26.1 Introduction


Experimental results indicate that it is impossible to accelerate matter to the speed of light.  The ultimate speed limit!


Combining Newtonian Mechanics with electrostatics does not produce the physics of moving charges  electrodynamics!


1905  Einstein  Special Relativity  Can solve these problems by re-deriving physical laws with 2 postulates:


  1. The laws of physics are the same in all coordinate systems at rest or moving at constant velocity with respect to one another.


  1. The speed of light in a vacuum has the same value, 3x108 m/s, regardless of the velocity of the observer or of the velocity of the source emitting the light.


This theory, supported by all known experiments, had a profound effect on the development of physics and the way we view the universe.
26.2 The Principle of Galilean Relativity


Newton’s 1st Law  Law of Inertia -  


Inertial Frame of Reference  one where  


Galilean Relativity  


The laws of mechanics must be the same in all inertial frames of reference.


Or (said another way)


There is no preferred frame of reference for describing the laws of mechanics.


Do the laws of physics suddenly take on new forms when you fly on an airplane? Of course not! BUT, the observed motion of objects might differ for observers in different inertial frames. The difference is what we call a “Galilean Transformation”


Simple example of a Galilean Transformation from everyday experience:



Here, the “stationary” observer sees the ball with a forward velocity. The observer on the airplane does not.  The actual trajectory of objects might differ, but the form of the laws would be the same (i.e. , etc).

26.3 The Speed of Light


Applying Galilean relativity to the speed of light relative to a moving observer and a stationary observer, suppose light emitted with speed c by source attached to S’:



Newtonian/Galilean Relativity predicts speed of light relative to S to be v+c. This contradicts experiments.


Water & sound waves require a medium of propagation. What about light?


One can get electromagnetism form mechanics & electrostatics only in the case there light traveled in a medium and the observer were stationary in this medium. The medium would be massless but infinitely rigid.  The Luminiferous Ether. This would be THE preferred frame of reference that would give correct results. Does it exist?


Due to the orbit of the Earth around the Sun, and the Sun through the Galaxy, we as observers ought to travel through the ether with different speeds at different times. Depending on the direction the light is propagating, it might have to go “upstream”, “downstream”, or “tack against the wind” of the ether:


Suppose the Earth were moving through the ether with a relative speed of :


Light propagation “downwind”:  








Light propagation “upwind”:  







Propagation “across wind”:  


All attempts to detect this ether drift failed.




26.4 The Michelson-Morley Experiment


Is there an absolute reference frame? A “luminiferous ether” medium of propagation of electromagnetic radiation?


1887  Michelson & Morley Experiment  speed of light in a “vacuum” is independent of the motion of the source!


The time it took the light to travel the horizontal path was the same as that of the vertical path. This should not be the case if light had to “swim upstream and downstream” along one of the paths, and “tack into the wind along the other path. (see p. 811-812 for mathematical details)


What goes?


There is no ether, and the speed of light is a constant regardless of the direction of propagation, the motion of the source, or the motion of the observer!


With no preferred frame (at rest w.r.t. ether) everyone must get the same laws of physics.


Einstein basically said to take it at face value, regardless of how it contradicts our intuition. (Our intuition is based on our experiences, which usually occur at . They need not be valid at high speeds!) Get over it!


26.5 Einstein’s Principle of Relativity


  1. The principle of relativity: All the laws of physics are the same in all inertial frames.


  1. The constancy of the speed of light: The speed of light in a vacuum has the same value,
    , in all inertial reference frames, regardless of the velocity of the observer or the velocity of the source emitting the light.


All the laws of physics, not just those of mechanics, are the same for all inertial observers, i.e. there is no preferred inertial frame of reference.



26.6 Consequences of Special Relativity


  1. There is no “absolute length” or “absolute time”. The values of measured Lengths and Time Intervals depends on the motion of the observer.


  1. Events that appear simultaneous to one observer may not appear so to another observer.



Here, the observer standing on the ground by the railroad tracks would see both lightning strikes at the same time, while the observer on the railroad car would not. Was the strike at the front of the car simultaneous with the strike at the rear of the car? Who decides? Both “conclusions” about the presence or lack of simultaneity are equally valid!


Time Dilation


Suppose you flew a rocket ship parallel to a perfectly reflecting mirror, and shown a flashlight beam out the widow so that it came right back to the flashlight.




Now suppose you were stationary on the ground where the mirror is mounted. What would you see?






