Gravitational potential energy, a thought experiment

[Ibix]

they do not treat mass and potential gravitational energy at all and are more concerned with the physicality of the geometry as far as i understood it.

The mass parameter of the exterior geometry does not change, even though the FLRW region starts instantaneously static and collapses into a black hole. Thus the measured mass does not change even as the matter accelerates. This is an analytically tractable example of what we've been telling you.

 

[Lok]

The mass parameter of the exterior geometry does not change, even though the FLRW region starts instantaneously static and collapses into a black hole. Thus the measured mass does not change even as the matter accelerates. This is an analytically tractable example of what we've been telling you.

Yet they do not discuss KE and GPE while falling even though these are present in their simplified model, thus it is mostly a looking at the box form outside and seeing/assuming that as the mass does not change.
Their case is also simplified in as KE of some gas entering the BH has an opposite analogue on the other side of the BH, effectively canceling it and taking it out of the equations.

 

[Lok]

Be aware, this should be 24% less mass, not more.

The process is this: the sun starts out far away. As it falls it gains KE and loses PE. When it is in the low PE high KE state the mass of the system is unchanged. The sun can collide with other objects, breaking apart, and thermalizing its KE. When it is in the low PE high thermal energy state the mass is still unchanged. The resulting hot mass can radiate energy away. Only after the radiation has left the box is the mass decreased. At that point it is a low PE and low thermal energy state. Only then is the mass lower.

There is a limit to how much the mass can drop, but if I recall correctly 24% is definitely possible.

More in comparison to m+M of initial state, as in the astronomically visually observable mass of the 2 body system at rest. Not the gravitationally lensed mass, as my assumption is that includes the 24% extra mass somewhere somehow. Therefore my conundrum.
I am liberal in my assumption KE has mass, as that mass can be created via thermalizing, and I assume it is not created instantaneously at thermalisation as that would violate conservation of mass, a law I try to keep in this case.

 

[Lok]

Not as you've stated the problem, no, because you have stated that the box is isolated. As has already been pointed out, if the box remains isolated the whole time (nothing goes in or out), then its externally measured mass cannot change. That is the only valid solution.

I totally agree via conservation of mass kind of reasoning. But if the Sun can attain a higher mass by transforming GPE into KE which can be transformed into rest mass, where was that mass in the initial state?

 

[PeroK]

The mass parameter of the exterior geometry does not change, even though the FLRW region starts instantaneously static and collapses into a black hole. Thus the measured mass does not change even as the matter accelerates. This is an analytically tractable example of what we've been telling you.

How can the initial geometry be described by a single mass parameter? In the vicinity of the black hole, we will have approximately a Schwarzschild geometry with mass $M$; and, in the vicinity of the star, we will have an approximately Schwarzschild geometry with mass $m$. I would expect the geometry to have two mass parameters. Why wouldn't it?

Asymptotically, I would expect there to be a single parameter of approximately $M + m$. I don't understand whether this is precisely $M + m$ or somehow includes the GPE of the two-body system? This is what I thought the Landau-Lifschitz pseudotensor was doing. Then it would be $M + m + \Delta$.

In any case, after the collision, I expect we have a single Schwarzschild geometry with mass $M + m + \Delta$. For this reason, I wasn't expecting conservation of mass to make sense in the full GR solution - unless we invoke the L-L pseudotensor. I also wasn't expecting "invariant mass" to make sense in a global GR scenario like this.

Finally, I don't understand how we can invoke the Newtonian GPE in a scenario where one body attains relativistic speeds.

 

[PeroK]

I don’t see that this is a GR question.

This response baffles me. See my questions above.

 

[Lok]

Finally, I don't understand how we can invoke the Newtonian GPE in a scenario where one body attains relativistic speeds.

I would not bother that much with relativistic effects as even if the speed is small and said effects are negligeble, there should be GPE to KE mass in the system that adds to the m+M.
I went close to Sag A* in my OP only to show the magnitude of the effect in comparison to a know intuitive quantity of mass like our Sun.

