Gravitational potential energy, a thought experiment

[PeterDonis]

I'm not sure I agree that the OP's scenario is simpler

It's simpler in that no computation whatever is required to get the correct answer; at most (if you want more than just applying the obvious fact of conservation of energy for an isolated system) you just need to read it off a simple one line formula that has already been posted.

 

[Lok]

It's simpler.

Negative energy. I cannot unsee that one.
Enough! A nice day to you all.

 

[PeterDonis]

Negative energy. I cannot unsee that one.

Negative gravitational potential energy. That has been a useful concept in physics for several centuries now.

 

[Lok]

Negative gravitational potential energy. That has been a useful concept in physics for several centuries now.

Negative gravitational potential energy. That has been a useful concept in physics for several centuries now.

The thing you create negative mass with.
I now understand.
Thank you!

 

[Nugatory]

Negative energy. I cannot unsee that one.
Enough! A nice day to you all.

Negative potential energy is one of the great simplifying concepts in classical physics - for example, you’ll find it very helpful when deriving Kepler’s Laws and solving orbital motion, which physics students will be doing no later than their first year of college. If you are not completely comfortable with the concept you really need to back up and review elementary classical mechanics before you try to take on relativity.

 

[Dale]

Negative energy. I cannot unsee that one.
Enough! A nice day to you all.

See https://en.wikipedia.org/wiki/Gravitational_energy
$$U=-\frac{GMm}{r}$$ Since $G$, $M$, $m$, and $r$ are all positive, $U$ is negative. This is ordinary Newtonian physics.

 

[pervect]

The thing you create negative mass with.
I now understand.
Thank you!

I'm not so sure you do understand - I rather suspect you do not, from your comments. I haven't read the entire thread in detail, as it is quite long, I've only skimmed it.

In Newtonian theory, if you have a planet, and you "dissassemble" it (my own term for lack of anything better) by cutting it into pieces and moving them all very far away from each other, this process requires energy.

This implies that the total energy of an assembled planet is lower than the energy of the disassembled one. This is called the "binding energy".

The total energy of a bound system is lower than the total energy of its parts.

Note that a planet orbiting a sun (or the sun orbiting Sag A) can also be considered a bound system in the same manner.

The gravitational binding energy of a system is the minimum energy which must be added to it in order for the system to cease being in a gravitationally bound state. A gravitationally bound system has a lower (i.e., more negative) gravitational potential energy than the sum of the energies of its parts when these are completely separated—this is what keeps the system aggregated in accordance with the minimum total potential energy principle.

Note particularly the remarks in wiki that the binding energy is negative.

This is standard usage - the potential energy of a bound system is negative because the energy of the bound system is lower than the energy of the unbound system.

As for how things actually work in GR (or even SR), I'd say it's too advanced for this thread.

I will say that SR has a concept of something called the "stress energy tensor", which GR borrows, and uses instead of "mass".

GR does also have several concepts of the mass of a system (which are derived from the stress-energy tensor) such as the "big three", Komar mass, ADM mass, and Bondi mass. The fact that it has three of them rather than one is a foreboding that things here are not simple.

However, other than dropping these names, I fear that a detailed discussion of these issues is beyond the scope of the thread.

 

[Lok]

See https://en.wikipedia.org/wiki/Gravitational_energy
$$U=-\frac{GMm}{r}$$ Since $G$, $M$, $m$, and $r$ are all positive, $U$ is negative. This is ordinary Newtonian physics.

I said that now I understand that negative mass cancels the positive one from KE. Problem solved.
In all honesty, how old is this mess?

 

[Dale]

In all honesty, how old is this mess?

About 300 years.

 

[Lok]

Negative potential energy is one of the great simplifying concepts in classical physics - for example, you’ll find it very helpful when deriving Kepler’s Laws and solving orbital motion, which physics students will be doing no later than their first year of college. If you are not completely comfortable with the concept you really need to back up and review elementary classical mechanics before you try to take on relativity.

About 300 years.

I appreciate the honesty. Best of luck as this will crash eventually.

 

[PeterDonis]

The thing you create negative mass with.

No. The total mass of any system is still positive.

Best of luck as this will crash eventually.

What do you mean by this? If you think the concept of negative gravitational potential energy is somehow going to "crash", I strongly suggest that you think again.

 

[Bosko]

In all honesty, how old is this mess?

1666 was a turning point for England. The outbreak of the Great Plague, the eruption of the Second Dutch War and the devastating Great Fire of London all hit the country in quick succession and with devastating consequences.

Isaac Newton took refuge in the countryside.
It is not known exactly what he was doing under that tree, but they say that an apple fell on his head.
In the following year and a half, the foundations of Newton's theory of gravitation were created.

I like another consequence of the same year 1666 > Iron Maiden - The Number Of The Beast

 

[Bosko]

I said that now I understand that negative mass cancels the positive one from KE.

I recommend that you start the calculations first using Newtonian mechanics.
You calculate the speed of the Sun after some time, the kinetic energy, how long it takes to travel a certain distance...
Interesting questions are:
How long will it take the Sun to reach Sag A?
What will be its speed at that moment?
...
When you clear everything up, you move on to the special theory of relativity and compare the results.

Then you do the same using the general theory of relativity.

 

[Bosko]

I appreciate the honesty. Best of luck as this will crash eventually.

If you use Newtonian mechanics, the potential and the kinetic energy will have the same value and the opposite sign.
You can start from
$$-E_P=\frac{GMm}{r}=\frac{mv^2}{2}=E_K$$
and calculate speed for various distances (r).

Then you can fix this formula a bit because at the beginning $E_P$ is small but not quite zero and the speed ( $v$ ) is zero so $E_P$ is also zero. ( The above formula is valid for falling from infinity )

 

[PeterDonis]

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

This is also true in GR for this scenario.

 

[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.