Basketball = the person dribbling the ball sees a different trajectory than others on the court. The main difference is that they also see the ball with different speeds, and this is not true for light!


Now, replace the simple beam of light generated in the spaceship with a light source (say, a clock face) pulsing every 1 sec for the observer in the ship.


Observer in the Rocket:          


Outside Observer:                     

The outside observer will see the clock time of the ship advance 1 sec for every 2.3 sec that his own clock ticks. The stationary observer sees the ship’s clock running slower!


Okay, now here is where intuition really fails us. Because there are no preferred inertial (v =  constant) frames of reference, the observer on the ship must see the same thing occurring to the clock of the outside observer! As long as the velocity of the ship is constant, both observers will see the same effects.


Sounds crazy! Can this really be correct?


Produced in upper atmosphere


Decay time  




If no SR, then it should be able to travel




Yet we see them at the Earth’s surface 4.8 km from region of origin!




So the distance traveled as measured by an observer on Earth is




Yes it really works!


Example: Problem #9


A muon formed high in Earth’s atmosphere travels at a speed v = 0.99c for a distance of 4.6 km before it decays into an electron, a neutrino, and an antineutrino (  ). (a) How long does the muon live, as measured in its reference frame? (b) How far does the muon travel, as measured in its frame?


The Twin Paradox


Start with 2 twins, one of whom travels to a distant star at relativistic speed, the other stays home on Earth. Due to time dilation, the traveling twin ages more slowly, returning to see their twin having aged more.


How is this possible? Wouldn’t the twin in the spacecraft see their twin age equally slowly?


Resolution  The trip includes noninertial motion for the spacecraft, which has to accelerate & decelerate along the way. (NOTE: You book doesn’t mention it, but wouldn’t the twin on the spacecraft see their twin on Earth do the same thing? However, having the spacecraft accelerate isn’t quite the same as having the rest of the entire universe accelerating!)


Length Contraction


If  is the length of an object measured at rest, then if the object is measured while in motion:

,  with the length contracting along the direction of motion.

Quick Quiz 26.3

You are observing a rocket moving away from you. Compared to its length when it was at rest on the ground, you will measure its length to be:  (a) shorter,  (b) longer  (c) the same. Now you see a clock through the window of the rocket. Compared to the passage of time on the watch on your wrist, you observe that the passage of time on the rocket’s clock is  (d) faster  (e) slower, or (f) the same. Answer the questions if the rocket turns around and comes toward you.


Example: Problem #13

A supertrain of proper length 100 m travels at a speed of 0.95c as it passes through a tunnel having proper length 50 m. As seen by a trackside observer, is the train ever completely within the tunnel? If so, by how much?

26.7 Relativistic Momentum


In Newtonian mechanics, . However, in order to have the law of the conservation of momentum apply in all inertial frames, the momentum equation must become







In summary:   




















Warning: when using the time dilation formula, keep in mind how it is defined (i.e., whose clock runs slow from whose viewpoint).


Example: Problem #19

An unstable particle at rest breaks up into two fragments of unequal mass. The mass of the lighter fragment is 2.50 x 1028 kg, and that of the heavier fragment is 1.67 x 1027 kg. If the lighter fragment has a speed of 0.893c after the breakup, what is the speed of the heavier fragment?



26.8 Relativistic Addition of Velocities (read)




Total Energy




There is rest energy.





Example: 1 kg of H:



Example: Problem #28

A proton moves with a speed of 0.950c. Calculate its (a) rest energy, (b) total energy, and (c) kinetic energy.


Example: Problem #32

Determine the energy required to accelerate an electron from (a) 0.500c to 0.900c and (b) 0.900c to 0.990c.


Thought for the day: Based on what you know now (and think of Problem #32), how much energy would be required to accelerate the electron from 0.990c to 1.000c?


Energy and Relativistic Momentum


For  and  it can be shown that:




For a particle at rest, , so the energy is just the rest energy.



Photons have zero mass, so  

Massless photons carry momentum!


Comets consist of an agglomeration of ice and “dust”, tiny bits of rock often as small as 10-3 mm in size. As they approach the Sun, the ice sublimes, and the outflowing water vapor carries the dust with it.


The momentum the photons carry is transferred to the dust to create the beautiful dust tails that we all know and love.


(Go out early some evening this month, look in the west, and see Comet C/2001 Q4 NEAT!)

Comet C/1995 O1 Hale-Bopp