Applications

1
Applications

• Completely symmetrical. It doesnt matter which
  observer is assigned to which frame (S or S), as
  long as you get the signs right.

• The Lorentz transformation equations reduce to
  the Galilean transformation for u ltlt c.
• Time dilation and length contraction follow as
  special cases.

2

• You have just certified my new rocket-speed
  record of 0.999 c from your stationary (!)
  judges stand. Unfortunately for me, 40 ns after
  I pass directly in front of you, there is a
  malfunction on my ship and an explosion occurs 4
  m from my seat toward the front. This triggers a
  second explosion 40 ns later, 4 m directly behind
  me. Because disaster is not rare in such events,
  your judges stand records everything (for the
  coroners inquest). What are the spacetime
  coordinates of the explosion in your reference
  frame?

• Requires full L.T. of spacetime coordinates to
  your frame (S) since events are separated in
  space and time in my frame (S).
• Usual coordinates choices all motion along x and
  x, with xx'0 at tt'0.

3
Comments

• The separations in space and time between the
  events in S are ?xx?-x?89.3 m and ?tt?-t?299
  ns.
• We see that the moving clock is slow, but the
  simple time dilation calculation would have given
  the wrong time separation for these two events.
• The order of these events, first ?, then ?, is
  preserved in the transformation between frames in
  this case.
• It does not have to be so for a pair of events.
  The order of two events can be reversed in one
  frame compared to another, subject to rules that
  are based on the nature of the interval that
  separates them. In particular, if there is a
  causal relationship (the occurrence of ? is the
  cause of ?), then there can be no frame in which
  the order is reversed.

4
How can two observers (Al and Hank) both be at
the center of spheres of expanding light fronts
from the same event when their positions only
coincided instantaneously at tt'0?

• S (Al)

• S' (Hank)

??
?
5
Relativistic Invariants
Specific combination of spacetime coordinates of
an event that has the same value (0) in any IRF.
More generally
Very useful for making and interpreting spacetime
diagrams.
6
More on drawing spacetime diagrams

• Plot ct vs. x instead of t vs. x
• No distinction between time and space
  coordinates!
• Light Lines (two directions) are x ?ct (at
  ?45?)

S