Building Spacetime Diagrams
PHYS 206 – Spring 2014
The “proper length” (radius)
of the circle in any rotated
frame of reference is:
t
But their coordinates
do not agree!
Δr = 4
x
Think of the red line as the
“rest” frame (inertial clock
with v=0).
y
Δr = 4
The vector pointing
purely in the ydirection is like the
worldline of a
motionless
observer, as
measured in their
own rest frame!
x
A frame of reference
“moving” (as measured in the
original frame) corresponds
to a rotation of the vector.
y
Δr = 4
These coordinates are
measured by the red
observer in the red
(motionless) reference
frame!
x
But if we transform into the
blue (moving) reference
frame, the vectors look like
this:
y
Δr = 4
These coordinates are
measured by the blue
observer in the blue (now
motionless) reference
frame! They are the
same as the red
coordinates from before
(as measured in the red
rest frame).
x
Now, as measured in the
blue rest frame, the red
frame looks like it is
moving (at the same
velocity in the opposite
direction).
t
The proper time in any frame
of reference is:
Vectors of the same length in
hyperbolic spacetime are not
the same length “on paper.”
Δτ = 4
At rest with
respect to red
coordinate
system
(“Earth”).
Moving at constant v as
measured in red (“Earth”)
coordinate system.
x
t
Moving with –v
as measured in
blue (“ship”)
coordinate
system.
Δτ = 4
The proper time in any frame
of reference is:
Δτ = 4
At rest in blue (ship)
coordinate system.
x
y
t
x
x
Spatial vectors lie on a circle, and have
the same length regardless of rotation
angle (but different coordinates).
Spacetime vectors (4-vecs) lie on a
hyperbola and have the same
“length”(proper time) regardless of
velocity (but but different coordinates).
The “frame of reference” where Δx = 0
is the “rest” frame.
The frame of reference where Δx = 0 is
the rest frame.
We can always make another vector
vertical (“at rest”) by rotating the
coordinate system by the corresponding
angle.
We can transform to the rest frame of
another vector by Lorentz transforming
to the corresponding velocity.
t
“Take a picture” = Line of constant t
Δτ = 4
The moving clocks (blue and green)
appear to run slowly as seen by the
red (rest) observer.
x
Time Dilation!
Time Dilation
Elapsed time (ship clock) in
moving frame as seen from
rest frame
Speed of moving frame
(ship) as seen in rest frame
Elapsed time in rest frame (Earth clock)
Spacetime Diagrams
t
Moving at velocity v (“ship”) with respect to rest
frame (“Earth”)
All x at specific t (photograph!)
Light cone represented by
45º lines (x=t) in all reference frames
Not moving (at
rest) with
respect to rest
frame (“Earth”)
x
t
We define distance by the round-trip
travel-time of a pulse of light (“radar method”)
D =½ T
T
Mirror
D
x
x
t
+3t

The line joining the
reflection points
defines the spatial
axis in the rest frame!
+2t 
+t
x
-t
-2t 
Sends light signal
-3t 
If light signal goes out some fixed distance in a time
t and reflects, then it will come back in the
same amount of time.
Moving Reference Frames
t
t
Line which represents something
moving at velocity v (spaceship as
seen from the Earth) is equivalent
to the time axis of the observer inside
the spaceship!!
x
t
t
+3t
+2t
The line joining the
reflection points
defines the spatial
axis in the moving
reference frame!
x
+t
x
-t
-2t
-3t 
Sends light signal

If light signal goes out a distance D in a time
t and reflects, then it will come back in the
same amount of time according to the moving
frame (ship)!
t
t
“Photographs” in rest frame
(events in all space at fixed time)
t
4
t3
t2
x
t1
x
Lines of constant t are parallel to x-axis!
t
t
“Photographs” in moving frame
t4
t3
x
t2
t1
x
Lines of constant t are parallel to x-axis!
Proper Time Calibration
t
t
c
t4 =4
t4 =4
t3 = 3
t2 = 2
t1 = 1
t3 = 3
t2 = 2
x
t1 = 1
x
In rest frame of Earth (t)…
t
t
c
t4
t3
x
t2
t1
x
At time t (parallel to x-axis)…
… t < t !!! (clocks on ship run slow)
In rest frame of ship (t’)…
t
t
c
t4
t3
x
t2
t1
x
At time t (parallel to x axis)… t < t !! (clocks on Earth appear slow)