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Space-Time Diagrams: Visualizing Special Relativity
Prof. Steuard Jensen
W.M. Keck Science Center, The Claremont Colleges
A space-time diagram shows the history of objects moving through space (usually in just
one dimension). A specific point on a space-time diagram is called an “event.”
• To make a space-time diagram, take many snapshots of the objects over time and set
them on top of each other. Lines in the diagram are like “contrails” through time.
• Given a diagram, to take a snapshot find the positions of each object on the appropriate
“time slice” (the dashed lines below). A time slice labels a set of “events” that
the observer considers simultaneous. We will draw time slices at equally spaced
intervals so that each slice corresponds to one “tick” of the observer’s clock.
t (yr)
Space-time diagram
Snapshots (time slices)
x (ly)
10
0
9
2
3
4
5
6
x (ly)
ly
/y
r
8
1
0
1
2
3
4
5
6
c
=
1
7
x (ly)
6
0
5
1
2
3
4
5
6
x (ly)
4
0
3
1
2
3
4
5
6
x (ly)
2
0
1
1
2
3
4
5
6
x (ly)
0
0
1
2
3
4
5
x (ly)
0
6
1
2
3
4
5
6
In these diagrams:
• The closer an object’s path is to vertical, the slower it is moving: slope = 1/v.
• A 45◦ angle corresponds to the speed of light: we will always measure time in years (yr)
and space in lightyears (ly) (or in seconds and light-seconds, etc.). Thus, c = 1 ly/yr.
A few examples:
Elastic collision
Totally inelastic
collision
t (yr)
t (yr)
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
x (ly)
0
0
1
2
3
4
5
6
x (ly)
0
0
1
2
3
4
5
There and
back again
t (yr)
x (ly)
0
0
1
2
3
4
5
Page 2 of 4
Thus far, this is just another way to visualize motion: it applies just as well to classical
physics as it does to relativity. Relativity adds two new ingredients:
• Nothing can travel at more than a 45◦ angle from vertical (even for a moment): v < c.
• Different observers see different time slices!
In particular, the time slices seen by the observer drawing the diagram (the observer “at
rest”) are the same horizontal lines as before. But the time slices seen by an observer
traveling at speedqv relative to the diagram’s reference frame have the following properties
(recall that γ = 1/ 1− vc22 , so for example, v = 21 c gives γ = 1.15):
• The time slices tilt up toward 45◦ as v increases: they have slope v/c2 = v ·
1 yr2
.
ly2
As long as we draw c at 45◦ , this means that a moving observer’s time slices tilt
up by the same angle θ that her path tilts down in our diagram. (Technically,
we are graphing the “Lorentz transformation” equations, plugging in various values for t0 .)
• Time slices get squeezed together as v increases. (Compare v = 21 c and 45 c below.)
A moving observer’s time slices spaced ∆t0 apart are separated vertically on the diagram
0
(e.g. for v = 12 c, 1 yr time slices intersect the t-axis every 0.87 yr).
by ∆t
γ
• A moving observer’s “clock ticks” occur when her time slices intersect her
path on the diagram. Thus, an observer “at rest” sees the moving clock ticking less
often: the intersections have larger vertical spacing, every ∆t = γ∆t0 (so the moving
observer’s ∆t0 = 1 yr time slice intersects her path at diagram time ∆t = 1.15 yr).
v=½c
t (yr)
4.62
5
5
r
4y
4
3.46
3
2.60
yr
3.33
r
2.31
2y
θ
1.73
1.67
r
1y
1.15
1
3
yr
4
3.46
3
2
v =⅘ c
t (yr)
0.87
0
1
3
2
1
2
3
1.80
1
θ
1.20
0.60
θ = 26.6°
0
2
x (ly)
4
5
yr
yr
θ = 38.7°
0
0
1
2
3
x (ly)
4
5
Example: Above, the diagram observer clearly sees moving clocks running slow: for each
“1 yr” intersection along the moving path, his dotted horizontal time slices come more than
1 yr apart. For example, in the v = 12 c diagram the moving observer’s 2 yr “clock tick” is
delayed until t = 2.31 yr (and when v = 45 c, it is delayed even more: until t = 3.33 yr).
But the moving observer would say the same thing about the diagram observer! In the
v = 21 c diagram, the diagram observer’s “clock tick” at x = 0 and t = 2 yr falls between the
moving observer’s 2 yr and 3 yr time slices. (In fact, it is at exactly ∆t0 = 2.31 yr.) Both
observers see the other clock running slow by the same amount. The page about the “twin
‘paradox’” will give you some idea of why this isn’t a contradiction.
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In relativity (including space-time diagrams), time and space are mixed together in two
Not-Quite-Pythagorean Theorems for Space-Time (a.k.a. the “space-time interval”):
• The time ∆t0 experienced along an observer’s path obeys (c∆t0 )2 = (c∆t)2 − (∆x)2 .
