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Does quantum tunneling happen slower at relativistic speeds?
It is not that quantum tunneling happens slower at relativistic speeds but rather the time for the event to happen does not exist. The universe in theory can be infinitely big or infinitely small. If it were any bigger then there would be an end and if it were any smaller then we could never get to its center. However, what this means is that every point in space has infinite mass because both a very large and a very small distance from it are equal (they have no direction). Therefore all points in space are equidistant from each other meaning they occupy the same place/time.
This means that space-time is not continuous but discrete. The events in time do not follow a linear progression, rather they appear to exist as if there was no time for them to occur; like the points on a line drawn between two points where nothing exists.
In this way space-time is actually a series of points with no time to connect them. In other words, the universe does not have any 'past', it just is and the future cannot exist until something happens in the present that gives meaning to its existence.
The point with the highest probability of existing is the one that exists and this point has no time to exist in. This means that if a particle were traveling fast enough to escape its own event horizon then it would not have any past and therefore could never reach a certain speed because it will always already be at that speed.
So, if the particle is traveling fast enough then it does not have an event horizon in which to change acceleration and therefore can never reach a speed of 0. This means that there is no such thing as quantum tunneling at relativistic speeds.
The event horizon is the point at which an object has reached its maximum speed and therefore can never reach a lower speed. When this happens then there is no such thing as time for that particle; it has always already reached 0 velocity.