Time dilation and length contraction

1
Time dilation and length contraction

• Moving clocks run slowly
• ?t ? ?t
• Moving rulers are contracted
• L L /?

• Use these formulas with caution! They apply to
  special cases
• The time interval between two events that happen
  at the same location in the other RF.
• The separation between two points at the same
  instant of time.
• The full transformations cover these cases, but
  are more general.

2
The Lorentz Transformation

• The Lorentz transformation video does a
  derivation of the transformation that is just
  hand-waving.
• Your text doesnt derive, but confirms that the
  given transformation equations work.
• We will derive it, imposing the requirement of c
  invariance.

• Simplify 1 space dimension

• Galilean Transformation

• L.T. must
• Maintain linearity and symmetry x ? x

3
Spacetime Diagrams (Minkowski Diagrams)

• Consider A, B, and C to be evenly spaced
  observers at rest. All events that occur at
  their locations fall on vertical lines (xA
  const, xB const, xC const).

• These are called the world lines of A, B, and C.

• Simultaneous events fall on horizontal lines
  (lines of simultaneity).

• Light flashes at B at t 0.

• Light arrives simultaneously at the equidistant
  points A and C.

4

• Consider A, B, and C to be evenly spaced
  observers in motion with constant velocity u.
• They are at rest in a different IRF S' moving
  w.r.t. S along x with speed u.

• World lines are inclined with slope 1/u.

• While simultaneous events in S fall on horizontal
  lines, such events are not simultaneous for the
  moving observers.

• Light flashes at B at t 0.

• Arrivals at A and C are not simultaneous in S,
  but they clearly must be simultaneous in S'
  (according to observers A, B, and C).

• Lines of simultaneity are sloped.

5

• Drawing the axes for S'
• S' is the rest frame of A, B and C. It moves at
  speed u toward x.
• Choose origins 0 and 0' to coincide at t t' 0.

• The t' axis is the world line of 0', which is x'
  0 (slope of 1/u).

• The x' axis is the line of simultaneity t' 0,
  that is, a line through 0' parallel to any line
  of simultaneity.

Note the tilting of the x' and t' axes
illustrates that they are mixtures of x' and t'.
No tilting of axes in three dimensions is implied.
6
Constructing the L.T.

• Generalized transformation

• We know that the origin 0' for RF S' moves in
  time in RF S as x ut (the equation for the x'
  axis is x - ut 0).

• Simplify 1 space dimension

• Galilean Transformation

• L.T. must
• Maintain linearity and symmetry x ? x'

7

• Next we impose the invariance of c by considering
  a light signal starting from 0 at t 0.

• Set the two expressions equal and solve for a

8
Half of the L.T.

• To get the other half, we need to solve for t and
  t. (Simplifies nicely when you use the full
  form of ?.)

• We live in a 3D world, so this is incomplete.
  Choose x and x to be in the direction of motion
  and the rest is trivial coordinates
  perpendicular to the motion are unaffected.

9
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.

10

• Two atomic clocks are synchronized. One remains
  in a laboratory in Minneapolis and the other is
  loaded onto a supersonic jet airplane and flown
  for 5 hours at 400 m/s, as measured by the clock
  left behind. What is the difference in the two
  clocks readings when they are brought back
  together?

• Simplified analysis
• Consider two IRFs, one in the lab (S) and one
  moving with constant speed u in a straight line
  (S' ).
• Not really IRFs, but close.
• Ignore accelerations at beginning and end (very
  little time).
• Both readings occur at rest in S, so ?t is the
  proper time, which differs from the elapsed time
  measured by the moving clock (?t' in S).

11

• In 2035, a spacecraft flies past a scientific
  outpost on the Moon at u 0.8 c. Scientists
  measure the length of the spacecraft to be 200 m.
  What length would they measure at a later time
  after the spacecraft lands at the station?

• No major simplifications needed here.
• Two IRFs one for the station (S) and one for the
  spacecraft, moving with speed u in a straight
  line (S' ).
• The length measured in S', the spacecrafts rest
  frame when its moving, which is the same as the
  length measured in S once the spacecraft has
  landed at the station, is the proper length.