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PHY3101 Detweiler
January 28, 2008
Relativity
Notes on lecture 9
Newtonian Doppler effect
Alice is at rest in the unprimed coordinate system, and Bob is moving with a speed v relative
to Alice along the positive x-axis. Bob points a light-source along the positive x-axis and he
turns the light-source on at event #1 just so that t1 = 0 and x01 = 0 in his coordinates, and
also t1 = 0 and x1 = 0 in Alice’s coordinates. Bob lets the light-source emit a pulse of N
waves of light, and he turns the light off at event #2 at time t2 where x02 is still x02 = 0. The
time between events #1 and #2 is ∆T = t2 − t1 . Remember this is Newtonian! When Bob
turns the light-source off at the time t2 , the leading edge of the pulse of light has traveled
to x3 = c∆T , because the light has been traveling for a time ∆T . So the length of the pulse
of light in Bob’s frame of reference is
x03 − x02 = c∆T,
and Bob would measure the wavelength of the light to be
λB =
x03 − x02
c∆T
=
.
N
N
In Alice’s frame of reference, Bob turns the light-source on at event #1 where
t1 = 0
and
x1 = 0.
And Bob turns the light source off at event #2 at time t2 , but where the position in Alice’s
frame of reference is
x2 = x02 + vt2 = vt2 = v∆T,
because the light-source moves in Alice’s frame of reference. Event #2 marks the trailing
edge of the pulse of light. Also, at time t2 in Alice’s frame of reference the leading edge
of the pulse of light is at x3 = ct2 = c∆T because the light has been traveling for a time
t2 = ∆T . So the length of the pulse of light in Alice’s frame of reference is
x3 − x2 = c∆T − v∆T
and Alice would measure the wavelength of the light to be
λA =
x3 − x2
c∆T − v∆T
c∆T (1 − v/c)
=
=
= λB (1 − v/c).
N
N
N
We can also use the relationship that λf = c to find the Doppler effect on frequencies,
fA =
c
c
=
= fB /(1 − v/c).
λA
λB (1 − v/c)
These results follow for the case that the light is being emitted by Bob in the same direction
that he is moving relative to Alice. If Bob were to move in the direction opposite to that of
the light, then the wavelengths and frequencies would be related by the same formulae as
above, except that v → −v.
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The Doppler effect for light makes the observed frequency of the light higher (wavelength
shorter, blue shift) if the source is moving in the same direction that the light moves (source
moves towards the receiver) and the observed frequency lower (wavelength longer, red shift)
if the source moves in the opposite direction as the light moves (source moves away from
the receiver).
Relativistic Doppler effect
The significant difference between the Newtonian and the Relativistic treatments of the
Doppler effect resides in the different time intervals measured by Bob and Alice, ∆T 0 and
∆T , between when the light-source is turned on and then turned off. The proper time
∆T 0 between these two events is measured by Bob, because the events occur at the same
place p
in his frame of reference. The difference in time measured by Alice is then ∆T =
0
∆T / 1 − v 2 /c2 .
Alice is at rest in the unprimed coordinate system, and Bob is moving with a speed
v relative to Alice along the positive x-axis. Bob points a light-source along the positive
x-axis and he turns the light-source on at event #1 just so that t01 = 0 and x01 = 0 in his
coordinates, and also t1 = 0 and x1 = 0 in Alice’s coordinates. Bob lets the light-source
emit a pulse of N waves of light, and he turns the light off at event #2 at time t02 where x02
is still x02 = 0. The proper time between events #1 and #2 is the time interval ∆T 0 = t02 − t01
that Bob measures because the events occur at the same place in his frame of reference.
When Bob turns the light-source off at the time t02 , the leading edge of the pulse of light has
traveled to x03 = c∆T 0 , because the light has been traveling for a time ∆T 0 . So the length
of the pulse of light in Bob’s frame of reference is
x03 − x02 = c∆T 0 ,
and Bob would measure the wavelength of the light to be
λB =
x03 − x02
c∆T 0
=
.
N
N
In Alice’s frame of reference, Bob turns the light-source on at event #1 where
t1 = 0
and
x1 = 0.