 

[PeroK]

I would not bother that much with relativistic effects as even if the speed is small and said effects are negligeble, there should be GPE to KE mass in the system that adds to the m+M.

You can't mix Newtonian mechanics with $E = mc^2$, which is a fundamentally relativistic concept.

 

[Lok]

You can't mix Newtonian mechanics with $E = mc^2$, which is a fundamentally relativistic concept.

You can calculate the KE via relativistic effects if you wish, would it be much different? As long as it is non-zero, it is a problem.

 

[PeroK]

You can calculate the KE via relativistic effects if you wish, would it be much different? As long as it is non-zero, it is a problem.

That approach makes no sense to me. How are you not going to get a contradiction somewhere?

 

[Lok]

That approach makes no sense to me. How are you not going to get a contradiction somewhere?

IMO this problem is a contradiction already. It either violates conservation of mass in the closed box or points to GPE as actual rest mass somehow.
As usual when a contradiction appears, eventually something new gets learned, even though the possibility of it not being new to anybody other than myself is forever present.

 

[Hill]

conservation of mass

What do you mean by this ^^^^?

 

[Lok]

What do you mean by this ^^^^?

Conservation of mass within the box.
As in the inital state has a mass of m+M observably, and m+M+KE in the final. I assume conservation of mass of the box, which leads to the requirement for the initial state to have an extra rest mass in the form of GPE, or the m+M to change somehow.
And if this is not valid then the box mass changes with time and it is not conserved.

 

[PeroK]

It either violates conservation of mass

Newtonian physics has conservation of mass.

In SR, we have conservation of invariant mass (for a closed system). But not conservation of rest mass.

In GR we have the local conservation of stress-energy. The extent to which this entails the global "conservation of mass" is not clear to me.

 

[Lok]

In SR, we have conservation of invariant mass (for a closed system). But not conservation of rest mass.

I only used Newtonian for the energy calculation, but honestly any value of KE is already problematic, so it does not matter.
I do not know what happens in SR with the mass of the Sun as it speeds up, but as I remember it is not going down to compensate for the extra KE. It usually looks like it incorporates it, which is problematic still.
GR is above me and I cannot state it has a solution to this, and thus here I am asking if someone has an idea if it does.

I strongly encourage anybody to do the most basic calculation and see how much GPE is in a as OP system or galactic if one has available data and modeling software. It is above my current setup.

 

[PeroK]

I only used Newtonian for the energy calculation, but honestly any value of KE is already problematic

We already know that mixing Newtonian gravity and SR is problematic. That's why GR was developed.

 

[Hill]

According to the MTW's Gravitation,

Asymptotic flatness [is] the key to the definability of M and S.

I understand that there is no defined meaning of mass of the system in question as an absolute value. But there

are the attractive possibilities of defining and measuring all three quantities [charge-energy-angular-momentum] in any space that is asymptotically flat. ... Surrounding a region where any dynamics, however complicated, is going on, whenever the geometry is asymptotically flat to some specified degree of precision, then to that degree of precision it makes sense to speak of the total energy-momentum 4-vector of the dynamic region, P, and its total intrinsic angular momentum, S. Parallel transport of either around any closed curve in the flat region brings it back to its starting point unchanged. Moreover, it makes no difference how enormous are the departures from flatness in the dynamic region (black holes, collapsing stars, intense gravitational waves, etc.); far away the curvature will be weak, and the 4-momentum and angular momentum will reveal themselves by their imprints on the spacetime geometry.

 

[Nugatory]

IMO the moving Sun should have more gravitationally measurable mass.

It does not.
One way of seeing this is to ask yourself “moving relative to what?”

