A thought experiment about the speed of light and breaking it

Jun 2016
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England
The amounts of energy required to accelerate anything toward the speed of light become ridiculously vast.
Theoretically (as the theory is currently formulated) the energy asymptotes toward infinity as the speed of light is approached.
Experimental evidence has (so far) shown no deviation from this theory.
This might be because we have not yet been able to pump in enough energy,
but what might happen if a jump (analogous to the hydraulic jump) were possible,
is too far outside any of the theoretical models currently available to be in any way even guessable.