Does gravitational mass change during collapse?

[nickyrtr]

When a body collapses under gravity, its initial gravitational potential energy is converted to kinetic energy and/or other forms of internal energy. At least that is how it is described in Newtonian gravity (not sure if this question is well-formed in GR, sorry).

Now say that the collapse is rapid enough that the internal, non-gravitational energy is increasing faster than it is radiated away into space. Since GR counts internal, non-gravitational energy as part of the gravitational mass, does this mean that the object's gravitational mass is increasing with time? In other words, would an orbiting test particle feel a stronger central attraction as the body it orbits collapses?

If the gravitational mass does increase during collapse, when does it stop increasing? ever? After it becomes a black hole with an event horizon? I have a feeling I am missing some basic concept because everything I have ever read about GR treats the gravitational mass of a body as constant. Any insight would be appreciated.

 

[John232]

GR isn't used to do particle physics. The gravitational pull of an object that collasped would only increase due to a higher density. The kinetic energy of a free fall is not converted to mass.

 

[bcrowell]

GR does not have a scalar mass that is globally conserved in all spacetimes. E.g., there is no way to define the mass of the entire universe in a cosmological model. However, there are definitions of scalar mass that are globally conserved in special cases. In the special case of an asymptotically flat spacetime, there are measures of mass called the ADM and Bondi masses. I believe they basically amount to the same thing when gravitational radiation is negligible. ADM mass: http://en.wikipedia.org/wiki/ADM_mass Basically the answer to your question is that the external field doesn't change at all.

If gravitational radiation isn't negligible, I think the field will decrease.

 

[pervect]

Defining mass in GR is tricky, because you do have to account for the internal energy. Currently, defintions are known only for special cases. However, there are a couple of related but slightly different concepts of mass in GR that could apply to collapse (and one notion that's excluded already because it requires a static system). Both of them require asymptotically flat space-times.

The Bondi mass of a system undergoing gravitational collapse will decrease as the system radiates energy away in the form of gravity waves. I don't believe it can increase, though I'm not sure where to confirm this half-remembered fact to get 100% assurance.

The ADM mass of a system won't change at all.

 

[nickyrtr]

... The kinetic energy of a free fall is not converted to mass.

This is exactly my question. Why does kinetic energy of a free falling object not count toward its gravitational mass?

Or another way of asking the same question: if you have a volume of non-interacting "dust" at finite temperature, why does the temperature not contribute to its gravitational mass? I thought that T_00 in the stress-energy tensor included all non-gravitational energy, including kinetic.

 

[bcrowell]

This is exactly my question. Why does kinetic energy of a free falling object not count toward its gravitational mass?

It does count. The ADM and Bondi masses include contributions from kinetic energy.

 

[bcrowell]

GR isn't used to do particle physics.

This isn't relevant because nobody is discussing particle physics in this thread.

The gravitational pull of an object that collasped would only increase due to a higher density.

This is incorrect. The OP referred to an "orbiting test particle," so the distance stays the same, and density is irrelevant.

The kinetic energy of a free fall is not converted to mass.

We're not talking about "converting" energy to mass. Energy is *equivalent* to mass.

 

[nickyrtr]

..We're not talking about "converting" energy to mass. Energy is *equivalent* to mass.

So, in your opinion, does the orbiting test particle feel a stronger attraction toward the collapsing body, as it collapses? (assuming the energy lost to radiation is insignificant)

 

[bcrowell]

So, in your opinion, does the orbiting test particle feel a stronger attraction toward the collapsing body, as it collapses? (assuming the energy lost to radiation is insignificant)

No. The ADM/Bondi mass is conserved.

 

[John232]

This isn't relevant because nobody is discussing particle physics in this thread.


This is incorrect. The OP referred to an "orbiting test particle," so the distance stays the same, and density is irrelevant.


We're not talking about "converting" energy to mass. Energy is *equivalent* to mass.

Seriously? How do you deal with "orbiting test particle" without particle physics and he was talking about kinetic energy from free fall... I can jump as many times as I want and not increase the mass of the Earth.

 

[bcrowell]

Seriously? How do you deal with "orbiting test particle" without particle physics

Use of the word "particle" is not the same as talking about particle physics.

I can jump as many times as I want and not increase the mass of the Earth.

Yes, this is correct.

 

[nickyrtr]

No. The ADM/Bondi mass is conserved.

Hmm. The ADM/Bondi mass must be something more complicated than integrating T_00 over space then. Do the other components of T somehow affect the ADM/Bondi mass to cancel out the increase in T_00 as the pieces of the object free-fall toward each other?

 

[bcrowell]

Hmm. The ADM/Bondi mass must be something more complicated than integrating T_00 over space then. Do the other components of T somehow affect the ADM/Bondi mass to cancel out the increase in T_00 as the pieces of the object free-fall toward each other?

