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Relativity 3
Disclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor.
Addition of Velocities
Now that we know that the Galilean transformation must be modified, it’s time to revisit the topic of adding velocities. Consider two inertial frames S and S’ with a relative velocity v.
u dx dt
u u v
u c u c
x
x x x x
in frame S'
in a Galilean transformation, which would imply if
Consider the inverse Lorentz Transformation:
x x vt y y z z
t t vx c
dx dx vdt dt dt v c dx
γ γ
γ γ
a f
c h
a f
c h
and
Take differentials:
2
2
x
y
z
S
x'
y'
z'
S'
v ux ′
Divide one by the other:
where we have divided by
Note that
u dx dt
dx vdt dt v c dx
u
dx dt
v
v c dx dt
dt
dx dt
u
x
x
x
= = ′ +^ ′
2
The velocity addition formulae are:
Note that even though y = y ′ and z = z ′ , that u (^) y ≠ u (^) y ′^ and u (^) z ≠ uz ′
Example : Consider a spacecraft that travels at 0.8 c from Earth and that launches a projectile with a relative velocity of 0.8 c. What is the velocity of the projectile from Earth? Galilean:!
Lorentz:
u c c c c
u c^ c c
x
x
If instead of a projectile we turned on a light beam, both observers on the spacecraft and on Earth would agree that the velocity of the light beam is c , as required by Einstein’s 2nd postulate.
The addition of velocity formulae tell us that nothing can exceed the speed of light.
u u^ v vu c
u
u vu c
u u vu c
x x x
y
y x
z z x
2
2
2
γ
γ
e j
e j
Now we substitute in for γ:
f v^ c v c
f
v c v c v c
= − f −
2 2 0 0
a fa f
So the Doppler shift equations are:
Thus, when source and receiver approach each other, the frequency is shifted higher. We say that the light is blue-shifted.
When source and receiver recede from each other, the frequency is shifted lower. We say that the light is red-shifted. Red light has a lower frequency than blue light.
Modern electronics allow us to determine frequencies very accurately, so we can measure relative velocities accurately as well using this effect. Examples include Doppler weather radar, police radar, and even the expansion of the universe!
Lorentz Invariance
We have seen that some quantities change from one inertial frame to another (length, time, velocity, frequency). A quantity which does not change after a Lorentz
transformation is said to be Lorentz Invariant. One special invariant is the
Space-time Interval:
s c t x y z t t t x x x
a f 2 a f 2 a f 2 a f 2 a f^2
2 1 2 1
= − etc.
This is the generalization of Cartesian distance for 4-dimensional space-time. The same value for ∆ s is obtained for any inertial frame. So although length and time separately are not invariant from one frame to another, this particular combination is.
We can prove that this is true by applying the Lorentz transformation. For example, consider a subatomic particle which decays in a time τ in its rest frame.
f f v^ c v c
f f v^ c v c
0
0
Source and receiver approaching
Source and receiver receding
In the rest frame S’: t t x x y s c
1 ′ =^0 2 ′ =^1 ′ =^2 ′ =^1 ′ =^ =^0
, τ , τ
L
Now make a Lorentz transformation to another frame S moving at velocity v :
x x vt y y z z t t vx c x v x x v t t t s c v c v c c v c
v c s c
γ γ γ τ γ τ γτ γτ γ τ γ τ γ τ τ
τ
a f
c h
a f c h
c h
as in the rest frame
2
2 1 2 1 (^2 2 2 2 2 2 2 2 2 2 2 ) 2 2 2 2
2 2
Some terminology:
∆
s
s
s
2
2
2
time - like
light - like
space - like
A frame exists where 2 events occur in one place, separated by time.
2 events are separated by the speed of light.
No light signal can connect the 2 events.
