# E=mc^2 What does this mean?

E=mc^2
Can someone explain this equation to me? Does this mean that energy has mass in the sense that energy will create gravity? For instance, if I have a massless ball and I somehow place (3*10^8)^2 Joules or c^2 joules of electromagnetic energy inside the ball, would that ball now have a mass of 1 kg (simple algebra) and thus have a gravitational pull? Even though the ball is merely a boundary in a sense and contains no mass itself?

E=mc^2
Can someone explain this equation to me? Does this mean that energy has mass in the sense that energy will create gravity? For instance, if I have a massless ball and I somehow place (3*10^8)^2 Joules or c^2 joules of electromagnetic energy inside the ball, would that ball now have a mass of 1 kg (simple algebra) and thus have a gravitational pull? Even though the ball is merely a boundary in a sense and contains no mass itself?

We placed some stuff into the ball, and the total mass of the ball and its contents is now 1 kg. Because m=E/c^2.

We can investigate the distribution of the mass by using a fishing line and a weight.

We would feel a force pulling the weight towards the center of the ball, I mean inside the ball the weight tends to fall towards the center.

Dale
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For instance, if I have a massless ball and I somehow place (3*10^8)^2 Joules or c^2 joules of electromagnetic energy inside the ball, would that ball now have a mass of 1 kg (simple algebra) and thus have a gravitational pull?
You have to consider both the energy and the pressure. See https://arxiv.org/abs/gr-qc/0510041

fresh_42
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Level adjusted, so answers up to now may not mirror the new assessment.

jtbell
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Even though the ball is merely a boundary in a sense and contains no mass itself?
In general, the mass of a system does not equal the sum of the masses of its components.

Does Potential energy work in this equation, E=mc^2? As in, if I hold a ball 1 meter above the ground, the PE=mgh. Does than mean that I have added mass to the system? Will the mass of the ball become PE/c^+m?

anorlunda
Staff Emeritus
Does Potential energy work in this equation, E=mc^2? As in, if I hold a ball 1 meter above the ground, the PE=mgh. Does than mean that I have added mass to the system? Will the mass of the ball become PE/c^+m?

No, the invariant mass does not change when you lift the ball. You need to read the references very carefully and pay attention to the wording.

Dale
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Does than mean that I have added mass to the system? Will the mass of the ball become PE/c^+m?
That depends on what you consider to be “the system”. If the system is just the ball then no. If the system is the ball and the earth then yes. If the system is the ball the earth and you then no.

No, the invariant mass does not change when you lift the ball. You need to read the references very carefully and pay attention to the wording.
According to what I have read I would say the mass does change. The added potential energy increases the mass thus the gravitational pull due to the ball would increase. Is this correct.

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Nugatory
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According to what I have read I would say the mass does change. The added potential energy increases the mass thus the gravitational pull due to the ball would increase. Is this correct.
No.
Neither the rest mass of the ball nor the earth changes.

The added potential energy increases the rest mass of the ball+earth system. That is, if you had the ball and the earth inside of a sealed box and you had some way of adding energy from outside to move them farther apart the added energy would increase the mass inside the box.

Look at #6 above by @jtbell if you haven't already.

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jtbell
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Returning to the original question about electromagnetic radiation in a box, first note that for a single particle ##mc^2 = \sqrt {E^2 - (pc)^2}##, where ##p## is the magnitude ##|\vec p|## of the momentum vector ##\vec p##. For a system of particles, ##m_{system}c^2 = \sqrt{E_{total}^2 + (p_{total}c)^2}##, where ##p_{total}## is the magnitude ##|\vec p_{total}|## of the total momentum vector ##\vec p_{total} = \sum {\vec p_i}##. If the box is at rest, the momentum of the box is zero. The photons in the box move in random directions, so their momentum vectors cancel out and ##\vec p_{total} = 0##. Therefore in this case ##m_{system}c^2 = E_{total} = m_{box}c^2 + E_{photons}##.

This ignores contributions from stresses in the walls of the box, but I expect we can consider them to be negligible under the right circumstances. Maybe someone who knows more about this can comment.

According to what I have read I would say the mass does change. The added potential energy increases the mass thus the gravitational pull due to the ball would increase. Is this correct.

I think you have some kind of point there. Although other posters seem to disagree.

Newton's law of gravitation has the same form as Coulomb's law.

So I would guess that two massive plates placed close together become more difficult to pull apart when we add energy to the system by pulling the plates apart.

(I know that with point masses the force always decreases as we pull the masses apart)

(A simple argument: According to a molecule on a plate every molecule pair located on different plates gains mass as the plates are pulled apart, the sum of forces from all those pairs increases.)

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Dale
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According to what I have read I would say the mass does change.
The mass of what? Please define what exactly is “the system”.