# Gravitational mass defect, weyl metric

## Main Question or Discussion Point

hello everyone,
following the book of Landau&Lifsitz I managed to understand the Schwarzchild solution.
At the end, it finds this formula for the mass of the spherical body generating the gravitational field:

$$M=\frac{4\pi}{c^2} \int^a_0 \epsilon(r) r^2 dr$$

in which $$\epsilon(r)$$ is the energy density of the spherical body and "a" is its radius.
This gravitational mass is smaller than the one calculated the "easy way", which is:

$$M=\frac{4\pi}{c^2} \int^a_0 \epsilon(r) r^2 \sqrt{\gamma} dr$$

in which $$\gamma$$ is the determinant of the 3-D spatial metric.
This is called "gravitational mass defect".

Can you suggest me some resources to do the same for the cylindrically symmetric case (weyl metric) ?

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Bill_K
Could be wrong, but I doubt that such a formula exists for Weyl metrics.

I really don't know, I'm having huge problems understanding the concept of gravitational mass defect. Is it related to the gravitational binding energy?
From the formulas above, can we deduce that for a spherical body of perfect fluid the inertial mass is different from the gravitational mass? And if the answer is yes, which one is bigger?
About the weyl metric, can you point me to some text which has a careful derivation? (I mean a derivation of the metric, not the mass defect)
Thanks a lot!

bcrowell
Staff Emeritus
Gold Member
From the formulas above, can we deduce that for a spherical body of perfect fluid the inertial mass is different from the gravitational mass?
I don't think so, because wouldn't that disprove the equivalence principle, thereby disproving GR itself? I suppose the equivalence principle isn't really violated unless the defect failed to approach 0 fast enough as M approached 0, but even so...it would certainly be an extremely surprising result (to me).

I really don't know, I'm having huge problems understanding the concept of gravitational mass defect.
Having never come across the concept before, my assumption is that it is not a really important concept in the greater scheme of things, but I could be wrong.

In a broader perspective, GR does not have a uniquely defined scalar mass-energy. However, it does have various definitions of scalar mass-energies (e.g., Komar mass, ADM mass, and Bondi mass) that work in certain specific cases, such as stationary or asymptotically flat spacetimes. It would surprise me if one of these differed from another by some factor 1+k, within their common realm of applicability.

If so, then I guess k would have to depend on M and a. I guess this is possible, since you can form a unitless ratio from M and a.

We know that all the scalar measures of mass are equal for a body like the earth, which has M<<a in geometrized units. We also know that for a black hole, with a=0, there is no unitless quantity that can be formed from its mass, and therefore again it seems that all scalar measures of mass must agree (because if they disagreed, how would their ratio be determined?).

Of course, I could just define a "bcrowell mass" that was the Komar mass multiplied by 7, but that would be silly.

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I don't think so, because wouldn't that disprove the equivalence principle, thereby disproving GR itself? I suppose the equivalence principle isn't really violated unless the defect failed to approach 0 fast enough as M approached 0, but even so...it would certainly be an extremely surprising result (to me).
It was a sort of milestone to me too. Gravitational mass equals inertial mass.
Until two weeks ago, when my professor asked me to review the concept. He pointed me to some papers like this:
http://arxiv.org/abs/gr-qc/0606077
in which the author clearly states that inertial mass is different from gravitational mass.
And, to my surprise, yesterday I found a similar statement in Weinberg's book "gravitation and cosmology", in chapter three he says that gravitational binding energy could contribute to the inertial mass of a sphere, making it smaller than gravitational mass.
I'm reading now about Komar's mass, do you think komar's integral is some way related to those two integrals I posted above?
I never thought this thing could bug me so much.

Bill_K
I wouldn't pay any attention to that archiv paper. His conclusions are simply wrong, plus he doesn't even know how to spell Schwarzschild!

The gravitational mass of an object is determined, naturally enough, by the gravitational field that surrounds it. For a stationary object this is easy to define - it's the coefficient of r-3 in the Riemann tensor, in an asymptotically flat coordinate system. Complications come in only if the object is radiating. The inertial mass is determined by how hard you have to push the object to make it accelerate. No one doubts that in general relativity the two are equal.

