Do Einstein's Theories of Relativity Contradict Each Other?

[DW]

DW- "One can not be blind to something that can not happen by definition. "
Thank you for proving my point.

I didn't prove your point true. I proved your point incorrect.

Should not science advance by observations/experiments challenging and possibly falsifying previous definitions and theories?

No. One never falsifies a definition. That is irrelevant to falsification of a theory.

Pete "Please explain what "varies cosmologically" means?"
That the mass of an object be a function of cosmological time, such as in Hoyle and Narlikar's conformal relativity theories (and self creation too I might add).

But you do need a standard, which by convention/definition is invariant, to compare it against. It is just a question of identifying the correct standard and the principles by which you think that it is invariant.

Invariant is not synonymous with conserved.

 

[Garth]

DW - The definition may be falsified in the sense that it is proven to be inadequate.

A standard of measurement is invariant; the principle by which it is defined so is a conservation law.

 

[quantumdude]

DW - The definition may be falsified in the sense that it is proven to be inadequate.

It is indeed true that definitions cannot be falsified. But this makes me wonder why DW thinks he has falsified the definition of relativistic mass.

 

[DW]

It is indeed true that definitions cannot be falsified. But this makes me wonder why DW thinks he has falsified the definition of relativistic mass.

I suppose falsified is a bad choice of words. What I clearly stated was that it was in the "sense" that I showed it to be inadiquate. How much more simply can I state that so you needn't wonder what I think?

 

[DW]

DW - The definition may be falsified in the sense that it is proven to be inadequate.

And the relativistic mass definition is inadiquate. In that "sense" the definition is falsified.

...invariant...the principle by which it is defined so is a conservation law.

Since when?

 

[quantumdude]

I suppose falsified is a bad choice of words. What I clearly stated was that it was in the "sense" that I showed it to be inadiquate. How much more simply can I state that so you needn't wonder what I think?

You can clearly state that all you like, but when you say (as you have repeatedly) that such a concept of mass is "blatantly wrong" or "proven incorrect", it sends an uniquivocal message that you believe that such a definition has been falsified in the fullest sense of the word.

 

[pmb_phy]

You can clearly state that all you like, but when you say (as you have repeatedly) that such a concept of mass is "blatantly wrong" or "proven incorrect", it sends an uniquivocal message that you believe that such a definition has been falsified in the fullest sense of the word.

Its quite the opposite in fact. I've already shown why mass = proper mass is inadequate in general, i.e. in more complex situations. Althought I don't recall mentioning it here, at least not recently. Any definition of mass should hold under all circumstances and be meaningful. If it is to be meaningful in special relativity then it should also be measurbable from all frames of references, not just measurable from the rest frame, e.g. you can't do that for a photon. The concept of mass = proper mass does not have that property.

In more complex systems the system itself is spatially extended and that complicates matters. When taking measurements of such a system then one needs to take measurements of parts which are spatially separated "at the same time." But "at the same time" in one frame is not "at the same time" in another frame. This implies that defining mass by adding the 4-momenta of all the particles in the system and then adding then will fail when the system is not closed. E.g. the mass of a rod in space which is cooling or the mass of two particles moving in a cyclotron.

For proof see the sections labeled Invariant Mass of a System of Particles - Non-Closed System and An Incorrect Application of Invariant Mass at
http://www.geocities.com/physics_world/sr/invariant_mass.htm

Tom - This is a good example of where mass = proper mass doesn't work and where a particle physicist would never run into this problem. After all, how many times does a particle physicist consider the mass of a moving rod which is cooling?

An acquantance of mine, who is an expert in GR, told me that this notion was originally spoken of by Pauli in his relativity text. I've just ordered it and will get back as far as what Pauli says about all this.

Pete

 

[Garth]

DW - My use of the word "falsified" when applied to definitions may have been an unwise one written in haste, but the meaning is clear, at least to me! Scientific definitions in the past have been proven inadequate (falsified?) as the result of further experiments and observations. For example, the definition of the aether as being the fluid through which light is transmitted.
If science is to progress then we have to allow the possibility that the definitions and theories that we work with today may be proven inadequate/falsified in the future.
Let us have open minds.

