7.1 Effects as One Approaches the Speed of Light
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
7.1 Effects as One Approaches the Speed of Light
To begin, consider two observers, A and B. Let A be here on Earth and
be considered at rest for now. B will be speeding past A at a highly
relativistic speed as he (B) heads towards some distant star. If B's speed
is 80% that of light with respect to A, then gamma for him (as defined in
Section 1.4) is 1.6666666... = 1/0.6. So from A's frame of reference, B's
clock is running slow and B's lengths in the direction of motion are shorter
by a factor of 0.6. If B were traveling at 0.9 c, then this factor becomes
about 0.436; and at 0.99 c, it is about 0.14. As the speed gets closer and
closer to the speed of light, A will see B's clock slow down infinitesimally
slow, and A will see B's lengths in the direction of motion becoming
infinitesimally small.
In addition, If B's speed is 0.8 c with respect to A, then A will see
B's energy as a factor of gamma larger than his rest-mass energy (Note, I
use an equation for energy here defined in Section 1.5, Equation 1:8):
(Eq 7:1)
E(of B in A's frame) = gamma*m(B)*c^2 = 1.666*[m(B)*c^2]
where m(B) is the mass of observer B. At 0.9 c and 0.99 c this factor is
about 2.3 and 7.1 respectively. As the speed gets closer and closer to the
speed of light, A will see B's Energy become infinitely large.
Obviously, from A's point of view, B will not be able to reach the
speed of light without stopping his own time, shrinking to nothingness in
the direction of motion, and taking on an infinite amount of energy.
Now let's look at the situation from B's point of view, so we will now
consider him to be at rest. First, notice that the sun, the other planets,
the nearby stars, etc. are not moving very relativistically with respect to
the Earth; so we will consider all of these to be in the same frame of
reference. Remember that to A, B is traveling past the earth and toward some
nearby star. However, in B's frame of reference, the earth, the sun, the
other star, etc. are the ones traveling at highly relativistic velocities
with respect to him. So to him the clocks on Earth are running slow, the
energy of all those objects becomes greater, and the distances between the
objects in the direction of motion become smaller.
Let's consider the distance between the Earth and the star to which B
is traveling. From B's point of view, as the speed gets closer and closer to
that of light, this distance becomes infinitesimally small. So from his
point of view, he can get to the star in practically no time. (This explains
how A seems to think that B's clock is practically stopped during the whole
trip when the velocity is almost c. B notices nothing odd about his own
clock, but in his frame the distance he travels is quite small.) If (in B's
frame) that distance shrinks to zero as his speed with respect to A goes to
the speed of light, and he is thus able to get there instantaneously, then
from B's point of view, c is the fastest possible speed.
From either point of view, it seems that the speed of light cannot be
reached, much less exceeded. This, then, is the "light speed barrier", but
most concepts people have in mind for producing FTL travel explicitly deal
with this problem (as we will see). However, the next problem isn't
generally as easy to get away with, and it probably isn't as well known
among the average science fiction fan.
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