# The problem of the mass of a body

• A
wLw
if we have known the density functionρ(r),and then we can calculate the total mass of a spherical body. M=integral of ρ. Now we will say that body has mass M, but I think it is wrong. according to special relativity, mass is equal to energy, so we can also say that body has total energy M,but i think it neglects the gravitational bind energy, which is negative, so the total energy(mass) of that body is smaller than M. and if your solve the Schwa. metric , there is a parameter (here we call Ms), and we define Ms as the mass(or energy) of central body , and I think it includes the energy of binding energy , so Ms is the total energy(mass )of central body, which is not defined by integral ρ, and maybe you can use integral ρ to calculate the mass(energy) of that, but it must larger than Ms. but i have read many papers and books, they all use that integral ρ to represent the mass of body, like a star How is that?、

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This is A level so you should be able to do the following:
• What is the mass of the sun?
• What is the gravitational binding energy of the sun?
• What is the fractional difference in mass when you consider gravitational binding?

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if we have known the density functionρ(r),and then we can calculate the total mass of a spherical body. M=integral of ρ. Now we will say that body has mass M, but I think it is wrong. according to special relativity, mass is equal to energy, so we can also say that body has total energy M,but i think it neglects the gravitational bind energy, which is negative, so the total energy(mass) of that body is smaller than M. and if your solve the Schwa. metric , there is a parameter (here we call Ms), and we define Ms as the mass(or energy) of central body , and I think it includes the energy of binding energy , so Ms is the total energy(mass )of central body, which is not defined by integral ρ, and maybe you can use integral ρ to calculate the mass(energy) of that, but it must larger than Ms. but i have read many papers and books, they all use that integral ρ to represent the mass of body, like a star How is that?、
If ##\rho## is the mass density then the calculation is correct. If it's a measure of only one aspect of mass, rest mass of the particles perhaps, then the integral will be the total of all the rest masses.

Note that in curved spacetime you will also have to use the correct volume element for your integral.

Mentor
I think it includes the energy of binding energy
In general relativity the concept of mass is not defined for all spacetimes. However, for certain specific classes of spacetimes there are a couple of definitions of mass that are used. One is the Komar mass and the other is the ADM mass. See here for an overview of the issues, limitations, and derivations:

https://en.m.wikipedia.org/wiki/Mass_in_general_relativity