Garth - "A standard of measurement is invariant; the principle by which it is defined so is a conservation law."

Since when?

Sorry if the following is pedantic, but here goes!
1. A principal conservation law on which GR is based is the conservation of energy-momentum.
2. The energy-momentum tensor (Weinberg convention) is conserved with respect to covariant differentiation if no external forces are acting on the system.
3. The EEP states that in an arbitrary gravitational field it is possible to choose a locally inertial coordinate system, the freely falling frame of reference, such that in a sufficiently small region the laws of nature take the same form as in an unaccelerated coordinate system.
4. There are no forces acting on such a sufficiently small system, thus the energy-momentum tensor is conserved.
5. By the principle of general covariance, generally covariant physical equations hold in a general gravitational field.
6. The process of transformation of coordinates from the freely falling frame into a general one can now be subsumed by a transformation of the affine connection. As a consequence Newtonian gravitational forces are "explained" by the curvature of the manifold.
7. The (rest) mass of a particle cannot be assumed or defined in these new coordinates but it has to be calculated from the energy-momentum tensor.
8. You will find this calculation in Weinberg G&C, culminating in equation 9.3.2, and the statement "this may be regarded as the law of conservation of mass" in the P.N.A. [At higher energies, in the P.P.N.A., particles may enter into energetic interactions and their mass not be conserved.]
10. If mass is 'conserved' then we can also say it is 'invariant' in the sense that the value we put on that mass does not vary. [I do not want a spat on the use of the two words invariant/conserved as it is a waste of time.]
11. Therefore the invariance of mass is a direct consequence of the law of the conservation of energy-momentum and the EEP.

However this is not the only approach.
If E = mc^2 is a fundamental relationship then there are two other possibilities:
One is that it is energy that is conserved, not energy-momentum, and the EEP is modified - this is the approach of self creation cosmology. [Note: although the EEP is violated in the theory, experimentally in Eotovos type experiments it would only be violated by one part in 10^-17, three orders of magnitude smaller than present experimental limits].
The other possibility is that it is c that varies - the VSL cosmologies, in which both energy and energy-momentum have to be carefully re-defined. As c is a comparison of length and time, measured by rulers and clocks, then it is the fine structure constant that would vary. [Note: There are a number of observations that claim to detect this, see a paper revised today, http://arxiv.org/abs/gr-qc/0403013, v3 28 Jul 2004.]

As I have said before if you insist on using the definition that mass is invariant then you would be blind to the fact that it might be otherwise.

Let us understand the present position but have minds open to other possibilities.

 

[russ_watters]

DW - "One can not be blind to something that can not happen by definition"
Thank you for proving my point.
Should not science advance by observations/experiments that challenge and possibly falsify previous definitions and theories?

I think you may be missing the difference between a definition and a postulate (assumption). A postulate is an assumption based on evidence and a definition is an entirely human construct with the purpose of being able to verbalize (or mathematize) a theory. A definition is not part of a theory. Ie, Einstein postulated that the speed of light is constant for all observers. He didn't define it that way. An awful lot of people think that it was a definition and he just as easily could have defined it as variable, eliminating uncomfortable realities such as time dilation. No so. Similarly:

Scientific definitions in the past have been proven inadequate (falsified?) as the result of further experiments and observations. For example, the definition of the aether as being the fluid through which light is transmitted.

That's not a definition, that's a postulate. Definitions must be invariant, otherwise two people could be saying the same thing and meaning two different things. Consider how much trouble the words "two," "too," and "to" cause for people learning English.

That all said, the argument on the first page was on a definition - but like Tom said, as long as people are using the same definition, both work fine. Postulates don't work that way (assuming something to be true doesn't make it true). Both parties in that argument know the other definition is functional, they just disagree on which is more often used.

 

[DW]

1. A principal conservation law on which GR is based is the conservation of energy-momentum.

Few people seem to know this, but GR does not generally conserve energy-momentum. It generally conserves an energy parameter and momentum parameters. It is merely that in many cases this corresponds to energy-momentum conservation. See-
http://www.geocities.com/zcphysicsms/chap5.htm#BM5_5

2. The energy-momentum tensor is conserved with respect to covariant differentiation if no external forces are acting on the system.

