Scott A.

Hughes Introduction to relativity and spacetime physics

Massachusetts Institute of Technology
Department of Physics
8.033 Fall 2021

Lecture 4
Spacetime, simultaneity, and the consequences of Lorentz

4.1 From space and time to spacetime

The Lorentz transformation shows us that the invariance of c requires space and time to
be mixed together; what is “space” for one observer is a mixture of “space” and “time” for
another. This should be familiar as far as spatial directions go — what is “left” for one
observer can be a mix of “left” and “forward” for another — but mixing time and space like
this likely feels somewhat odd. We can no longer think of space and time as purely separate
things; we instead describe them as a new, unified entity: spacetime. Each inertial observer
splits spacetime into space and time; however, how they split into space and time differs.
This is fundamentally why different inertial observers measure different intervals of time and
different interval distances.
One of the tools we will use to examine the geometry of spacetime is the spacetime
diagram. This is a figure that illustrates how space and time are laid out, as seen by an
observer in some particular inertial frame. The convention in making this figures is that
time is used for the vertical axis, and space for horizontal axes.

Worldline:
Object at rest

Worldline:
Single Object moving
event at constant
positive v

Worldsheet: Extended
x
object moving at
constant negative v

Figure 1: Example of a spacetime diagram. An event is a single point. A worldline is the

sequence of events swept out by an event as it moves through space and time, with a slope
that depends on its velocity in the frame. A worldsheet is the set of events swept out by an
extended set of events as they move through space and time.

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The units of a spacetime diagram’s axes are usually chosen so that light moves on 45◦ lines
with respect to the axes of the rest frame:

t
3 m/c

2 m/c

Light cone: Opening

of trajectories de ned
1 m/c by motion of light.

1m 2m 3m 4m x
fi

With such units, a pulse of light, moving through time and projected onto 1 spatial dimen-
sion, makes a lightcone with an opening angle of 90◦ . As we will discuss shortly, the lightcone
plays an important role in helping us to figure out how events are related to one another.
When making a spacetime diagram, one draws axes corresponding to some particular
observer. Suppose we draw the axes of some observer O who uses coordinates (t, x). How
do we represent the coordinates (t0 , x0 ) of an observer O0 who moves with v = vex according
to O? In other words, what do the (t0 , x0 ) axes look like as seen by O?
To figure this out, let’s look at the transformation rule:

ct0 = γ(ct) − βγx (4.1)

x0 = −βγ(ct) + γx (4.2)

The t0 axis is defined as the set of events for which x0 = 0:

x x
0 = −βγ(ct) + γx −→ t= = . (4.3)
βc v

The x0 axis is defined by the events for which t0 = 0:

βx vx
0 = γ(ct) − βγx −→ t= = 2 . (4.4)
c c

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 √
Figure 2 illustrates the (t0 , x0 ) axes as seen by O for an observer moving with v = 3c/2.

t
t’

x’

Figure 2: Axes of observer O0 as they appear in the frame of O. The dot represents a
particular event.

In this figure, we show a particular event. This event is a geometric object, a single point in
spacetime. Although both observers agree on where it is in spacetime, they assign it rather
different space and time coordinates. (We will analyze the different labels observers attach
to coordinates in some detail shortly.)
We could equally well ask how the axes (t, x) appear according to O0 — we simply use
the inverse transformation rule, which yields

x0 −vx0
t0 = − for the t axis , t0 = for the x axis. (4.5)
v c2

t t’

x’

Figure 3: Axes of observer O as they appear in the frame of O0 .

Drawing transformed axes in this way illustrates why length contraction and time dilation
arise: Events which are simultaneous — occurring at the same time — in one frame of
reference are not simultaneous in another frame; events which occur in the same location in
one frame do not occur in the same location in another frame. This is the essence of how
“space” and “time” are mixed, but “spacetime” remains unified. Different observers agree
on “spacetime,” but they split it into “space” and “time” in different ways.