Say the sun is moving past me but at rest relative to you (so you consider me to be flying rapidly past a stationary sun while I consider you and the sun to be moving while I am at rest). But we both have to agree on the sun’s gravitational effects - for example, the sun is compressed by its own gravity and we have to agree about the pressure at the core - and this is only possible if the gravity is the same. For an extreme example, consider that if the moving sun did have more gravitationally measurable mass we would have the ridiculous situation where if I’m moving fast enough relative to the sun it would collapse into a black hole for me but not you.

 

[Lok]

It does not.
One way of seeing this is to ask yourself “moving relative to what?”

Say the sun is moving past me but at rest relative to you (so you consider me to be flying rapidly past a stationary sun while I consider you and the sun to be moving while I am at rest). But we both have to agree on the sun’s gravitational effects - for example, the sun is compressed by its own gravity and we have to agree about the pressure at the core - and this is only possible if the gravity is the same. For an extreme example, consider that if the moving sun did have more gravitationally measurable mass we would have the ridiculous situation where if I’m moving fast enough relative to the sun it would collapse into a black hole for me but not you.

Interesting.
Would we agree about the same time-frame in which said gravity acts in the Sun's reference frame?

 

[Nugatory]

Interesting.
Would we agree about the same time-frame in which said gravity acts in the Sun's reference frame?

I have no idea what you’re trying to ask here, but I do know from the timestamps that you spent no more than fifteen minutes thinking before posting, and that’s not enough.

 

[Lok]

I have no idea what you’re trying to ask here, but I do know from the timestamps that you spent no more than fifteen minutes thinking before posting, and that’s not enough.

True, sorry for the hasty reply.
I mean to say that in your reference you would measure a higher total gravity of the moving Sun but also a slower time for clocks in the moving reference frame. And for everyone to experience the same outcomes these 2 effects can cancel out, or should. IMO.

Conversely, if you see time being slower in a Sun moving past you, would you be able to calculate it's gravity from observing an object free falling into it? And if gravity is slower by virtue of relative motion, would the Sun have less gravity acting upon your reference frame?
I hope this made some sense.

 

[Dale]

More in comparison to m+M of initial state, as in the astronomically visually observable mass of the 2 body system at rest

I don’t know how I can be more clear about this. It is less, not more. It starts at M+m and remains M+m until radiation leaves the box. When radiation leaves the box the mass becomes less than M+m. At no point is it ever more.

As in the inital state has a mass of m+M observably, and m+M+KE in the final.

This is false. At no point is the mass ever greater than m+M. After the radiation leaves the box it becomes m+M-E where E is the energy of the radiation that leaves.

I assume it is not created instantaneously

There is no delay. The mass is M+m the entire time until the radiation leaves at which point it is without delay M+m-E.

 

[Lok]

I don’t know how I can be more clear about this. It is less, not more. It starts at M+m and remains M+m until radiation leaves the box. When radiation leaves the box the mass becomes less than M+m. At no point is it ever more.

This is false. At no point is the mass ever greater than m+M. After the radiation leaves the box it becomes m+M-E where E is the energy of the radiation that leaves.

I explicitly avoided radiation leakeage by stopping the problem right before the merger. And ask what the weight of the box is initially versus Sun close to A*.
My gripe is with the system in the final state having a mass of m+M+ a big chunk of Kinetic energy that does something.
And I do concur KE having mass is an assumption that is most evident at an impact, but that would mean mass is generated at impact, which makes just as little sense if trying to conserve the mass in the box.

 

[Dale]

I explicitly avoided radiation leakeage by stopping the problem right before the merger.

Then there is never any change in mass. Not at any time. The only way for the mass in the box to change is for something to leave the box.

that would mean mass is generated at impact

Mass is never generated in this scenario. In other related scenarios, the box will only gain mass if something enters from outside and it will only lose mass if something exits from inside. No internal state change of any kind changes the mass.

I hope that is sufficiently clear to avoid misunderstanding.

 

[Lok]

Then there is never any change in mass. Not at any time. The only way for the mass in the box to change is for something to leave the box.

