How long would it take to get to Andromeda Galaxy at the speed of light?

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Light would reach it in 2.2 million years, but according to Einstein's relativity, a spaceship can only get close to the speed of light , but never reach it. At that sort of speed, due to time dilation , a spaceship would get there almost instantly, but it would take an almost infinite amount of energy.

So a spaceship very very close to the speed of light could reach it in minutes, whereas on Earth 2.2 million years would have passed.

Not only does time slow down, but mass increases and length contracts at relativistic velocities. So as an object approaches the speed of light, its mass becomes greater and greater, until eventually it begins pushing up towards the mass of a star. The amount of energy needed to push the mass higher becomes correspondingly greater. Another effect, however, is that as you approach the speed of light, your own escape velocity increases. Before you reach the speed of light you enter your own Schwarzchild radius, effectively pinching off from the rest of the universe as a black hole. So you can't even theoretically get "infinitely close" to the speed of light. There are limiting factors precluding it.
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