I think you mean stress-energy tensor. That does correspond to a "local" conservation of energy-momentum, but in fact that it is covariant differentiation does lead to what is globally not a true conservation of energy-momentum because the nonzero affine connections set this apart from an ordinary divergence of zero. See-
http://www.geocities.com/zcphysicsms/chap6.htm#BM78

4. There are no forces acting on such a sufficiently small system, thus the energy-momentum tensor is conserved.

I think you mean a free fall system. It is local free fall frames according to which the affine connections vanish and the stress-energy tensor obeys what's interpereted as a conservation equation.

7. The (rest) mass of a particle cannot be assumed or defined in these new coordinates but it has to be calculated from the energy-momentum tensor.

Since when? "Particles" are not ever calculated from the stress-energy tensor. It is the other way around. The stress-energy tensor is a density tensor which means that it is made of local averages from a virtually continuous distribution of particles. It sounds like you are now actually referring to the energy-momentum four-vector, but then your statements above don't make sense. Anyway, the mass can be calculated from the momentum four-vector for arbitrary gravitation with ease.

10. If mass is 'conserved' then we can also say it is 'invariant' in the sense that the value we put on that mass does not vary. [I do not want a spat on the use of the two words invariant/conserved as it is a waste of time.]

Maybe you should because it does sound like in places you are using the word invariant to mean conserved.

However this is not the only approach.
If E = mc^2 is a fundamental relationship

It is not. You are missing the momentum term.

As I have said before if you insist on the invariance of mass by definition then you would be blind to the fact that it might be otherwise.

And you have been told in responce already why that statement was wrong.

 

[DW]

You can clearly state that all you like, but when you say (as you have repeatedly) that such a concept of mass is "blatantly wrong" or "proven incorrect", it sends an uniquivocal message that you believe that such a definition has been falsified in the fullest sense of the word.

As you subtly admit here, I am not really arguing against a definition at all. I am arguing against a "CONCEPT". It is the "relativistic mass" paradigm that is blatantly wrong as I have already proven.

 

[DW]

Its quite the opposite in fact. I've already shown why mass = proper mass is inadequate in general, i.e. in more complex situations.

No you haven't.

If it is to be meaningful in special relativity then it should also be measurbable from all frames of references, not just measurable from the rest frame, e.g. you can't do that for a photon.

Since when? Oh you must be missrepresenting my definitions again, pretending that I define momentum in terms of four-vector velocity again instead of in terms of a wavelength k-vector that applies to both massless and massive particles as I do.

The concept of mass = proper mass does not have that property.

Yes, unlike mass as invariant, it does. You state here by the term proper that you define it in terms of the "proper" frame for which none exists for a photon.

This implies that defining mass by adding the 4-momenta of all the particles in the system and then adding then will fail when the system is not closed. ...

No it doesn't. It just implies that the sum is not simultaneous. But, one doesn't "define" it that way for a system anyway.

For proof see the sections labeled Invariant Mass of a System of Particles - Non-Closed System and An Incorrect Application of Invariant Mass at
http://www.geocities.com/physics_world/sr/invariant_mass.htm

Your site is not a reference.

 

[Garth]

I think you may be missing the difference between a definition and a postulate (assumption).

as long as people are using the same definition, both work fine. Postulates don't work that way (assuming something to be true doesn't make it true). Both parties in that argument know the other definition is functional, they just disagree on which is more often used.

Thank you for that. I was picking up on the contributions already posted.
There has seemed to have been a general confusion between the two.
Garth

 

[Garth]

For the record, as there does seem to be confusion in this thread:

I use the word "invariant" as an adjective to qualify a quantity that does not vary.

I use the word "an invariant or an invariance" as a noun to mean those quantities (such as the speed of light) or geometric objects (such as a body's energy-momentum) that are not coordinate or frame dependent.

I use the word "conserved" to mean without source or sink, such that a conserved quantity is an invariance in a physical process, or under a transformation of frame of reference.

