  High precision tests of the light speed constancy for all observers as of
empirical basis of the Special Relativity have continuously been among the
goals of advanced experimental studies. Based on the Compton Edge method
proposed by us [1], a constraint on the one-way light speed isotropy and the
Lorentz invariance violation has been obtained at the dedicated GRAAL
experiment at European Synchrotron Radiation Facility (ESRF, Grenoble) [2-5].
Using the GRAAL's data we now get a new constraint on one of key tests of
Special Relativity - the Kennedy-Thorndike experiment [6] - in probing the
light speed invariance with respect to the velocity of the observer
(apparatus). Our analysis takes advantage of GRAAL's setup where two separate
energy scales are involved: first, via the position of the Compton Edge
determining the light speed in the reference frame of incident 6 GeV electrons
within the tagging system, second, in the calorimeter via the 1.27 MeV photons
of the ^22 Na source. The two energy scales are engaged to each other through
production of $\eta$ mesons by tagged laser Compton backscattered
$\gamma$-rays. Accuracy of the calibration and stability of energies reached in
each section enable us to obtain the limit 7 10^-12 for the Kennedy-Thorndike
test, which improves the currently existing limits by three orders of
magnitude.