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4.2 The invariant interval
Imagine two events, labeled A and B. Compute their separation in time and space in some
given frame:

∆t = tB − tA , ∆x = xB − xA , ∆y = yB − yA , ∆z = zB − zA . (4.6)

From these quantities, compute

∆s2 ≡ −c2 ∆t2 + ∆x2 + ∆y 2 + ∆z 2 . (4.7)

Theorem: All inertial observers, in all reference frames, agree on the value of ∆s2 .
This theorem is easily proved by simply examining (∆s0 )2 , the invariant interval computed
using the coordinate separation of the events as measured in some other frame:

∆t0 = t0B − t0A , ∆x0 = x0B − x0A , ∆y 0 = yB0 − yA0 , ∆z 0 = zB0 − zA0 . (4.8)

Let us relate these “primed” separations to the “unprimed” ones using the Lorentz transfor-
mation along x we’ve been using:

c∆t0 = γ(c∆t) − γβ∆x , (4.9)

∆x0 = −γβ(c∆t) + γ∆x , (4.10)
∆y 0 = ∆y , (4.11)
∆z 0 = ∆z . (4.12)

Let us now compute (∆s0 )2 :

(∆s0 )2 = −(c∆t0 )2 + (∆x0 )2 + (∆y 0 )2 + (∆z 0 )2 (4.13)

= −γ 2 (c∆t)2 + 2γ 2 β(∆x)(c∆t) − γ 2 β 2 (∆x)2
+ γ 2 β 2 (c∆t)2 − 2γ 2 β(c∆t)(∆x) + γ 2 (∆x)2
+ ∆y 2 + ∆z 2 (4.14)
= −c2 ∆t2 γ 2 (1 − β 2 ) + ∆x2 γ 2 (1 − β 2 ) + ∆y 2 + ∆z 2
   
(4.15)
= −c2 ∆t2 + ∆x2 + ∆y 2 + ∆z 2 (4.16)
= ∆s2 . (4.17)

The first line of this is just the definition of (∆s0 )2 . To go to the second line, we’ve used the
Lorentz transformation to express the primed-frame quantities in terms of unprimed-frame
quantities. To go to the third line, we gather terms together, canceling out the terms that
involve (∆x)(c∆t), and gatheringp common factors of ∆x2 and c2 ∆t2 . To go to the fourth
line, we used the fact that γ = 1/ 1 − β 2 . That line reproduces ∆s2 , demonstrating1 that
this quantity is a Lorentz invariant.
We are going to do a lot with ∆s2 , a quantity that we call the invariant interval (of-
ten abbreviated to just the “interval”). To start, it’s worth noting that perhaps the most
important property of this quantity is whether it is negative, positive, or zero:
1
It is easy to verify that this works for the transformation along any axis. In another lecture or two, we
will introduce notation that makes proving the invariance of quantities like this really easy for any Lorentz
transformation.

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 • ∆s2 < 0: in this case, the interval is dominated by ∆t. We say that the two events
have timelike separation. When ∆s2 < 0, it means that we can find some Lorentz
frame in which the events A and B have the same spatial position (i.e., in that frame
xA = xB , y A √
= yB , zA = zB ); the events are only separated by time in that frame. We
define ∆τ ≡ −∆s2 /c to be the time elapsed between events A and B in that frame.
We call ∆τ the proper-time interval — it is the interval of time measured by the
observer who is at rest in the frame in which A and B are co-located.
It’s worth noting that if the interval between two events is timelike, then one can
imagine a signal which travels with speed v < c that connects them.
• ∆s2 > 0: the interval here is dominated by ∆x2 + ∆y 2 + ∆z 2 , and we say that the
two events have spacelike separation. In this case, we can find a Lorentz frame in
which events A and B are simultaneous; ∆s is the distance between these events in
that frame. We call ∆s the proper separation of A and B.
• ∆s2 = 0: in this case, we find that c∆t = ∆x2 + ∆y 2 + ∆z 2 — events A and B have
p

a lightlike or “null” separation. If ∆s2 = 0, then these events can be connected by a

light pulse.

