Some examples with the invariant spacetime interval

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Some examples with the invariant spacetime
interval

• Ch 1.6

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Example 1
Two explosions are seen to occur simultaneously
on Earth at the two positions x1 1700 km
(explosion 1) and x2 3500 km (explosion 2). At
the same time, Superman goes flying by us moving
in the positive-x direction with a speed of 0.800
c.
S?
v
S
x
x2
x1
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(a) What is the distance between the two events
as seen by us? What time difference do we measure
between the two events?
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(b) Does Superman see the two events as occurring
simultaneously? If so, prove it. If not,
determine how much time Superman measures between
the two explosions and state which event
(explosion 1 or 2) Superman sees first.
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(c) What is the distance between the two
explosions as measured by Superman?
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(d) Calculate the value of the spacetime interval
(?s)2 as seen by us and by Superman.
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Be very careful with distances and lengths in
special relativity. Make sure to understand
exactly what the situation is, and what the
question is asking.
You can only use the length contraction and time
dilation equations if you are dealing with proper
lengths and proper times.
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Example 2
A space station is floating stationary with
respect to Earth in deep space at a distance of
0.240 l.y. (light year the distance traveled by
light in a vacuum in one year) from Earth. A
rocketship is traveling at 0.630 c directly away
from Earth toward the space station.
S
S?
v
In the following, distances should be kept in
l.y. cy, time in y (years), and speeds in units
of c.
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(a) How long does it take for the rocketship to
reach the space station as measured by someone on
Earth?
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(b) What is the distance between the Earth and
the space station as measured by someone on the
rocketship?
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(c) What is the position interval between the
points where the rocketship leaves the earth and
reaches the space station as measured by someone
on the rocketship?
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(d) How long does it take for the rocketship to
reach the space station as measured by someone on
the rocketship?
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(e) Verify by calculation that your results above
are correct.