In the film Contact, after building the giant machine on Hokkaido, Arroway 'travels' to Vega, landing at the beach where her father is.
However, the capsule never left earth. It's not clear if Arroway had a dream, or that it really happened (as 18 hours of noise was recorded, which is 1 second on earth).
Did Arroway have a dream or was it real?
The entire point of the movie is that we don't know, since there was no evidence. If you want to believe that she went to Vega (and beyond) then you have to rely on faith. That is what the movie was about.
Personally I definitely choose to believe that Ellie's journey was real. The final lines of the movie do hint that this was the intention, with the government officer admitting that, although Ellie's camera recorded only static, it recorded static for the duration that she claimed to be travelling; rather than for the subjective zero seconds that her hypothetical dream would have lasted from everybody else's perspective.
Don't know if this helps, but something similar happens in the book: outside observers see the Machine spin up, then immediately spin down. However the travelers (in the book there's a team of five) have an amazing journey and all share the same experience. The Being Arroway speaks with tells her of a message hidden in the digits of pi, which would sound impossible to anyone well versed in math. Nevertheless, when she gets home she indeed finds such a message after a large amount of number crunching.
So in the book, the experience was likely real.
This is left entirely up to the viewers's interpretation (as pointed out by KSmarts in comments).
The most that can be proven with physical evidence is that the machine definitely dilated time at it's core, as the static recording proved. Since no other evidence arose that ever showed the pod disappearing, other than the briefest of half-seconds as the rings obscured it, if you rely only on verifiable proof that's all you can take away. Even the damage to the chair inside the pod can be attributed to just the space-time dilation. Ellie admits as much, that it's possible nothing happened, in the hearing.
If you allow that Ellie's memory is accurate, it is highly likely she traveled. The details she picked up along her journey are not flights of fancy that she has had before. A completely contained metallic pod becoming translucent isn't something she's likely to invent. Nor in all her history has she had the kind of emotional response as she did at simply seeing the beauty in space that she did (allowing for that she was a very driven individual, and thus was open to it).
In addition, nothing eliminates both scenarios. The mechanics of the pod becoming translucent aren't really applied to any known science, just the potential of wormhole travel, and it's assumed in movie that it's just a container. It is entirely possible that the only thing that made the trip was Ellie's mind, the pod acting as a conductor or mental conduit, supplying all the images she witnessed. Through the wormhole connection, this could allow the other race to access her mind, and arrange a scenario that would allow them to communicate. Even if you don't buy that scenario, the aliens accessing her memories implies they could do pretty much what they wanted with them, and all of it could be retroactively implanted to say... well, whatever they wanted Ellie to get across to the rest of humanity.
The most obvious debate the movie has in itself is the pressure between evidence-based science and emotional-based faith. It intentionally never answers that question, to its characters or to its viewers. So as stated, it's up to you.
In the book, there was lots of evidence. There was a lot of wear and tear to the pod and she brought back a souvenir - a palm frond. Not to mention the shared experience of five people. The people who were trying to discredit her had to reach really far to try to construct a plausible alternative scenario to explain those things. They tried anyway because - well that's just politics.
The 18 hours of static recording is what the movies gives you as it's way of saying, yes it really happened. But again, it's just as open for discussion as the top spinning scene at the end of Inception.
IMO, the one giveaway in the movie that never gets discussed is the fact that she was out of her chair when the IPV came through. If you remember the beginning when she first got in to the chair she was secured to the chair with magnetic locks on her back to hold her in place. If the IPV really passed through the machine in a matter of seconds from when it was dropped as the observers thought it did, then how did she get out of the chair that quickly and wind up laying on the floor? Since she hit both the safety net AND the water, you'd think that chair would have been strong enough to hold her in place. Right?
But in her space travel sequence, we know she gets out of the chair because of the vibration. Even that takes her a few good seconds to do. It's not like there was an eject button or anything. From the moment of the drop in real time and also given the amount of G's pulling on her from the fall, she wouldn't have been able to get out of it unless she was in fact in space and had the appropriate time.
I would like to bring up something that both the movie and the book deliberately skips over.
In the movie it is implicit while in the book it is very explicit; The transmission we recieved was very complex but gave us clear instructions on how to build a machine that was far beyond any technology we had.
In the film we get the appearence that it was something you just build, while in the book it took many many years to understand, develop and construct. It made the Apollo program look like a walk in the park.
That Hadden was just sitting on all this new tech and used it as a practical joke on humanity is Carl Sagan satire. That Arroway found a circle in pi must have been a laugh at every mathematician he ever knew.
But the book and the film is not about contact. It's about being numinous. And Sagan left it wide open. I for one say everything happened, as I don't think even Hadden could hide such advanced technology from other stakeholders.
I realize this has already been mentioned, but as Angela Basset's character (forget her name) points out, Ellie's equipment recorded 18 hours of footage in what everyone else tells her was only 1 second- no amount of delusion or trickery can cram that much time in an instant even if it was 18 hours of static (could've sworn she said 14 but that's beside the point)- I'm gonna go with the trip really happened.
Ellie does go to Vega, but she travels through a wormhole, as she states she thinks it was in the film.
To the people on Earth it looks like the capsule only fell through to the sea, but in between the time as it fell space time bent and she traveled to Vega in "approximately 18 hours" as she says, and about 18 hours of static were recorded on her camera, too. She also sees in the capsule (The material is somehow transparent in some places as she moves around) a planet of Vega with lights and a civilization.
She somehow falls asleep, though, and the Vega aliens go virtually in her dream and make her dream she is on a beach she made up when she was a kid, and the aliens pretend to be her dad and they explain why they did everything to help the humans. She then wakes up injured right when she arrives to Earth from the capsule falling down. So she does go near the Vega planet, but she dreams with the aliens. So in some ways she does go to Vega, and in some ways she does not.
