Electron-Positron Annihilation in a Gravitational Field When an electron and positron annihilate what happens to any gravitational energy they have?
In a way the energy is 'shared' between these two particles and all the matter in space.  It must take a finite time for it to vanish. I realise of course that the gravitational energy of the two particles is very small.
 A: We all learned in school that gravity is associated with mass. For example if we have two bodies with masses $m$ and $M$ then Newton's law tells us that the gravitational force between them is:
$$ F = \frac{GmM}{d^2} $$
So if either object has zero mass the force goes to zero. And since photons are massless it's entirely reasonable to ask if the gravity just disappears when an electron and positron annihilate into two photons.
However Newton's law is only an approximation and to understand gravity better we need to move to general relativity. In general relativity Einstein's equation tells us:
$$ G_{\alpha\beta} = 8\pi G T_{\alpha\beta} $$
The $G_{\alpha\beta}$ on the left hand side is related to the curvature of spacetime and from it we can work out how freely falling particles will move. So the $G_{\alpha\beta}$ is sort of related to the gravitational force. On the right hand side the $T_{\alpha\beta}$ is called the stress-energy tensor and it takes the place of mass.
In the stress-energy tensor we don't distinguish between mass and energy - we consider them equally and interconvert between them using the well known equation $E = mc^2$. So before the annihilation the masses of the electron and positron would go into the stress-energy tensor and after the annihilation the energies of the two photons would go into the stress-energy tensor. The result is that the gravitational field doesn't just vanish the moment the electron and positron annihilate.
However the stress-energy tensor for two photons moving in opposite directions is very different to the stress energy tensor for two (effectively stationary) 511keV mass particles. This means that as soon as the particles annihilate the spacetime curvature will start to change and that change will propagate outwards at the speed of light.
