Relativity on rotated graph paper

Roberto B. Salgado

Citation: American Journal of Physics 84, 344 (2016); doi: 10.1119/1.4943251

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 Relativity on rotated graph paper
Roberto B. Salgadoa)
Department of Physics, University of Wisconsin, La Crosse, Wisconsin 54601
(Received 4 November 2015; accepted 21 February 2016)
We demonstrate a method for constructing spacetime diagrams for special relativity on graph paper
that has been rotated by 45 . The diagonal grid lines represent light-flash worldlines in Minkowski
spacetime, and the boxes in the grid (called “clock diamonds”) represent units of measurement
corresponding to the ticks of an inertial observer’s light clock. We show that many quantitative
results can be read off a spacetime diagram simply by counting boxes, with very little algebra. In
particular, we show that the squared interval between two events is equal to the signed area of the
parallelogram on the grid (called the “causal diamond”) with opposite vertices corresponding to
those events. We use the Doppler effect—without explicit use of the Doppler formula—to motivate
the method. VC 2016 American Association of Physics Teachers.
[http://dx.doi.org/10.1119/1.4943251]

I. INTRODUCTION discussion in Sec. II and a more advanced development in

Secs. V and VII.
Soon after Einstein’s paper1 on special relativity revolu- Unlike specialized hyperbolic graph paper or two-
tionized our understanding of space and time, Minkowski2 observer graph paper, our rotated graph paper method can
introduced the “spacetime diagram” (akin to a position vs simultaneously manage numerous piecewise-inertial observ-
time graph) that can visually display the relationships ers. In addition, one can easily convert a spacetime diagram
between spacetime events. However, because spacetime is drawn for one inertial observer to a diagram for another ob-
not Euclidean, it is not obvious how draw the coordinate server, vividly illustrating a Lorentz transformation.
axes corresponding to different inertial observers or where to Because a goal of this paper is to provide a spacetime-
place tickmarks representing units of elapsed time and spa- geometrical intuition for relativity, we have taken care to use
tial distance along those axes. But one needs such axes and and develop aspects of Minkowskian spacetime geometry
tickmarks to successfully interpret a spacetime diagram, and (and not Euclidean geometry) throughout. We adopt the
methods involving invariant hyperbolas or the Lorentz trans- usual conventions for spacetime diagrams: we assume that
formation equations can be unnecessarily challenging for time runs upwards on the diagram and we choose units so
novices.
that light flashes follow worldlines that make an angle of 45
In this paper, we present an approach that allows us to
to the vertical. Rotating a sheet of ordinary graph paper by
draw and calibrate a (1 þ 1)-dimensional Minkowski space-
45 makes it easy to draw the worldlines of light flashes con-
time diagram using ordinary graph paper rotated by 45 . The
necting events.
rotated lines represent light-flash worldlines in Minkowski
spacetime, and the boxes in the grid (called “clock dia-
monds,” introduced in Sec. II) represent units of measure- II. CLOCK DIAMONDS AND CAUSAL DIAMONDS:
ment modeled on the ticks of a light clock for an inertial A FIRST LOOK
observer Alice, as shown in Sec. III. In Sec. IV, we use a
geometric construction motivated by the principles of special Because the key geometrical figure is the “causal dia-
relativity to create an analogous grid of clock diamonds that mond,” we begin with definitions that will be physically
define axes and tickmarks for another inertial observer Bob. motivated and further developed later. Although some
When the relative velocities between inertial observers are aspects of relativity2,13–20 are assumed here, the important
rational numbers, one can read quantitative results from the aspects will be derived as needed later in the article.
diagram simply by counting clock diamonds and doing sim- On a spacetime diagram (Fig. 1), consider event O and an
ple arithmetic (without the explicit use of traditional relativ- event Q in O’s causal-future (i.e., in the future light-cone of
istic formulas). O and its interior). The causal diamond3–5 of OQ is the set of
In Sec. VI, we apply such diagrams to standard examples events that can receive causal signals (that is, timelike or
from special relativity. This approach allows us to place em- lightlike signals) from event O, and then send causal signals
phasis on the physical interpretation first, followed by the de- to event Q. It is the intersection of O’s causal-future and Q’s
velopment of the relativistic formulas (to handle more causal-past. On rotated graph paper, this causal diamond is
general situations), if desired. The simplicity of the method drawn as a parallelogram OPQR with timelike diagonal OQ
is such that it could be useful in an introductory course. The and edges parallel to the lightlike gridlines.
distinctive feature of the approach discussed here is that the Given an underlying regular grid, we can describe proper-
“area” of a clock diamond (or, more generally, a “causal dia- ties of that causal diamond in units of one grid box—the
mond,”3–5 to be defined in Sec. II) is proportional to the causal diamond of OT (see Fig. 1). These grid boxes will be
squared interval of its timelike diagonal.6–12 Emphasizing called “clock diamonds”21 since we will associate them with
this area promotes the notion of Lorentz-invariance, which is the ticks of a light clock in Sec. III. For convenience in
more fundamental than the coordinate-dependent time and counting, we use coordinates (u, v) along the gridlines
space measurements associated with the diagonals of an (called light-cone coordinates22–24). In Fig. 1, the causal dia-
observer’s clock diamond. We provide an introductory mond of OQ has width Du ¼ 8 and height Dv ¼ 2.

