Do Einstein's Theories of Relativity Contradict Each Other?

[pmb_phy]

In such a case lift a test particle up from a lower rest position to a higher rest position. According to GR its mass, or rest energy, is invariant, so where does the potential energy go?

It is always incorrect to think of energy has having a location. It is just a convenience at times, e.g. it is a convenience to think of the energy of an EM wave as being where the EM wave is, the stronger the field intensity, the higher the energy density. But that is merely a convenience.

But I thought rest energy was invariant?

Depends on what you're calling "rest energy". If you're referring to the energy of a particle at rest in a G-field then that is different than the proper energy of the particle. It is the proper energy which is invariant. That's why you'll rarely see me use the term "rest energy" or "rest mass".

Pete

 

[Garth]

It is always incorrect to think of energy has having a location. It is just a convenience at times, e.g. it is a convenience to think of the energy of an EM wave as being where the EM wave is, the stronger the field intensity, the higher the energy density. But that is merely a convenience.

True - but we would still like to be able to account for it, especially in exchanges of energy, in order not to be defrauded! See my post above.

Depends on what you're calling "rest energy". If you're referring to the energy of a particle at rest in a G-field then that is different than the proper energy of the particle. It is the proper energy which is invariant. That's why you'll rarely see me use the term "rest energy" or "rest mass".

By rest energy I mean rest mass, the mass as measured in the co-moving frame in which the body is at rest, or just plain mass in four-momentum speak. Is that what you are referring to as "proper energy"?
Garth

 

[sal]

If you added energy from "outside the system" to raise the particle then the far field should certainly change: it should increase.

...

Case i. If the energy required to lift the shell has come from outside the system, by some ‘sky-hook’, then from a co-moving frame of reference outside the system that energy has simply disappeared in the GR equations. GR does not conserve energy.
...

This sounds wrong. I think you are using the wrong value for mass/energy when you integrate over the spherical space to obtain the value for M.

In essence, you have assumed the total mass/energy didn't change when you added energy to raise the outer shell, and then you plugged your assumption into the stress/energy tensor.

Consider this: Let the shell fall back. It will collapse back down onto the inner sphere, and the whole system will warm up as a result of the impact. That heating will certainly contribute to the mass/energy, and will increase the far field you detect. So, what you've got is a system which can change internally, with no interaction with the outside world, and as a result of that change, it can increase the strength of its gravitational field.

That sure sounds wrong to me!

 

[Garth]

That sure sounds wrong to me!

Thank you, that is an excellent example of the problem; it sure sounds wrong to me as well but it is correct according to GR! [See Weinberg Gravitation & Cosmology Chapter 8.2 The Schwarzschild Solution ending with equation 8.2.16.] That is why the theory of Self Creation Cosmology defines mass to include gravitational potential energy. See the thread "Self Creation Cosmology - a new gravitational theory " https://www.physicsforums.com/showthread.php?t=32713 .

In the example above if you let the shell fall back in again then the system would not be static, and to be absolutely rigorous you have to treat the hoisting of the shell roof to also be a violation of the static condition no matter how slowly it is done. Whether that explains the anomaly within GR I don’t know. The input of energy changes the energy-momentum of the system but not its total energy; the gravitational field within the shell is being re-arranged but not that outside the spherically symmetric mass. The total system energy remains at M until a wave-front of radiation from the impact reaches the distant observer.

 

[sal]

Thank you, that is an excellent example of the problem; it sure sounds wrong to me as well but it is correct according to GR! [See Weinberg Gravitation & Cosmology Chapter 8.2 The Schwarzschild Solution ending with equation 8.2.16.] That is why the theory of Self Creation Cosmology defines mass to include gravitational potential energy. See the thread "Self Creation Cosmology - a new gravitational theory " https://www.physicsforums.com/showthread.php?t=32713 .

In the example above if you let the shell fall back in again then the system would not be static...

Static vs dynamic is not the point.

Hoist the shell, and wait a billion years. Measure the field.

Let the shell fall, and wait another billion years (assume the extra heat generated isn't radiated away -- outer shell is a perfect mirror). Now measure the far field.

Propagation delay has nothing to do with it -- conservation of mass/energy is being violated bigtime here, as well as the principle that all energy is to be counted in solving the field equations, not just so-called "mass".

This does not agree with what I've read elsewhere.

I'm not particularly familiar with Weinberg, but I have a copy here. I'm looking at eq. 8.2.16 in Weinberg (p. 182). He says

P^0 = M

but I don't see a definition for "P" -- what's he use uppercase Latin for? I can't help noticing that this is not the object being integrated over -- this is the result of the integral, which tells me nothing of what actually went into it.

Can you find another reference which explicitly states that gravitational potential energy doesn't affect the (far) field? I was certainly under the impression that it did, based on a once-through reading of Schutz.

 

[sal]

In the example above if you let the shell fall back in again then the system would not be static, and to be absolutely rigorous you have to treat the hoisting of the shell roof to also be a violation of the static condition no matter how slowly it is done. Whether that explains the anomaly within GR I don’t know. The input of energy changes the energy-momentum of the system but not its total energy; the gravitational field within the shell is being re-arranged but not that outside the spherically symmetric mass. The total system energy remains at M until a wave-front of radiation from the impact reaches the distant observer.

If I recall correctly, spherical collapse doesn't generate gravitational radiation -- it doesn't affect the far field.

