# Thread: Potential energy and gravity

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## Potential energy and gravity

When I lift an object, where does the energy I put in go? Does the object actually become slightly more massive, as it gains potential energy?

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Heat

It all ends up as heat in the end. Your arms get warmer.

3. Originally Posted by Anders Starmark
When I lift an object, where does the energy I put in go? Does the object actually become slightly more massive, as it gains potential energy?
Chemical Energy in your body -------------> Gain in Potential Energy of object + Some Heat in your muscles.

No mass gained/lost.

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Originally Posted by Anders Starmark
When I lift an object, where does the energy I put in go? Does the object actually become slightly more massive, as it gains potential energy?
Effectively, you are warping space-time. You are creating a gravitational gradient which then tries to restore itself by pumping energy back into the object; forcing it downward. As long as you are applying upward force, you are maintaining that gradient.

The heat in your arm is only the mechanical waste from an inefficient operation; i.e. extra energy that you had burn in order to sustain the act of manipulating the space-time curve.

Pete Hurst

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Originally Posted by randompete
Effectively, you are warping space-time.
I like that.
The work you do moves the object and the Earth slightly farther apart. You can get much the same effect by doing work to stretch a spring. In the case of the spring, the energy goes into distorting the chemical bonds that hold the spring together; in the case of gravity, you are distorting the "gravitational bond".

Grant Hutchison

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Originally Posted by grant hutchison
The work you do moves the object and the Earth slightly farther apart. You can get much the same effect by doing work to stretch a spring. In the case of the spring, the energy goes into distorting the chemical bonds that hold the spring together; in the case of gravity, you are distorting the "gravitational bond".
Ok, but where is the energy stored? What I am getting at is - if I increase the energy of a system, Einstein teaches us that the mass of the system increases. So does the mass of a dumbbell increase if I raise it above my head?

7. Originally Posted by Anders Starmark
Ok, but where is the energy stored? What I am getting at is - if I increase the energy of a system, Einstein teaches us that the mass of the system increases. So does the mass of a dumbbell increase if I raise it above my head?
You haven't increased the energy of the system Anders Starmark. Your body has lost, and continues to lose, chemical energy. Don't forget that. You're just changing one form of energy into another.

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Originally Posted by Anders Starmark
Ok, but where is the energy stored? What I am getting at is - if I increase the energy of a system, Einstein teaches us that the mass of the system increases. So does the mass of a dumbbell increase if I raise it above my head?
The only conceivable place the energy can be stored is in the fabric of space-time itself.

Imagine pinching a point under a trampoline, and pulling it. The potential energy is stored in the fabric of the trampoline itself, ready to be released.

Or, it could be that Einstein's system is simply a good approximation that closely fits observable results, and there is something deeply more complex (or perhaps unifyingly simpler) underlying everything we currently understand.

But the trampoline analogy works for me.

Pete Hurst

9. Originally Posted by randompete
The only conceivable place the energy can be stored is in the fabric of space-time itself.

Imagine pinching a point under a trampoline, and pulling it. The potential energy is stored in the fabric of the trampoline itself, ready to be released.

Or, it could be that Einstein's system is simply a good approximation that closely fits observable results, and there is something deeply more complex (or perhaps unifyingly simpler) underlying everything we currently understand.

But the trampoline analogy works for me.

Pete Hurst
This analogy suggests that space-time is basically either filled with or made up of potential energy? would that be possible?

thanks David

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Originally Posted by cosmocrazy
This analogy suggests that space-time is basically either filled with or made up of potential energy? would that be possible?

thanks David
If not, what is going on?

I'd say:
- Space-time is the medium in which everything we know of exists
- A medium capable of supporting a mechanical/energetic system as vast and complex as we have observed, must in itself be of significant power and energy, and therefore capable of storing that kind of potential

Then, what can we observe?

If you lift an object up, energy has been stored somewhere. Yet the ball's mass is the same, its temperature is the same, any observable property is the same; other than its weight, which is reduced slightly because it is now on a shallower gradient farther from the Earth.

Yet when you let go, energy is released from somewhere. An invisible force accelerates the ball downward. For conservation of energy to be addressed, the energy must have been converted to another form during that time. Where has it gone?

