4. Paradoxes and Solutions (Relativity and FTL Travel)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
4. Paradoxes and Solutions (Relativity and FTL Travel)
One misleading statement many people hear in connection with relativity
is something like this: "Time moves slower for you as your speed increases."
It is misleading because it implies some incorrect concepts. It implies that
there is an ABSOLUTE way to decide whether or not someone is truly at rest
or moving (at a constant, non-zero velocity) when in reality this depends on
your frame of reference. It implies that if you are moving at a constant
velocity, then your clock is moving slower than some sort of "correct" clock
which is truly not in motion. It also implies that you yourself might find
your clock ticking slower than usual.
However, as I have mentioned earlier, motion is relative. There is no
way to say that one object is truly at rest and another is truly moving at a
constant velocity. You can only say that one object is moving at a constant
velocity RELATIVE TO another object. You can say that in the frame of
reference of one observer (call him Joe) another observer (call her Jane) is
moving at a constant velocity. Then, in Joe's frame of reference, Jane's
clock is running slowly, and she is length contracted in the direction of
her motion. However, in Jane's frame of reference, JOE is the one who is
moving at a constant velocity relative to her. Because the laws of physics
are the same for all inertial frames, we must be able to apply the same laws
to Jane as we just applied to Joe. Thus, in Jane's frame, Joe's clock is the
one which is running slowly, and Joe is length contracted in the direction
of his motion.
This leads one to question whether or not relativity contradicts
itself. If all motion is relative, we have concluded that each observer
believes that the other observer's clock is running slowly, and each
believes that the other observer is length contracted in the direction of
motion. Isn't that a contradiction? For example, how can Jane's clock be
running slower than Joe's AND Joe's clock be running slower than Jane's?
Well, these questions lead to various solvable paradoxes in special
relativity.
As a note, the word "paradox" has a few different meanings, and when I
use it here, I will have this meaning in mine: "a paradox is a statement
that seems contradictory or absurd but that may in fact make sense." A
"solvable paradox" is then a paradox that does in fact make sense when
explained correctly, while an "unsolvable paradox" is a paradox for which
the statement "may in fact make sense" doesn't hold (i.e. an unsolvable
paradox is truly self-contradictory).
The paradoxes in special relativity are solvable, and below I will
present two of these paradoxes along with their solutions.
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