Exhaust velocity and relativity?

[Artlav]

I've been thinking, for a reaction engine with exhaust velocity in relativistic values, what will be the specific impulse?

For example, some kinds of fusion and antimatter thrusters give exhaust velocities around a third of the speed of light, and from classical point of view, DV achieved will be calculated by multiplying the numbers straight.
But in relativistic view the energy needed to accelerate the exhaust to that speed will be more than in classic solution, and therefore the reaction should be stronger -> more DV.

How does it work in that region?

 

[Urwumpe]

Yes, you should have lambda as factor in the equation. A beam core engine does only emit impulse directly, so for particles with existing resting mass, lambda should apply, but possibly on both sides of the equation...