The Lorentz transformation

Problem:

In the year 2310 construction of the large space colony “Heavy Metal“ is finally completed.  The distance between the “East” and “West” edges of the colony is 400 km in the rest frame of the colony.  To celebrate the end of construction, two flares are lighted at midnight during a New Year's celebration at the “East” and “West” edges.  The light from the flares reaches an observer at the center of the colony at the same time.  A fast spacecraft flying with a constant speed of v = 0.95 c “East” to “West”.  Let Event 1 be flare is lighted on “East” edge and Event 2 be flare is lighted on “West” edge.  In the reference frame of the spacecraft, does Event 1 occur before, after, or at the same time as Event 2?  Explain your reasoning!

Solution:

Problem:

A space traveler, moving with velocity vi with respect to Earth synchronizes his clock with a friend on Earth at t' = t = 0.  The earthman then observes both clocks and simultaneously reads the times t and t' (t directly and t' through a telescope).  What does t read when t' reads 1 hour?

Solution:

Problem:

For an observer on Earth, the star Sirius is 8.8 ly away.  Assume Sirius is stationary with respect to Earth.  At t = 0, a spaceship moving with velocity v = 0.95 c i towards Sirius passes Earth.  Earth and the ship synchronize clocks. 
An observer on Earth tells the following story at a much later time:
At t = 0, just as the spaceship passed Earth, a flare erupted on Sirius.  The light from the flare reached Earth at t = 8.8 y.  At t = 0.9 years a comet hit Earth.  At t = 9.26 years the spaceship passed Sirius and sent us a light signal.  The signal reached us at 18.06 years.
How does an observer on the ship tell the same story?

Solution:

Problem:

A pyramid in the Egyptian desert (see the figure) has sides of length l inclined at an angle θ according to Egyptian references.  An archeologist at the base of the pyramid begins moving up the side of the pyramid and reaches the top of the pyramid in time T according to observers on the ground.  A spaceship, moving in the positive x-direction, is approaching the pyramid at a relativistic velocity v.
(a)  According to observers in the spaceship, how far did the archeologist move and how long did it take him to reach to top of the pyramid?
(b)  Give two calculations of the invariant space-time interval ∆s2 for the archaeologist's ascent: one, using the inertial frame of the spaceship, and the other, using the inertial frame of the ground.
For part (c), assume that the archeologist moved up the pyramid at a constant velocity v0 = l/T.  Do not neglect v0 compared to c.
(c)  Calculate the proper time required for the archeologist's ascent.

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Solution:

Problem:

Consider three reference frames.  A meter stick is at rest in frame K2.  It is positioned on the x-axis, from x = 0 to x = 1m.  Frame K2 moves with velocity v = v2j in the positive y-direction with respect to frame K1.  Frame K3 moves with velocity v = v3i in the positive x-direction with respect to frame K1.
(a)  Find the velocity of the stick in K3.
(b)  Find the length of the stick in K3.
(c)  Find the angle θ the stick makes with the x-axis in K3.

Solution:

Problem:

A spaceship travels with velocity v = vi  with respect to a space station.  In the frame of the space station a linear structure of length L = 10 km moves with velocity u = (c/2)j in the positive y direction.  The structure lies in the xy-plane and makes an angle θ = 7.2o with the x axis?  When viewed from the spaceship, the structure is aligned with the x-axis.
(a)  What is the velocity v of the spaceship (magnitude and direction)?
(b)  What is the length of the structure in the frame of the spaceship'?
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Solution:

Problem:

A spaceship has length 100 m in its own reference frame.  It travels at v = 0.95 c with respect to earth.  Suppose that the tail of the spaceship emits a flash of light.
(a)  In the reference frame of the spaceship, how long does it take the light to reach the nose?
(b)  In the frame of the earth, how long does it take the light to reach the nose?

Solution:

Problem:

(a)  Observer A is stationary in a laboratory on Earth's surface and observer B is in a laboratory moving at high velocity parallel to Earth's surface (neglect the curvature of the earth in the subsequent considerations) and parallel to the x axis.  Assume that the observers each witness two separate lightning strikes on the surface of the earth lying along the x axis.
(i)  Describe concisely a practical scheme whereby observer A can determine whether the two lighting strikes occurred simultaneously in her reference frame.
(ii)  Show that if observer A concludes from her observations that the strikes were simultaneous, observer B generally must conclude from his observations of the same events that the strikes are not simultaneous, thus demonstrating that simultaneity is a relative and not absolute concept.
(b)  Einstein was strongly inspired by Maxwell's theory of electromagnetism and viewed Galilean invariance for transformations between reference frames as being seriously flawed because it was inconsistent with Maxwell's theory.  The special theory of relativity removed this inconsistency.  Explain the meaning of the preceding two statements.

Solution:

Problem:

(a)  Show that if two events are separated in space and time so that no signal leaving one event can reach the other, then there is an observer for whom the two events are simultaneous.
(b)  Show that the converse is also true: if a signal can get from one event to the other, then no observer will find them simultaneous.
(c)  Show if a signal can get from one event to the other, then there is an observer for whom the two events have the same space coordinates, i.e. for whom the two events “happen at the same place”.

Solution:

Problem:

Two events "occur" at the same place in the laboratory frame of reference and are separated in time by 3 seconds.
(a)  What is the spatial distance between these events in a rocket frame moving with respect to the laboratory frame, in which the events are separated in time by 5 seconds?
(b)  What is the relative speed of the moving frame and the laboratory frame?

Solution: