Special Relativity
Classical Newtonian Mechanics – its limitations
•At the beginning of the 20th century, the only theory of motion was that of
Newton, based on his three laws – ‘classical’ Newtonian mechanics.
•Newtonian mechanics was known to work well in describing the motions of
objects in the macroscopic world, whose velocities are generally slow
compared to the speed of light (c=3x105 km/s).
•But what about objects traveling at speeds approaching the speed of light?
Such objects do exist in the microscopic world, for example an electron
accelerated through a 10 MV potential difference (and hence given an
energy of 10MeV), will attain a speed of 0.9988c. [And note that in
Newtonian mechanics, there is no reason why a particle cannot have any
speed it like, with v>c!!]
•However, if we accelerate an electron through a 40MV potential difference,
giving it 4x the energy, experiment shows its does not have 2x the speed
(as expected from K=½mv2). So instead of having v~2c, it is found to have
v=0.9999c.
Experiment shows that Newtonian mechanics does not predict the
correct answers when it is applied to such fast particles; it fails in
the limit of v  c (with c appearing to be an upper limit)!
Galilean Transformations – describing events from
different reference frames
First some definitions:
A physical event – something that happens independent of the reference
frame we might use to describe it; e.g. a collision between two particles, a
flash of light. The event occurs at a point in space (x,y,z) and at an instant
of time t.
An inertial system – a frame of reference in which the law of inertia –
Newton’s 1st Law – holds. In such a system, which may also be referred to
as an unaccelerated system, a body that is acted on by zero net external
force will move with constant velocity.
While an unaccelerated frame on earth is not truly ‘inertial’ due
to the accelerating effects of the earth’s rotation and its orbital
motion around the Sun, the accelerations involved are so small
that it approximates an inertial frame.
The Theory of Special relativity deals only with the description of events
by observers in inertial reference frames; the Theory of General Relativity
concerns itself with all reference frames, including non-inertial ones.
Galilean Transformations
y’
y
S
x
v
S’
x’
vt
z
z’
●P
x’
•Consider two inertial frames S and
S’, with S’ moving at a constant
velocity v with respect to S along
common x, x’ axis. An observer in S
measures P to occur at coordinates
x,y,z and at time t, while an observer
in S’ measure P to occur at
coordinates x’,y’,z’ and at time t’.
•Fundamental assumption in making measurements in different frames
using ‘classical’ procedures: length intervals and time intervals are the
same for all inertial observers of the same events
if meter sticks are of the same length when measured and
compared at rest, they are of the same length when compared in
relative motion to one another;
if clocks are calibrated and synchronised when at rest, their
readings and rates will agree even if they are in relative motion to
one another.
Galilean Transformations
y’
y
S
x
v
S’
x’
●P
•If the clocks of each observer read
zero at the instant that the origins of
the frames S and S’ coincide, then the
Galilean coordinate transformations
are:
x’
vt
z
x’ = x – vt
y’ = y
z’
z’ = z
t’ = t
•It follows from these transformations that the time interval between two
events, P and Q, is the same for each observer:
tP’ – tQ’ = tP – tQ
and that the distance between two points, A and B, measured at a given
instant, is the same for each observer:
xB’ – xA’ = xB - xA
Einstein turned these concepts completely on their head by arguing that
the two observers would measure different lengths and time intervals in
this situation!
Galilean Transformations and Newtonian Relativity
•We can differentiate the Galilean transformations to determine the velocities
and accelerations that different inertial observers measure:
starting with:
x’ = x – vt
and differentiating with respect to t: dx’/dt = dx/dt – v
and because t’ = t, we can write
hence
d/dt = d/dt’
dx’/dt’ = dx/dt – v, dy’/dt’ = dy/dt, dz’/dt’ = dz/dt
which we can write as
ux’ = ux – v, uy’ = uy, uz’ = uz
•To obtain the acceleration transformation, we differentiate the above
velocity relations:
dux’/dt’ = d(ux-v)/dt, duy’/dt’ = duy/dt, duz’/dt’ = duz/dt
and since v = constant, d(ux-v)/dt = dux/dt
hence we have:
ax’ = ax, ay’ = ay, az’ = az
•Hence we see that the different observers (in S and S’) will assign different
velocities because of their relative motion, but will measure the same
acceleration!