The gravitational mass defect of Landau and Lifgarbagez is apparently an attempt to define gravitational potential energy. It's defined not on the vacuum Schwarzschild solution but on the class of interior Schwarzschild solutions. They get it by integrating over the mass density while ignoring any curvature, which is like asking "how much total energy would this thing have if the gravitational field were turned off." Being able to do this unambiguously hinges on the fact that for spherical symmetry the radial coordinate can be uniquely defined, by saying spheres have area 4πr2. So I'd be surprised if it can be generalized.

Thanks for explaining Landau and Lifgarbagez, it's still mysterious why they never mentioned gravitational potential energy in defining their gravitational mass defect (nor they mentioned anything else, actually) but you convinced me.
This is the passage in Steven Weinberg's book which troubles me (chapter 3, page 69-70):

Certainly the experiments of Eötvös and Dicke are not accurate enough to say whether gravitational binding energies affect inertial and gravitational masses in the same way. This question might be settled by studying the motion of a small body in orbit about a large body that is itself in free fall in a gravitational field. For instance, the gravitational binding energy of the earth contributes a fraction -8.4x10^-10 of its total mass, whereas the gravitational binding energy of an artificial satellite contributes a very much smaller fraction of its mass. Thus, if (to take an extreme case) the (negative) gravitational binding energy contributes fully to the inertial mass but not at all to the gravitational mass, then the ratio of gravitational to inertial mass of the satellite would be greater than that for the earth by a fraction 8.4x10^-10 .

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bcrowell
Staff Emeritus
Gold Member
It was a sort of milestone to me too. Gravitational mass equals inertial mass.
Until two weeks ago, when my professor asked me to review the concept. He pointed me to some papers like this:
http://arxiv.org/abs/gr-qc/0606077
in which the author clearly states that inertial mass is different from gravitational mass.
And, to my surprise, yesterday I found a similar statement in Weinberg's book "gravitation and cosmology", in chapter three he says that gravitational binding energy could contribute to the inertial mass of a sphere, making it smaller than gravitational mass.
I'm reading now about Komar's mass, do you think komar's integral is some way related to those two integrals I posted above?
I never thought this thing could bug me so much.
Well, yes and no.

First off, it's infamously difficult to give a perfectly rigorous definition of the equivalence principle. Nobody has ever managed to do it: Sotiriou et al, "Theory of gravitation theories: a no-progress report," http://arxiv.org/abs/0707.2748

In the Weinberg quote, he's not saying that gravitational and inertial mass *are* unequal, he's saying that there's one way in which they *could* be unequal, violating GR, but we might not have been able to detect the violation. This falls in the category of strong-field tests of GR: http://relativity.livingreviews.org/Articles/lrr-2006-3/articlese4.html#x18-380004 [Broken] A body's gravitational binding energy varies as $m^2/r$, where m is its rest mass and r is its radius. That means that the ratio of gravitational binding energy to rest mass varies as m/r. When m/r is on the order of unity, you have a black hole. What year was Weinberg writing in? We're getting more strong-field tests of GR (supermassive black holes, binary pulsars, gravitational wave detectors, ...), so the lack of evidence he refers to may already have been filled in, or may be filled in in the near future. Note that if gravitational and inertial mass are unequal, you get violations of conservation of momentum. So, e.g., if gravitational binding energy didn't contribute to inertial mass but did contribute to gravitational mass, then you'd expect the center of mass of a binary star to accelerate anomalously if one or both of its components was a relativistic object like a black hole or a neutron star.

I think the situation is much more settled, both theoretically and observationally, when it comes to weak-field tests of the equivalence principle. Here is an example of a formulation of the e.p. that has been rigorously proved based on GR plus an energy condition: arxiv.org/abs/gr-qc/0309074v1

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Thanks a lot bcromwell, you have been extremely helpful. I'll read all of the articles you suggested and probably come up with another question, but it's going to take me some time to study the subject.
By the way, Weinberg wrote that book about forty years ago, thing may have changed in the meanwhile as you say.