4))
ν∗e = γ νe = γ νo. (8)
Therefore, a theoretical expression Ro that can be evaluated
in terms of measurable frequencies νo, νp , νa (in the labora-
tory frame FL) can be found from (6), (8) as
Ro = νpνa
(νo)2
= γ
2νpνa
(ν∗e )2
= 1
1 − β2 ≈ 1 + β
2. (9)
3 Lorentz violation
Experimental observations find this ratio Ro,exp of the prod-
uct of the laser frequencies νp,expνa,exp to the transition fre-
quency νo,exp of the stationary ion to be a constant
Ro,exp = νp,expνa,exp
(νo,exp)2
= 1 (10)
that is independent of the speed β [11] where the subscript
exp indicates an experimentally obtained value. This results
in a difference between theoretical (Ro) and experimental
(Ro,exp) predictions of this ratio (from (9), (10))
Ro = Ro,exp. (11)
Eur. Phys. J. C (2010) 69: 343–346 345
This β2-order deviation in Ro using SR time dilation ex-
pressions (between (9) and (10)) indicates potential Lorentz
violation in high-energy ions.
The difference between the theoretical ratio (Ro in (9))
and the experimental ratio (Ro,exp in (10)) should be exper-
imentally discernible as the speed β increases and, there-
fore, the difference β2 increases. For example, at speed
β = 0.064, the experimental accuracy of Ro,exp (in (10)) is
estimated in [10] as
|Ro,exp − 1| ≤ 2β2αˆ = 1.8 × 10−9 (12)
with αˆ = 2.2 × 10−7 [10] whereas the theoretical ratio (Ro
in (9)) yields
‖Ro − 1‖ ≈ β2 = 0.0041. (13)
3.1 Analysis without Doppler shift in emission
If the Doppler shift in the photon emission is neglected,
i.e., emission frequency ν∗e (in the ion frame FI ) is approx-
imated by the emission frequency νe = νo (in the laboratory
frame FL), i.e., ν∗e ≈ νe = νo then the constant Ro (in (9))
can be approximated as
Ro = νpνa
(νo)2
≈ νpνa
(ν∗e )2
= 1, (14)
which would match the experimental ratio Ro,exp.
The SR predictions are matched exactly by the experi-
mental results only if the transverse Doppler effect in the
observed emissions from the ions are neglected in the analy-
sis. However, such an approximation (ν∗e ≈ νe = νo) is not
reasonable because the Doppler effect in the emission is of
the same order (γ ) as the time dilation being measured and
is, therefore, not negligible.
3.2 Other effects
It is possible that other effects (such as variations in the
observation angle) might explain or reduce the apparent
Lorentz violation; further work is needed to investigate such
effects. For example, previous work has shown that reso-
nance fluorescence can be affected by the observational an-
gle [15]. Therefore, the experimental results would be af-
fected if the PMT is not measuring emissions that are ex-
actly perpendicular to the ion beam. Further analysis would
be needed to evaluate such angle-deviation effects.
Future experiments could include the Doppler shift in the
emissions in their design. For example, it might be possible
to place filters, centered at the frequency νo/γ, before the
PMT. In this case the emissions measured in the laboratory
frame FL at frequency νo/γ (and the associated Lamb dip)
would correspond to νo (a known value) in the ion frame.
This would guarantee that there is no shift in the transition
frequency in the ion frame FI , i.e., emissions measured cor-
respond to frequency νo in the ion frame, which is the known
value for stationary ions. Such experimental efforts could
clarify and quantify, better, the potential Lorentz violation
identified in this article and its potential effect on systematic
errors reported in other related experiments, e.g., when the
ions are moving perpendicular to the lasers [8].
4 Conclusions
This article analyzed experiments that investigate Lorentz
violation by measuring deviations in the time dilation pre-
dicted by special relativity (SR) using high-energy ions. It
was shown that including the Doppler effect on the emission
(which is of the same order as the time dilation effect) in the
analysis leads to differences between experimental and the-
oretical predictions that indicate potential Lorentz violation
in high-energy ions.
Open Access This article is distributed under the terms of the Cre-
ative Commons Attribution Noncommercial License which permits
any noncommercial use, distribution, and reproduction in any medium,
provided the original author(s) and source are credited.
