Length Contraction and Time
Dilation
Edmar G. Pantohan
Newtonian Relativity
• Laws of mechanics must be the same in all inertial frames.
• Time is the same in all inertial frames.
Galilean Transformation
• Galilean transformation relates measurements made in different
inertial frames.
• Time interval in both inertial frames is the same.
• ux – velocity of the object w.r.t. at rest inertial frame
• u’x – velocity of the object w.r.t. moving inertial frame
• ax – acceleration of the object w.r.t. at rest inertial frame
• a’x – acceleration of the object w.r.t. moving inertial frame
ux = u’x + v
ax = a’x
Galilean Transformation
• Ex. You are on the ground in frame S and the baseball pitcher is in the
pickup in frame S’. Find v, the velocity of S’ relative to S.
Given: ux= -10 m/s, ux’= -40 m/s
Find: v
Solution:
ux = u’x + v
v= ux - u’x = -10m/s-(-40m/s)
= -10m/s+40m/s
v= 30m/s
Therefore the baseball pitcher is moving
to the right with speed 30m/s relative to
frame S.
The Speed of Light
• However, Galilean transformation is inconsistent with Maxwell’s
theory of electromagnetism.
• In the 19th century physicist thought that Electromagnetic (EM) waves
need a medium to propagate.
• This medium is called as luminiferous ether.
• The laws of electricity and magnetism take their simplest form in
absolute frame, special frame of reference at rest w.r.t the ether.
Michelson-Morley Experiment
• If v is the velocity of the ether w.r.t.
Earth then velocity of light (c) w.r.t
to Earth for different cases is shown
to the right.
• Experimental set-up done by
Albert A. Michelson (1852-1931)
and Edward W. Morley (1838-1923)
is shown to the left.
• However their experiment fails to
detect the existence of ether.
• Conclusion: Light propagates
without a medium.
Postulates of SR
• In 1905, Albert Einstein published his special theory of relativity (SR).
• Two basic postulates:
• The laws of physics are the same in all inertial frame. For example,
Newton’s laws (Σ 𝐹Ԧ = 𝑑𝑝Ԧ) have the same mathematical form for all
inertial observers; 𝑑𝑡
• The speed of light in vacuum is always equal to 3 x 108 m/s (c=
3x108 m/s). This speed is the same with respect to all inertial
observers.
• Inertial frame is one in which an object acted by no net force will
obey Newton’s first law (Law of inertia).
• Law of inertia- when no net force applied, object will remain at rest if
initially at rest or remain in motion with constant velocity if initially in
motion.
Simultaneity
• Two events that are simultaneous in one inertial frame are in general
not simultaneous in a second inertial frame moving relative to the
first.
Time Dilation
• Time interval (Δt) measured by an at rest observer is longer (dilated)
compare to the time interval (Δtp) measured by the moving observer.
Time Dilation
• Proper time (Δtp) is the time interval between two events as
measured by an observer who sees the events occur at the same
position.
Time Dilation
Solution: (a) By observer on the ground,
Solution: (b) By observer in the plane,
Binomial Theorem
The Twin Paradox
• Speedo and Goslo were both 20 years old when they were separated.
• Goslo remained on Earth while Speedo traveled to Planet X, 20 light
years from Earth, reaching a speed of 0.95c relative to inertial frame
of Goslo.
• Upon the return of Speedo, Goslo is now 62 years old while Speedo is
33 years old.
Length Contraction
• Proper length (Lp) is the length of the object
as measured by an observer at rest relative
to the object.
• Length (L) w.r.t. at rest observer is shorter
than the length(Lp) w.r.t. the moving
observer.
• Length contraction takes place only along
the direction of motion.
Length Contraction
Length Contraction
Given: v=0.990c and lp=400m
Solution:
𝑣 2 (0.990𝑐)2
=
=
𝑐2
𝑐2
(0.990)2