The last point helps us to see that the value of ∆s2 is very closely connected to the properties
of the lightcone mentioned earlier. Suppose a flash of light is emitted from event A. If the
interval between A and another event is negative, ∆s2 < 0, then the other event must be
inside the lightcone. If the interval is positive, then the event must be outside the lightcone.
And if ∆s2 = 0, then the other event must be on the light cone itself. Figure 4 illustrates
how these notions connect to the lightcone.
t

F
L

O
A

x
Figure 4: The intervals between events A and F and events A and P are timelike: ∆s2AF < 0,
∆s2AP < 0. In all frames, event F has time coordinate greater than the time coordinate of
event A: tF > tA . Event F is unambiguously in the future of event A. Likewise, event P has
time coordinate less than the time coordinate of event A: tP < tA in all frames. Event P is
unambiguously in the past of event A. Events A and O have a spacelike interval: ∆s2AO > 0.
Event O is neither in the future nor the past of A; it is “elsewhere,” so the time-ordering of
these events is not invariant. Events A and L have a lightlike or null interval: ∆s2AL = 0.
These events are connected by a light beam in all reference frames.

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4.3 The geometry of spacetime
The relationship ∆s2 = −c2 ∆t2 + ∆x2 + ∆y 2 + ∆z 2 essentially expresses the Pythagorean
theorem for spacetime. For intuition, consider the Pythagorean theorem purely in space. On
a flat two-dimensional surface, a right triangle whose sides are ∆x and ∆y has a hypotenuse
whose length is determined from ∆s2 = ∆x2 + ∆y 2 . In three dimensions, the distance from
(x, y, z) to (x + ∆x, y + ∆y, z + ∆z) is given by ∆s2 = ∆x2 + ∆y 2 + ∆z 2 .
In spacetime, it turns out to be extremely useful to regard ∆s2 = −c2 ∆t2 + ∆x2 + ∆y 2 +
∆z 2 as expressing an invariant notion of “distance squared” between two events. Students
usually want to know “Why does the c2 ∆t2 have a minus sign?” The best answer I can give
is that this is how the geometry of the universe works. The fact that time enters ∆s2 with a
different sign from space reflects the fact that time is fundamentally quite different from the
other directions of spacetime. We can forward and backward; we can move left and right;
we can move up and down. But we can only move toward the future — we cannot step back
to the past.
Indeed, the whole notion of “past” and “future” depends on events’ separation in space-
time. If two events are timelike or lightlike separated, then one can describe one event as
being the future, and one in the past. Although the specific time coordinates assigned to
these events will vary by reference frame, the time ordering of these events is invariant: if
tF > tA in one frame, and if the interval between events A and F is timelike or lightlike,
then tF > tA in all reference frames. However, if two events are spacelike separated, then
their time ordering depends on reference frame. Consider the situation shown in Figure 5:

t
t’

x’
B

Figure 5: Observer O measures coordinates for events A and B using the (t, x) axes. Observer
O0 , who travels with velocity v = (c/2)ex according to O, measures coordinates for these
events using the (t0 , x0 ) axes.

Suppose observer O measures these events at the coordinates (tA , xA ) = (2 sec, 2 lightsec),
(tB , xA ) = (3 sec, 5 lightsec). So, for observer O, event A happens first. However, the invari-
ant interval between these events,

∆s2 = −c2 ∆t2 + ∆x2 = −(1 lightsec)2 + (3 lightsec)2 = 8 lightsec2 , (4.18)

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is positive — these events are spacelike separated, so different observers may very well order
them differently.
Let’s use the Lorentz transformation to compute√the events’ coordinates according to
0
O . Given the relative speed c/2, we have γ = 2/ 3, β = 1/2. Applying the Lorentz
transformation, we find
 √ √  2
ct0A = γtA − βγxA = 4/ 3 − 2/ 3 lightsec = √ lightsec , (4.19)
3
 √ √  2
x0A = −βγtA + γxA = −2/ 3 + 2/ 3 lightsec = √ lightsec ; (4.20)
3
 √ √  1
0
ctB = γtB − βγxB = 6/ 3 − 5/ 3 lightsec = √ lightsec , (4.21)
3
 √ √  7
x0B = −βγtB + γxB = −3/ 3 + 10/ 3 lightsec = √ lightsec . (4.22)
3

 
2 2
−→ (t0A , x0A )
= √ sec, √ lightsec
3 3
' (1.15 sec, 1.15 lightsec) (4.23)
 
0 0 1 7
−→ (tB , xB ) = √ sec, √ lightsec
3 3
' (0.577 sec, 4.04 lightsec) . (4.24)

Notice that t0A > t0B : the order of the events is reversed according to observer O0 . Using
these numbers, it is not difficult to show that O0 nonetheless finds ∆s2 = 8 lightsec2 .

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