There’s a few ways to approach this topic, all related but coming from different perspectives. And each is grounded in the best understanding of physics we have, tested to amazingly precise accuracy.
First, we know that the faster an object (with mass) moves, the less time it experiences relative to slower moving things (this is really tricky to explain properly, but it’s well-established experimentally). As something approaches the speed of light, an observer who is not moving at that speed would see the clock on that something tick slower and slower (and a clock doesn’t just mean an actual clock – it’s anything that changes at regular intervals). The closer it gets to the speed of light, the more the clock appears to slow down. And if it ever reached the speed of light (which is not possible for any object with mass – more on this below), its clock would stop and it would experience no passage of time at all.
Second, it takes energy to move an object faster and faster, and the faster you want it to go, the more energy you need. As an object approaches the speed of light, the energy needed to increase its speed even a tiny tiny bit is huge. And if it could reach the speed of light (which it can’t), infinite energy would be required. This is simply not possible.
Third (and this is the flip side of the second point), the faster an object moves, the more “massive” it becomes – in a sense, it weighs more and more as it moves faster and faster. As it approaches the speed of light, its mass approaches infinity. Infinite mass is impossible.
All of this stems from Einstein’s special relativity. It’s always possible that this model of how the universe works is wrong, but all evidence we have to date indicates otherwise.
With respect to things without mass (technically “rest mass”) like photons or gravitational waves, they can move only at the speed of light. They do not experience the passage of time. A photon that was created in the big bang and travels through the universe for billions of years would not experience any time.
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