I'm more suggesting that all matter has a kinetic potential across space/time to all other matter, and that this potential is what changes as the matter moves relative to all other matter in the universe. The potential can be changed either by releasing energy, or applying force to increase the potential. Release of potential is seen as gravity.

Really I see this as nothing more than a handy visualisation of what the accepted equations say. You can call it "potential energy" and make the numbers balance and ensure energy is conserved; or you can imagine the energy is actually stored somewhere (which makes more sense to me), as actual perturbations in the space-time curve... by stretching a particular part of it you are creating extra tension in the medium which can be released again as kinetic force.

Something like that!

11. Yes i can see your point and i like your analogies.

thank you.

12. Going back to the simpler spring analogy.
My understanding is that a compressed spring weighs-more/has-more-mass (than an uncompressed spring of equal characteristics).

How you can trace/explain this mass difference in the molecular level is a mystery to me.

I have brought it up before but do not recall a resolution.

Once you dissolve both springs in acid the compressed-spring solution will be warmer.

13. Originally Posted by a1call
Once you dissolve both springs in acid the compressed-spring solution will be warmer.
Would that not be the result of the stored potential/kinetic energy converted to heat energy dissolved into the acid?

14. Originally Posted by cosmocrazy
Would that not be the result of the stored potential/kinetic energy converted to heat energy dissolved into the acid?
Precisely,
Now if you could also explain the mass increase before dissolving (in molecular or atomic level ), we would be in business.

15. Originally Posted by a1call
My understanding is that a compressed spring weighs-more/has-more-mass (than an uncompressed spring of equal characteristics).
I am pretty sure that this is not true.

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Originally Posted by Anders Starmark
When I lift an object, where does the energy I put in go? Does the object actually become slightly more massive, as it gains potential energy?
If you were a perfect machine, no losses due to friction or entropy, your arms would not get warmer. The energy does not "all end up as heat in the end."

The energy goes into the object, as potential energy, according to the following equation:

PE=m*g*h, where m is the mass of the object in kg, g is standard gravity in m/s2, and h is the height in meters.

There is no change in the mass of the object.

Originally Posted by randompete
Effectively, you are warping space-time. You are creating a gravitational gradient which then tries to restore itself by pumping energy back into the object; forcing it downward. As long as you are applying upward force, you are maintaining that gradient.
This is closer to what's going on, but not quite. More to the point, it's the integrated differential between two gravimetric potentials. But the m*g*h is the simplified Newtonian case for relatively short differences in altitude.

Originally Posted by randompete
The only conceivable place the energy can be stored is in the fabric of space-time itself.
Yes.

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Originally Posted by a1call
Precisely,
Now if you could also explain the mass increase before dissolving (in molecular or atomic level ), we would be in business.
Well, we can dodge around it by pointing out that at some point the dissolving spring will break and its parts will recoil to their rest length, dissipating energy into the acid and so warming it.
But the potential energy of the compressed spring is stored in the distortion of its chemical bonds, which are shifted away from their minimum-energy configuration. So it is stored in the electromagnetic field, just as gravitational potential energy is stored in the gravitational field.

At the deepest level, "mass" is another word for energy in a particular form, and "energy" is just book-keeping. It's a conserved quantity we have defined in a useful way, but we've no idea what it actually is.

Grant Hutchison

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Originally Posted by PraedSt
I am pretty sure that this is not true.
It kind of has to be, if energy is conserved.

Grant Hutchison

19. Originally Posted by a1call
Precisely,
Now if you could also explain the mass increase before dissolving (in molecular or atomic level ), we would be in business.
Would love to but i have no idea. I realize that mass and energy are considered to be interchangeable but how this would be achieved to increase the mass of the spring, pass??

has it been verified that the actual mass increases in the form of measurable weight?

what if the spring was compressed such that it was squashed to a singularity? how much would the mass increase? perhaps this is totally a different situation though?

20. Originally Posted by PraedSt
I am pretty sure that this is not true.
It is quite literally true, though the magnitude of the effect is minuscule on the scale of mechanically or chemically stored energy. The bonds between atoms are displaced from their minimum-energy lengths, energy is stored, and mass is increased.

