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A distant quasar would be less massive in its frame of reference than our observations would suggest. Are such highly red-shifted objects corrected for relativistic mass dilation?

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You say:

A distant quasar would be less massive in its frame of reference than our observations would suggest

and this refers to the notorious expression for the relativistic mass:

$$ m = \gamma m_0 $$

The trouble is that relativistic mass is a troublesome concept that causes more problems than it solves. For example, the gravitational field of a fast moving body is exactly the same as a stationary body, so the apparent mass increase doesn't affect gravity.

These days the only mass used by physicists is the rest mass. So the mass quoted for distant astronomical objects is the rest mass, and this is a constant that remains the same for all observers.

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Apart from the fact that the concept of relativistic mass is best avoided, as John Rennie mentioned, it is also a concept of special relativity: it can only be defined in an inertial frame (a Minkowski spacetime) where special relativity is valid. However, the expansion of space is a consequence of general relativity. There is no global inertial frame between us and distant quasars.

Also, the recession of these quasars is not a "true" velocity: they are not travelling through space, instead they are receding because the amount of space between them and us increases. They are (almost) at rest in their own local inertial frames (actually quasars do move within their local inertial frames, as they orbit inside their parent galaxy clusters, but these local velocities are non-relativistic). The expansion of space merely gives us the impression of a recession velocity, which in fact can exceed the speed of light for very distant galaxies.

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