References
1. D. Colladay, V.A. Kostelecky, Lorentz-violating extension of the
standard model. Phys. Rev. D 58(11), 116002 (1998)
2. M.E. Tobar, P. Wolf, A. Fowler, J.G. Hartnett, New methods of
testing Lorentz violation in electrodynamics. Phys. Rev. D 71(2),
025004 (2005)
3. D. Anselmi, Standard model without elementary scalars and high
energy Lorentz violation. Eur. Phys. J. C 65(3), 523–536 (2010)
4. K. Nozari, S.D. Sadatian, Late-time acceleration and phantom
divide line crossing with non-minimal coupling and Lorentz-
invariance violation. Eur. Phys. J. C 58(3), 499–510 (2008)
5. J.L. Chkareuli, Z. Kepuladze, G. Tatishvili, Spontaneous Lorentz
violation via QED with non-exact gauge invariance. Eur. Phys.
J. C 55(2), 309–316 (2008)
6. L.P. Colatto, A.L.A. Penna, W.C. Santos, Charged tensor matter
fields and Lorentz symmetry violation via spontaneous symmetry
breaking. Eur. Phys. J. C 36(1), 70–87 (2004)
7. J.J. Snyder, J.L. Hall, A New Measurement of the Relativistic
Doppler Shift, in Laser Spectroscopy, Lecture Notes in Physics,
vol. 43 (Springer, Berlin, 1975)
8. M. Kaivola, O. Poulsen, E. Riis, S.A. Lee, Measurement of the
relativistic Doppler shift in neon. Phys. Rev. Lett. 54(4), 255–258
(1985)
9. R.W. McGowan, D.M. Giltner, S.J. Sternberg, S.A. Lee, New mea-
surement of the relativistic Doppler shift in neon. Phys. Rev. Lett.
70(3), 251–254 (1993)
10. G. Saathoff, S. Karpuk, U. Eisenbarth, G. Huber, S. Krohn, R.M.
Horta, S. Reinhardt, D. Schwalm, A. Wolf, G. Gwinner, Improved
test of time dilation in special relativity. Phys. Rev. Lett. 91(19),
190403 (2003)
346 Eur. Phys. J. C (2010) 69: 343–346
11. C. Novotny, G. Huber, S. Karpuk, S. Reinhardt et al., Sub-Doppler
laser spectroscopy on relativistic beams and tests of Lorentz in-
variance. Phys. Rev. A 80, 022107 (2009)
12. H.E. Ives, G.R. Stilwell, An experimental study of the rate of a
moving atomic clock. J. Opt. Soc. Am. 28(7), 215–219 (1938)
13. M. Mansuripur, Doppler shift, stellar abberation, and convection
of light by moving media. Opt. Photon. News, 52–56 (2003)
14. S.B. Reinhardt, Measurement of time dilation by laser spec-
troscopy on fast stored lithium ions. PhD dissertation, Ruperto-
Carola University of Heidelberg, Germany (2005)
15. D.-Z. Wang, J.-Y. Gao, Effect of Doppler broadening on resonance
fluorescence. Phys. Lett. A 228, 25–28 (1997)


Lorentz violation in high-energy ions

Crossref

Eur. Phys. J. C (2010) 69: 343–346DOI 10.1140/epjc/s10052-010-1437-9LetterLorentz violation in high-energy ionsSantosh DevasiaaU. of Washington, Seattle, WA 98195-2600, USAReceived: 20 June 2010 / Revised: 28 July 2010 / Published online: 15 September 2010© The Author(s) 2010. This article is published with open access at Springerlink.comAbstract The theoretical interest in small Lorentz viola-tions has motivated experiments that investigate it by mea-suring deviations in the time dilation predicted by specialrelativity (SR) using high-energy ions. The main contribu-tion of this article is to show that including the Doppler ef-fect in the emission (which is of the same order as the timedilation effect) in the analysis leads to differences betweenexperimental and theoretical predictions that indicate poten-tial Lorentz violation.1 IntroductionRecent theoretical efforts allow for small Lorentz viola-tions, see, e.g., [1–6]. For example, Doppler shift experi-ments such as the Ives–Stillwell experiment are sensitive toLorentz violating terms in standard model extension [1, 2].This renews the interest in experiments on high-speed ionsthat check for Lorentz violations by experimentally measur-ing potential deviations in the time dilation predicted by spe-cial relativity [7–11].The time dilation predicted by special relativity (SR) hasbeen verified at low speeds with a number of experimentsstarting with the classical experiment using hydrogen canalrays by Ives and Stilwell [12] at speed v = 0.005c where