Growing theoretical evidence suggests that the first generation of stars may have been quite massive ( \sim 100 - 300 { \mathrm { M } _ { \odot } } ) .
If they retain their high mass until death , such stars will , after about 3 \mathrm { Myr } , make pair-instability supernovae .
Theoretical models for these explosions have been studied in the literature for about four decades , but very few of these studies have included the effects of rotation and none ever employed a realistic model for neutrino trapping and transport .
Both turn out to be very important , especially for those stars whose cores collapse into black holes ( helium cores above about 140 \mathrm { M } _ { \odot } ) .
We consider the complete evolution of two zero-metallicity stars of 250 and 300 \mathrm { M } _ { \odot } .
Despite their large stellar masses , we argue that the low-metallicities of these stars result in negligible mass-loss .
Evolving the stars with no mass-loss and including angular momentum transport and rotationally induced mixing , these two stars produce helium cores of 130 and 180 \mathrm { M } _ { \odot } respectively .
Products of central helium burning ( e.g .
primary nitrogen ) are mixed into the hydrogen envelope , which can dramatically change the expansion of the envelope , especially in the case of the 300 \mathrm { M } _ { \odot } model .
Explosive oxygen and silicon burning cause the 130 M _ { \scriptscriptstyle \odot } helium core ( 250 \mathrm { M } _ { \odot } star ) to explode , but explosive burning is unable to drive an explosion in the 180 \mathrm { M } _ { \odot } helium core and it collapses to a black hole .
For this star , the calculated angular momentum in the presupernova model is sufficient to delay black hole formation and the star initially forms a 50 M _ { \scriptscriptstyle \odot } , 1000 km core within which neutrinos are trapped .
Although the star does not become dynamically unstable , the calculated growth time of secular rotational instabilities is shorter than the black hole formation time , and such instabilities may develop .
The estimated gravitational wave energy and wave amplitude would then be E _ { GW } \approx { 10 ^ { -3 } } { \mathrm { M } _ { \odot } } c ^ { 2 } and h _ { + } \approx { 10 ^ { -21 } } / d ( Gpc ) , but these estimates are very rough and depend sensitively on the non-linear nature of the instabilities .
After the black hole forms , accretion continues through a disk .
The mass of the disk depends on the adopted viscosity , but may be quite large , up to 30 \mathrm { M } _ { \odot } when the black hole mass is 140 \mathrm { M } _ { \odot } .
The accretion rate through the disk can be as large as 1-10 { \mathrm { M } _ { \odot } } { \mathrm { s } } ^ { -1 } .
Although the disk is far too large and cool to transport energy efficiently to the rotational axis by neutrino annihilation , it has ample potential energy to produce a 10 ^ { 54 } \mathrm { erg } jet driven by magnetic fields .
The interaction of this jet with surrounding circumstellar gas may produce an energetic gamma-ray transient , but given the redshift and time scale , this is probably not a model for typical gamma-ray bursts .