How time dilation affects a star's lifespan and luminosity near a black hole

[neh4pres]

I would like to know, how much a stars life would appear to lengthen relativisticly as observed from earth, if the star was in a close orbit around a supermasive black hole. If the stars lifespan is longer from our perspective than a star of its size should be, would it also appear to be outputting less energy than it actually is in its own reference frame?
If time dilation does not make the star dimmer from a far away reference frame, would that not be a paradox in mass energy conversion? Where a far away observer sees more energy output over the stars life than was actually produced?

The question comes from a twist on the scene in interstellar, where I'm replacing people, or the planet with a star, if that helps to understand the question.

 

[PeterDonis]

I would like to know, how much a stars life would appear to lengthen relativisticly as observed from earth, if the star was in a close orbit around a supermasive black hole.

It depends on the black hole and how close the orbit is. In the movie Interstellar, which you reference, the supermassive black hole is rotating rapidly and the orbit is about as close as it's possible to have an orbit without falling into the hole. Kip Thorne (the physicist who was asked to provide the actual parameters for the scenario in the movie) chose this situation because it gives pretty much the maximum possible time dilation for an object in a stable orbit. That time dilation is the same regardless of what kind of object is orbiting--planet, star, whatever.

If the stars lifespan is longer from our perspective than a star of its size should be, would it also appear to be outputting less energy than it actually is in its own reference frame?

Yes, because the light we see from the star will be strongly redshifted, as compared to the light that would be seen by someone orbiting close to the star and at rest relative to it.