cis the speed of light. More recent experiments [7–11] haveevaluated the time-dilation effect by using Doppler-shiftedlasers to excite transitions in high-energy neon (Ne) andlithium (Li) ions and then observe the emissions at high ionspeeds—as high as v = 0.338c in [11].The main contribution of this article is to show that in-cluding the Doppler effect on the emission from high-energyions (which is of the same order as the time dilation effect) inthe analysis leads to differences between experimental andtheoretical predictions that indicate potential Lorentz viola-tion.a e-mail: devasia@u.washington.edu2 High-energy ion spectroscopy2.1 Doppler effect on laser and emission frequenciesThe main idea is to use Doppler-shifted lasers to excitetransitions in high-energy ions and then observe the result-ing emissions to evaluate time dilation. In particular, anion moving with speed v = βc with respect to a laboratoryframe FL can be excited by using parallel (co-propagating)or anti-parallel (counter-propagating) lasers as in Fig. 1.The relations between associated laser frequencies νp , νa(parallel and anti-parallel to the ion velocity with respectto the laboratory frame FL) and ν∗p , ν∗a (parallel and anti-parallel to the ion velocity with respect to a frame FI at-tached to the moving ion) are given by relativistic Dopplerexpressions as (see, e.g., [10, 13])ν∗p = νpγ (1 − β), (1)ν∗a = νaγ (1 + β), (2)where the time dilation term isγ = 1/√1 − β2. (3)Similarly, the emission frequency ν∗e of photons from theions (with respect to the ion frame FI ) and the detectionfrequency νe perpendicular to the moving ions (with respectto the laboratory frame FL) are related by (see, e.g., [13])ν∗e = νeγ, (4)with angle cos θ = −β in the ion frame FI as shown inFig. 1.2.2 Evaluating potential Lorentz violationThe ratio R, given byR = νpνaν∗pν∗a= 1γ 2(1 − β2) = 1, (5)344 Eur. Phys. J. C (2010) 69: 343–346Fig. 1 Laser frequencies νp , νa in the laboratory frame FL areDoppler shifted to frequencies ν∗p , ν∗a in a frame FI that is movingwith the ion (circle) at speed v = βc. Photons emitted at frequencyν∗e in the ion frame FI are observed at frequency νe in the laboratoryframe FL perpendicular to the moving ionsis independent of the speed of the ions. Potential dependenceof the measured ratio R on speed β is used to evaluate thetime dilation predicted by SR and thereby, to evaluate po-tential Lorentz violation.The frequency terms ν∗p , ν∗a needed to evaluate the ex-pression for R (in (5)) are not directly measurable. This in-ability to measure ν∗p , ν∗a can be avoided if the lasers excitea known transition, say at frequency ν∗e in the ion frame FI .For example, in saturation spectroscopy [10, 11], one of thelaser frequencies is kept constant and the other frequency isvaried to observe the Lamb dip in the fluorescence spectrum,which indicates that both lasers are acting on ions with thesame speed, i.e.,ν∗p = ν∗e and ν∗a = ν∗e .Then, the ratio R (in (5)) can be rewritten asR = νpνa(ν∗e )2= 1γ 2(1 − β2) = 1. (6)2.3 Transition frequency shiftIdeally, the transition frequency ν∗e in the moving ion frameFI should be the same as the transition frequency νo forions that are stationary in the laboratory frame FL, and istherefore known—the transition frequency νo can be deter-mined with high accuracy using stationary ions in the labo-ratory frame FL. However, the transition frequency for themoving ions can get shifted (i.e., ν∗e = νo) due to externalfields and charged particles causing Stark and Zeeman ef-fects, e.g., [11].The potential shift in the transition frequency implies thatthe transition frequency ν∗e excited in the moving ion can-not be assumed to be exactly the same as the transition fre-quency νo measured for stationary ions under different ex-perimental conditions. Moreover, this shifted frequency ν∗eis not directly measurable (in the moving ion frame FI )—although the transition