Physics 228
Today: Special Relativity: Continued
Principle of Relativity: Physical laws, including speed of light,
are the same in every inertial reference frame.
Results in:
•
•
•
•
relativity of simultaneity
time dilation
length contraction
twin “paradox”
Transverse Dimensions Unchanged
While the train appears to Stanley shortened in the direction it is
moving, it is the same size in the “transverse” directions.
This is required by the principle of relativity.
iClicker
A Star Wars photon torpedo explodes, emitting a spherical shell of
debris, if measured in its rest frame. In your frame, the torpedo’s
speed was 0.8 c to the right when it exploded. What would be the
shape of the debris shell in your frame? (The dark areas represent
the debris shell at an earlier time, the light areas at a later time.)
a)
d)
b)
e)
c)
(Hang on, there will be one more iClicker!)
iClicker
Instead of the shape of the debris shell, let us now consider the
shape of the light front that is emitted by the exploding torpedo. In
your frame, the torpedo’s speed was 0.8 c to the right when it
exploded. What would be the shape of the light front in your frame?
(The dark areas represent the light front at an earlier time, the
light areas at a later time.)
a)
d)
b)
e)
c)
Review: Galilean Transformation
It the primed frame moves at velocity u in the x-direction, in nonrelativistic physics the coordinates of an event in the two frames
are related by:
x' = x - ut
y' = y
z' = z
t' = t
x = x’ + ut
y = y’
z = z’
t = t’
Valid for small velocities u only!
Lorentz Transformation
In special relativity, the coordinate transformation becomes:
x' = γ (x – u t)
x = γ (x‘ + u t')
y' = y
y = y'
z' = z
z = z'
t' = γ (t – u x/c2)
t = γ (t‘ + u x'/c2)
Space-Time Diagrams
iClicker
light cone
A
B
time
Starting from event 1,
which other events could
be reached by travel?
1
C
space
a)
b)
c)
d)
e)
A
Requires faster-than-light travel
B
Requires traveling into the past
C
More than one
None
Space-Time Diagram of Moving Frame
(Galilei Transformation)
x' = x – u t
y' = y
z' = z
t' = t
x = x‘ + u t'
y = y'
z = z'
t = t‘
Space-Time Diagram of Moving Frame
(Lorentz Transformation)
x' = γ (x – u t)
y' = y
y = y'
z = z'
z' = z
t' = γ (t – u
x = γ (x‘ + u t')
x/c2)
t = γ (t‘ + u x'/c2)
world line of Mavis
Mavis moving at
u = 0.6 c
(lines of simultaneity are
horizontal in Stanley’s frame)
world line of Stanley
lines of simultaneity
for Mavis
Clocks in Space
•
To simplify the discussion, we are
going to pretend that space is filled
with an infinite array of clocks, all
synchronized with each other.
•
Since time is relative to the
observer, we need two sets of
clocks: One fixed to Stanley’s
frame, and one traveling with Mavis
in her frame.
•
As we have seen, the two sets of
clocks will show different times.
•
In particular, Stanley will observe
Mavis’s clocks as not synchronized,
and vice versa.
•
In this way, any “event” can be
assigned definite x, y, z, and t
coordinates, as well as definite x’,
y’, z’, and t’ coordinates.
Space-Time Diagrams of Clocks in Space
Mavis
Stanley
Stanley’s clocks
Mavis
Stanley
Mavis’s clocks
Twin Paradox Revisited
Bert
Al
Al
Bert
Twin Paradox Revisited
time (years)
Space trip starts at birth
u = 0.6 c
world line of space twin
γ = 1.25
8
7
6
world line
of earth twin
lines of simultaneity
after turnaround
5
4
age of
space twin
3
age of
earth twin
2
1
distance (lightyears)
lines of simultaneity
before turnaround
Space-Time Diagram from Space Twin’s
Point of View
Lorentz transformed
space-time diagram
using space twin’s
inertial frame (outbound
trip) as the rest frame.
Signaling Between Twins
Space trip starts at birth.
u = 0.6 c
γ = 1.25
Each year on his birthday,
each twin sends the other a
snapchat greeting.
Birthday greetings travel
at the speed of light
(45 degree diagonal lines).
Ladder in Garage Paradox
Ladder won’t fit into garage!
What to do?
If the ladder is shoved in fast
enough, it will (momentarily) fit
due to length contraction!
Yet, from the point of view of the
ladder, the garage is moving and
is contracted. So the ladder
doesn’t fit after all! What gives?
iClicker
a) The ladder fits in the garage due to ladder’s length
contraction.
b) The ladder does not fit in the garage because of
contraction of garage.
c) Ladders will never move fast enough for this to
work.
d) The ladder fits into the garage as seen in the
garage’s frame of reference, but not in ladder’s
frame.
e) This is a logical contradiction that disproves
special relativity.
Ladder in Garage Paradox
Stationary garage,
moving ladder
Stationary ladder,
moving garage