High School Negative potential energy and negative mass

[DrStupid]

Then switch it off gradually while they move apart. Gravity will then slow them down until they reach a maximum separation distance before they start to approach each other.

Gravity needs to be very strong in the scenaro we are talking about and would therefore slow them down very fast. What makes you sure that the external field can be sitched off gradually without letting the objects moving too fast and emitting gravitational waves?

 

[PAllen]

Then switch it off gradually while they move apart. Gravity will then slow them down until they reach a maximum separation distance before they start to approach each other.

Yep, that's a good way. But you could also just posit it as initial conditions. To use ADM methods in GR, all you need is conditions specified on one Cauchy surface. So it is not actually necessary to answer the question at all.

 

[DrStupid]

all you need is conditions specified on one Cauchy surface.

Is that the case for your initial conditions? Apart from the question for with observer the bodies come to rest for the same time - can you simply define that the space-time is static as well?

 

[PAllen]

Then you need to explain how to reach such a state. Fixing the system with an external field and then releasing it doesn't work, because this field cannot be switched off instantaneous.

No I don't. ADM methods in GR only require specification of conditions on some Cauchy surface. In this case, the most direct precursor history, if you insist, is not physically plausible, but is mathematically consistent in GR: just time reverse the forwrard evolution. This woul describe strong GW incoming from infinity, splitting a BH in two, with BF slowing and incoming GW decreasing until the Cauchy surface is reached. Then, outgoing GW start as the BH approach each other. ADM mass is constant the whole time, but implicitly has different componts due to potential energy, GW, and kinetic energy at different times.

 

[PAllen]

Is that the case for your initial conditions? Apart from the question for with observer the bodies come to rest for the same time - can you simply define that the space-time is static as well?

I don't define that it is static or stationary. I have stated it is not quite a few times already. Initial conditions must include first derivatives of metric quantities, so I simply posit, in some chosen harmonic coordinates, conditions such that the first derivative of separation between the BH is zero on Cauchy surface.

 

[DrStupid]

That sounds like we could get an answer this way but it would be limited to a Cauchy surface that includes your initial conditions. This would be sufficient if the answer is yes, negative total energy is possible. If we get the answer No, negative total energies are not possible with these special conditions we would need to check other conditions as well (e.g. stable systems).

 

[DrStupid]

I simply posit, in some chosen harmonic coordinates, conditions such that the first derivative of separation between the BH is zero on Cauchy surface.

That's what I mean with "static".

 

[PAllen]

That sounds like we could get an answer this way but it would be limited to a Cauchy surface that includes your initial conditions. This would be sufficient if the answer is yes, negative total energy is possible. If we get the answer No, negative total energies are not possible with these special conditions we would need to check other conditions as well (e.g. stable systems).

My claim is any stable system would have a harder time achieving negative total energy because it would have additional positive components, which could then be removed to create an instance of my approach.

 

[PAllen]

That's what I mean with "static".

Oh, static has a very precise meaning in GR, and that is not it. Using it in a loose way is particularly confusing in a GR context.

 

[PAllen]

It occurs to me, that the question of this thread is covered by the Positive Energy Theorem, the best IMO proof of which was provided by Witten. The answer is then, no, you cannot do this without violating the dominant energy condition. At least in classical GR, one does not want to give this one up, because then geodesic motion no longer follows from the EFE, and it has even been shown that violation of the dominant energy condition makes possible to have small body that moves tachyonically. Many people interpret the dominant energy condition as simply saying physics alway looks consistent with SR locally.

 

[PeterDonis]

That sounds like we could get an answer this way but it would be limited to a Cauchy surface that includes your initial conditions.

No, it isn't. Data specified on a Cauchy surface is sufficient to determine the entire spacetime geometry.

That's what I mean with "static".

As PAllen said, that is not the correct definition of "static". The class of spacetimes that have a Cauchy surface includes many spacetimes which are not static, or even stationary.

Hawking & Ellis lays all of this out in detail. It is advanced, but definitely worth reading if you want to understand the most general theorems we have on global properties of spacetimes.

 

[Herman Trivilino]

What makes you sure that the external field can be sitched off gradually without letting the objects moving too fast and emitting gravitational waves?

It doesn't matter what happened in the past to create the condition described. There could have been waves created.

I was addressing your objection that the scenario required instantaneous switching by describing one way in which it didn't.