How Is Energy Conserved in General Relativity and Cosmology?

[paweld]

My question concerns the gravitation self-energy of a body. For example a uniform sphere of
mass m and radiuos r has in Newtonian limit gravitation energy equal to:
$$E_g = - \frac{3}{5} \frac{G m^2}{r}$$
Should this energy be included in inertial or gravitational mass (if so probably in both
since they are believed to be equal). How to do it? Then what does m in the above formula
stand for?

 

[bcrowell]

GR doesn't have a conserved scalar energy that can be defined in all spacetimes. In special cases like asymptotically flat spacetimes there is more than one definition. Assuming that you intend this body to be stuck in an asymptotically flat spacetime, then you can, for example, use the ADM mass or the Bondi mass.

If your body has spherical symmetry and is basically alone in an asymptotically flat universe, with no gravitational radiation, then the metric is the Schwarzschild metric, and the m appearing in the S. metric is the same as the ADM and Bondi masses.

Since the ADM and Bondi masses are conserved in an asymptotically flat spacetime, the m of the body after it has formed is equal to the total mass of the universe before you put it together. For example, if there were gravitational waves as an ingredient, then their mass energy contributes to the mix. Since GR becomes Newtonian gravity in the weak-field limit, clearly it has to be reasonable to compute a self-energy of the type you're talking about in the weak-field limit, and expect it to be approximately equal to the Schwarzschild m. But just as clearly this must break down in the strong-field limit, since, e.g., the Newtonian expression is negative and infinite for r=0, but a black hole doesn't have infinite negative mass.

The more general question I think you're getting at is whether GR allows us to find the energy contained in a gravitational field. The general answer to that is no. For example, in a cosmological spacetime, which is not asymptotically flat or static, there is no conserved scalar measure of the energy.

 

[paweld]

Thanks for answer.
So the conclusion is that although (at least theoretically) we can divide inertial mass
into rest energy, electromagnetic energy, weak-interaction energy etc. we cannot
indentive the part coming from gravitation interaction.

 

[bcrowell]

Thanks for answer.
So the conclusion is that although (at least theoretically) we can divide inertial mass
into rest energy, electromagnetic energy, weak-interaction energy etc. we cannot
indentive the part coming from gravitation interaction.

Well, you can in some cases but not others.

 

[pervect]

There are some results in MTW's "Gravitation" on pg 603-604 that illustrate that the mass "before assembly" minus the mass "after assembly" is equal to the Newtonian binding energy in the weak field limit.

To make the problem tractable one imagines assembling a star out of some hypothetical material that doesn't change its density with pressure. This means that during the assembly process, there's no internal work done compressing the pieces being assembled, which otherwise would have to be accounted for, and would make the calculation unclear.

This is not a realistic scenario, in actuality one expects realistic gasses to compress and heat up, resulting in energy transfers that would need to be modeled and would confuse the calculation.

If one assumes the necessary asymptotic flatness (something that doesn't exist in our universe, but exists in the simple Schwarzschild geometry), one has the tools to compute the mass "before assembly", and the mass "after assembly", as the various sorts of masses that Ben Crowell talks about are all defined.

One can compare the difference in mass "before asssembly" and "after assembly" and conclude that it must be "binding energy" since one has taken care that work isn't done on any of the pieces during assembly, due to the fact that their volume doesn't change with pressure.

A different method of computing mass - the Komar mass - provides some additional insight, though it isn't the approach that MTW uses. The Komar mass is defined for any static space time, and when the space-time is also asymptotically flat, it will match up with the ADM and Bondi mases.

In lay terms, the Komar mass formula says that if you slice the space-time up into pieces, and you use some "local" coordinate system in each piece to measure the density and the volume, you get the total mass by integrating

K * [rho(local) + 3*pressure(local)] * dV(local)

where K is the gravitational redshift factor of that slice to infinity - equal to the square root of the metric coefficient g_00.

This setup is easy to understand, not so easy to carry out, because one has to imagine slicing the mass up into pieces first, then adopting different coordinate systems for each piece to perform the integration.

The mathematically tractable formulation of this integral, which has little intuitive significance is the surface integral form:

$$- \frac{1}{8 \pi} \int_S \epsilon_{abcd} \nabla^c \xi^d$$

where $\xi$ is the "timelike Killing vector" associated with any static space-time, and K is defined as $\sqrt \xi^a \xi_a$.

Thus, with the Komar perscription, one basically says that matter deep in a gravity well contributes less to the total mass than matter that's not-so-deep - with the addition that pressure , as well as mass, causes gravity.

BTW - a warning - the Komar formula won't work in any space-time that isn't static or at least stationary - don't attempt to apply it to dynamic systems!

 

[bcrowell]

Thanks, pervect, for the excellent, informative post!

 

[Antiphon]

Cosmologically speaking, if you can't compute the energy in the gravitational field from within GR, how can you know that energy is being conserved in the theory?

In electromagnetics you can integrate the energy in E and B over all of space and definitively compute the field energy. Why can't this be done in GR?

 

[bcrowell]

Cosmologically speaking, if you can't compute the energy in the gravitational field from within GR, how can you know that energy is being conserved in the theory?

You don't. See the FAQ below.

In electromagnetics you can integrate the energy in E and B over all of space and definitively compute the field energy. Why can't this be done in GR?

Because Gauss's theorem fails in curved spacetime for a flux that is a vector rather than a scalar.

FAQ: How does conservation of energy work in general relativity, and how does this apply to cosmology? What is the total mass-energy of the universe?

Conservation of energy doesn't apply to cosmology. General relativity doesn't have a conserved scalar mass-energy that can be defined in all spacetimes.[MTW] There is no standard way to define the total energy of the universe (regardless of whether the universe is spatially finite or infinite). There is not even any standard way to define the total mass-energy of the *observable* universe. There is no standard way to say whether or not mass-energy is conserved during cosmological expansion.

Note the repeated use of the word "standard" above. To amplify further on this point, there is a variety of possible ways to define mass-energy in general relativity. Some of these (Komar mass, ADM mass [Wald, p. 293], Bondi mass [Wald, p. 291]) are valid tensors, while others are things known as "pseudo-tensors" [Berman 1981]. Pseudo-tensors have various undesirable properties, such as coordinate-dependence.[Weiss] The tensorial definitions only apply to spacetimes that have certain special properties, such as asymptotic flatness or stationarity, and cosmological spacetimes don't have those properties. For certain pseudo-tensor definitions of mass-energy, the total energy of a closed universe can be calculated, and is zero.[Berman 2009] This does not mean that "the" energy of the universe is zero, especially since our universe is not closed.

One can also estimate certain quantities such as the sum of the rest masses of all the hydrogen atoms in the observable universe, which is something like 10^54 kg. Such an estimate is not the same thing as the total mass-energy of the observable universe (which can't even be defined). It is not the mass-energy measured by any observer in any particular state of motion, and it is not conserved.

MTW: Misner, Thorne, and Wheeler, Gravitation, 1973. See p. 457.

Berman 1981: M. Berman, unpublished M.Sc. thesis, 1981.

Berman 2009: M. Berman, Int J Theor Phys, http://www.springerlink.com/content/357757q4g88144p0/

Weiss and Baez, "Is Energy Conserved in General Relativity?," http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

Wald, General Relativity, 1984