Calculating Mass-Energy of a System Without Using Pseudotensors

[pmb_phy]

Systems of point masses can be handled if and only if one can assign an energy E and a momentum P to each particle, and if there are no divergences in the field energy.

It just occurred to me that this is generally not true. Given a well-defined value of both E and P it does not always follow that E² - p² = m². If you are speaking of a mathematical point particle then yes, that's true. But if you're speaking of an object so small that for all practical purposes may be though of as a point when compared to all the dimensions of the object then that is not always true.

E.g. consider a small drop of water placed in a uniform E field. The total force on the drop will be zero. The E field will exert a stress inside the drop and thus elongating it slightly and forming an electric dipole, even though the total force on the particle will be zero. If the dipole moment vector is parallel to the field lines then it will have a definite value of E and P but E² - p² = m² will not be valid for this drop of water. However this is a situation in which there is a field present and thus is only part of a larger system.

Pete

 

[pervect]

I forsee the possibility of a longish discussion here - therefore I want to check to make sure that tim_lou feels that his questions have been answered and that we are not "hijacking" the thread out from under him.

Continuing on until some objection is raised...

The problem with your example is that the water drop is not an isolated system. E^2 - p^2 is not an invariant for a non-isolated system, this is just another example. (The "box" of moving particles is another example - one must include the "box" to get a conserved mass.) We expect an invariant E^2 - p^2 only for an isolated system (no external fields, no external pressures).

In a Minkowski space-time, we can take the energy of the system E as the intergal of T⁰⁰, and the momentum p as the intergal of T⁰¹.

E^2 - p^2 then gives us the invariant mass of the system, and using the above formula we can express E and P (and hence m) in terms of the stress-energy tensor of the system.

To anticipate a bit, one might ask - what happens to the pressure terms?

A formula exists for the mass of a static system: (Call it formula #2)

m = \int T^{00 }+ T^{11} + T^{22} + T^{33}

where the intergal is performed over the volume of the system.

When the system is static, the intergal of the pressure terms (Tⁱⁱ ) for an isolated system is zero and they don't contribute to the mass of the system.

Thus the formula for the mass of a static system in a Minkowski space-time will give the same answer as E^2 - p^2 as long as the system is static. This is the same as the intergal of just T⁰⁰

It can be seen that the above formula for the mass of a static system only works when the system is static. When the system is not static, formula #2 will give the wrong answer.

For example, consider a container of hot gas in which the container is allowed to explode by allowing the container fail. The tension in the box disappears when the container fails. The pressure and energy density inside the box do not change instantaneously. Formula #2 suddenly changes its value (because the tension in the box disappears, but the pressure in the box remains). But we know the mass of the system does not change just becuase the container failed.

Formula #2 was derived using the assumption that the system was static, and we can see that it gives nonsensical results when applied to a non-static system. The formula must be applied properly.

The computation using E^2 - p^2 doesn't have this problem. It works before and after the container fails, giving the same answer.

One additional point needs to be made. The assumption that space-time is Minkowskian implicitly removes from consideration the issue of gravitational self-energy.

Formula #2, by the introduction of metric coefficients, can be easily made to handle static space-times with non-Minkowskian metric coefficients, giving it a huge advantage in this application.

It is not so easy to find the energy E and momentum p in terms of the stress-energy tensor, the previous intergals don't work right anymore, i.e in non-flat spacetimes, E is not the intergal of T^00, p is not the intergal of T^01.

Finding E and p for an isolated system (i.e. a system in an asymptotically flat spacetime) is a job for pseudo-tensors, and/or the ADM approach. It's unfortunately not trival. The results are usually expressed by partial derivatives of the metric, but I believe they can be converted into non-unique intergals of the stress energy tensor as well. (I havea references for how to compute it using the derivatives of the metric, but not how to do the last step of the above conversion).

By non-unique I mean that there are several different intergals which yield equivalent results. I think there is more on this on livingreviews, I'll go take a look. Right now I have to leave.

 

[pervect]

Here is a reference for what I call "formula 2"

Wald, "General Relativity", pg 289, eq 11.2.10

The volume-intergal form of the Komar mass is given as:
<br /> 2 \int_{\Sigma} \left( T_{ab} - \frac{1}{2} T g_{ab} \right) n^a \xi^b dV<br />

n^a is a unit future vector. \xi^b is a Killing vector with a normalization condition of unit magnitude at infinity.

If in addition, none of the metric coefficients is a function of time, it can be shown that \xi^b = n^a.

Then the expression above reduces to

<br /> 2 \int_{\Sigma} \left( T_{00} - \frac{1}{2} T g_{00} \right) dV<br />

Now
<br /> T = T^0{}_0 + T^1{}_1 + T^2{}_2 + T^3{}_3 = g^{00} T_{00} + g^{11} T_{11} + g^{22} T_{22} + g^{33} T_{33}<br />

Substituting and re-aranging, we get

<br /> \int_{\Sigma} (2-g^{00} g_{00}) T_{00} - g_{00}g^{11} T_{11} - g_{00} g^{22} T_{22} - g_{00} g^{33} T_{33}<br />

which works in any metric where the g_{ij} are not functions of time, and can be seen to be of the form I described earlier for "formula 2".

Specifically, if g_{00} = -1 and g_{11} = g_{22} = g_{33} = g^{11} = g^{22} = g^{33} = 1 then all of the coefficients turn out to be unity, and one gets the flat-space result I quoted earlier.

As complicated as it looks, this is by far the simplest formula for mass when the metric isn't Minkowskian as long as the necessary conditions are met (i.e. the metric coefficients are not function of time). I don't think this formula is given in MTW though.

The name of the mass calculates is the "Komar mass", and I want to stress again that one must have the metric coefficients be independent of time for the formula to be applicable at all.

The volume-intergal form of the mass for an asymptotically flat space-time in terms of pseudotensors is given in MTW (along with the surface intergal form).

I've personally only ever done the surface intergal form. (But the volume intergal form is given - I've just never calculated it.)

The result is given on pg 462-465. Because the expressions being integrated are not gauge-invariant (as remarked on pg 463), there are many versions of the formulas (as many as there are gauge choices).

However, any gauge choice will still give the same end result for an asymptotically flat space-time.

An equivalent result (I believe) is given in Wald on pg 293 in a form much harder to work with, as the formula for the ADM mass. Thus the above formula is for the ADM mass, which is applicable in any asymptotically flat space-time. Wald, however, does not give the volume intergal form, just the surface intergal form.

The moral of the story:

Sticking with Minkowskian space-times and using the defintion E^2 - p^2 for isolated systems is by far the simplest approach to defining mass. This works fine in SR where space-times are always Minkowskian.

Sources for the next simplest approach, the Komar mass, can be cited, howver understanding the justification for the formulas is difficult. I have not seen any elementary treatments of the topic (but then I could have missed them).

 

[tim_lou]

"therfore I want to check to make sure that tim_lou feels that his questions have been answered."

certainly, my problem is solved and I'm still learning the things that were given. go on with your discussion, these all seem very interesting to me although i have no idea what you guys are talking about. hehe :)

 

[pmb_phy]

pervect - Please post the definition (rather than the equality) of the Komar mass and ADM mass. Thank you.

Pete

 

[pervect]

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)

Thus we are led to the following defintion of the total mass of a static, asymptotically flat space-time which is a vacuum in the exterior region

<br /> M = \frac{1}{8 \pi} \int_S \epsilon_{abcd} \nabla^c \xi^d<br />

Here \xi^d 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:

<br /> 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<br />

where i,j range from 1..3 (i.e. over the spatial dimensions only). h_ij 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.