  The search for extra dimensions is a challenging endeavor to probe physics
beyond the Standard Model. The joint detection of gravitational waves (GW) and
electromagnetic (EM) signals from the merging of a binary system of compact
objects like neutron stars (NS), can help constrain the geometry of extra
dimensions beyond our 3+1 spacetime ones. A theoretically well-motivated
possibility is that our observable Universe is a 3+1-dimensional hypersurface,
or brane, embedded in a higher 4+1-dimensional Anti-de Sitter (AdS$_5$)
spacetime, in which gravity is the only force which propagates through the
infinite bulk space, while other forces are confined to the brane. In these
types of brane-world models, GW and EM signals between two points on the brane
would, in general, travel different paths. This would result in a time-lag
between the detection of GW and EM signals emitted simultaneously from the same
source. We consider the recent near-simultaneous detection of the GW event
GW170817 from the LIGO/Virgo collaboration, and its EM counterpart, the short
gamma-ray burst GRB170817A detected by the Fermi Gamma-ray Burst Monitor and
the INTEGRAL Anti-Coincidence Shield spectrometer. Assuming the standard
$\Lambda$-Cold Dark Matter ($\Lambda$CDM) scenario and performing a likelihood
analysis which takes into account astrophysical uncertainties associated to the
measured time-lag, we set an upper limit of $\ell \lesssim 0.535\,$Mpc at
$68\%$ confidence level on the AdS$_5$ radius of curvature $\ell$. Although the
bound is not competitive with current Solar System constraints, it is the first
time that data from a multi-messenger GW-EM measurement is used to constrain
extra-dimensional models. Thus, our work provides a proof-of-principle for the
possibility of using multi-messenger astronomy for probing the geometry of our
space-time.