Galilean Transformations and Newtonian Relativity
Important conclusions to be drawn from the above relations:
•The acceleration of a particle is the same in all reference frames which
move relative to one another with constant velocity.
•Since in classical physics, the mass is unaffected by the motion of the
reference frame, the product of mass x acceleration (ma) will be the same
for all inertial observers.
•Since F = ma is the definition of force, this will also be the same for each
inertial observer.
Newton’s laws of motion and the equations of motion of a particle are
exactly the same in all inertial frames  the laws of mechanics are the
same in all inertial frames.
Electromagnetism and Newtonian Relativity
We have just shown that the laws of Newtonian mechanics are invariant
under a Galilean transformation. Is this the case for the laws of
electromagnetism??
•Consider a pulse of light (an electromagnetic wave) traveling to the right
with respect to the medium through which it is propagated at a speed c.
The “medium” of light propagation was given the name “ether”,
historically, since the mechanical view of light that emerged from the
physics of the 19th century was inconsistent with it traveling through
empty space.
•Let’s call the “ether” frame S, and assume it to be an inertial one in which
the observer measures the speed of light to be exactly c.
•In a frame S’ moving at a constant speed v with respect to the ether frame,
an observer would measure a different speed for the light pulse, ranging
from c+v to c-v, depending on the direction of relative motion, according to
the Galilean velocity transformation.
Electromagnetism and Newtonian Relativity
Hence the speed of light is certainly not invariant under a Galilean
transformation, and there is only one inertial frame in which the speed of
light is measured to be c, and hence one frame in which Maxwell’s laws of
electromagnetism apply  what would be an absolute frame.
Attempts to locate the Absolute Frame – The
Michelson-Morley Experiment
The question as to whether the “ether” frame existed and the speed of light
was measured to be different in different inertial frames was deemed to be
so important that in 1887 Michelson and Morley devised an experiment to
determine whether this was the case.
•If an ether exists, the spinning and orbiting earth should be moving through
it. An observer on earth would sense an “ether wind” whose velocity is v
relative to earth, where v is ~30 km/s (v/c ~ 10-4) the earth’s orbital speed
about the Sun.
•It was Michelson’s invention of the optical interferometer that provided a
device sensitive enough to make the measurement (1 in 10,000 effect).
The Michelson-Morley
Experiment
The interferometer consists of a beam
of light (from source S) that is split by a
partially silvered mirror M into two
coherent beams, beam 1 being
transmitted through M and beam 2
being reflected off M. Beam 1 is
reflected back to M by mirror M1 and
beam 2 by mirror M2. Then the
returning beam 1 is reflected and the
returning beam 2 is transmitted by M
back to a telescope T where they
interfere. The interference is
constructive or destructive depending
on the phase difference of the beams.
If M1 and M2 are very nearly but not
quite at right angles, a fringe pattern is
observed in the telescope (similar to a
thin wedge of air between two glass
plates).
Suppose v is the velocity of the ether
with respect to the interferometer.
The Michelson-Morley Experiment
•The phase difference between beams 1 and 2 can arise from two causes:
(i)different path lengths travelled, (ii)different speeds of travel with respect
to the instrument because of the “ether wind”. The latter is the crucial one
with beam 1 being a ‘up-stream’ and ‘down-stream’ situation, and beam 2
being a ‘cross-stream’ situation.
•If the instrument is rotated by 90°, thereby making beam 1 the crossstream beam and beam 2 the up-stream/down-stream beam, it can be
shown that this will cause a shift in the fringe pattern, since it changes the
phase relationship between the two beams. The key to detecting this is to
make the lengths of the paths the two beams travel to be as long as is
practically possible; M-M were able to achieve path lengths of ~11m.
