# When calculating absolute magnitudes of distant objects, do you use the light-travel distance or the comoving distance?

I want to calculate the absolute magnitude of GN-z11. Its light-travel distance is 13.4 billion ly (4.1 Gpc), its comoving distance is 32 billion ly (9.8 Gpc), and its apparent magnitude is $$25.8$$. Calculating its absolute magnitude, we get $$M=-76.86$$ for light-travel, and $$M=-94.09$$ for comoving. Which one do I use? I am assuming it is the former. Is this correct, or is there a different formula for this?

Neither. You use the Luminosity Distance.

$$M = m -5\log D_L + 5$$

This assumes bolometric magnitudes. If you are trying to estimate it in some photometric band then you must also calculate and apply a K-correction that will depend upon the intrinsic spectrum of the source.

For $$z=11.09$$ this cosmology calculator gives a model-dependent luminosity-distance modulus of 50.39 (corresponding to $$D_L=1.2\times 10^{11}$$ pc).

NB: The distances you quote in the question do not give the absolute magnitudes quoted.