frequency νo of the stationary ions(in laboratory frame FL) is known. Therefore, the ratio R(in (6)) cannot be evaluated directly from the measurements,and an expression in terms of the directly measurable fre-quencies (νa , νp , νo) is sought.2.4 Effect of PMT pre-filtersEmissions from the ions are used to identify when theDoppler-shifted laser frequencies match the transition fre-quencies in the moving ions, e.g., using the Lamb dip in theobserved fluorescence spectrum and to optimize the experi-ment [8–10].In general, measurements of the number of photons emit-ted by the moving ions (to determine excitation of the iontransition) using photomultiplier tubes (PMTs) should be in-dependent of the photon frequency. However, the measure-ments will not be independent of photon frequency if opti-cal pre-filters are used before the PMTs. For example, theemission is observed using an interference filter (before thePMT) centered at the transition frequency νo (of stationaryions) with a narrow (10 nm) halfwidth in [14] to preciselydetect the laser frequency where the Lamb dip occurs. In thiscase, the observed Lamb dip corresponds to emitted photonsof a specific frequency,νe = νo, (7)with respect to the laboratory frame FL, which is relatedto the emitted photon frequency ν∗e in the ion frame FI by(from (4))ν∗e = γ νe = γ νo. (8)Therefore, a theoretical expression Ro that can be evaluatedin terms of measurable frequencies νo, νp , νa (in the labora-tory frame FL) can be found from (6), (8) asRo = νpνa(νo)2= γ2νpνa(ν∗e )2= 11 − β2 ≈ 1 + β2. (9)3 Lorentz violationExperimental observations find this ratio Ro,exp of the prod-uct of the laser frequencies νp,expνa,exp to the transition fre-quency νo,exp of the stationary ion to be a constantRo,exp = νp,expνa,exp(νo,exp)2= 1 (10)that is independent of the speed β [11] where the subscriptexp indicates an experimentally obtained value. This resultsin a difference between theoretical (Ro) and experimental(Ro,exp) predictions of this ratio (from (9), (10))Ro = Ro,exp. (11)Eur. Phys. J. C (2010) 69: 343–346 345This β2-order deviation in Ro using SR time dilation ex-pressions (between (9) and (10)) indicates potential Lorentzviolation in high-energy ions.The difference between the theoretical ratio (Ro in (9))and the experimental ratio (Ro,exp in (10)) should be exper-imentally discernible as the speed β increases and, there-fore, the difference β2 increases. For example, at speedβ = 0.064, the experimental accuracy of Ro,exp (in (10)) isestimated in [10] as|Ro,exp − 1| ≤ 2β2αˆ = 1.8 × 10−9 (12)with αˆ = 2.2 × 10−7 [10] whereas the theoretical ratio (Roin (9)) yields‖Ro − 1‖ ≈ β2 = 0.0041. (13)3.1 Analysis without Doppler shift in emissionIf the Doppler shift in the photon emission is neglected,i.e., emission frequency ν∗e (in the ion frame FI ) is approx-imated by the emission frequency νe = νo (in the laboratoryframe FL), i.e., ν∗e ≈ νe = νo then the constant Ro (in (9))can be approximated asRo = νpνa(νo)2≈ νpνa(ν∗e )2= 1, (14)which would match the experimental ratio Ro,exp.The SR predictions are matched exactly by the experi-mental results only if the transverse Doppler effect in theobserved emissions from the ions are neglected in the analy-sis. However, such an approximation (ν∗e ≈ νe = νo) is notreasonable because the Doppler effect in the emission is ofthe same order (γ ) as the time dilation being measured andis, therefore, not negligible.3.2 Other effectsIt is possible that other effects (such as variations in theobservation angle) might explain or reduce the apparentLorentz violation; further work is needed to investigate sucheffects. For example, previous work has shown that reso-nance fluorescence can be affected by the observational an-gle [15]. Therefore, the experimental results would be af-fected if the PMT is not measuring emissions that are ex-actly perpendicular to the ion beam. Further analysis wouldbe needed to