•The fringe shift is given by N = (2l/)(v/c)2, where l is the path length,
and  the wavelength of the light. Putting in the values for M-M’s
experiment, we get N = (22/5.5 x 10-7)(10-4)2 = 0.4  four-tenths of a
fringe!!!
•M-M mounted their interferometer on a massive stone slab for stability
and floated it in mercury for smooth rotation. The fringes were observed
under a continuous rotation, and observations were made day and night.
The Michelson-Morley Experiment
•A precision of 0.01 of a fringe was achieved in measuring their shift!
•Despite making observations during all seasons of the year (and hence
with the earth at different locations about the Sun and at different
orientations with respect to the “ether”, NO FRINGE SHIFT WAS
OBSERVED! M-M concluded from their experiment that there was no
fringe shift at all.
•This null result (N = 0) was such a blow to the ether hypothesis that the
experiment was repeated many times by different workers over a 50-year
period. The null result was amply confirmed!
Postulates of Special Relativity
Einstein published his Theory of Special Relativity in 1905 (hence this year
is being celebrated as the World Year of Physics in acknowledgement that it
is 100 years since Einstein’s “Annus Mirablis”). The assumptions or
postulates on which his theory was based are:
•The laws of physics are the same in all inertial reference frames (The
Principle of Relativity)
• The speed of light in free space has the same value c in all inertial
reference frames (The Principle of the Constancy of the Speed of Light)
The latter principle contradicts the Galilean velocity transformation, but is
consistent with the Michelson-Morley experiment. However, it is the ability
of a theory to explain experimental results upon which its success is judged,
and here Special Relativity was not only able to explain existing results, but
it also predicted new effects which were confirmed later by experiment.
Furthermore, no experimental objection to Special Relativity has yet been
found.
Simultaneous Events
Suppose one observer S notes that two independent events occur at the
same time. Suppose another observer S’, who is moving at constant velocity
v with respect to S, also records the same two events.
Q: Will S’ observe the two events to occur simultaneously?
A: No!!
Simultaneity is not an absolute concept but a relative one, depending
on the state of the observer.
It was this very fact that led Einstein to recognise that time is not absolute for
all observers (as assumed in the Galilean transformations) – since timing is
intimately related to timing events – and of all the features of the classical
theory, it was the measurement of time that was the most suspect.
Lets look at this more closely via an example:
•Consider two spaceships which provide inertial frames for their observers Sally
and Sam who are located at the midpoint of each ship. Sally’s ship is moving
with a velocity v relative to Sam’s ship. Two large meteorites strike the ships,
and leave a permanent mark on each ship at positions R, R’ and B, B’.
•Suppose the wavefronts from the
two events happen to reach Sam
at the same time (c). Sam also
determines by measurement that
he is stationed halfway between
the marks B and R on his ship. He
will rightly conclude that the two
events occurred simultaneously.
•For Sally, the wavefront from R’
reaches her (b) before the
wavefront from B’ does (d) . And
having determined that she is
halfway between B’ and R’, she
concludes the two events were not
simultaneous.
Simultaneous Events
•Sam and Sally do not agree on whether the events were simultaneous, but
both are correct  the relativity of simultaneity!
•Of key importance in considering what is observed in this situation is the
Second Postulate of Special Relativity in that the speed of light, c, is the
same in both reference frames.
The Relativity of Time
The relativity of simultaneity has an important implication for measuring
time: if two observers measure the time interval between a given pair of
events, they will not agree as to how long that interval is.
Consider another situation:
Sally is on a train that is moving with uniform velocity v with respect to the
station. She has an electronic clock, which she uses to measure the time
to between two events:
1. Turning on of flash bulb B
2. Arrival of light back at its source after reflection from a mirror on the
ceiling
The Relativity of Time
•For the time interval between these two events Sally finds:
to = 2D/c
where D is the distance between the source and the mirror. For Sally, these
two events occur at the same place and she can time the interval between
them with a single clock C located at that place.
