Do Einstein's Theories of Relativity Contradict Each Other?

[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.

 

[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.