evaluate such angle-deviation effects.Future experiments could include the Doppler shift in theemissions in their design. For example, it might be possibleto place filters, centered at the frequency νo/γ, before thePMT. In this case the emissions measured in the laboratoryframe FL at frequency νo/γ (and the associated Lamb dip)would correspond to νo (a known value) in the ion frame.This would guarantee that there is no shift in the transitionfrequency in the ion frame FI , i.e., emissions measured cor-respond to frequency νo in the ion frame, which is the knownvalue for stationary ions. Such experimental efforts couldclarify and quantify, better, the potential Lorentz violationidentified in this article and its potential effect on systematicerrors reported in other related experiments, e.g., when theions are moving perpendicular to the lasers [8].4 ConclusionsThis article analyzed experiments that investigate Lorentzviolation by measuring deviations in the time dilation pre-dicted by special relativity (SR) using high-energy ions. Itwas shown that including the Doppler effect on the emission(which is of the same order as the time dilation effect) in theanalysis leads to differences between experimental and the-oretical predictions that indicate potential Lorentz violationin high-energy ions.Open Access This article is distributed under the terms of the Cre-ative Commons Attribution Noncommercial License which permitsany noncommercial use, distribution, and reproduction in any medium,provided the original author(s) and source are credited.References1. D. Colladay, V.A. Kostelecky, Lorentz-violating extension of thestandard model. Phys. Rev. D 58(11), 116002 (1998)2. M.E. Tobar, P. Wolf, A. Fowler, J.G. Hartnett, New methods oftesting Lorentz violation in electrodynamics. Phys. Rev. D 71(2),025004 (2005)3. D. Anselmi, Standard model without elementary scalars and highenergy Lorentz violation. Eur. Phys. J. C 65(3), 523–536 (2010)4. K. Nozari, S.D. Sadatian, Late-time acceleration and phantomdivide line crossing with non-minimal coupling and Lorentz-invariance violation. Eur. Phys. J. C 58(3), 499–510 (2008)5. J.L. Chkareuli, Z. Kepuladze, G. Tatishvili, Spontaneous Lorentzviolation via QED with non-exact gauge invariance. Eur. Phys.J. C 55(2), 309–316 (2008)6. L.P. Colatto, A.L.A. Penna, W.C. Santos, Charged tensor matterfields and Lorentz symmetry violation via spontaneous symmetrybreaking. Eur. Phys. J. C 36(1), 70–87 (2004)7. J.J. Snyder, J.L. Hall, A New Measurement of the RelativisticDoppler Shift, in Laser Spectroscopy, Lecture Notes in Physics,vol. 43 (Springer, Berlin, 1975)8. M. Kaivola, O. Poulsen, E. Riis, S.A. Lee, Measurement of therelativistic Doppler shift in neon. Phys. Rev. Lett. 54(4), 255–258(1985)9. R.W. McGowan, D.M. Giltner, S.J. Sternberg, S.A. Lee, New mea-surement of the relativistic Doppler shift in neon. Phys. Rev. Lett.70(3), 251–254 (1993)10. G. Saathoff, S. Karpuk, U. Eisenbarth, G. Huber, S. Krohn, R.M.Horta, S. Reinhardt, D. Schwalm, A. Wolf, G. Gwinner, Improvedtest of time dilation in special relativity. Phys. Rev. Lett. 91(19),190403 (2003)346 Eur. Phys. J. C (2010) 69: 343–34611. C. Novotny, G. Huber, S. Karpuk, S. Reinhardt et al., Sub-Dopplerlaser spectroscopy on relativistic beams and tests of Lorentz in-variance. Phys. Rev. A 80, 022107 (2009)12. H.E. Ives, G.R. Stilwell, An experimental study of the rate of amoving atomic clock. J. Opt. Soc. Am. 28(7), 215–219 (1938)13. M. Mansuripur, Doppler shift, stellar abberation, and convectionof light by moving media. Opt. Photon. News, 52–56 (2003)14. S.B. Reinhardt, Measurement of time dilation by laser spec-troscopy on fast stored lithium ions. PhD dissertation, Ruperto-Carola University of Heidelberg, Germany (2005)15. D.-Z. Wang, J.-Y. Gao, Effect of Doppler broadening on resonancefluorescence. Phys. Lett. A 228, 25–28 (1997)

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