A time interval between two events at the same location, as measured
by a stationary clock at that location, is called a proper time interval,
identified by the subscript zero.
•For Sam, who is standing on the station platform watching the train go by,
these two events will appear different. As shown in the diagram, while the
light beam still appears to travel at speed c, it travels a larger distance given
by 2L (cf. 2D for Sally). Hence the time interval between the two events is:
t = 2L/c
where L =  (½vt)2 + D2)
And we can express D in terms of the proper time: L = (½vt)2 + (½cto)2
The Relativity of Time
• Now L = ½tc, so substituting into the previous equation and solving for
t, gives:
t = to/1 – (v/c)2
Hence the time interval between the two events that Sam measures is
longer than the proper time that Sally measures  TIME DILATION
The key difference here is that Sam sees the two events occur at
different places in his reference frame, and so to measure t he must
use two synchronised clocks, one at each place. Thus the interval he
measures is not a proper time. Hence the situations for Sam and Sally
are not symmetrical.
•Time dilation is very real and has nothing to do with any mechanical
change that takes place in a clock because of its motion; it is simply the
nature of time.
Lorentz Factor: this is the dimensionless 1/1 – (v/c)2 term that appears in
the time dilation equation, and which gets increasingly larger as v  c. This
factor appears in almost all the relativity equations and is denoted by .
The Relativity of Length
Measuring length of object at rest (with respect to observer): note the
position of the two ends on a long stationary scale and subtract the two
readings – no worries!
Measuring length of a moving object: need to measure the position of the
two ends simultaneously (in observer’s reference frame).
 Because simultaneity is relative and it enters into length measurements,
then length is also expected to be a relative quantity and differ for observers
in different inertial frames.
Because of this relativity, we need to define proper length – this is
the length Lo of an object, that is measured in an inertial reference
frame in which the object is at rest.
Length Contraction
Consider once more the two observers Sally and Sam, Sally seated on a
train moving through a station and Sam on the station platform. Suppose
they both want to measure the length of the platform.
Length Contraction
First step: we need to ask the question, which of the two observers
measures (or can measure) the proper length? In this case it is Sam, since
the platform is at rest with respect to him.
Sam also notes that a marker on the train covers this length in a time:
t = Lo/v
where v is the speed of the train. That is Lo = v t.
The time interval t is not a proper time interval because the two
events that define it – that is, when the marker passes the beginning of
the platform and when the marker passes the end of the platform –
occur at different places and Sam must use two synchronised clocks to
measure the time interval t.
For Sally, however, the platform is moving, and she sees it approach and
then recede at speed v, and finds that the two events measured by Sam
occur at the same place in her reference frame. So she can time them with
a single stationary clock, so the time interval she measures is the proper
time, to. To her, the length L of the platform is given by L = v to.
Length Contraction
So we have Lo = v t (Sam), and L = v to (Sally). Dividing one by the other:
L/Lo = (vto)/(vt) = 1/
(since t =  to). Hence:
L = Lo/
and so the length L that Sally measures is shorter than the proper length,
and so this is called the length contraction equation.
Lorentz Transformations
y’
y
S
x
v
S’
x’
vt
z
z’
●P
x’
•To conclude this section, it is
appropriate to revisit the Galilean
Transformations and see what form
they take under Special Relativity.
The transformations that describe the
space and time coordinates of events
in different inertial reference frames in
this case are called the Lorentz
Transformation.
•Yet again, we consider two inertial frames S and S’, with S’ moving at a
constant velocity v with respect to S along common x, x’ axis. An observer
in S measures P to occur at coordinates x,y,z and at time t, while an
observer in S’ measure P to occur at coordinates x’,y’,z’ and at time t’.
•The Galilean transformations were x’ = x – vt, y’ = y, z’ = z, and t = t’.
•The Lorentz transformations are:
x’ =  (x – vt)
y’ = y
z’ = z
t’ =  (1 – vx/c2)