Announcements
What we found so far:
• Reading for Wednesday: Review 1.9-1.12
and 1.13
Simultaneity of two events depends on the choice of
the reference frame
• HW2 due Wednesday noon.
L
R
v
Joel
…-3 -2 -1
0
1
2
3...
Tom
G1B30
G1B20
Phys help room
(G2B90)
Joel concludes:
Light hits both ends at
the same time.
Drop-off HW here
Today’s class
Time Dilation
Time Dilation: Two observers (moving relative to each
other) can measure different durations between two
events.
h
Joel
v
h
Tom
Joel measures:
Δt’ = 2h/c
Here: Δt’ is the proper time
Tom concludes:
Light hits left side first.
• Spacetime diagrams
• Clocks in different reference
frames.
• Some evidence for time dilation
Crow
1
Tom and Crow:
 
v2
Δt = γ 2h/c , with
1 2
c
Spacetime Diagrams (1D in space)
In PHYS 1110:
v
Spacetime diagrams
x
(also called “Minkowski diagrams”)
(very useful in SR!)
Position of the cart as
a function of time
x
Δx
Δt
v = Δx/Δt
t
1
In 2130: Spacetime (1-d)
To describe objects and events we
need (x,y,z,t)
ct
Spacetime Diagrams (1D in space)
c· t
Different objects
in PHYS 2130:
object moving with 0<v<c.
‘Worldline’ of the object
For 1-d space we can describe all
events with 2-d spacetime.
x
As that time progresses the object
moves. Its “world line” shows how
it moves through spacetime both
in the future and the past.
The world line represents reality to
this object. We hall have them and
they may be quite complicated
Note: time axis scaled by c
At t=t’=0 the origins of S and S’ coincide. S’ is moving
with velocity v<c relative to S. Which shows the world
line of the origin of S’ as viewed in S?
ct
a
b
-2 -1
0
1
object moving with 0>v>-c
c·t
c·t
object at rest
at x=1
-2 -1
0
1
object moving with
v=c. x=0 at time t=0
2 x
-2 -1
0
1
2
x
x  x1  x2
y  y1  y2
z  z1  z 2
d
45°
Spacetime interval
x
ct
x
Say the difference between two events is given by
ct
x
c
2
This is a weird kind
of “distance” in
spacetime.
t  t1  t 2
ct
Then the spacetime interval
x
x
s 2  ct 2  x 2  y 2  z 2
Has the same value in all reference frames.
Spacetime (1-d)
ct
Here is an event in
spacetime.
x
Any light signal that
passes through this event
has the dashed world
lines. These identify the
light cone of this event.
Spacetime
ct
B
Here is an event in
spacetime.
The blue area is the future
on this event.
A
x
The spacetime interval
between events A and B
is
a) Positive
b) Negative
c) zero
2
Causality
ct
B
Spacetime
Here is an event in
spacetime.
ct
If A precedes B in all
frames moving less
than c, then
s2 >0.
The blue area is the future
on this event.
A
The pink is its past.
x
If s2 >0 in one reference
frame, then it’s positive
in all reference frames.
x
So, causality is maintained
in special relativity.
All events that must come
before (in all reference
frames) are in pink.
Recall:
Spacetime
All events that must come
after it (for all reference
frames) are in blue.
Joel plays with a fire cracker in the train.
Tom watches the scene from the track.
Here is an event in
spacetime.
ct
The yellow area is the
elsewhere of the event.
No physical signal can
travel from the event to
its elsewhere.
A
L
R
v
x
Joel
These are all points with
negative spacetime
intervals. Events here
may come before or
after.
Tom
Cannot cause A
Example:
Tom on the tracks
ct
Example: Joel in the train
ct
Light reaches both
walls at the same time.
Light travels to both walls
Tom concludes:
Light reaches left
side first.
45°
45°
x
L
R
Joel concludes:
Light reaches both
sides at the same time
In Joel’s frame: Walls are at rest
L
x
R
In Tom’s frame: Walls are in motion
3
Origin of S’ viewed from S
For all points on
this axes: x=0
S
... -3
-2
-1
0
1
2
3 ...
... -3
-2
-1
0
1
2
3 ...
S’
v=0.5c
All points on this axes:
x’=0 (origin of S’)
ct
Frame S’ is moving to the right at v = 0.5c. The origins
of S and S’ coincide at t=t’=0. Which shows the world
line of the origin of S’ as viewed from S?
A
B
ct
C
ct
x
D
ct
x
ct
x
Frame S’ as viewed from S
For all points on
this axes: x=0
These angles
are equal
ct
x
All points on this axes:
x’=0 (origin of S’)
For all points on
this axes: t=0
x
Both frames are adequate for describing events,
but will generally give different spacetime
coordinates. In S: (x,t), or in S’: (x’,t’)
ct
ct’
ct’
x’
x
x’
All points on this axis:
t’=0 (x’ axes of S’)
For all points on
this axes: t=0
x
In S: (3,3)
In S’: (1.8,2)
In classical physics we had
something similar: The
Galileo transformations.
x’ = x - v∙t
t’ = t
(We know that they are no
good for light. We’ll fix
them soon!)
4
Where am I?
Notable events?
•
• Galileo and the Jovian moons
1614 found their orbits.
Precise location can be used for
timing didn’t really work on boats
(they rock)
•
•
•
•
This question has economic,
survival implications
Latitude is easy Just look at the
sun and know the time of year.
But to get longitude you need
time
The history of this problem
spanned hundreds of years.
Timing was everything
Notable events?
Notable events?
• Lunar position relative to the stars
can tell you the time.
Extremely difficult calculations
(lots of trig and other
corrections)
•
•
•
•
Now how do we do
it?
• Send light waves with their time
encoded. (used to be land based)
• Now from satellites (can see
anywhere)
Galileo and the Jovian moons
Lunar position relative to the stars
Longitude prize (1714 England)
Accurate clocks
(1765 John Harrison)
• Not accurate enough until 1850’s
Now how do we do
it?
• Send light waves with their time
encoded. (used to be land based)
• Now from satellites (can see
anywhere)
• By comparing to your local time
you know when the signals
originated
Radius of sphere R=ct
5
Now how do we do it?
• Send light waves with their time
encoded. (used to be land based)
• Now from satellites (can see
anywhere)
• By comparing to your local time
you know when the signals
originated
• With 3 satellites location is known
(4 removes uncertainty)
This is the
reverse of how
we synchronize
clocks in one
frame.
Each satellite makes a sphere and
the intersection is circle, then 2
points, then 1.
6