 ## Time dilation experiment at 34%c using laser spectroscopy Table of contents

### 1. Abstract

There are many open questions in the description of nature like how to combine gravity and quantum mechanics or what is dark matter/dark energy. Also CPT-symmetry and Lorentz-Transformation are under investigation if they are correct. In the following an experiment at GSI is presented as a continuation of a former experiment at MPI for nuclear physics which delivers one of the best test of Lorentz violation. At the experimental storage ring ESR lithium ions are accelerated to a velocity of 34% speed of light and laser spectroscopic methods are used to measure the time dilation and compare it with theory of Special Relativity (SR).

### 2. Basics of the experiment

The basic effect of the experiment is the Doppler effect as it is given in Special Relativity. The relation between the frequency \nu_0 in the rest of the moving ion and laser frequency \nu_l in the lab frame is

\nu_0=\nu_l\gamma(1-\beta\cos(\theta)).

\theta is the angle between the laser and ion direction. \beta is the velocity in units of speed of light and \gamma is the time dilation factor. In Special Relativity the time dilation factor is given by

\gamma_{SR}=1/\sqrt{1-\beta^2}.

For \theta=0 or \theta=\pi the cosine is \pm 1. The case \theta=0 is called parallel (signed with p) and \theta=\pi (signed with a) is called antiparallel. Mulitpling these two cases you obtain

\nu_0=\nu_p\gamma(1-\beta)
\nu_0=\nu_a\gamma(1+\beta)
\Rightarrow \nu_0^2=\nu_p\nu_a\gamma^2(1-\beta^2)

If \gamma=\gamma_{SR} it reduces to

\nu_0^2=\nu_p\nu_a.

To verify the last equation only frequencies must be measured and no velocities must be known. Frequencies can be measured to a high accuarcy.

For comparision of different experiments the last relation can be written as (s. Testtheories)

\frac{\nu_p\nu_a}{\nu_0^2}=1+2\alpha\beta^2
.

### 3. Results from earlier storage ring experiments

At the storage ring TSR in Heidelberg laser spectroscopy with ions at 3 percent and 6.4 percent speed of light were carried out. As transition a closed two level system of the metastable triplet and saturation spectroscopy was used. The transition wavelength is 548 nm. At 3% the parallel laser was a solid state laser at 532 nm and as parallel laser a dye laser at 565 nm was used. The laser for 6.4% were a dye laser at 585 nm and an argon ion laser 514 nm. An complete describtion of the experiment is given in the PhD thesis of Sascha Reinhardt. The final results are published in (SaschaReinhardtNaturePhysics2007):

\alpha=(-4.8\pm 8.4)\cdot 10^{-8}

and for the transition frequency \nu_0 (if Special Relativity is ok)

\nu_0=(546\ 466\ 918\ 531\pm 73)\mbox{ kHz.}

### 4. Illustration how the ions are accelerated to 0.34c at GSI

At GSI the lithium ions ^7Li^+ are produced in an ECR ion source and is mainly accelerated by a linear accelerator (UNILAC) and a synchrotron (SIS). The final velocity is 0.338c or a kinetic energy of 409 MeV. The ions are stored in the experimental storage ring ESR and prepared for the laser spectroscopy.

### 5. Laser spectroscopy at ESR

For the laser spectroscopy at ESR the \Lambda-system of the lithium ion is used. Energy level of lithium ion 7Li+

To excite this transition the parallel laser has wavelength of 386 nm and the antiparallel laser has a wavelength of 780 nm. The antiparallel laser can be realized by a laser diode and the parallel laser by a frequency doubled of a titanium sapphire laser. The overlap between the laser beams and the ion beam is done by mechanical apperaturs and the fluorescens is detected by photomultipliers distributed over the interaction region. Setup ESR experiment for beamtime August 2007 Example for a saturation signal with Li ions at 6.4%c

### 6. Determination of alpha and alpha2 parameters from Robertson-Mansouir-Sexl (RMS) test theory

In RMS several test parameters are introduced by expanding a function in velocity. For the experiment the relation between the test parameters and the measured values are:

measured values

• nua=frequency light antiparallel to ion movement
• nup=frequency light parallel to ion movement
• nu1=rest frequency transition 1 of lambda system
• nu2=rest frequency transition 2 of lambda system
• nu0=rest frequency transition of saturation two level system
• beta=velocity of ions

test parameters

• epsilon=nua*nup/(nu1*nu2) (lambda) or epsilon=nua*nup/(nu0^2) (saturation)
• alpha=(epsilon-1)/(2*beta^2)
• alpha2=1/2*((epsilon-1-2*alphal*beta^2)/beta^4-alphal)
(alphal is a value for alpha obtained by another low velocity experiment)

ion velocity in units of speed of light beta=

Frequency parallel laser beam nup=(+/-) MHz

Frequency antiparallel laser beam nua=(+/-) MHz

Rest frequency transition lambda nu1=(+/-) MHz

Rest frequency transition lambda nu2=(+/-) MHz

Rest frequency transition saturation nu0 =(+/-) MHz

alpha obtained from low velocity alphal=(+/-)

SaturationLambda

epsilon=+/-
alpha=+/-
alpha2(alphal)=+/-