How Does a Neutron Star Affect the Mass of an Object Falling Into It?

[Ans]

I try to find answer to quite basic question.
Let's imagine neutron star and object with mass of 1 kg located far from the neutron star. Total energy of the object is $E = U_g + mc^2$, for case when its velocity is zero and and $U_g$ is potential energy of gravitation.
The neutron star have such mass and radius, what $U_g =\frac{1}{10} mc^2$
The object failed to the star, energy was dissipated in the star, nothing emitted into space.
How mass of the star will be changed?
Will it be increased by 1 kg or by 1.1kg, from point of view of distant observer?
And how mass of the star will be changed if observer is located at surface of the star?
Is any easy way to find answer from GR equations?

 

[jbriggs444]

Why do you consider the potential energy of gravity to be positive?

 

[Nugatory]

Under the conditions you've specified, with no energy being radiated away so that the star-plus-object can be treated as a single closed system, the total mass of that system will be the same whether the object is floating around outside the star or squashed onto the surface of the star. In principle you could put the whole thing inside a giant opaque box so you couldn't even see what the object was doing, and the mass inside the box would remain constant.

 

[Ans]

Why do you consider the potential energy of gravity to be positive?

Really, forgot minus

 

[Ans]

Under the conditions you've specified, with no energy being radiated away so that the star-plus-object can be treated as a single closed system, the total mass of that system will be the same whether the object is floating around outside the star or squashed onto the surface of the star. In principle you could put the whole thing inside a giant opaque box so you couldn't even see what the object was doing, and the mass inside the box would remain constant.

I think at least from point of view of observer on surface of star mass of star should increase on 1.1 kg.
Energy of impact is equivalent to 0.1 kg. The energy may create, for example, photons, photons may create electron/positron pairs, etc. And that additional particles have non zero mass.
Is something incorrect in it?

 

[Nugatory]

I think at least from point of view of observer on surface of star mass of star should increase on 1.1 kg.

For a carefully chosen definition of what counts as "the mass of the star" before and after the object falls to the surface of the star, yes.