Mass is never generated in this scenario. In other related scenarios, the box will only gain mass if something enters from outside and it will only lose mass if something exits from inside. No internal state change of any kind changes the mass.

I do understand the mass conservation problem, but this does not explain what happens to m, M and the 24% solar mass equivalent energy in the final state so that it would show up as a total mass of m+M of the initial.

 

[Dale]

this does not explain what happens to m, M and the 24% solar mass equivalent energy in the final state so that it would show up as a total mass of m+M of the initial

Does this explain it for you? If not, please ask about the part of the explanation that is confusing:

The process is this: the sun starts out far away. As it falls it gains KE and loses PE. When it is in the low PE high KE state the mass of the system is unchanged. The sun can collide with other objects, breaking apart, and thermalizing its KE. When it is in the low PE high thermal energy state the mass is still unchanged.

 

[Lok]

Does this explain it for you? If not, please ask about the part of the explanation that is confusing:

Dale's:
So box starts out as m and an unspecified hard wall plus gravity from another source, Sun starts to attain kinetic energy losing PE, thermalizes it by hitting the wall, and creates actual rest mass via heat, fusion and pair production. What is the mass of the final thermalized Sun?

 

[Dale]

creates actual rest mass via heat, fusion and pair production

The mass does not change at any point. Heat, fusion, and pair production do not change the mass inside the box.

What is the mass of the final thermalized Sun?

To determine that you would have to draw a box around the sun and keep track of anything that enters or leaves that box. If nothing enters or leaves then the mass would remain m. But of course in that case there would be no way for the KE to thermalize.

 

[PeroK]

I do understand the mass conservation problem, but this does not explain what happens to m, M and the 24% solar mass equivalent energy in the final state so that it would show up as a total mass of m+M of the initial.

It may be worth reading this:

https://en.wikipedia.org/wiki/Mass_in_general_relativity

I think it says what I was trying to say nearly a hundred posts ago!

 

[PeterDonis]

they do not treat mass and potential gravitational energy at all and are more concerned with the physicality of the geometry as far as i understood it.

That is correct. The scenario they are analyzing is different from the scenario you are asking about. Their paper, as I have already said, does not give any useful information about your scenario.

 

[PeterDonis]

I used the Sun and Sag A* as the initial bodies to underline the massive amount of energy that such a system holds as potential energy.

Again you are missing the key point: in the starting configuration of the system, it holds zero gravitational potential energy. The total mass is just $m + M$; there is no extra "potential energy" term in there.

It is true that as the Sun falls, it gains kinetic energy, and to compensate it loses gravitational potential energy, i.e., as I have said, its gravitational potential energy becomes negative, and gets more negative as it falls. But viewing this as somehow using gravitational potential energy that was "stored" in the initial configuration is not helpful, because it makes you think there is some kind of issue involved with the total mass just being $m + M$. There isn't. Once you properly understand this, the question you think you are asking simply evaporates.

In the OP I used the as in Wiki weirdly formulate value for Gravitational potential energy.
U=-GMm/R, where R is the smallest distance that can be attained by the masses

That's wrong. The formula does not depend on the "smallest distance", it depends on the actual distance. At the start it is zero because the actual distance is effectively infinity. At the end it is a negative value that exactly cancels the Sun's kinetic energy.

I would not state that KE cancels out GPE, but rather one transforms into another.

And your insistence on looking at it this way is why this thread keeps going on--the correct answer is staring you in the face and you are refusing to accept it.

either the extra 24% mass is a real thing, and this leaves me wondering where it is distributed in the initial state.

Which, again, is because you are refusing to accept the correct answer that is staring you in the face: the total mass is just $m + M$, always, and what happens internally can't change it because the system is isolated. There is no "extra mass" anywhere.

Or it is a figment of equations and there should be no difference in outcome whether there is or there isn't KE in the final state.

