The sophomore puck-stopper suffered the injury during Tuesday's practice, head coach Alain Vigneault confirmed. He made 26 saves in a 6-5 victory over the Boston Bruins one night prior.
Philadelphia has recalled netminder Alex Lyon from the AHL's Lehigh Valley Phantoms on an emergency basis. In addition, forward Joel Farabee has been loaned to the Phantoms and winger Chris Stewart has been placed on waivers.
Hart, 21, owns a 15-11-3 record with a 2.61 goals-against average and a .905 save percentage through 29 starts this season.
Backup netminder Brian Elliott will likely carry most of the load in Hart's absence. The veteran has authored an .896 save percentage across 17 starts in 2019-20.
Philadelphia will play just five games over the next 16 days, in large part due to the All-Star break that begins Jan. 23.