How does energy convert to matter? To my understanding, matter and energy are one and the same.  Shifting from $E$ to $M$ in Einstein's famous equation requires only a large negative acceleration.  If $M$ really is $E/c^2$, does that make matter the solid state of energy?  I've read a lot about positron-electron collisions at high energies creating larger particles, and there is obvious matter conversion in fusion and fission reactions, but I can't find anything describing the physics of the conversion from energy to matter, rather than the interactions of what is already matter.
Specifically, the thing I'm getting hung up on is the reason energy would take on a solid state in the first place.  If energy is represented by waves, how does it become particles?  If gravity is determined by mass, and mass is nothing more than static energy, does that make gravity a static-electromagnetic force?
 A: Energy and matter are not the same. Matter is a type of thing, whereas energy is a property of a thing, like velocity or volume. So your premise is flawed. In particular:


*

*there's no such thing as "a solid state of energy" - hopefully it makes sense that a property of something does not have states

*energy is not represented by waves, though it is a property of a wave. It's also a property of a particle (which, in quantum field theory, is really just a tightly bunched wave).


Note that mass can be converted to energy, because mass actually is energy. It is one of various types of energy: kinetic energy, potential energy, mass energy, and so on. Different types of energy get converted into each other all the time.
I'd suggest looking at several of the questions under the "Related" heading at the right for more information about this. (I actually thought this had been asked here before, but I didn't find an exact duplicate.)
A: The thing about energy becoming particles is not entirely true.  Quantum mechanics explains that particles themselves are waves.  The energy that forms mass, however, is not a part of the particles themselves. For subatomic particles such as electrons and quarks, their mass is caused by their interaction with the Higgs field.  The energy itself is stored in the Higgs field, much like how electric potential energy is stored in electric fields.  As for gravity, gravity is not determined by mass.  General relativity explains that gravity is the curvature of spacetime caused by the presence of energy, be it in the form of mass, electric potential energy, or electromagnetic radiation.  So gravity is not a static force, rather, gravity is the curvature of spacetime caused by the presence of energy.
A: What is conserved is energy and momentum, not mass.
For a massive particle, mass is the energy at rest (when the particle is not moving)
For instance in a 2 particles => 2 particles reaction, you will have: 
$E_1 + E_2 = E_3 + E_4$
$\vec p_1 + \vec p_2 = \vec p_3 + \vec p_4$
In the case of massive particles collisions at high energies, you have to use, for each particle , the relativistic values, that is : 
$$E = \frac{m c^2}{\sqrt{1 - \frac{v^2}{c^2}}}$$
$$\vec p = \frac{m \vec v}{\sqrt{1 - \frac{v^2}{c^2}}}$$
You can consider that the energy is the sum of the energy at rest, which is nothing that the mass of the particle ($mc^2$), and the kinetic energy ($K = E - mc^2$).
(For massless particles like photons, you use only $E$ and $\vec p$, of corse).
Waves is a classical point of view. In this point of view, waves can carry energy.
But the correct point of view, the quantum point of view, used in Quantum Field Theories, is that there are quanta of fields, that we call "particles"
The source of the gravitation is the energy/momentum tensor, so it is not a mass, nor a mass density.
