Professor Dave again, let's talk about
length contraction.
Einstein's special relativity was a huge step in
physics, and even though time dilation
seems like the weirdest thing you've
ever heard, there's plenty more to get
through with this theory. As it happens,
it's not just time that's relative, it's
length as well, and as we will find out
the math tells us that when you approach
the speed of light, your measurement of
space changes just like time. This is
because if an observer on earth and an
observer in a fast spaceship register
different spans of time for an event
they must also be recording different
distances since both observers agree on
the same relative velocity between them.
Specifically, the faster you go, the
smaller objects seem to be, and the
shorter the distance you perceive
yourself to be traveling. This is a
phenomenon called
length contraction. Here we can see two
versions of a spaceship traveling very
fast to a faraway celestial object.
In one, we see things from the perspective
of an observer on earth, and in the other
we see things from the perspective of
someone on board the spaceship.
The earthbound observer sees the spaceship
moving at some velocity v, notices
time moving at the familiar rate, and can
measure some distance for the journey
which we label as L0, and refer to as the
proper length, as this is the length that
is measured by an observer that is at
rest with respect to the objects
demarcating the distance. But on the
spaceship the only thing that is the
same is the relative velocity v, though
in this case it represents the velocity
of Earth and the destination as they
move relative to the ship. And because
this velocity must be the same as for
the earthbound observer, everything else
must be different. Because of time
dilation, the time interval will be
different, delta t0 rather than delta t,
and we label the
contracted length of the journey as L.
The relationship between L zero and L is
given by this equation, which can be
derived from the time-dilation equation.
Not only do the two observers arrive at
different values for the length of the
journey, they even arrive at different
values for the length of the spaceship,
as this dimension is parallel to the
direction of travel. The earthbound
observer would see the ship as being
much shorter than the astronaut does.
In fact, this discrepancy in length neatly
explains how the two observers could
perceive different rates for the passage
of time. It also explains how fast-moving
particles can defy certain expectations,
because at such speeds, time slows down
and the distance traveled contracts, so a
particle like a muon with a half-life of
a millionth of a second at rest, is able
to exist for longer and travel further
than expected when moving near the speed
of light, due to relativistic effects.
At this point let's take a moment to make
sure we understand how to assign delta t
and L values. Delta t zero is the proper
time interval, which is the time interval
as measured by the inertial reference
frame where the two events occur in the
same location. For a space journey that's
the spaceship, because the Earth leaves
and the destination arrives while the
ship goes nowhere. Everyone else, like
someone on earth will register a longer
time interval, delta t, for this journey.
L0, the proper length however, is length
as measured by an observer that is at
rest with respect to the objects in
question, like the earthbound observer is
in this case. Anyone in motion with
respect to the objects, like the person
on the spaceship, will register some
shorter length for this distance, L. This
means that the proper time and length
may not always be measured by the same
observer, and we need to know how to
assign these in order to do the math correctly.
Let's say a ship is traveling from earth
to another system at 90 percent light
speed. The person on board measures the
journey as being 8.2 light-years in length.
How far away is the destination
according to someone on earth? Well we
can plug in 8.2 for L and 0.9c for v,
then we just do some algebra and we
should get 18.8 for L0, or the proper
length between Earth and the destination.
We still have more to go with special relativity, but first let's check comprehension.
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