Gravitational potential energy, a thought experiment

[Nugatory]

If you use Newtonian mechanics, the potential and the kinetic energy will have the same value and the opposite sign.

We do want to be a bit careful not to overstate our case here. We can always add or subtract an arbitrary value from the potential energy, so we can always make the potential energy at any point come out positive, negative, or zero as we please. It just so happens that often (central force problems, two otherwise isolated interacting bodies as in this thread, …) it is very convenient to take the potential energy at infinity to be zero; that is the maximum of potential energy so potential energy at any finite distance will then be negative.

But in other problems we may choose different conventions. If I am standing in my front yard it may be most convenient to take the zero of gravitational potential energy to be ground level, increasing upwards. Now the mass I am holding above my head will have positive potential energy $mgh$ (reflecting the work needed to lift it to height $h$ off the ground) while the mass I dropped into the well will have negative kinetic energy (also $mgh$, but $h$ is negative, the depth of the well below ground level).

Kinetic energy on the other hand will always be non-negative. We can choose a frame in which the kinetic energy of a given object is zero, but we can’t make it come out negative.

 

[Dale]

Best of luck as this will crash eventually.

There is nothing wrong with the science here, no imminent danger of a "crash". It is your expectation that energy is always positive which needs to be adjusted. The ideas of binding energy and mass deficit (or mass defect) are well known, theoretically sound, and experimentally validated.

You asked a question and you received the correct answer from multiple experts. Now it is up to you to learn. Sometimes the result of asking a question is the opportunity to learn something surprising. That is the opportunity you have now.

 

[PeterDonis]

We can always add or subtract an arbitrary value from the potential energy

But we can't change direct observables, and as I have already pointed out, the externally measured mass of the system is a direct observable. So any convention for gravitational potential energy that does not have it go to zero at infinity means, as I pointed out, that there is now an arbitrary constant in the math that ends up dropping out of the analysis as soon as you try to evaluate any direct observable. So the only convention that actually makes physical sense for this scenario is that GPE goes to zero at infinity.

in other problems we may choose different conventions

Agreed, that's why I qualified my response to @Bosko with "in this scenario".

 

[PeroK]

In all honesty, how old is this mess?

It's over 120 posts now.

 

[bigbear73]

So to get on track…

Initial system: Total E: (rest)Mass + Ekin + Ep
Final system: Total E: (rest)Mass + Ekin + Ep has not changed (as long as no Energy left the system).
Ep converted to Ekin

But I stil have a question,
When we talk about the Mass of a galaxy like the Milky way, I get the impression only the (rest)Mass is meant, not the Ekin + Ep part.

But when we talk about the gravitational effects of galaxies we have to consider the Total E. Can someone please explain to me why the Ekin+ Ep part is different from dark Mass/Energy.

 

[PeroK]

So to get on track…

Initial system: Total E: (rest)Mass + Ekin + Ep
Final system: Total E: (rest)Mass + Ekin + Ep has not changed (as long as no Energy left the system).
Ep converted to Ekin

But I stil have a question,
When we talk about the Mass of a galaxy like the Milky way, I get the impression only the (rest)Mass is meant, not the Ekin + Ep part.

But when we talk about the gravitational effects of galaxies we have to consider the Total E. Can someone please explain to me why the Ekin+ Ep part is different from dark Mass/Energy.

It might be better to start a new thread and reference this one if you need to. The mass of the Milky Way is almost entirely the rest mass of the stars and dark matter. The internal KE and GPE are negligible. The stars are generally too far apart and moving too slowly relative to each other for these other quantities to be significant.