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In some ways, yes. In other ways, no.

First of all, the Sun dying scenario is ridiculous. Sun has enough fuel to keep shining for another 5 BILLION years. The film is set in 2057, at that time, death of the Sun will be the least of our worries.

Making a stellar bomb the size of Seattle, ambitious, but not impossible. Because harvesting the huge amount of heavy metals will be a pain. Especially when you consider that most of them are found under the sea.

Restarting the Sun, ridiculous. The idea was to jump-start the fusion process of the Sun by blowing a bomb in it's core, so powerful that it is like a mini Big Bang. That's just not how it works. I won't go into the details, but the Sun's birth was a process that took billions of years and had unimaginable amounts of energies involved. It will much more than the bomb to recreate this scenario.

Getting so close to the Sun. Surprisingly, this may just be possible. The closest we have gotten to the Sun is around 45 million kilometers, slightly inside the Mercury's orbit, which is really close. So, if you also take into account that the ship in Sunshine was using a giant heat shield (kind of like a tough mirror), they may have been able to get so close without melting away. Also, the ship was traveling fast enough so it could be possible to go through against the solar winds. You have to remember that in the movie, the ship never went inside the Sun (which is ridiculous) but only released the bomb by going REALLY close to it.

While falling in the Sun, time slows down. Absolutely true! Einstein's theory of relativity has already proven that when you travel really fast (close to the speed of light) time slows down. Sun has enormous gravity, and can possibly accelerate you near the speed of light (that is of course if you don't burn up before that happens). We can only imagine how reality will feel like when we approach the speed of light. With zero time, laws of maths and physics just simply don't work, so science can't tell you how it will feel like. So what Capa saw near the end of the movie was what Danny Boyle thought reality feels like when you approach the speed of light.

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