In the case of lifting an object, the mass of the object-Earth system is increased, and whatever provided that energy decreases in mass in the same amount, ignoring losses.

21. Originally Posted by grant hutchison
It kind of has to be, if energy is conserved.

Grant Hutchison
Really? Mass? Hmm. Maybe I'm staying at my school-grade classical physics level. You'd store potential energy in the spring, but there would be no change in mass. You're taking it into, and onto, Einstein's level, right?

Originally Posted by cjameshuff
It is quite literally true, though the magnitude of the effect is minuscule on the scale of mechanically or chemically stored energy. The bonds between atoms are displaced from their minimum-energy lengths, energy is stored, and mass is increased.
Yeah, same answer cjameshuff. You two are operating beyond my level.
I was going with: Chemical energy used in compressing the spring------------->Potential energy of compressed spring.
Simple stuff!

22. In a sense when you consider mass to be nothing more than concentrated energy then when you input energy into a system and that system stores that energy, until any energy is released the mass has to increase? But how is that energy measurable in the form of mass rather than potential energy? (in the case of the spring weighing more under compression)

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Originally Posted by cosmocrazy
But how is that energy measurable in the form of mass rather than potential energy?
In principle, by any other test that you could use to measure mass: weigh the thing on scales, test how its gravitation deflects light, apply a force and measure the acceleration. In practice, you just don't have the precision to see an effect experimentally at the level of compressed springs.

But when you consider the structure of something as massive and compressed as a neutron star, you need to take into account the "extra" gravity produced by the internal pressure. And, if memory serves, the self-gravity of pressure in the early Universe altered the containment of Big Bang nucleosynthesis, making a detectible difference to the proportion of light elements produced at that time.

Grant Hutchison

24. Thanks Grant for you explanation.

I understand that the effects in the spring scenario would be extremely difficult to measure.

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Originally Posted by cjameshuff
In the case of lifting an object, the mass of the object-Earth system is increased, and whatever provided that energy decreases in mass in the same amount, ignoring losses.
I have expended chemical energy, my mass has lowered, I can accept that. But when you say the mass of the object-Earth system is increased, really is this just another way to say "some energy has gone somewhere and we can't see it"? Because the mass of the Earth has not increased, otherwise every time we sent a rocket into space we would have permanently increased the Earth's mass. The object I am lifting does not increase in mass, otherwise everything would increase in mass as you lifted it, which can't be the case. So really, to say that "the system has increased in mass" is really just another fancy way to dress up the fact that energy has been stored somewhere and we don't know where. It becomes a virtual property of the system, or a mystical "potential energy" of the object.

I looked a bit further into the mass of a compressed string and found this fascinating thread:

As I understand, by compressing the spring, we've pushed the atoms closer together, increasing the potential energy in the chemical bonds (this potential presumably made by the strong or weak force interactions). This increase in energy means the spring actually has higher mass for the purposes of all mechanical calculations (although the increase is so small as to be unmeasurable, so this theory appears unproven).

Whilst the atoms have the same mass, there is additional mass now in the energy potentials operating at the atomic scale, and so the system as a whole is considered to have increased mass.

If we apply this to graviational potential, would be see the same thing? If we were some galaxy-striding gigabeing, would we be able to observe that when a planet was pulled farther away from its star, that whole solar system would appear to have a higher mass when you observed its orbit relative to galactic core? It seems unlikely.

A question then: if my potential energy is part of the summed mass of the me/Earth graviational system, and I fly into space, and keep on flying to Alpha C, then allowing myself to fall into it under its gravity... Where is that force coming from? Somehow the me/Earth mass system is now leaking energy into a me/Alpha C system. It just doesn't seem to fit.

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Originally Posted by randompete
If we apply this to graviational potential, would be see the same thing? If we were some galaxy-striding gigabeing, would we be able to observe that when a planet was pulled farther away from its star, that whole solar system would appear to have a higher mass when you observed its orbit relative to galactic core? It seems unlikely.
Yet true, I do believe, though probably to all intents and purposes undetectable in the example you give.
A very similar thing accounts for the "mass defect" that occurs when you assemble an atomic nucleus from an assemblage of nucleons. The individual nucleons are attracted to each other by the strong force, and therefore a system of separate nucleons has a higher potential energy than the assembled nucleus. And the sum of the masses of the separate nucleons is demonstrably greater than the mass of the final nucleus.
We deal with the book-keeping by setting the potential energy to zero when nucleons are infinitely far apart, and giving them more negative potential energy as they approach each other: that is, the mass defect increases.