• A length ∆x0 measured along an observer’s time slice obeys (∆x0 )2 = (∆x)2 − (c∆t)2 .
0
These are equivalent to ∆t = γ∆t0 and L = Lγ (just plug in ∆x = v∆t, etc.). WARNING!
Never measure lengths at an angle on a spacetime diagram with a ruler: lines on
paper obey the ordinary Pythagorean theorem, not these special space-time rules.
To find the length of an object, an observer must measure between points on the SAME time slice, as shown.
(Why? At right, if you located the tail at x = 0 when t = 0
and then located the nose at x = 4.5 ls (“light-seconds”) when
t = 5 s, you shouldn’t conclude that the ship is 4.5 ls long!)
Measuring length
t (s)
5
L′
3
Motion in the −x direction: The observer’s path tilts
down to the left toward 45◦ and her time slices tilt up, as shown
at right. (Noteworthy aside: This would be the diagram drawn by
L = Δx
diagram frame
4
2
=Δ
diagram on the previous page.
the moving observer in the v =
To her, the formerly “at rest” observer is moving to the left with
velocity v = − 21 c. The original observer’s formerly horizontal time
slices are tilted in this reference frame.)
Δt
Δx
moving frame
(different Δx!)
1
1
2c
x′
x (ls)
0
0
1
2
4
3
5
v = −½ c
Multiple moving objects: The diagrams below show
three observers with different velocities. In both, the time
slices shown are those for the middle observer: we change
from the left observer’s perspective to his. After the change,
the middle observer’s time slices become horizontal and the
velocities {v0 = 0, v, u} change to {v00 , v 0 = 0, u0 } as shown.
t (yr)
5
4y
r
4
3y
r
3
2y
r
You cannot simply add and subtract velocities
1y
r
when changing reference frames except in a couple of
special cases. Instead, use the following result (derived by x (ly) 26.6° = θ
measuring a moving object’s position along tilted time slices
3
2
1
).
using the methods above): u0 = (u − v)/(1 − uv
c2
2
θ
1
0
0
Changing to another reference frame
t (yr)
v0 = 0 v = ½ c
u = ⅘c
5
4
2y
3
1y
2
v0′ = −½ c
r
3y
t′ (yr)
v′ = 0
u′ = ½ c
5
r
4
r
3
2
θ
1
1
θ
0
0
1
2
x (ly)
3
4
5
x′ (ly)
3
2
1
0
1
2
3
Page 4 of 4
The Twin “Paradox” in a Space-Time Diagram
Our textbook describes a 20 year old man
named Speedo who takes a trip in a spaceship while his identical twin brother Goslo
stays on Earth. If Speedo travels at v = 12 c
for five years (as measured on Earth) and
then quickly turns around and comes home
at the same rate, his path on Goslo’s spacetime diagram is as shown (Goslo’s path is
straight up the t axis). The diagram at right
shows Speedo’s time slices (dashed lines) at
one year intervals (by his measurement). It
also includes extra (dotted line) time slices
at (Speedo’s) quarter-year intervals between
his years 4 and 5 as he turns around.
The first important observation is that
when Speedo gets home, Goslo has aged 10
years (he is now 30), but Speedo has only
aged 8.66 years (he is not yet even 29).
As long as Speedo moves at constant
speed (away from Earth or toward it), he and
Goslo each see the other as aging less quickly.
(When Speedo celebrates his 24th birthday
he thinks that Goslo is still only 23.46, but
Goslo thinks the very same thing on his own
24th birthday.)
t (yr)
10
9
Twin “Paradox”
8.6
v = ±½ c
6y
r
8y
r
8
7y
r
7
6
5
6y
r
4.7
5y
5y
r
4.5
r
yr
r
4y
4.25 yr
4
r
3y
3
2
1
r
2y
θ
r
1y
θ = 26.6°
But that leads to the second important
x (ly)
0
point: the difference between motion in an
0
4
1
2
3
inertial frame (without acceleration) and motion that includes acceleration. During the
brief period around t = 5 yr on Earth (or
around t0 = 4.33 yr on Speedo’s spaceship) when Speedo reverses direction, the angles of his
time slices change because his velocity is changing. Because those angles change while he is
far away from Earth, he sees Goslo age very quickly during the brief time it takes him to
turn around (by an extra 2.5 years). The faster he changes direction, the faster he sees those
extra years pass back on Earth. Meanwhile, Goslo sees no such extra time pass for Speedo
because Goslo’s time slices never change their direction.
For this reason, we could not draw a correct space-time diagram based on Speedo as the
observer “at rest.” Special relativity is only valid when the observer drawing the
space-time diagram moves with constant velocity.
As a side note, it might be possible to draw a similar kind of space-time diagram from
Speedo’s perspective using techniques from general relativity, but that’s beyond the scope of
this class. Doing so would probably require us to draw the picture on a curved surface rather
than on a flat sheet of paper: essentially, Speedo would move straight along the time axis at
x = 0, but Goslo’s path through time would have to pass over a hill, making it longer.