| MadSci Network: Astronomy |
Indeed, if one does a simple calculation, a 1 g acceleration (~= 10 meters per second per second, i.e. a gain of velocity of 10 meters per second over a period of time of one second) gets you to the speed of light (300 million meters per second) in a time of 30 million seconds or about a year.
The problem is this assumes that space-time is Gallilean in nature. What this means is that this assumes that if one has measured the velocity of object 1 with respect to object 0 to be V10, and the velocity of object 2 with respect to object 1 to be V21, then the velocity of object 2 with respect to 0 is given by:
V20 = V10 + V21
Where I have assumed that all objects were moving along the same line. If they are not, one has to add vectors.
We now know through experiments that this expression is not correct even though this is a very good approximation when the speeds involved are much less than the speed of light. The most extreme case where the assumption breaks down is when photons (moving at the speed of light) are involved. Photons always move at the same speed with respect to an observer, regardless of his/her velocity. Photons pass you by at the same speed regardless of whether you are moving towards or away the light bulb (but their energy changes through red and blue shift Doppler effects). This was the point that the famous Michelson Morley experiment was trying to make.
As a result of these discrepancies, Einstein along with others (Lorentz, Minkowski) constructed the theory of special relativity which gives new rules for adding velocities. These can be looked up in any special relativity textbook.
Getting back to your spaceship, suppose that after a bit more than half a year, the spaceship has reached velocity of half the speed of light. The spaceship drops of a capsul which continues at constant velocity of half the speed of light with respect to Earth. The spaceship then accelerates for another half a year and gains a velocity of half the speed of light over the capsule. Intuitively, using Gallilean rules, one would then say that the speed of the spaceship with respect to the earth would now be the simple addition of spaceship/capsule velocity and capsule/Earth velocity = speed of light. This is not true, the spaceship/Earth velocity is only about 3/4 the speed of light.
This phenomenon gets worse as the ship's velocity gets closer to the speed of light and this is why even at 1 g, the ship will never reach the speed of light.
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