The textbook by Wald discusses the Bondi mass in ch. 11. All I know about Bondi mass is some general ideas. Bondi and ADM mass basically just measure how much of a field you get from a certain object when you're far away from it. An easier place to start would be with stationary spacetimes. Then there's a pretty straightforward argument that there's a conserved mass-energy. Hawking and Ellis have a discussion of this on pp. 62-63.

 

[pervect]

There are ways of expressing the Bondi mass as the integral of a coordinate dependent pseudotensor over some volume. (And the pseudotensor isn't unique, there are a number of different ones you could integrate, though the Lanadau Lifschitz pseudotensor is a popular choice).

And - it's not just T_00, obvioiusly.

But the motivation for the approach is more like applying Gauss's law from E&M and using a surface integral to define the mass. I.e. one draws a large sphere around the enclosed mass, using an accelerometer "at rest" on the surface of the sphere to measure the "force of gravity" - and multiply and integrate force*area. Unlike E&M, the resulting number won't be constant, but as the size of the sphere gets larger and larger, the idea is that under the right circumstances it will approach a constant value in the limit of a large sphere.

The non-unique choice of the pseudotensor volume integral is in fact derived from the surface area integral by using Stoke's theorem and Einstein's field equations.

The approach only works for certain special metrics (ones that are asymptotically flat). The above sketch is very non-rigorous, you'd need a textbook to provide something specific and detailed enough for you to actually calculate anything, much less show that it was correct.

 

[nickyrtr]

Thanks all for the responses, and suggestions on topics to study further. Discussion of the components of the stress-energy tensor, other than T_00, brings to mind the role of pressure in GR. Increasing the pressure of a spherically symmetric fluid increases its gravitational mass, if I understand correctly.

So can we perhaps view the motion of a fluid (or of discrete particles) as a kind of negative pressure? In classical fluid physics, a moving fluid has lower pressure than it would if sitting still ... for example a garden hose seems less rigid when the water is flowing than when the faucet is closed off.

In that sense, maybe we could qualitatively say that a body collapsing in free fall gains kinetic energy but loses pressure, keeping the overall gravitational mass constant (as measured by a distant observer in asymptotically flat spacetime).

 

[yuiop]

Since GR counts internal, non-gravitational energy as part of the gravitational mass, does this mean that the object's gravitational mass is increasing with time? In other words, would an orbiting test particle feel a stronger central attraction as the body it orbits collapses?

I don't think anyone has mentioned that Birkhoff's theorem http://en.wikipedia.org/wiki/Birkhoff's_theorem_(relativity ) agrees with what poster's here have concluded, that the external attraction remains constant and would not affect the orbiting test particle. To quote Wikipedia:

"The conclusion that the exterior field must also be stationary is more surprising, and has an interesting consequence. Suppose we have a spherically symmetric star of fixed mass which is experiencing spherical pulsations. Then Birkhoff's theorem says that the exterior geometry must be Schwarzschild; the only effect of the pulsation is to change the location of the stellar surface. This means that a spherically pulsating star cannot emit gravitational waves."

 

[pervect]

If you have a static system, a third sort of mass becomes important - the Komar mass. And for a small enough system, you can mostly think of gravity as being the volume integral of rho+3P. While in general volume integrals don't make sense in GR, when you have a static metric, you create a preferred frame of reference, so it's at least possible to talk about them without throwing away tensor notation as the pseudotensor approach requires.

Actually the rho+3P picture I gave above isn't quite right. It's true when g_00 doesn't vary significantly over the region in question, but you'll additionally have to multiply (inside the integral) by the redshift factor K = sqrt(g_00) for larger static systems to get the Komar mass. You can think of this as accounting for the gravitational self-binding energy, which must be negative.

Wald is a good reference to read more about the topic, but his approach is highly abstract and technical, though he does discuss the analogy with Gauss's law in the earlier sections when he discusses the Komar mass. Later on, when he talks about asymptotic time translation symmetries, he mentions that if you impose the right gauge conditions (which he specifies in more detail) you can use the Komar formula by replacing the actual time translation symmetries present in a static system with the asymptotic ones in the asymptotically flat space-time.

Wald doesn't explain the ADM derivation though, he refers the reader to the original paper and presents the results.

You'll find only a small bit about the topic in MTW, which uses mainly the pseudotensor approach. IT also mentions some of the pitfalls, which we haven't mentioned in the post here. Basically, while one _needs_ to include the gravitational binding energy to have a conserved global energy, it's a silly idea to attempt to actually locate it in any specific spot. The difficulty you run into is that any such attempt to localize gravitational binding energy can not transform properly (i.e. as a tensor). It's failure to transform properly leads one to serious questions as to physical significance of the results - though it's certainly handy for book-keeping purposes.