  The discovery of two neutron stars with gravitational masses $\approx
2~M_\odot$ has placed a strong lower limit on the maximum mass of nonrotating
neutron stars, and with it a strong constraint on the properties of cold matter
beyond nuclear density. Current upper mass limits are much looser. Here we note
that, if most short gamma-ray bursts are produced by the coalescence of two
neutron stars, and if the merger remnant collapses quickly, then the upper mass
limit is constrained tightly. If the rotation of the merger remnant is limited
only by mass-shedding (which seems probable based on numerical studies), then
the maximum gravitational mass of a nonrotating neutron star is $\approx
2-2.2~M_\odot$ if the masses of neutron stars that coalesce to produce
gamma-ray bursts are in the range seen in Galactic double neutron star systems.
These limits would be increased by $\sim 4$% in the probably unrealistic case
that the remnants rotate at $\sim 30$% below mass-shedding, and by $\sim 15$%
in the extreme case that the remnants do not rotate at all. Future coincident
detection of short gamma-ray bursts with gravitational waves will strengthen
these arguments because they will produce tight bounds on the masses of the
components for individual events. If these limits are accurate then a
reasonable fraction of double neutron star mergers might not produce gamma-ray
bursts. In that case, or in the case that many short bursts are produced
instead by the mergers of neutron stars with black holes, the implied rate of
gravitational wave detections will be increased.
