Please go to View, then select Slide Show.
To progress through a slide, hit the down arrow key. There
are some animations in this example. If you’d like to see
something again, hit the up arrow followed by the down
arrow. The first slide in a set asks the questions, while the
following slide(s) provides a detailed example. Answer the
questions as best you can before moving to the next slide(s).
You should have printed out a spacetime diagram. This
spacetime diagram has the standard axes for the rest frame
of the Sun and the Earth (assume an inertial frame). The
Sun is located at x = 0; the Earth is located at x = 8 lt-min.
The Jupiter 2 is a spaceship.
Spacetime Diagram Worksheet
This worksheet gives you practice constructing and
interpreting a spacetime diagram
1) Experience with Events
2) Working with Worldlines
3) Finding the t’ and x’ axes for
different reference frames
5) Graphically determining if the
interval between pairs of events is
time-like, light-like, or space-like
6) Determining if you can reverse the
time ordering or space ordering of
pairs of events
4) Finding the time and space order of
events
7) Problems for Practice
1) Experience with Events
a) Label an event C that
occurs at x = 15 lt-min and
t = 6 min.
t (min)
20
b) Event A is the launch of a
satellite from the Earth to
the Sun. Where and when
is the satellite launched, as
measured by observers in
the Sun/Earth (unprimed)
frame?
15
10
c) Five minutes after launch,
the satellite malfunctions at
a distance of 2 lt-min from
its launch point (still
measured in the unprimed
frame). Label this event B
on the spacetime diagram.
A
5
0
0
5
10
15
x (lt-min)
20
a) Shown on diagram.
1) Experience with Events
(answers)
b) Draw line through event A
parallel to x and find
intersection with t.
20
t (min)
tA= 5 min
Draw line through event A
parallel to t and find
intersection with x
15
xA= 8 lt-min
B
10
c) Five minutes after launch,
tB= tA+ 5 min = 10 min
C
A
5
a distance of 2 lt-min from
its launch point
xB= xA – 2 lt-min
= 6 lt-min
0
0
5
10
Why subtract? Recall the
satellite is moving towards the
15
20
x (lt-min)
Sun.
2) Working with Worldlines
a) Draw and label the
worldline of the Earth,
located at x = 8 lt-min.
t (min)
20
b) Determine the speed of the
Jupiter 2, as measured in
the Sun/Earth (unprimed)
reference frame.
15
c) At the same time as event
B, a comet passes by the
Earth at 0.5c (as measured
in the unprimed frame),
headed away from the Sun.
Draw and label the
worldline of the comet.
B
10
C
A
5
0
0
5
10
15
x (lt-min)
20
a) Shown on diagram.
2) Working with Worldlines
(answers)
15
B
10
5
0
0
5
Dt = 5 min
t (min)
Earth
20
b) Jupiter 2 trajectory shown
on diagram is Jupiter 2’s
worldline. Slope of Jupiter
2’s worldline is Dt/Dx =
5 min/4 lt-min =
1.25 min/lt-min. Recall
that speed is Dx/Dt, or that
Dx = 4 lt-min
slope of worldline is inverse
Note: magnitude of
of velocity to get speed =
slope > 1 as required.
0.8c.
If mag. of slope < 1,
c) “Same time as event B”
object moves faster
means 10 min; “passes by
than light!
Earth” means x = 8 lt-min,
C
so know the Comet passes
A
through that point. Slope
of worldline is 1/v = 1/(0.5)
= 2 min/lt-min. Comet
moves away from Sun. Put
together for worldline,
10
15
20
x (lt-min)
shown.
3) Finding the t’ and x’ axes for
different reference frames
a) Draw/label the t’ axis for
the comet.
t (min)
Earth
20
b) Draw/label the x’ axis for
the comet.
15
B
10
C
A
5
0
0
5
10
15
x (lt-min)
20
a) Worldline IS t’ axis!
3) Finding the t’ and x’ axes for
different reference frames
t’
t (min)
Earth
20
b) The x’ axis is “mirror
image” of t’ axis (about
+45o). Or, since t’ line is “2
up, 1 over”, then x’ line is
“1 up, 2 over.” In other
x’ words, the magnitude of
the numerical value of the
slope of the x’ line is the
speed of the object.
