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HallsofIvy

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The equations refer to reference systems and no reference system can reach the speed of light - and infinites would not be very useful. But if one could closely approach the speed of light, indeed one would be at one's destiny "in no time" or instantly. So, perhaps there is nothing that you missed, and you discovered the concept of "time dilation".

Einstein phrased it as follows:

"For v=c all moving objects—viewed from the “stationary” system—shrivel up into plane figures. [..] we shall [..] find in what follows, that the velocity of light in our theory plays the part, physically, of an infinitely great velocity."

and

"the time marked by the clock (viewed in the stationary system) is slow by 1-sqrt{1-v^2/c^2} seconds per second".

- http://www.fourmilab.ch/etexts/einstein/specrel/www/

Note: it could be that there's a thingy with your formulation "get contracted": you should have written "appears to be contracted for the light". It's essential to keep track of who/which system (although light isn't either) measures what and when!