# How can the overall potential energy be the same in these two examples?

Imagine a universe A, containing only two identical heavy planets situated at an almost infinite distance from each other.

Imagine another universe B, containing only two identical lighter planets situated at an almost infinite distance from each other.

In both cases the gravitational potential energy between the planets are said to be almost zero.

How can that be?

I think there must be a bigger overall potential in universe A.
Don’t you?

Two bigger planets “falling together” trough billions of years, must lead to a bigger fireworks, than two lighter planets “falling together” trough billions of years.

How do we describe this actual difference in the overall potential between universe A and B?

## Answers and Replies

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almost zero? who says that? what does 'almost' mean?

rock.freak667
Homework Helper
Well...one is bigger. You just approximate both as zero.

Example one potential could be 0.000000000000001 J/kg
and the other could be 0.0.0000000000000000001 J/kg.

Yes both are approx. zero yet one is bigger than the other.

Yes. And as times goes by this small little difference grows, and turns into a huge difference.

If I were a salesman of universes, I would take different prices for universes with different overall energies.

How do I sell universe A and B, what do I tell my customers?

“Well universe A is more expensive, because after billions of years the collision of the heavier planets will lead into much bigger fireworks.
I know the declaration on the package says, that there is only a very small little difference in potential energy between universe A and B. A difference that is 0,0000000000000000001 J/kg compared to 0,000000000000001 J/kg. But for some reason the factory decided, that people should calculate this overall potential themselves, by using the law of potential energy.”

That seems strange to me to give that kind of an explanation, because collisions between masses in universes must be a very common thing. So there must be a better and more simple way of describing the final difference in energy between universe A and B.

How do I declare the packages in a better way;-)

russ_watters
Mentor
I think you are missing something here: gravitational potential energy increases with distance.

For a small object a huge distance from earth, the GPE will approach a maximum of 61,605,000 j/kg.

Regardless, even if you were right about it being "almost zero", the other guys are right: "almost" isn't good enough. If there is a difference, there's a difference and you can't simply ignore it. Just in case you run into small numbers somwhere else and decide you want to make them equal...

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Dale
Mentor
Imagine a universe A, containing only two identical heavy planets situated at an almost infinite distance from each other.

Imagine another universe B, containing only two identical lighter planets situated at an almost infinite distance from each other.

In both cases the gravitational potential energy between the planets are said to be almost zero.

How can that be?

I think there must be a bigger overall potential in universe A.
Don’t you?

Two bigger planets “falling together” trough billions of years, must lead to a bigger fireworks, than two lighter planets “falling together” trough billions of years.

How do we describe this actual difference in the overall potential between universe A and B?
First, I agree with the other comments about "almost equal" is not "equal" etc. Basically $$\lim_{x\to \infty } \, a(x)=\lim_{x\to \infty } \, b(x)$$ does not imply $$\lim_{x\to \infty } \, (a(x)=b(x))$$

In addition, using zero for the potential at infinity is just a matter of convenience. The universes don't care where you set your zeros and you are actually free to set your zero point wherever you want. That is because potentials are physically meaningless, only the differences in potentials are physically meaningful.

If I were a salesman of universes, I would take different prices for universes with different overall energies.

How do I sell universe A and B, what do I tell my customers?
In your brochure you don't talk about potentials, you talk about energy. You can especially talk about KE at collision. That also allows you to mark up the price on universes with the identical masses but more dense planets or universes with initial velocities. If a competitor uses potentials in their literature you educate the customer and make your competitor look like they are trying to manipulate the customer.

"almost" isn't good enough. If there is a difference, thee's a difference and you can't simply ignore it. Just in case you run into small numbers somwhere else and decide you want to make them equal...
I understand that, and will take note on that. What I meant was that the difference is negligible small.

For a small object a huge distance from earth, the GPE will approach a maximum of 61,605,000 j/kg.
What happens to these 61,605,000 j/kg if this small object annihilates with another small object of antimatter, and their masses converts into electromagnetic radiation?

Gone with the annihilation? No mass, no potential?

Even if a spaceship did work on bringing the small object to this huge distance from Earth?

Dale
Mentor
It goes into reducing the gravitational redshift of the radiation.

russ_watters
Mentor
I understand that, and will take note on that. What I meant was that the difference is negligible small.
Well it's not. In the example you gave, one is ten thousand times bigger than the other. That's a huge difference, not a negligible difference. If you don't see that, you're not going to be a very good universe salesman. However, if you have a universe to sell that you think is worth $10,000, I'll certainly be willing to buy it from you for$1!
What happens to these 61,605,000 j/kg if this small object annihilates with another small object of antimatter, and their masses converts into electromagnetic radiation?

Gone with the annihilation? No mass, no potential?
The energy released by such a collision is vast and the pre-collision potential energy of the objects isn't lost, but it is truly insignificant compared to the energy released in the annihilation.
Even if a spaceship did work on bringing the small object to this huge distance from Earth?
I don't see what that has to do with anything.

mheslep
Gold Member
...What happens to these 61,605,000 j/kg if this small object annihilates with another small object of antimatter, and their masses converts into electromagnetic radiation?

Gone with the annihilation? No mass, no potential?
Not gone at all. Remember E=mc^2, energy and mass are equivalent; more importantly here they are both subject to space time curvature caused by gravitational fields, as famously shown by the deflection of light passing near the Sun. http://en.wikipedia.org/wiki/Tests_of_general_relativity#Perihelion_precession_of_Mercury