I have a simple question regarding matter-antimatter gravity interaction.
Consider the following though experiment:
If we imagine a mass $m$ and an antimass $m^-$, revolving around a large mass $M$ the potential energy of mass $m$ should be:
$$ U_1=-\frac{GmM}{R} $$
and the potential energy of mass $m^-$ should be:
$$ U_2=-\frac{GmM}{R} $$
or:
$$ U_2=\frac{GmM}{R} $$
depending on the sign of the gravity interaction between matter and antimatter.
If the two particles annihilate to energy, then the gravitational field of $M$ will interact with the emitted photons and will change their frequency.
But, as the interaction between gravity and the photons has nothing to do with the question of the gravity between matter and antimatter, can't we simply use the interaction between gravity and photons, and the energy conservation to establish the nature of the gravity interaction between matter and antimatter?