# Does potential energy change mass

1. Aug 24, 2012

### roboticmehdi

Hello everybody. Imagine a box in which two spheres are separated by some distance. nothing is moving inside. Einsteins E=m*c^2 must be always valid. Since nothing moves the energy of box is E=m°*c^2 where m° is rest mass. Since the spheres exert gravitational force on each other they will be ruching towards each other after some time. just before the collison the energy of box is E=m*c^2 and it is higher than previous energy since the motion os spheres has increased their mass and made it m. Now you can see that law of conservation of energy is violated although no energy is given/take out of system. Thinking of this experiment I could not find the answer to the question, JUST WHERE DOES THE POTENTIAL ENERGY FIT IN THE E=MC^2 EQUATION ? OR DOES IT AT ALL ?

2. Aug 24, 2012

### serp777

Potential energy doesnt fit inside, because its just that--potential. By the same argument, everything has stupid amounts of potential energy, since everything moves relative to a black hole. However, just because the earth, for example, could enter a black hole and gain infinite energy in the singularity, doesn't mean that it has infinite energy now. Potential refers to the future and things that have not yet happened, so it wouldn't make sense if future events could affect the current energy/mass of particles. Thats paradox city.

3. Aug 24, 2012

### Staff: Mentor

The gravitational potential energy of the two spheres must be included in E. As the kinetic energy of the spheres increases, their potential energy decreases correspondingly. The total energy remains constant, and therefore so does the mass of the system.

4. Aug 24, 2012

### roboticmehdi

But E=mc^2 where C is constant, only m is variable and depends on speed only m=ym° where y is lorents factor and m° is rest mass. nothing depends on relative positions of objects but at the end energy increases out of nowhere.

5. Aug 24, 2012

### zonde

Amount of potential energy depends from position of objects.

6. Aug 24, 2012

### roboticmehdi

- I want to find the energy of this system. Does energy depend on mass ? Yes it does, energy is equal mass times c squared.

- What does mass depend on? on lorentz factor, m=mo*y.
-What does lorentz factor depend on? speed only.

So energy depends on mass, mass on lorentz factor and lorentz factor on speed. So energy depend on speed only( as a variable I mean, of course mo and c are constant at all times). So see energy does not depend on relative position.

7. Aug 24, 2012

### pervect

Staff Emeritus
Jtbell is basically right. Which isn't surprising, since he's a Mentor (and the only one to contribute to this thread so far). If you don't mind a bit of a lecture (Or even if you do), I will point out that one can USUALLY trust the answers of Mentors and science advisors here on Physics Forums. It never hurts to get references from them about the point in question and do some reading and cross-checking for onself.

Just blowing SA's answers off because you think you know better isn't a good way to learn, however.

Anyway -the mass of a bound system is not, in general, equal to the sum of the masses of its component parts as you have assumed. However, the difference will be negligible.

I should add that this is the basic answer, which glosses over some subtle points peculair to gravitation as a binding energy source.

If we change the problem to nuclear binding nergy, not only do we not have to worry about some of these subtle issues, we have well-known measured resutls

See for instance the wiki article on binding energy http://en.wikipedia.org/w/index.php?title=Binding_energy&oldid=506322211

8. Aug 24, 2012

### stevendaryl

Staff Emeritus
For a composite system, E=mc^2 is more useful in computing the mass of the composite system, given the energy, than it is computing the energy, given the mass. If you have two particles that are bound by some attractive force, the total mass of the two-particle system is not going to be equal to the sums of the masses of the particles. For two particles moving nonrelativistically, the total mass will roughly be given by:

mtotal = m1 + KE1/c2 + m2 + KE2/c2 + U/c2

where m1 is the mass of the first particle, and KE1 is the kinetic energy, m2 and KE2 are the mass and kinetic energy of the second particle, and U is the potential energy (with the zero shifted so that U=0 when the particles are infinitely far apart).

9. Aug 24, 2012

### roboticmehdi

Thank you stevendaryl ! Thats the answer I was looking for.

10. Aug 24, 2012

### stevendaryl

Staff Emeritus
What you wrote is just not true. There is nothing in SR that says that mass only depends on the lorentz factor. What SR says is that if a system has total mass M0 as measured in its rest frame (that is, the frame in which the total momentum is zero), then in that frame, it has total energy M0 c2, and in any other frame it has total energy γ M0 c2. SR does not say anything about how to calculate the total mass of a composite system, it only says how that mass relates to the energy in various frames.

11. Aug 24, 2012

### Kraflyn

Hi.

Yes, particles have potential energy, and just like in high school, potential energy and kinetic energy are conserved. The issue here is then: why potential energy didn't add some mass too? Kinetic energy did add mass, through factor $1/ \sqrt{1-u^2}$.

We have gravity in the mix? We are not within the realm of special theory of relativity anymore, then! Space is no longer Minkowski space. We need general theory of relativity here. And what does it say? It says: clocks run slower in strong gravitational field. So how do we calculate speed $u$? Well, we must know what time it is... So things become more complicated in gravitational field. Furthermore, the reference frame is no longer inertial. You can't just swap observers. They see world differently one from another.

Furthermore, You know about analogy between accelerated elevator and gravitational field? Object standing still in gravitational field and object accelerated in elevator feel the same force acting. It's the same thing. It's called equivalence principle. And accelerated object is moving. So it has some velocity. So it has some extra mass. This picture is equivalent to object not moving, but being in gravitational field. Two pictures are equivalent. So: gravity adds to mass.

Keyword: equivalence principle.

This is how Einstein himself described it for general audience: http://www.marxists.org/reference/archive/einstein/works/1910s/relative/relativity.pdf

I hope this helps a bit.

Cheers.

12. Aug 24, 2012

### Kraflyn

Hi.

Finally, changing a referent potential energy level in general theory of relativity impacts the measurable forces, unlike the change of referent energy level in classical physics. So the original question is quite a complex one.

Cheers.

Last edited: Aug 24, 2012
13. Aug 24, 2012

### bcrowell

Staff Emeritus
Some experimental evidence:

For electrical and nuclear potential energy, see http://www.livingreviews.org/lrr-2006-3 , section 2.1.1. For gravitational potential energy, see 3.6.1.

And of course we need the usual warning that the source of gravity is not scalar mass-energy but the stress-energy tensor. For experimental evidence that pressure is a source of gravity, see the discussion of the Kreuzer experiment here: http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.1 [Broken]

Last edited by a moderator: May 6, 2017