Likewise for gravitational potential energy: in principle the Earth has a lower mass, by a tiny amount, than all the spread-out stuff it originally formed from. (In reality, the process of formation is rather profligate with energy, so it's not something that's going to show up as a real effect.)

Grant Hutchison

27. Originally Posted by randompete
I have expended chemical energy, my mass has lowered, I can accept that. But when you say the mass of the object-Earth system is increased, really is this just another way to say "some energy has gone somewhere and we can't see it"? Because the mass of the Earth has not increased, otherwise every time we sent a rocket into space we would have permanently increased the Earth's mass.
And you do, assuming the rocket is boosted to escape velocity. The Earth gains mass consistent with the potential energy gain of it moving out of the gravity well of the rocket. Its gain is far less than the mass of the rocket, but it is there.

Originally Posted by randompete
The object I am lifting does not increase in mass, otherwise everything would increase in mass as you lifted it, which can't be the case. So really, to say that "the system has increased in mass" is really just another fancy way to dress up the fact that energy has been stored somewhere and we don't know where. It becomes a virtual property of the system, or a mystical "potential energy" of the object.
Nothing any more virtual or mystical than kinetic energy, velocity, or mass.

Another example: a cloud of gas collapses under its own gravity. As it does so, it moves deeper into its own gravity well, losing potential energy, and compresses into a smaller and smaller volume. It heats up, and radiates photons. These photons are not inherently different from the gamma rays that can produce electron-positron pairs. How do you balance that out *without* a loss of mass as the gas cloud loses potential energy and emits EM radiation?

28. This is really good stuff you three. Thanks!

29. Ah, interesting subject. Let's look at the (very) basic GR bookeeping of all this stuff. In GR, a stationary (or is it the more restrictive static, I forget) space-time contains no energy at all. This is in contrast to EM fields where we associate energy-density with the field itself. In GR, a static/stationary "gravitational field" (space-time) contains no energy. Now, this gets insanely complex, more high powered 'rithmetic than you can shake a stick at, but dynamic gravitational fields do carry away (or bring in) energy via gravitational radiation. Indeed, if the space-time is no asymptotically flat, there is no notion of invariant globally conserved energy at all! Any observer can say the "field is carrying away" energy, but that difference cannot be made invariant. That's a shocker to many people, but that's the way it is.

In nice simple stationary/static cases, all is well though.

There is no notion of Newtonian gravitational potential energy at all. The work you do on lifting an object actually goes into the object, not the "field". Your work speeds up its clock. In the simple cases, this is the expression for what we'll call the "coordinate rest energy":

E = mc^2 *sqrt(g_00), where g_00 is the upper left (time-time) component of the metric. In a simple Schwarzschild field, our g_00 is 1 - R/r where R is the Schwarschild radius, 2GM/c^2 and 'r' is the radial coordinate.

So, when we drop a mass m in from infinity and bring it to rest at some 'r' down in the well, how much coordinate rest energy do we loose.

Well, subtract them:

delta_E = mc^2[ 1 - sqrt(1 - R/r) ]

In the weak field, where R/r is small, we can use the binomial theorem to expand the square root:

delta_E ~= mc^2[1 - (1 - R/2r)] = mc^2*[R/2r]. Now R = 2GM/c^2, so we have

delta_E ~= m * GM/r, which is exactly our Newtonian potential.

When we drop something that is the energy we gain. Normally that goes into kinetic energy. In GR, the total energy, coordinate rest + kinetic remains mc^2 for something dropped from infinity. When it hits the ground and stops, that difference is transferred to something else.

Note the Newtonian potential is only approximate in the weak field. In Newton, total energy E is Potential + Kinetic. We see that in GR, our "coordinate rest" energy is what plays the role of potential energy.

-Richard

30. Well put Richard.

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