  We review the tests of general relativity that will become possible with
space-based gravitational-wave detectors operating in the ~0.01mHz - 1Hz
low-frequency band. The fundamental aspects of gravitation that can be tested
include the presence of additional gravitational fields other than the metric;
the number and tensorial nature of gravitational-wave polarization states; the
velocity of propagation of gravitational waves; the binding energy and
gravitational-wave radiation of binaries, and therefore the time evolution of
binary inspirals; the strength and shape of the waves emitted from binary
mergers and ringdowns; the true nature of astrophysical black holes; and much
more. The strength of this science alone calls for the swift implementation of
a space-based detector; the remarkable richness of astrophysics, astronomy, and
cosmology in the low-frequency gravitational-wave band make the case even
stronger.
