Suppose you're traveling on a space ship at 9/10 the speed of light (.9c). You have a high-powered rifle that shoots bullets at the same speed. Suppose you shoot the bullet perpendicular to your direction of travel.
It appears that the bullet would travel at a 45-degree angle (northeast, if the ship is traveling north and the bullet is shot eastward), at about 1.2728c which is faster than light. Why is this wrong, and what would the actual speed and direction be?
As TomM and Danny suspected, the angle is indeed no longer 45 degrees and length contraction is the reason for this effect.
According to special relativity, an if an object is moving at high
speed relative to an observer, that object appears be contracted in the
direction of its movement (it looks shorter). The factor by which
an object appears to be shortened is equivalent to sqrt(1-v^2/c^2)
where c is the speed of light. For a speed of 0.9c, this gives us
a factor of about 0.4359. Thus, to an observer on earth, the
spaceship appears to be only 43.59% of its original length.
Likewise, to observers on the spaceship,
the earth appears to be 43.59% of its original length. This is because to the observer on the spaceship, the earth appears to be moving past at 9/10 the speed of light.
Now consider the bullet. To an observer on the ship, the bullet
appears to be moving away from the ship at 9/10 the speed of
light. Call the shortened length of earth one unit of
distance. As the spaceship moves from one side of the shortened
earth to the other, it has moved one unit of distance, and the bullet
(since it is moving away from the ship at 0.9c) has moved one unit away
from the ship. To an observer on the earth, however, the bullet
has moved one unit away from the ship as it goes past the earth, but it
has gone 1/0.4359 or about 2.2942 units in the direction of travel of
the ship (since observers on the earth think the earth is wider).
Thus, according to the earthling, the bullet's trajectory makes an
angle of arctan(1/2.2942) or about 23.55 degrees with the trajectory of
the ship. Therefore, the earthling sees the bullet as moving at a
speed of 0.9c / cos(23.55) or 0.9818c, which is still less than the
speed of light.
From this analysis, we see that if increasing
the speed of the ship and the bullet decreases the perceived
angle. As the ship and bullet speed approach c, the angle
approaches zero, though to the ship, the bullet still appears to be
moving away with the given speed. At the speed of light, the
angle is indeed zero, which doesn't seem to make sense until we
remember that time dialation is infite (the earthling sees time as
stopped on the ship, so the bullet would never get around to leaving
the ship!)
The questions as to why time dialation and length
contraction occur in the first place, and why the factor is
sqrt(1-v^2/c^2) are a little more complicated, but are ultimately
understandable with high school algebra and geometry. I urge
anyone who's interested to read Einstein's own book on the subject
(titled, interestingly,
Special Relativity) which contains finer explanitions than mine, geared towards the casual audiance.
Edit: TomM: you posted your comment just as it wrote this :) Yes, your analysis is correct!
Edited on January 18, 2005, 9:02 am
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