I don't have time to dig this out just now; maybe tomorrow, or maybe somebody else can pursue this...

 

[pmb_phy]

I'm not particularly familiar with Weinberg, but I have a copy here. I'm looking at eq. 8.2.16 in Weinberg (p. 182). He says

P^0 = M

but I don't see a definition for "P" -- what's he use uppercase Latin for?

P⁰ is the time component of the 4-momentum P as measured in the rest frame.

Pete

 

[Garth]

Static vs dynamic is not the point.

Can you find another reference which explicitly states that gravitational potential energy doesn't affect the (far) field? I was certainly under the impression that it did, based on a once-through reading of Schutz.

I agree that static/dynamic is not the point, which is that GR is an "improper energy theorem" (Noether's phrase) that does not in general conserve energy. It conserves energy-momentum instead. However I included those caveats because our discussion was about the Schwarzschild solution which is the strictly static and spherically symmetric case. Your example of the shell infalling is just the reverse of mine; put the energy in and wonder where it goes, take it out and wonder where it has come from! An answer is to say the gravitational field has absorbed/released the energy, to be accurate as I said before the energy-momentum vector has changed rather than the (rest) energy, i.e. mass, of each particle.

For another reference try Misner Thorne and Wheeler, Gravitation, page 655 and following. You can easily see the problem by considering their equation 25.12, the standard spherically symmetric and static line element, given that it is the external solution of the GR field equation for a static and spherically symmetric mass, it says nothing about how that mass is distributed so long as the density is just a function of r. However these references will not say, "gravitational potential energy doesn't affect the (far) field" because in GR there is no such thing as gravitational potential energy in the classical sense of the word, because curvature has replaced the concept of the gravitational force. Although for the brick, or electron sitting on your desk feels the inertial force of the desk supporting it against its 'natural' freely falling state that force does no work because your brick/electron is going nowhere! See my second question on the thread "Self Creation Cosmology - a new gravitational theory."
"According to the EEP a stationary electron on a laboratory bench is accelerating w.r.t. the local Lorentzian freely falling
inertial frame of reference. According to Maxwell's theory of electromagnetism an accelerating electric charge, such as an electron, radiates. So why doesn't it? Or, if it is thought that such an electron actually does radiate, what is the source of such radiated energy? However, note that in the preferred CoM frame of reference the electron is not accelerating."

 

[sal]

P⁰ is the time component of the 4-momentum P as measured in the rest frame.

Pete

Yes, of course, but the 4-momentum of what? What's "P" the 4-momentum of, in this case? Not a single particle, that's for sure.

My question was shallower than you realized, I think

 

[sal]

For another reference try Misner Thorne and Wheeler, Gravitation, page 655 and following. You can easily see the problem by considering their equation 25.12...

That's just the Schwarzschild metric expressed as a line element. In that section, they're not addressing the issue you raised at all. They're assuming a massive, static star, and then considering the effect on a single particle whose mass is ignorably small.

The question you have raised is what happens to the far field of a system of particles if one adds energy in order to "spread them apart" (in 3-space). It is more general than a simple question about the Schwarzschild metric, and in fact the issue you raise is not typically addressed in discussions of stars, planets, and other objects where the Schwarzschild metric is used. Such situations are usually extremely asymmetric: the test particle is much less massive than the object generating the field and the perturbation of the field by the test particle can be ignored.

In fact, discussions of the Schwarzschild metric and non-rotating stars typically include a tacit assumption that you are wrong: If the star collapses, the field outside the star doesn't change. By your assumption, it must change, because the collapse itself doesn't affect the field (you assume!) but the energy released by the collapse must affect the field, so the result is that collapse should result in a net increase in the external field.

Here's an example of the sort of reference I was asking for. Schutz, "first course", page 233, first line (italics are his):

"...the following fundamental fact: spherically symmetric motions do not radiate."

In other words, spherical collapse does not affect the field outside the shell. Yet kinetic energy of some sort must be released in that case, and in the absence of any other effect, that would affect the field! Conclusion: The total energy is the cause of the field, and in spherically symmetric motions within an isolated system, the total energy is conserved. If you add energy from the outside in order to expand a spherical mass, you will affect the gravitational field; otherwise spherical collapse in which all energy remains within the system would have to affect the field, and would therefore have to radiate, and it does not.

If you can find a reference which actually contradicts that conclusion I'll be interested in seeing it. In particular, try to find something about gravitational radiation produced when a star collapses into a black hole -- if you are correct, there should be a wicked big burst of G-radiation. But as I understand the situation, there is in fact no radiation produced by the event.

 

[Garth]

The question you have raised is what happens to the far field of a system of particles if one adds energy in order to "spread them apart" (in 3-space). It is more general than a simple question about the Schwarzschild metric, and in fact the issue you raise is not typically addressed in discussions of stars, planets, and other objects where the Schwarzschild metric is used.

Thank you, we could keep quoting references at each other, but they themselves may be wrong of course.
The question is, does the external field of a static spherically symmetric gravitational field depend on the radial distribution of the mass? I believe it does not as is obvious from the Schwarzschild metric.
Therefore a redistribution of that mass by expanding a shell will not change the external field. I believe it is not me that is saying that but the Schwarzschild solution as the distribution of density is absorbed into the parameter M. Yet such a redistribution will use/generate energy that is 'lost to the system'; GR conserves energy-momentum and not in general energy.