McKay Hologram

[Davey]

How exactly is a hologram able to communicate in real time with a living person, and what did the McKay hologram do for 700 years waiting for Shepard?
And how exactly did Shepard get home? Did he gate back to the planet where he was originally and then re-gate back to the correct Atlantis? How did McKay originally know exactly when Shepard was going to arrive in the future Atlantis? How did McKay plant the hologram to exist there tens of thousands of years in the future? Finally, if Shepard had successfully returned home, why didn't the hologram tell him that part of the history?

[Xaeden]

How exactly is a hologram able to communicate in real time with a living person,

Mckay created a simulation designed to mimic his personality and respond to various variables. He in-bedded this computer program into the city's computer system. That program than tapped into the hologram chamber to create a likeness of Mckay.

and what did the McKay hologram do for 700 years waiting for Shepard?

Likely the same thing he did for the first forty-eight thousand years; nothing. The most reasonable course of action would have been for him to deactive himself. He didn't need to be active that whole time as the city systems could have easily been automated (so when the city's sensors detect the sun has reached a certain point the shield's kick in), and then when the time came to greet Sheppard, the city would simply reactive Mckay as it did when Sheppard first came through the gate.

And how exactly did Shepard get home? Did he gate back to the planet where he was originally and then re-gate back to the correct Atlantis?

Probably...

McKAY: Theoretically, we can send you back exactly the way you got here, using the Gate, the right address and a solar flare. The only problem is, we're waiting for something very specific – a prominence with exactly the right shape, size, characteristics and relative position in space so that it will interact with the wormhole in exactly the right manner and send you back exactly the right amount of time.

How did McKay originally know exactly when Shepard was going to arrive in the future Atlantis?

It's possible to mathematically figure out when a person will travel to by analyzing a particular solar flare...

McKAY: Huh, an interesting question – and one that was not easy to figure out. I had to determine the exact characteristics of the solar flare in question ...

SHEPPARD (irritated): Rodney!

McKAY: Forty eight thousand years, give or take.

This is how Mckay was able to figure out how to send Sheppard back to his time; he needed to wait until a specific solar flare would appear that would result in Sheppard traveling approximately 48,000 years in the past. This is the crux of the episode; while many solar flares would have been forthcoming, intersecting with this one might've sent him back 50 years, intersecting with that one might've sent him forward 200,000 years, etc. The precise solar flare they needed to get Sheppard back to his own time wouldn't form until 700 years later...

McKAY: Theoretically, we can send you back exactly the way you got here, using the Gate, the right address and a solar flare. The only problem is, we're waiting for something very specific – a prominence with exactly the right shape, size, characteristics and relative position in space so that it will interact with the wormhole in exactly the right manner and send you back exactly the right amount of time.

SHEPPARD: And that doesn't happen every day.

McKAY: Exactly.

SHEPPARD: How long are we talking about?

McKAY: Oh, seven, eight hundred years. A thousand, tops.

If you recall, in the movie, "Continuum," Ba'al sets up a network of satellites to track solar flares so he could similarly predict when one would happen that would send him back ~70 years and Sg-1 then uses that same network to find a solar flare that will send Mitchell back before Ba'al arrived.

How did McKay plant the hologram to exist there tens of thousands of years in the future?

Again, Mckay put a computer program into Atlantis computer system that was designed to activate when the city's sensors detected Sheppard's arrival through the Stargate. Sheppard then manually activated the hologram projector (at Mckay's auditory instruction), allowing Mckay's computer program to interact with Sheppard via a hologram.

Finally, if Shepard had successfully returned home, why didn't the hologram tell him that part of the history?

We discussed this is another thread. Stargate abandoned the idea that the predestination paradox controls time travel after "1969." The retcon from that point on established that the timeline is rewritten when someone travels through time. This means that Mckay existed in a timeline where Sheppard never managed to get back. Knowing this, he worked to create a way to rescue Sheppard and, upon successfully sending Sheppard to the present again, that Mckay's timeline was wiped out of existence and replaced with the one that was guided by Sheppard's actions.