How Is Total Mass Calculated in General Relativity for a Finite Body?

[Garth]

In GR you need to have invariant particle (rest) mass in order to make measurements of mass, length and time. If an atom increases in mass, for example, it will shrink and 'vibrate' more quickly. Rulers will likewise shrink and clocks speed up relative to the original standards of measurement.

Garth

 

[blue_sky]

In GR you need to have invariant particle (rest) mass in order to make measurements of mass, length and time. If an atom increases in mass, for example, it will shrink and 'vibrate' more quickly. Rulers will likewise shrink and clocks speed up relative to the original standards of measurement.

Garth

Actually my thoght is that an integral over a finite 3 volume in full GR doen't make sense because c is finite and there is no rule which say how to relate 2 different point of the 3 volume (the 3 volume integral assume to make the integral at the same time for all the points... but this can't be done in full GR due to the finite speed of light, at least can be done only in particular cases)... so in general doen's make sense the total mass of a body.
What you think about that?

 

[Garth]

What I think is that it is GR that needs to be modified, but that is a personal (and how!) opinion!
There are two issues here for GR, or any physical theory, the first is how to apply the theory's equations consistently, the question you raise above, i.e. 'doing the mathematics', and the second is how to relate the terms and definitions in those equations to physical objects and measurements. In order to have any correlation between the GR theory world and the real world the theory demands that particle (rest) masses have to be invariant, otherwise we would be in an 'Alice Through The Looking Glass' world in which the mathematical terms used would lose, or at least change, their meaning.

If we now consider an extended gravitating body with particle number conservation then in GR the total rest mass of all those particles has to be constant. The question is then how to add on the mass equivalent of all the energy fields present, especially the binding energy of the gravitational field itself. It is here that the going gets tough!
- Garth

 

[pervect]

Actually my thoght is that an integral over a finite 3 volume in full GR doen't make sense because c is finite and there is no rule which say how to relate 2 different point of the 3 volume (the 3 volume integral assume to make the integral at the same time for all the points... but this can't be done in full GR due to the finite speed of light, at least can be done only in particular cases)... so in general doen's make sense the total mass of a body.
What you think about that?

It sounds like you've read the sci.physics.faq on energy in GR :-) Of course there are some important special cases where this issue can be resolved.

 

[Garth]

In relating the total mass of a body to the equations used in GR we insert a mass M into the gravitational potentials of the metric describing the gravitational field around a spherical body, the Schwarzschild solution. But where does this M come from, what value do we attribute to it? It is actually defined by the fitting of the GR solution to the Newtonian, the normalisation of the Robertson parameter alpha to be unity. Hence M is defined to be the mass of the body as calculated from the orbit of a distant satellite (where space-time is 'nearly' flat) using Kepler's laws.

It is also necessary, as you have obviously been thinking, to make the rest of the theory consistent, hence the need to define by different routes the mass of a gravitating body in a consistent way. Its not easy away from that asymptotic null infinity!

Garth

 

[pmb_phy]

I'am startint to think thaf in full GR the integral over a finite volume don't have any phisical meaning; what is describing the full GR world are only the local equations.
So in full GR is a non sense to define the mass of a body with finite dimentions.
Any of you support this?

Not I.

In GR you need to have invariant particle (rest) mass in order to make measurements of mass, length and time.

Sorry Garth but that makes no sense to me.

I know that in SR that's not true at all. E.g. If a charged particle of known charge, q, is moving in a uniform magnetic field (of known strength B) in a plane which is perpendicular to the B-field then the particle will move in a circle. The speed, v, can be measured from observing the position as a function of time. This then determines gamma. The radius, R, of the circle is also measurable quantity. Then measurement of q, B and R thus gives you a measurement of momentum since p = qBr. Since p = mv = gamma*m₀ and since you know p and v you then know m = p/v and this m is not an invariant quantity.

Pete

 

[blue_sky]

It sounds like you've read the sci.physics.faq on energy in GR :-) Of course there are some important special cases where this issue can be resolved.

Nope, but if you can tell me where to look, I would appreciate.

blue

 

[Stingray]

Whether or not you would like to define something in GR with the name "mass," it is not in any way required. GR cares about the stress-energy tensor, and only the stress-energy tensor. And that is certainly a measurable object.

The situation is complicated a bit by singularities, which may be called a form of "topological mass," but then you just specify the metric (almost) everywhere, and you've described your spacetime. Since knowing the metric implies knowing the stress-energy tensor, you can also do this with matter. Of course that's not very pretty, but there are no issues of principle.

 

[blue_sky]

What I think is that it is GR that needs to be modified, but that is a personal (and how!) opinion!
There are two issues here for GR, or any physical theory, the first is how to apply the theory's equations consistently, the question you raise above, i.e. 'doing the mathematics', and the second is how to relate the terms and definitions in those equations to physical objects and measurements. In order to have any correlation between the GR theory world and the real world the theory demands that particle (rest) masses have to be invariant, otherwise we would be in an 'Alice Through The Looking Glass' world in which the mathematical terms used would lose, or at least change, their meaning.

If we now consider an extended gravitating body with particle number conservation then in GR the total rest mass of all those particles has to be constant. The question is then how to add on the mass equivalent of all the energy fields present, especially the binding energy of the gravitational field itself. It is here that the going gets tough!
- Garth

I agree with the exception of the total mass rest. Why we do need a total mass rest? This looks like a concept we are "importing" from classical phisycs. The question is: we do need it?

blue

 

[blue_sky]

Whether or not you would like to define something in GR with the name "mass," it is not in any way required. GR cares about the stress-energy tensor, and only the stress-energy tensor. And that is certainly a measurable object.

The situation is complicated a bit by singularities, which may be called a form of "topological mass," but then you just specify the metric (almost) everywhere, and you've described your spacetime. Since knowing the metric implies knowing the stress-energy tensor, you can also do this with matter. Of course that's not very pretty, but there are no issues of principle.

I tend to agree. Any 1 as a different view?

blue

 

[Garth]

"In GR you need to have invariant particle (rest) mass in order to make measurements of mass, length and time. "

Sorry Garth but that makes no sense to me.

I was referring to remote observations, i.e. extending a metric using units defined here in a laboratory out to the far reaches of the universe.

Weyl’s hypothesis (Weyl, H.: 1918, ‘Gravitation und Electriticitat’ Sitzungsberichte der Preussichen Akad. d. Wissenschaften, English translation, 1923, in: The Principle of Relativity, Dover Publications.) was that a true infinitesimal geometry should recognize only a principle for transferring the magnitude of a vector to an infinitesimally close point, and not throughout the space-time manifold as in GR.

This led to the concept that the space-time manifold M is equipped with a class [g_µν] of conformally equivalent Lorentz metrics g_µν and not a unique metric as in GR. Conformal gravity theories use this insight in conformal transformations, in which one metric transforms into a physically equivalent alternative.

The problem with these theories is the mass of a particle varies with the transformation, together with units of length and time. It is a measurement problem, how do we know the terms in our equations concerning a distant object (mass, length, time) are the same as for a similar object in the laboratory? The adoption of the conservation of energy-momentum in GR provides a solution. Define mass to be constant and rulers will be fixed and clocks regular, we then interpret red shift as recession, the universe is expanding but have to invoke inflation, dark matter and dark energy to make it work.
Perhaps it is otherwise?

 

[pervect]

Nope, but if you can tell me where to look, I would appreciate.

blue

The link is:

Is energy conserved in general relativity