Let us have an two objects (in a box) both of mass $m_0$ far from each other.

They attract each other and at some moment their distance is close and they have substantial kinetic energy. Something was answered here - that we should include potential energy to keep the energy balance close = far.

$2m_0 c^2 + T + U = 2m_0 c^2, \quad where \quad U=-T$

Or I better write: $2 \gamma m_0 c^2 +U = 2m_0 c^2$

Now the same, let's have three states: far = close = bound state: a state where the kinetic energy is removed from the system.

$2 m_0 = 2\gamma m_0 + U/c^2 = 2m_0 + U/c^2 + E_{out}/c^2 $

What happens with the mass of the system?

Far - it should be simply sum of both $m_0$.

Close - I can clearly see (as an observer in rest) kinetic energy ($\gamma m_0$), which increases the mass of the system.

Bound - for simplicity - the two balls stop - surface to surface, I remove $E_{out}$ and I my system weights $2m_0$ minus binding energy ($U$ is negative).

Supposing it was correctly formulated - While I am ok with far&bound - What mechanism in $U$ compensates the system weight when you come close? Imagine two neutron stars in gravitational field or two charged particles in Coulomb field or even some short range interaction...?

Reaction on comment user7027: Thanks. Generating momentum should go at expense of mass, now I see a kind of GTR answer, as far as I understand $\lambda$ has a sense of mass loss/binding energy (per mass or so). So what is the answer for other (non gravity) interactions?

  • $\begingroup$ Where was it answered elsewhere? $\endgroup$
    – Kyle Kanos
    Mar 10 '15 at 17:13
  • $\begingroup$ It takes energy to get masses moving. The masses are a good source of energy for getting the masses moving. We have nuclear fusion, and nuclear fusion fuel, which is mass, right? And then we have gravitational fusion, and gravitational fusion fuel, which is also mass. $\endgroup$
    – stuffu
    Mar 10 '15 at 21:48
  • $\begingroup$ Case 1: There's a charged plate capacitor with mass 1 kg and 2 liters of electric field between the plates. Move the plates closer while resisting the attractive force. Now the capacitor has 1 liter less electric field, and 0.1 kg less mass. Electric field gave some of its mass away. Case 2: You start running around, your bones become more massive, your total mass is unchanged, your muscles gave some of their mass to your bones. This is trivial stuff :) $\endgroup$
    – stuffu
    Mar 12 '15 at 8:46
  • $\begingroup$ @user7027- I understand the principle of energy conservation, this is not a problem. Case2- change of mass is detected by an observer, not you, but it is a simple STR, not exactly a question. $\endgroup$
    – jaromrax
    Mar 12 '15 at 12:17
  • $\begingroup$ Consider carefully the trivial things or be confused forever ;) After accelerating yourself to great speed you don't feel extra mass in your bones, you only feel reduced mass in your muscles. $\endgroup$
    – stuffu
    Mar 12 '15 at 17:11

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