15
Note: Doesn’t matter
where x’, t’ intersect (unless
origin in primed reference
frame indicated.)
B
10
C
A
5
0
0
5
10
15
x (lt-min)
20
a) Order the events A, B, and
C according to observers in
the Sun/Earth reference
frame, from earliest to
latest. Be clear in
indicating if any events
occur at the same time.
4) Finding the time and space
order of events
t’
t (min)
Earth
20
x’
b) Order the events A, B, and
C according to observers at
rest with respect to the
Comet, from earliest to
latest. Be clear in
indicating if any events
occur at the same time.
15
B
10
C
A
5
0
0
5
10
15
x (lt-min)
20
a) To find the time order of
events in the unprimed frame,
draw a line THROUGH each
event PARALLEL to the x
axis, and see where it intersects
the t axis.
4) Finding the time and space
order of events (answers)
t’
t (min)
Earth
20
A, then C, then B
x’
15
b) To find the time order of
events in the primed frame,
draw a line THROUGH each
event PARALLEL to the x’
axis, and see where it intersects
the t’ axis.
B
10
C, then A, then B
C
A
5
0
0
5
10
15
x (lt-min)
20
c) To find the space (left-right)
order of events in a frame,
draw a line THROUGH each
event PARALLEL to that
frame’s time axis, and see
where the line intersects that
frame’s position axis.
5) Graphically determining if the
interval between pairs of events is
time-like, light-like, or space-like
20
t (min)
a) Determine if the spacetime
interval between the events A
and B is space-like, time-like,
or light-like.
15
b) Determine if the spacetime
interval between the events A
and C is space-like, time-like,
or light-like.
B
10
C
A
5
0
0
5
10
15
x (lt-min)
20
5) Graphically determining if the
interval between pairs of events is
time-like, light-like, or space-like
a) Can calculate the value for the
spacetime interval, using the
definition:
(Ds)2 = (Dt)2 – (Dx)2. Here,
DtAB = 5 min, and DxAB =
2 lt-min, so (Ds)2 = 21 min2.
By definition, a positive (Ds)2 is
a time-like event. Also, (Ds)2 =
0 is light-like, and (Ds)2 < 0 is
space-like. Note the units.
(answers)
t (min)
20
15
However, can be done graphically without any calculations:
Just draw a line that connects
the two events.
B
10
Compare the numerical value
of the magnitude of the slope of
this line to 1:
C
A
5
If mag. slope > 1  time-like
If mag. slope = 1  light-like
0
0
5
10
15
x (lt-min)
20
If mag. slope < 1  space-like
5) Graphically determining if the
interval between pairs of events is
time-like, light-like, or space-like
(cont.)
Why does this work?
Go back to definition of interval:
(Ds)2 = (Dt)2 – (Dx)2.
Divide by (Dx)2:
( Ds)2 ( Dt )2 ( Dx )2


( Dx )2 ( Dx )2 ( Dx )2
2
 Dt 
 Dt 
if 
 1
 1  0
D
x
D
x




 Ds 2  0  light  like
So if numerical value of magnitude of the slope
of the line that connects the two events = 1, the
interval is LIGHT-LIKE.
2
 Dt 
 Dt 
if    1     1  0
 Dx 
 Dx 
 Ds 2  0  time  like
So if numerical value of magnitude of the slope
of the line that connects the two events > 1, the
interval is TIME-LIKE.
As (Dx)2 > 0, this won’t
change the sign of I2.
2
2
(Dt )2 (Dx)2  Dt 

   1
(Dx)2 (Dx)2  Dx 
(cDt/Dx) IS the numerical value of
the slope of the line connecting the
two events. Since it is squared,
doesn’t matter if slope is positive
or negative.
 Dt 
 Dt 
if 
1
 1 0
 Dx 
 Dx 
 Ds 2  0  space  like
So if numerical value of magnitude of the slope
of the line that connects the two events < 1, the
interval is SPACE-LIKE.
Alternate explanation in part 6)
5) Graphically determining if the
interval between pairs of events is
time-like, light-like, or space-like
t’
20
(answers)
t (min)
Earth
a) Draw line connecting events A
and B.
x’
15
|Slope of line| > 1
 Time-like.
B
10
b) Draw line connecting events A
and C.
