  The observation of the time delay between the soft emission and the
high-energy radiation from cosmological gamma ray bursts can be used as an
important observational test of multi-dimensional physical theories. The main
source of the time delay is the variation of the electromagnetic coupling, due
to dimensional reduction, which induces an energy dependence of the speed of
light. For photons with energies around 1 TeV, the time delay could range from
a few seconds in the case of Kaluza-Klein models to a few days for models with
large extra-dimensions. Based on these results we suggest that the detection of
the 18-GeV photon $ \sim $4500 s after the keV/MeV burst in GRB 940217 provides
a strong evidence for the existence of extra-dimensions. The time delay of
photons, if observed by the next generation of high energy detectors, like, for
example, the SWIFT and GLAST satellite based detectors, or the VERITAS
ground-based TeV gamma-ray instrument, could differentiate between the
different models with extra-dimensions.
