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I don't have a terse defintion, other than the mathematical formula, though I have a longish description of the Komar mass.
For the Komar mass:
You integrate the force, considered to be applied from a string "at infinity", over the area of a sphere enclosing a mass M (the area of the sphere is measured using local rulers - the force is measured using the "string at infinity").
You find that this number (force*area) is a constant, regardless of the radius of the sphere - much like Gauss's law. An important difference is that this isn't local force * local area, it's "force at infinity" * "local area". This is somewhat of an odd mix, but it's the mix that gives a constant number.
This turns into a formal defintion (Wald)
[tex]
M = \frac{1}{8 \pi} \int_S \epsilon_{abcd} \nabla^c \xi^d
[/tex]
Here [itex]\xi^d[/itex] is a Killing vector field, normalized to have a unit magnitude at infinity.
In the case where the metric coefficients are not functions of time, it can be shown that the Killing field is just a unit, timelike vector.
This can be turned into a volume intergal rather than a surface intergal, which is the form I quoted earlier, by using Einstein's equation.
As far as the ADM mass goes, a derivation is given in appendex E of Wald which I don't follow particularly well, but is based on a Hamiltonian formulation of relativity.
Basically, I just use the resulting formula without fully appreciating where they came from. The formula are the same as those derived via the pseudotensor approach in MTW - i.e. the energy is defined as:
[tex]
E=\frac{1}{16 \pi}\int \left( \frac{\partial h_{ij}}{\partial x^i} - \frac{\partial h_{ii}}{\partial x^j} \right) N^j dA
[/tex]
where i,j range from 1..3 (i.e. over the spatial dimensions only). hij are the metric coefficients, N is a normal vector to the surface S.
Other intergals give the momentum.
You can see that this is in the form of a surface intergal. The pseudotensor approach also allows one to construct an equivalent volume intergal.
Unfortunately, MTW doesn't explictly say that the pseudotensor formula give the ADM energy and momentum, I am inferring this from the fact that the formula are the same.
Note that the ADM mass would be the invariant of the ADM energy-momentum 4-vector. If one choses a coordinate system in which the momentum is zero, the ADM energy is just mc^2.
The Komar mass must always be computed in a coordinate system in which the object is at rest, so it only gives an energy - the momentum will be zero.
For the Komar mass:
You integrate the force, considered to be applied from a string "at infinity", over the area of a sphere enclosing a mass M (the area of the sphere is measured using local rulers - the force is measured using the "string at infinity").
You find that this number (force*area) is a constant, regardless of the radius of the sphere - much like Gauss's law. An important difference is that this isn't local force * local area, it's "force at infinity" * "local area". This is somewhat of an odd mix, but it's the mix that gives a constant number.
This turns into a formal defintion (Wald)
[tex]
M = \frac{1}{8 \pi} \int_S \epsilon_{abcd} \nabla^c \xi^d
[/tex]
Here [itex]\xi^d[/itex] is a Killing vector field, normalized to have a unit magnitude at infinity.
In the case where the metric coefficients are not functions of time, it can be shown that the Killing field is just a unit, timelike vector.
This can be turned into a volume intergal rather than a surface intergal, which is the form I quoted earlier, by using Einstein's equation.
As far as the ADM mass goes, a derivation is given in appendex E of Wald which I don't follow particularly well, but is based on a Hamiltonian formulation of relativity.
Basically, I just use the resulting formula without fully appreciating where they came from. The formula are the same as those derived via the pseudotensor approach in MTW - i.e. the energy is defined as:
[tex]
E=\frac{1}{16 \pi}\int \left( \frac{\partial h_{ij}}{\partial x^i} - \frac{\partial h_{ii}}{\partial x^j} \right) N^j dA
[/tex]
where i,j range from 1..3 (i.e. over the spatial dimensions only). hij are the metric coefficients, N is a normal vector to the surface S.
Other intergals give the momentum.
You can see that this is in the form of a surface intergal. The pseudotensor approach also allows one to construct an equivalent volume intergal.
Unfortunately, MTW doesn't explictly say that the pseudotensor formula give the ADM energy and momentum, I am inferring this from the fact that the formula are the same.
Note that the ADM mass would be the invariant of the ADM energy-momentum 4-vector. If one choses a coordinate system in which the momentum is zero, the ADM energy is just mc^2.
The Komar mass must always be computed in a coordinate system in which the object is at rest, so it only gives an energy - the momentum will be zero.