And Bob turns the light source off at event #2 where
t02 + vx02 /c2
t0
∆T 0
t2 = p
=p 2
=p
1 − v 2 /c2
1 − v 2 /c2
1 − v 2 /c2
and
x0 + vt02
vt02
v∆T 0
x2 = p 2
=p
=p
,
1 − v 2 /c2
1 − v 2 /c2
1 − v 2 /c2
from the Lorentz transformations. Event #2 marks the trailing edge of the pulse of
light. Also, at time t2 in Alice’s
frame of reference the leading edge of the pulse of
p
light is atp
x3 = ct2 = c∆T 0 / 1 − v 2 /c2 because the light has been traveling for a time
t2 = ∆T 0 / 1 − v 2 /c2 . So the length of the pulse of light in Alice’s frame of reference is
c∆T 0
v∆T 0
x3 − x2 = p
−p
1 − v 2 /c2
1 − v 2 /c2
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and Alice would measure the wavelength of the light to be
s
s
(c − v)∆T 0
1 − v/c
(1 − v/c)2
1 − v/c
x3 − x2
= p
= λB p
= λB
= λB
.
λA =
N
(1 − v/c)(1 + v/c)
1 + v/c
N 1 − v 2 /c2
1 − v 2 /c2
We can also use the relationship that λf = c to find the Doppler effect on frequencies,
s
s
c
c
1 + v/c
1 + v/c
fA =
=
= fB
.
λA
λB 1 − v/c
1 − v/c
These results follow for the case that the light is being emitted by Bob in the same direction
that he is moving relative to Alice. If Bob were to move in the direction opposite to that of
the light, then the wavelengths and frequencies would be related by the same formulae as
above, except that v → −v.
The Doppler effect for light makes the observed frequency of the light higher (wavelength
shorter, blue shift) if the source is moving in the same direction that the light moves (source
moves towards the receiver) and the observed frequency lower (wavelength longer, red shift)
if the source moves in the opposite direction as the light moves (source moves away from
the receiver).
Definition: Interval
The interval between two events is defined to be
±interval2 = −c2 ∆t2 + ∆x2 + ∆y 2 + ∆z 2
The interval between two events is the same when measured in any frame of reference:
±interval2 = −c2 ∆t2 + ∆x2 + ∆y 2 + ∆z 2
(∆t0 − v∆x0 /c2 )2 (∆x0 − v∆t0 )2
= −c2
+
+ ∆y 02 + ∆z 02
1 − v 2 /c2
1 − v 2 /c2
2 2
2 2
2v∆t0 ∆x0 − 2v∆x0 ∆t0
2
0 2 (1 − v /c )
0 2 (−v /c + 1)
= −c (∆t )
+
+ (∆x )
+ ∆y 02 + ∆z 02
2
2
2
2
2
2
1 − v /c
1 − v /c
1 − v /c
2
02
02
02
02
= −c ∆t + ∆x + ∆y + ∆z
So the value of the interval is independent of which coordinate system is in use.
Compare the concept of the distance between two points in Euclidean geometry to the
concept of the interval between two events in special relativity: The distance is unchanged
under a rotation of coordinates, and the interval is unchanged under a Lorentz transformation, which is also known as a boost.
It appears as though time behaves as a fourth dimension, in addition to height, length
and width. The c is required in the −c2 ∆t2 term to match the units with the other terms
in the interval. Also, the minus sign in −c2 ∆t2 implies that time might be thought of as an
imaginary direction — that is imaginary in the mathematical sense of i2 = −1. From this
same point of view, a Lorentz transformation (boost of the t and x coordinates is similar to
a rotation of the x-y plane. Mathematically this similarity is more striking if v/c for the
boost is considered to be an imaginary angle (in the i2 = −1, sense) and the coefficients
in the Lorentz transformations may then be written in terms of hyperbolic sine and cosine
functions—if you know what hyperbolic trig functions are.
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Definition: Timelike interval
We say that the interval between two events is timelike if the right hand side of the interval
squared is negative. If the interval between two events is timelike then there is some frame of
reference where the events occur at the same place, and there is no frame of reference where
the events occur at the same time. And the event that occurs first in one frame of reference,
occurs first in all frames of reference. The value of a timelike interval determines the proper
time between the two events—that is the the time between the events as measured in a
frame of reference where they occur at the same position.
Definition: Spacelike interval
We say that the interval between two events is spacelike if the right hand side of the interval
squared is positive. If the interval between two events is spacelike then there is some frame of
reference where the events occur at the same time, and there is no frame of reference where
the events occur at the same place. The value of a spacelike interval is actually the proper
distance between the two events—that is the distance as measured in a frame of reference
where the two events occur at the same place.
Definition: Light-like interval
We say that the interval is light-like or null if the interval is zero. If the interval is light-like
(or null) then a ray of light can go from one event to the other. If the interval between two
events is light-like then there is no frame of reference where the events occur at either the
same time or at the same place. And the event that occurs first in one frame of reference,
occurs first in all frames of reference.