Quantum Gravity : Mathematical Models and Experimental Bounds

18 Claus Lämmerzahl
In the case the particle is charged or possesses a spin, then standard GR
shows that these particles couple to curvature. For a charged particle we have an
interaction term of the form D u u =
e m R(·,u) [10], where D u is the Christoffel
covariant derivative along u and R is the Ricci tensor. A particle with spin experiences
an acceleration D u u = λ C R(u, S)u, whereS is the spin vector of the particle
and R(·, ·) the curvature operator. Therefore, in principle UFF is always violated
for charged particles and particles with spin. However, nevertheless it makes sense
to look for an anomalous coupling of spin and charge to the gravitational field. For
a charged particle one may look, on the Newtonian level, for extra coupling terms
of the form κqU where q is the charge, U the Newtonian potential and κ some
parameter of the dimension 1/(specific charge). This necessarily leads to a charge
dependent gravitational mass and, thus, to a charge dependent Eötvös coefficient
[11]. Until now only one test of the UFF for charged particles has been carried out
[12] with a precision of approx. 10% only.
Motivations for anomalous spin couplings came from the search for the axion,
a candidate for the dark matter in the universe [13] and from general schemes of
violation of Lorentz invariance, e.g. [14]. In these models spin may couple to the
gradient of the gravitational potential or to gravitational fields generated by the
spin of the gravitating body. The first case can easily be tested by weighting
polarized bodies what gave that for polarized matter the UFF is valid up to the
order 10 −8 [15].
Until now, no tests of the UFF with antimatter has been carried through.
However, corresponding experiments are in preparation [16] with an anticipated
accuracy 10 −3 on ground and possibly 10 −5 in space.
Physical system Experiment Method Accuracy
neutral bulk matter Adelberger et al [1] torsion balance 5 · 10 −13
polarized matter Ritter et al [15] weighting 10 −8
charged particles Witteborn & Fairbank [12] time–of–flight 10 −1
Quantum system Chu & Peters [8] atom interferometry 10 −9
antimatter
not yet carried through
3.2. Tests of the universality of the gravitational redshift
For a test of this principle the run of clocks based on different physical principles
has to be compared during their common transport through a gravitational
potential. Clocks that have been used are:
• Light clocks which frequency is defined by standing electromagnetic waves
between two mirrors separated by a length L.
• Atomic clocks based on electronic hyperfine transitions which are characterized
by g(m e /m p )α 2 f(α) whereg and f are some functions, and m e and
m p are the electron and proton mass, respectively, and where α is the fine
structure constant.
• Atomic clocks based on electronic fine structure transitions characterized by
α 2 .