It is impossible for the Sun to not have KE in the final state. As I have already said, there is only one valid solution to this scenario. Talking as though there were other possibilities is simply wrong. And this error appears to be part of what is confusing you and keeping you from accepting the obvious correct answer.

would those $m+M+E_K+E_P$ energy terms add to the mass of the box?

Yes, because $E_P = - E_K$ so $m + M + E_K + E_P = m + M$. This is the obvious correct answer that you are refusing to accept.

If yes, how is that mass distributed in the initial state as the final state is less of an issue.

In the initial state, $E_K = E_P = 0$. Which is just a special case of $E_P = - E_K$; the latter is always true in this scenario.

KE does have a measurable mass via e.g. heat

Sure, but in this scenario that is irrelevant, because PE exactly cancels out KE and there is no net contribution to the total mass. For KE to appear as "extra mass" in the total mass of the system you have to look at a different scenario in which PE and KE do not exactly cancel. And for such a scenario the total mass would not be $m + M$, even at the start (although even here calling the KE "extra mass" can be misleading, since the total mass will be less than $m + M$). If you want to discuss such scenarios, you can start a new thread to ask about one (for example, you could consider having the Sun in a circular orbit around Sag A at some finite radius).

I am going for dark matter of course

That is because you keep talking as if there were some "extra mass" in this scenario, when there is none.

 

[PeterDonis]

Not the gravitationally lensed mass, as my assumption is that includes the 24% extra mass somewhere somehow.

It doesn't. There is no "24% extra mass" anywhere in the system as measured from outside.

if the Sun can attain a higher mass by transforming GPE into KE which can be transformed into rest mass

If you want to consider the Sun's extra KE as contributing to the total mass of the system, you also must consider the Sun's negative GPE as contributing to the total mass of the system--and the two contributions exactly cancel in this scenario. So there is no "extra mass".

Basically, you are trying to consider the KE as "extra mass" but ignore the negative GPE as a negative contribution to the mass. That's wrong. You have to consider both.

where was that mass in the initial state?

Nowhere; it doesn't exist in this scenario. See above.

IMO this problem is a contradiction already.

It's not. You are just refusing to analyze it correctly. See above.

I strongly encourage anybody to do the most basic calculation

No such calculation is needed to give the obvious correct answer I gave above, and have given already multiple times in this thread.

 

[PeterDonis]

How can the initial geometry be described by a single mass parameter?

Please, please, let's not derail the thread any further with irrelevant confusions about the Oppenheimer-Snyder model. The fact is that that scenario can be described with a constant external mass parameter: this follows from the fact that the scenario is spherically symmetric and asymptotically flat and an obvious application of Birkhoff's theorem. But discussion of that is off topic here; we're having enough trouble with the even simpler scenario the OP has posed.

 

[Vanadium 50]

I strongly encourage anybody to do the most basic calculation and see how much GPE is in a as OP system or galactic if one has available data and modeling software. It is above my current setup.

Are you saying that you cannot do this calculation yourself? Yet you are sure we're wrong? That's a tough, tough position to hold.

Further, the actual numbers are unnecessary. You did disregard ny suggestion to eschew the cute stories, but had you followed it, we could discuss this all symbolically.

I have no idea what you’re trying to ask here, but I do know from the timestamps that you spent no more than fifteen minutes thinking before posting, and that’s not enough.

If you're not thinking about what we are writing, why are we writing it? Skimming and reacting will not teach you anything. You have to spend the time thinking.

I suggest you stop posting for some period of time - a few hours, a day, whatever it takes - and reread the entire thread, thinking about every response carefully. If after the endn of this period, there is something you still do not understand, post as short and concise a question as you can about it, with minimal filler.

 

[Ibix]

we're having enough trouble with the even simpler scenario the OP has posed.

I'm not sure I agree that the OP's scenario is simpler, but will leave it for now. Happy to discuss in another thread.