If the statement "the mass of an object is invariant" is a definition then that creates a convention that is to be used consistently throughout. It does not rule out the use of other definitions of mass that create other conventions, so long as they are used consistently throughout too. An argument in one language is still the same argument if translated into another. However there may be nuances that appear in one language and not the other and likewise different definitions may have different merits.

If the statement "the mass of an object is invariant" is a postulate then it is to be tested and possibly falsified, it is a matter of observation not dogma. Of course that raises the all-important question, "How is it measured?" The answer is inevitably theory dependent and that may bring us back to the statement being a definition again. This is where the definition/postulate confusion has arisen. So long as consistent theories define how such a measurement may be made the statement remains a testable postulate.

The idea that invariances are intimately bound up with conservation laws is not new; in fact it is self-evident. A popular account by a front-line cosmologist may be found in John Barrow's 'The World within the World', "Unchanging properties are called conserved quantities and the statements that these quantities remain unchanged in Nature are called conservation laws." (page 114) He is prepared to consider other possibilities, other conventions than the standard one, such as a varying speed of light VSL theory. Such theories may just be playing with words', rather 'equations', but when they lead to testable, or observed, predictions then they enter the realm of hard science.

I do continue to argue for open minds able and willing to examine and critically discuss these possibilities. It is a way to understand the standard model more deeply and who knows? It might even lead to new physics.

 

[pmb_phy]

For the record, as there does seem to be confusion in this thread:

I use the word "invariant" as an adjective to qualify a quantity that does not vary.

That still doesn't define what you mean by the term. You have to say with what it does not vary with respect to. Does it vary with respect to a general coordinate transformation? Does it vary with respect to time/proper time?

Pete

 

[Garth]

Pete - Thank you, I agree the definiton is not complete, the qualifier is deliberately left unqualified so it can be used in different circumstances. If you look in each use I specify the conditions under which the quantity is invariant, otherwise I have thought it was obvious, if that is not the case then you are correct to ask for further qualification.

The question is always, "How do you measure it?"

In SR & GR the speed of light, proper time and energy-momentum are invariant under boosts and translations. In VSL theories c is not constant under translations and in the Jordan frame of self creation energy-momentum is not invariant under translations of space and time, but energy is, (In its Einstein frame it is as GR). Other concepts, such as G, have to be carefully defined as a consequence in those theories. Note: The Brans Dicke theory (BD) introduces Mach's Principle into GR and varies G but keeps the EEP. In that I think it is inconsistent. To include Mach's Principle is to violate the EEP, which deep down is invariably connected to the principle of no preferred frames. BD is popular now but doesn't seem to work, that is why I have modified it in my work.

Behind the conservation laws of physics lay the principles of least Action. However it has to be realized that the conservation law thus defined depends on the Action chosen and whether it is being applied to general space-time with no preferred frame of reference or to a preferred foliation of it.

Garth

 

[pmb_phy]

The question is always, "How do you measure it?"

It depends on what the object whose mass you wish to measure is. If the body has mass m₀ and you measure its speed then it can be said that you've measured its mass since they are directly related. But it always depends on what it is your measuring. You can't measure the mass of a neutrino in the same way that you measure the mass of a super black hole.

Pete

 

[Garth]

True - there are two parts to the question "How do you measure it?". The first is the method of measurement, the apparatus and the theory used to interpret its results. The second is the nature of the measurement. What are you comparing it with? This is more difficult in an astronomical/cosmological situation than particle physics because 'we are inside and part of the experiment'. Einstein (perhaps you know the reference) wondered whether the (rest) mass of an object should increase with potential energy but dismissed the question by asking, "How would you measure it". That is, "How do you transfer a standard from part of space/time to another?" The increase of mass of an object lifted up a mountain would be undetectable as the standard would also have to be taken from its Paris safe and lifted up the mountain too. It is only by introducing another method of measurement, using a different standard of comparison, that any such increase could be detected, otherwise we would be blind to it.

 

[DW]

It depends on what the object whose mass you wish to measure is. If the body has mass m₀ and you measure its speed then it can be said that you've measured its mass since they are directly related.

