We identify SDSS J010657.39 - 100003.3 ( hereafter J0106 - 1000 ) as the shortest period detached binary white dwarf ( WD ) system currently known .
We targeted J0106 - 1000 as part of our radial velocity program to search for companions around known extremely low-mass ( ELM , \sim 0.2M _ { \sun } ) WDs using the 6.5m MMT .
We detect peak-to-peak radial velocity variations of 740 km s ^ { -1 } with an orbital period of 39.1 min .
The mass function and optical photometry rule out a main-sequence star companion .
Follow-up high-speed photometric observations obtained at the McDonald 2.1m telescope reveal ellipsoidal variations from the distorted primary but no eclipses .
This is the first example of a tidally distorted WD .
Modeling the lightcurve , we constrain the inclination angle of the system to be 67 ^ { \circ } \pm 13 ^ { \circ } .
J0106 - 1000 contains a pair of WDs ( 0.17M _ { \sun } primary + 0.43M _ { \sun } invisible secondary ) at a separation of 0.32R _ { \sun } .
The two WDs will merge in 37 Myr and most likely form a core He-burning single subdwarf star .
J0106 - 1000 is the shortest timescale merger system currently known .
The gravitational wave strain from J0106 - 1000 is at the detection limit of the Laser Interferometer Space Antenna ( LISA ) .
However , accurate ephemeris and orbital period measurements may enable LISA to detect J0106 - 1000 above the Galactic background noise .