Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

On the experimental foundation of Maxwell’s equations 299
2.3. THE GENERAL STRUCTURE OF
MODIFIED DISPERSION RELATIONS
In both cases, that is in Eqs.(4) and (11), the general structure of
dispersion relation for the propagation in one direction reads [10]
where E is the energy of the photon. The parameter depends on the
underlying theory and can be derived to be for string theory [11]
and ~ 4 for loop gravity [8]. The quantum gravity energy scale
is, of course, of the order of the Planck energy From the above
dispersion relation we derive the velocity of light
Therefore, the difference of the velocity of light for high energy photons
and low energy photons, is given by
Exactly this quantity will be measured in, e.g., astrophysical observations,
see below.
3. TESTS OF MAXWELL’S EQUATIONS
In order to make statements about the experimental status of Maxwell’s
equations, one can just look experimentally for violations of predictions
from Maxwell’s equations. A better strategy is first to build up a theoretical
model capable to describe as many deviations form the standard
Maxwell theory as possible and afterwards to analyze the experimental
results and astrophysical observations with this model. Only in the latter
case it is possible to make quantitative statements about the order of
agreement of the Maxwell equations with experiments and observations.
One theoretical frame which provides a complete description of all
tests of Maxwell’s equations and which allows for a violation of all
the distinctive features characteristic for the validity of the standard
Maxwell equation, is given by the generalized Maxwell equations
For the usual Maxwell equations we have and
where is the Minkowski metric. The homogeneous equations are not