Gravitational potential energy, a thought experiment

[PeterDonis]

Weight is the force that occurs when you place a mass in a gravitational field.
You shouldn't use it because then you would need a third object in whose gravitational field these two (the Sun and Sagittarius A) would be.

This is a valid point; I think everyone commenting in the thread has assumed that the OP actually meant "invariant mass of the system" instead of "weight", but it is good to be precise.

Classical Newtonian mechanics when velocities are small compared to the speed of light and when space-time is approximately flat

This obviously won't apply at the end state of the scenario, where neither of these things are true.

Special relativity when the speed of the Sun is not negligible compared to the speed of light, but the Sun is not close to Sag A (space-time of significant curvature).

This won't tell us anything useful. In fact no dynamical analysis is required at all; the question can be answered purely in terms of energy conservation. As has already been done in the thread.

Relativistic mass is not "real" mass, but only the body's resistance to acceleration (in this case, the gravitational force of Sag A)

This is wrong. "Acceleration" due to gravity is not proper acceleration; it is not felt as a force, and therefore relativistic mass as "inertia" (resistance to an applied force) does not apply.

General relativity.

This is implied by the use of a black hole in the scenario, yes. However, in this particular case the analysis looks exactly the same as a Newtonian analysis using energy conservation.

 

[Bosko]

This obviously won't apply at the end state of the scenario, where neither of these things are true.

Of course you are right but OP is talking about masses, but in fact he is interested in speed, kinetic energy stored as relativistic mass and the energy of the collision that will happen at the end.
I thought he should practice Newtonian mechanics and then move on to more complex relativistic formulas.

This is wrong. "Acceleration" due to gravity is not proper acceleration; it is not felt as a force, and therefore relativistic mass as "inertia" (resistance to an applied force) does not apply.

It is also true, IMO, but you speak from the point of view of GR and I from a static observer in SR relative to Sag A

This is implied by the use of a black hole in the scenario, yes. However, in this particular case the analysis looks exactly the same as a Newtonian analysis using energy conservation.

Of course, but reading the OP, it seems to me that he is still interested in final speed and kinetic energy.

 

[PeterDonis]

kinetic energy stored as relativistic mass

Is, as I have already said, irrelevant to the scenario under discussion (and relativistic mass is an outdated concept anyway, as you will see if you search for relevant PF Insights articles and previous threads).

the energy of the collision that will happen at the end

There is no collision. Sag A is a black hole; the Sun just falls through its horizon.

I thought he should practice Newtonian mechanics and then move on to more complex relativistic formulas.

As I have already said, there is no need to dynamically analyze the situation at all. The analysis that is actually needed looks the same in Newtonian mechanics as it does in GR. You can't do the analysis in SR (see further comments below).

a static observer in SR relative to Sag A

There is no such thing in this scenario since Sag A is a black hole and the spacetime around it is curved, and the Sun does not stay a very large distance away for the entire scenario so you cannot ignore the curvature of the spacetime.

reading the OP, it seems to me that he is still interested in final speed and kinetic energy.

Only because he didn't understand when he posted the OP that the Sun's kinetic energy is exactly canceled by negative gravitational potential energy, so the externally measured mass of the system is constant even though the Sun's kinetic energy changes during the scenario.

 

[Lok]

The energy in your case is already in the system and already accounted for in the mass measurement. So the mass does not increase.

Again, look up Oppenheimer-Snyder collapse for an analytically tractable case.

Nifty little paper with nice heart warming historical notations.
I see where the similarity begins, but 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.

 

[Lok]

Heat is internal energy that is effectively frame independent. That contributes to the rest mass or invariant mass of an object. And, that contributes to its gravitational mass.

The object's KE is entirely frame dependent. There is always a frame of reference in which the centre of mass of the object is at rest. You can't simply take that KE to be gravitating mass. That's why you need to be a lot more precise about "mass-energy equivalence". And, especially, once you go beyond SR into GR and cosmology.

There is an argument to be made that heat is randomly distributed KE vectors in a body.

 

[Lok]

Of course, but reading the OP, it seems to me that he is still interested in final speed and kinetic energy.

Not really interested in final speed and real value of KE as long as it is >0 and can in theory be counted as a gravitationally observable mass.

 

[Lok]

There is no collision. Sag A is a black hole; the Sun just falls through its horizon.

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. It could have been an electron and a star, were the collision can easily produce rest mass, but then I would get quantum arguments of fields holding whatever mass somewhere, I wanted to avoid this and make it cosmological because IMO the amount of mass that comes about is of dark matter proportions.

 

[Lok]

Only because he didn't understand when he posted the OP that the Sun's kinetic energy is exactly canceled by negative gravitational potential energy, so the externally measured mass of the system is constant even though the Sun's kinetic energy changes during the scenario.

I still do not understand this. 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, in my case the radius of Sag A* or close by, and that is the final state, it does not matter what happens after.
Thus I would not state that KE cancels out GPE, but rather one transforms into another.
So either the extra 24% mass is a real thing, and this leaves me wondering where it is distributed in the initial state.
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.

 

[Lok]

I don’t see that this is a GR question. It’s basically about energy conservation and how we choose the zero point of total energy.
SR was dragged in only because OP has to convert energy to mass before they can add it to $m+M$ to work with the total mass/energy of the system. But because $m$ and $M$ are both constant there’s no need to do that; they could just be working with the sum of the potential and kinetic energy. Comparing $m+M+E_K+E_P$ when one or the other energy terms is zero doesn’t show us anything we won’t see just by looking at $E_K+E_P$.

[edit to add - I posted this before it was fully baked, then finished it, which is what the exchange with @Vanadium 50 below was about]

Still would those $m+M+E_K+E_P$ energy terms add to the mass of the box?
If yes, how is that mass distributed in the initial state as the final state is less of an issue.

 

[Lok]

Clear up the concepts:
$m$ - mass of the Sun
$M$ - mass Sag. A
$(t,r,\theta,\phi)$ -spherical coordinates centered at Sag. The coordinates $\theta$ and $\phi$ are constant for the Sun.

I used the Sun Sag A* masses and radii as found on Wiki. But they do not matter as long as they are non zero, assuming initial state distance is greater than the final state.

Relativistic mass is not "real" mass, but only the body's resistance to acceleration (in this case, the gravitational force of Sag A)

I agree up to a point as KE does have a measurable mass via e.g. heat (non uniform energy vectors) can produce rest mass via nuclear or pair production (thus I am a bit liberal with their use interchangeably) so having uniform KE in one specific direction does not seem to me to be different.
IMO the moving Sun should have more gravitationally measurable mass.

I am going for dark matter of course, as that 24% is only the contribution of GPE of the SMBH Sag A* and the total GPE should account for all masses and distances in the Milky way once the Sun ends up in the final state somewhere in the center.

As a side note I find it interesting to calculate this for more mundane celestial bodies.
So Moon Earth system ends up 6.9e-10% (basically a rounding error).
Earth Sun system ends up as 6.36e-10% (less of a basic rounding error).
So if true, this physically shows up only for some pretty extreme cases of BH or galaxies.

 

[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.