So they created a time machine with a microwave and a cellphone. As far as I've seen in the anime series, something it is explained about how to operate it, but do the characters ever explain something about what makes it work? Do they give any Physics related explanation about what made the microwave and cellphone capable of sending things into the past?
I think the original visual novel goes into more detail on this. Basically, time travel in Steins;Gate universe requires the use of a black hole. Sern is able to create them (though very tiny ones) using their large hadron collider (this was theorized as possible in real life with CERN's LHC, though I'm not sure if that theory was ever proven or disproven.)
So basically, the explanation for the phone microwave was that by lucky accident, they created their own particle collider. Remember one of the key components of their phone microwave was the large CRT television in the downstairs shop, which they referred to as the "lifter" as in ion-propelled aircraft. Cathode ray tube televisions function by shooting a beam of electrons at the screen, and thus they handwaved it to be the same as the beams of particles used in the LHC (obviously it's not). Also, as to how this combined with the microwave to create mini Kerr blackholes was as I recall unclear.
The time-leap machine was not a separate machine, but an addition to the phone microwave, that converted memories into data that was sent instead of a normal SMS. This was even more handwave-y because there's no way all a person's memories could be contained in the small number of bytes being sent by the phone microwave. They excused it by the blackhole physically "compressing" the data, as I recall, which is a very fundamental misunderstanding of how data compression even works. The black hole's "compression" effect was also what was turning matter that went into the black hole into green goo.
Anyways, that's my recollection, since I was unable to find very good resources on it, and I'm not about to replay the visual novel right now to jog the memory.
(Note: I originally posted this on MAL. If I'm missing something, please point it out.)
How the Microwave works (chronological order):
The microwave works the same as SERN’s Large Hadron Collider. It accelerates protons at 99.99999% the speed of light. By doing so, they compress a mass into an extremely tiny space. This process creates a micro-singularity or a mini-black hole. [Note: John Titor’s and SERN’s (or even Okabe’s microwave) time machine uses two micro-singularities to distort the gravitational field around the machine enough to “replicate the effects of Kerr blackhole” thus making time travel possible.]
If you inject electrons into the micro-singularity to rotate it at an extremely high velocity, the resulting black hole exhibits what’s called Kerr Effect.
If you keep injecting electrons into the ring of the Kerr black hole’s singularity, the black hole will rotate faster and faster, and when its angular momentum exceeds a certain threshold the event horizon disappears, generating a naked singularity.
When the event horizon disappears (or naked singularity occurs), there’s no more reason for time and space to change places, meaning that you can enter a naked singularity without being trapped (killed even).
NOTE: “Kurisu: On the other side of the event horizon, moving (your body) freely though space-time becomes impossible, while moving freely through time becomes possible.”
Long story short: the microwave works the same way as the LHC (creation of two micro-singularities, lifters, etc.) The visual novel mentioned the use of lifters to stabilize the immense mass of the singularity and gravitational fields, thus making it possible to travel through black holes without getting crushed.
Read more below about how time travel works through Kerr black holes (not completely related to the anime).
The main obstacle to using black holes as a means of time travel is singularities – an infinitely dense region of space-time from which nothing can escape.
This is where Kerr black holes come in. The creation is due to (refer to a.). Instead of collapsing into the common, infinitely-dense singularity, it would create a ring of rotating neutrons, their centrifugal force preventing the singularity from forming at all. This would, theoretically, allow safe passage through a wormhole within the black hole avoiding the aforementioned spagehettification and opening the way for a very interesting possibility:
On the other side somewhere out there, would be a “white hole”, ostensibly the opposite of a black hole, spewing out the contents that had been sucked in. It would act as a bridge that would allow travel through both time and space or, perhaps, into another universe (worldline).