What, no "rest"? About time. Now you can stop subscripting with the zero for the same reason, and no mass isn't related to speed.

 

[DW]

True - there are two parts to the question "How do you measure it?". The first is the method of measurement, the apparatus and the theory used to interpret its results. The second is the nature of the measurement. What are you comparing it with? This is more difficult in an astronomical/cosmological situation than particle physics because 'we are inside and part of the experiment'. Einstein (perhaps you know the reference) wondered whether the (rest) mass of an object should increase with potential energy but dismissed the question by asking, "How would you measure it". That is, "How do you transfer a standard from part of space/time to another?" The increase of mass of an object lifted up a mountain would be undetectable as the standard would also have to be taken from its Paris safe and lifted up the mountain too. It is only by introducing another method of measurement, using a different standard of comparison, that any such increase could be detected, otherwise we would be blind to it.

There is no "rest" there. As for the introduction of a potential, I have already answered that.

 

[pmb_phy]

Einstein (perhaps you know the reference) wondered whether the (rest) mass of an object should increase with potential energy ...

I never heard of him wondering that. Where did you hear of this?

Pete

 

[Garth]

I never heard of him wondering that. Where did you hear of this?

I definitely read it in an authoritative work but for the life of me I cannot recall which one. However even if Einstein did not say it, it is still a valid question that has to be addressed.
Garth

 

[Garth]

Let me restate the original problem "Einstein's Inconsistency?" in another way.

In the static spherically symmetric case of the Schwarzschild solution energy is conserved in the frame of reference of the Centre of Mass/Momentum. This is because energy is the time component of four-momentum and the components of the metric do not depend on time when observed in that frame.

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?

The standard answer is into the system energy of the whole gravitational field, but the gravitational field has not changed so how does that work?

As an observer considers the elevated test particle she realizes that "time at the top is passing at a different rate, more quickly, than at the bottom". And, if mass is fundamentally a form of energy because the most fundamental particles are vibrating strings(?), then, as they are vibrating more quickly at the top, the mass is greater than at the bottom, and here is our potential energy!
But I thought rest energy was invariant?

 

[sal]

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?

The standard answer is into the system energy of the whole gravitational field, but the gravitational field has not changed so how does that work?

Not so fast. What gravitational field are you talking about, and where did the energy to raise the particle come from?

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

OTOH, if you used energy from within the system, then some component of the system lost enough mass/energy to balance the energy you pumped into the particle, and the far field won't change.

 

[Garth]

Not so fast. What gravitational field are you talking about, and where did the energy to raise the particle come from?

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

OTOH, if you used energy from within the system, then some component of the system lost enough mass/energy to balance the energy you pumped into the particle, and the far field won't change.

Thank you for those points, I will try to illustrate in more detail!

Take the classical Schwarzschild solution of the gravitational field around a static and spherically symmetric mass, a planet. The solution does not depend on how the mass is distributed, only that it is spherically symmetric and static. The system mass M that enters into the components of the metric tensor is the total energy of the body and its gravitational field.

Admittedly we have to go all the way out to infinity, which might be a practical problem!

To be rigorous, and to keep spherical symmetry, lift the top surface shell of the body of thickness h, at radius r, so that it radially expands as a spherical shell out to a greater radius r' where it is fixed as some roof around the planet. Outside this shell the gravitational field has not changed, the Schwarzschild solution only depends on the mass M being spherically symmetric, its density can be any function of r and the new total energy including the gravitational field is still M.

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.

Case ii. If that energy has come from a source within the system, from an inner shell to keep everything spherically symmetric, then it is true that not only has the “far field” not changed, but also the inner shell has retained its initial total energy (its original and final states are at rest) and the outer shell, the roof, has retained its initial total energy (its original and final states are also at rest). There is no record of the energy being transferred from one shell to the other! GR does not locally conserve energy.

It is a bit like the fraudulent accounting of an investment company when your money is transferred from your account to another's, the total of both accounts remains the same, its just that you can no longer get hold of what is yours – it has been lost to the system.

 

[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.