C
|Slope of line| < 1
A
5
 Space-like.
0
0
5
10
15
x (lt-min)
20
6) Determining if you can reverse the
time ordering or space ordering of
pairs of events
20
t (min)
a) Consider the events A and B.
Can you reverse the time order
or the space order of these
events?
15
b) Consider the events A and C.
Can you reverse the time order
or the space order of these
events?
B
10
C
A
5
0
0
5
10
15
x (lt-min)
20
6) Determining if you can reverse the
time ordering or space ordering of
pairs of events (answers)
a) In part 5), showed that the
interval between A and B was
time-like. Time-like means
that the time order of events is
important. The time order of
time-like events CANNOT be
reversed.
t (min)
20
This means that in ALL
reference frames, the order of
these events remains the same
(though the time between the
events of course may change.)
15
However, the space (left to
right) order of time-like events
CAN be changed. There exist
reference frames where the left
to right order is reversed.
B
10
C
A
5
b) Opposite holds for space-like
events. A and C are space-like,
so CANNOT change space
order, but CAN change time
order.
0
0
5
10
15
x (lt-min)
20
Light-like events preserve their time AND space order in all frames.
6) Determining if you can reverse the
time ordering or space ordering of
pairs of events (further explanation)
How does this come about?
As before, when classifying
intervals, draw a line
connecting the two events:
t (min)
20
This line COULD be a worldline, as the magnitude of its
slope > 1. So this could be the
t’ axis of some reference frame.
15
Events A and B occur on a line
parallel to the t’ axis (they
actually occur on the t’ axis).
Events which occur on lines
parallel to the time axis of a
reference frame must occur at
the SAME PLACE in that
frame.
B
10
C
A
5
Event B occurs to the LEFT of
A in the unprimed frame. In
the proposed frame, they occur
in the SAME place.
0
0
5
10
15
x (lt-min)
20
Space order can be changed by switching reference frames.
6) Determining if you can reverse the
time ordering or space ordering of
pairs of events (more explanation)
What about time order?
(similar argument as before)
A and C are space-like, for
which the time order is not
unique. Again, draw a line
connecting the two events:
t (min)
20
This line can NOT be a worldline, as the magnitude of its
slope < 1. But this could be the
x’ axis of some reference frame.
15
B
10
Events A and C occur on a line
parallel to the x’ axis. Events
which occur on lines parallel to
the position axis of a reference
frame must occur at the SAME
TIME in that frame.
C
A
5
0
0
5
10
15
x (lt-min)
Time order can be changed by switching reference frames.
20
Event A occurs BEFORE event
C in the unprimed frame. In
the proposed frame, they occur
at the SAME time.
7) Problems for Practice (gives you practice with most concepts and calculations
related to our work in special relativity)
a) Event D occurs on the Comet when it is
halfway between the Sun and the Earth (in
the unprimed frame). Label the event D.
What time does event D occur, in the
unprimed frame?
e) In the Sun/Earth frame, the event D occurs
before event A. According to observers
zipping by in the Millenium Falcon, A occurs
before D. What is the minimum speed of the
Millenium Falcon (as measured in the
unprimed frame) for this to occur?
b) Assume the satellite moves with constant
velocity from its launch point, and even after f) According to observers on board Jupiter 2,
how fast is the comet moving?
the malfunction. Draw/label the worldline of
the satellite, and determine its speed, as
g) What is the time between events A and B,
measured in the unprimed frame.
according to an atomic clock on board the
satellite?
c) Order the events A, B, C, and D in time
according to observers on board the Jupiter
h) According to observers on board Jupiter 2, is
2. Assume the Jupiter 2 always travels with
the interval between events A and C spacethe constant velocity shown in the diagram.
like, light-like, or time-like?
d) Determine whether you can reverse the time
i) According to an alien hitching a ride on the
order or the space order of events B and C.
comet, what is the distance between events A
and B? (Tough question. Hints: find
relative speed between comet and satellite.
Use answer to question g)
a) D occurs at 2 min, 4 lt-min; b) Worldline of satellite starts at A, goes through B. Speed is 0.4c; c) D, A, B, C;
d) Space-like, so time order can be reversed; e) v > 0.75c; f) 0.9286; g) 4.58 min; h) Space-like; i) 5.19 lt-min