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A key plot device of avengers: Endgame is

Time travel. A central point is that they have to be very careful with time travel, as explained by some team members when Rhodes asked why can't they simply go back in time and kill baby Thanos. Someone mentions that "all the rules apply, no interaction with our past selves, no trying to change the future, etc...". When Hulk meets The Ancient One, she explains that timelines and realities can be damaged, etc.



The only thing they can really pull off without messing up history is retrieving the Infinity Stones, using them in the present, and then returning them to the proper place in history to avoid any disruptions to the time line.

But then in the final scene, we discover that Captain America

went back in time and had the life that he always wanted to have with Agent Carter.

Doesn't this contradict the above mentioned dangers

of time travel? Wouldn't his getting together with Agent Carter at the very least completely alter the history of the S.S.R and S.H.I.E.L.D (if not the entire U.S military) and hence completely mess up the time line?

Not necessarily. Two things to bear in mind:

First, it was revealed that Peggy Carter had been married sometime after World War 2, but her husband was, notably, not named nor was he seen at her funeral, not was it ever mentioned if he was still living or not.

Second, there's really been nothing shown about SHIELD and Carter from the 1950s to the 1990s, aside from brief flashbacks and references.

So, you're looking at a closed loop.

Rogers returns the Infinity Stones but he doesn't go back to right after his disappearance in 1945; he goes back sometime later, after the years shown in the Agent Carter series, because he knows what she was up to during those years and that his being missing is part of the reason she did the things she did, and what she did was important for events later. Then he shows up, say, in 1950 or so. Maybe he tells her the truth, maybe he just tells her he miraculously survived and just made his way back to civilization from the crash, but he wants to get away from the life and retire.

So he assumes a false ID and marries her.

He knows about HYDRA infiltrating SHIELD, but doesn't say anything because he knows events have to play out, and he carefully plays the role of husband not involved with the secret agent life.

Peggy starts developing Alzheimer's and is well into the dementia stage when

his past self, still frozen in ice, is found. At that point, he has to be careful not cross paths with his younger self, so he avoids visiting her when he knows his younger self will be there, sits quietly in the back at her funeral, and then waits to reveal himself to Bucky and Sam after his younger self has made the jump back in time.

I do not think Captain America lives out his life in our reality. Similar to how the stones were taken in a different reality he just went there and messed further with their timeline.

Here are the reasons I think this must be true.

1. Anything else would go against all the rules of time-travel established in the movie


2. Captain America knows Becky is alive and SHIELD is interrelated by Hydra. I am sure that in his new reality he fixes both of these things, but not in ours.

Afterwards he comes back to us by some means. Maybe he gets more Pim [particles from his version of Dr Pim at some point. Maybe the Sorcerer Supreme helps him. He meets her to give back the Time Stone.

The only question is why does he not come back to the platform? That must be just for dramatic effect. He had a lifetime to plan this moment.

EDIT:

Before anyone says that Cap would not mess with the timeline, if he did go back to our time, he clearly has shown that he does not care about it. If he risked changing something just for himself, he defiantly would have changed things to stop his best friend from being tortured.

Confusing situation

This situation is confusing because Endgame uses a version of time
travel that's not often seen in science fiction, especially movies or
television. Endgames time travel is based on the multi universe
theory and isn't really time travel, it's more like jumping to a
universe that's running a bit behind your own

How it works

This version of time travel is the one used in the book Timeline
written by the late Michael Crichton. I highly recommend it, and it
does a very good job of explaining how multi universe "time travel"
works. It's also a movie Timeline Movie, but if I remember
correctly they screw up how Crichton's time travel works.

What that means in endgame

So a short explanation of the time travel in Endgame. When the team
goes back to 2012 they didn't travel back in time in their own
universe, they traveled to a different universe that happens to be
identical to the one they started from, except it's 7 years earlier
there. This is how removing the stones from that time can't effect
the present in the original universe. Interestingly this means that
the stones taken from Asgard and the ones that Rhodes, Nebula, Barton,
and Natasha take are from different universes as well. So one
universe only loses a max of 2 stones (note that Tony and Cap get the
Tesseract from a different universe).

Answering your Question

Now that's out of the way, what's going on with old Captain America. This is >!where it gets a bit tricky. As Keith Morrison mentioned in his answer Peggy >!Carters husband is never named in previous films. Which means that the old >!Captain America seen in Endgame was probably her husband all along in the >!movie universe and avoided notice as Keith describes. However this is not a >!closed loop. Old Captain America is from a different universe. One that was >!exactly identical to the movie universe but ~70 years ahead of the one in the >!movie. In fact if the characters were to check his "quantum signature" or >!some such thing they'd probably be able to identify he wasn't from the movie >!universe.


The Cap who went back in time will never return to the movie universe. He'll >!live out his life in the universe he jumped to for Peggy.

His gift

Regarding the shield that old Captain America has with him. Perhaps in the >!universe he came from it either wasn't destroyed or was rebuilt before he went >!back, so he took it with him when he returned the stones. Of course he >!wouldn't know that it was destroyed and not present in the movie universe.

What banner mentions

To tie this back to what Banner said earlier in the movie about killing baby >!Thanos. He was right and wrong. They could kill baby Thanos and stop the >!snap, it just wouldn't be in their universe. In fact they could spend years >!committing baby Thanos genocide in many other universes and saving them from >!the snap, but never their own. Of course that might have other undesirable >!outcomes for those universes, for instance if Thanos never existed, would the >!Guardians of the Galaxy have formed? And if they didn't could Quill of >!stopped his father Ego? All life becoming Ego is arguably worse than the >!snap.

end spoilers

After some thought about this, my personal theory is that

The elderly Captain America is from a different (though similar) timeline

We know that

the timeline Captain America went back to already has changes made to it that simply returning the infinity stones won't change (Loki's escape being the most obvious, but also things like Peter Quill being knocked out, Thor's hammer being pulled away from him while on Asgard, or HYDRA agents hearing Cap say "Hail Hydra"). But these changes are all relatively minor - they won't cause the millions of deaths that the Ancient One warned would result from actually removing an infinity stone from the timeline.

So,

that timeline will be slightly altered, but its events will largely be the same. Thanos or someone similar will still gather the stones, and in 2019 heros will likely travel back in time themselves to set things right. This will change events in other ways, creating yet another timeline slightly altered from their own. In essence, we see a cycle of slightly-different timelines being forked off over and over.

But

how do we know our timeline is the first one in the cycle? More likely, heroes from years the future came back to this one, stealing the infinity stones and returning them, and in doing so changed things in subtle ways from their "original" timeline. Maybe in their timeline Yondu's still alive, or Vision is monochrome, or heck, maybe the earth was saved at some point by a talking duck from outer space.

The main point here is

the elderly Steve we see at the end is from that timeline, and has come back to our, forked timeline to live out his life, just like our Steve went back to the newly-forked timeline where Loki escaped. This is consistent with the MCU's rules of time travel - you still can't change your present, only create alternate timelines. But since people in the alternate timelines can themselves go back in time, there's no reason not to think that we were in an "alternate timeline" to begin with.

To quote another franchise,

All of this has happened before, and all of this will happen again.