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V 344

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 III. ALICE’S LONGITUDINAL LIGHT CLOCK
A. Alice’s clock diamonds
We begin the construction by interpreting the unit boxes
in the rotated grid. Consider an inertial observer, Alice, at
rest in her reference frame, carrying a mirror a constant
distance D away. Alice emits a light flash (traveling with
speed c) that reflects off the distant mirror and returns
(at speed c) to her after a round-trip elapsed time 2D=c. If
this returning light flash is immediately reflected back,
this functions like a clock, called the light clock.1,12,25–28
Since we wish to regard time as a more primitive concept
than space, let us declare the round-trip travel time to be
1 “tick,” so that D ¼ ð1=2Þc tick ¼ 1=2 “light-tick” (analo-
gous to the light-year as a unit of distance). For conven-
ience, let us define d ¼ ðD=cÞ so that the spatial
displacement d is also measured in “ticks.” Thus, d meas-
Fig. 1. The causal diamond of OQ is the set of events that can receive signals from ures the duration of time for light to travel the desired spa-
event O, and then send signals to event Q. On rotated graph paper, this is a paralle- tial displacement. In these units, c ¼ ð1 tickÞ=tick, or
logram OPQR with timelike diagonal OQ, with edges parallel to the lightlike grid- simply c ¼ 1. More generally, velocities will be dimension-
lines, labeled by coordinates u and v. From counting grid boxes (congruent to the
causal diamond of OT), the area of the causal diamond of OQ is AreahOQi
less and denoted by the symbol b.
¼ Du Dv ¼ ð8Þð2Þ ¼ 16 area-units, and the aspect ratio is Du=Dv ¼ 8=2 ¼ 4. On the rotated grid in Fig. 2, we draw the spacetime dia-
gram of Alice and two such mirrors, one to the right (the
direction in which Alice faces) and the other to the left. The
For a causal diamond OPQR (with timelike diagonal OQ
parallelogram OMTN represents one tick of Alice’s longitu-
and spacelike diagonal PR) of width DuOQ and height DvOQ ,
dinal light clock, where the spatial trajectories of the light-
we state and interpret two key formulas that will be developed
rays are parallel to the direction of relative motion.
later. The “aspect ratio formula” (developed in Sec. VII A)
Henceforth, we will refer to this parallelogram as Alice’s
relates the width-to-height ratio of the causal diamond10 to the
clock diamond.
relative Doppler factor k between the timelike diagonal OQ
By tiling spacetime with copies of her clock diamond,
and the timelike-diagonal of a clock diamond OT by
Alice sets up a coordinate system (see Figs. 3 and 4). She
DuOQ measures displacements in time along a parallel to her world-
k2 ¼ : (1) line (along diagonal OT, which happens to be vertical on our
DvOQ rotated grid). We will show in Sec. III B that she measures
These observer-dependent factors, Du and Dv, are related
to the elapsed-times between emissions and receptions in
radar-measurements of events O and Q. The “area formula”
(developed in Sec. VII B) relates the area of a diamond to the
squared interval of its diagonal

Ds2OQ ¼ DuOQ DvOQ ¼ AreahOQi; (2)

which implies that the clock diamonds of different inertial

observers have equal areas.
Since causal diamond OPQRpffiffiffiffiffi in Fig. 1 has area ð8Þð2Þ
¼ 16, we find that DsOQ ¼ 16 ¼ 4 ticks elapsed along the
diagonal OQ, which we will use to mark off another grid of
clock diamonds for that inertial observer along OQ. In addi-
tion, because the aspect ratio of the causal diamond is
Du=Dvpffiffiffi¼ 8=2 ¼ 4, we find that the relative Doppler factor is
k ¼ 4 ¼ 2. From the Doppler formula and its inverted
form (to be derived in Sec. VII A),
sffiffiffiffiffiffiffiffiffiffiffi
1þb
k¼ ; (3)
1b

k2  1 Fig. 2. Alice’s clock diamonds from her longitudinal light clock on her
b¼ ; (4) spacetime diagram. At event O, Alice emits light flashes which reflect
k2 þ 1 off her two mirrors (the dotted worldlines at x ¼ d and x ¼ d) to be
received by Alice after an elapsed time Dt ¼ 2d. The resulting paralle-
we find the dimensionless relative velocity factor b along di- logram OMTN (called Alice’s clock diamond) defines 1 “tick” of
agonal OQ is ð4  1Þ=ð4 þ 1Þ ¼ 3=5. Alice’s clock, which can be used to set up a coordinate system for
We now proceed to systematically motivate these ideas. Alice.

345 Am. J. Phys., Vol. 84, No. 5, May 2016 Roberto B. Salgado 345

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 diagonal MN of her clock diamond is parallel to her line of
constant time. Later, a moving inertial observer Bob will
apply this same procedure.
Rather than somehow extending a long ruler into space,
Alice supplements her clock with a light-signaling setup to
perform the following radar measurement (see Fig. 5; for
now, we will dispense with the details of the light clock’s
construction, and in particular, the mirror worldlines.) To
assign coordinates to a distant event Q, she emits at a certain
time te the light flash (traveling at speed c) that will be
reflected back by the distant target event, (returning at speed
c) to be received by her at a later time tr. (Events e and r are
the intersections of Alice’s worldline with the light-cone of
event Q.)
Fig. 3. Using her clock-diamonds, Alice counts the number of diagonals to
measure temporal and spatial displacements and the number of edges to
With her clock-readings te and tr at the emission and
measure lightlike displacements. reception events, she assigns a time coordinate tQ to be the
halftime clock-reading

displacements in space along her “line of constant time” 1

tQ ¼ ðtr þ te Þ; (5)
(parallel to diagonal MN, which happens to be horizontal on 2
our rotated grid). Lightlike displacements are measured par-
allel to the edges of her clock diamond. and a spatial coordinate xQ to be half of the round-trip time
multiplied by the signal-speed c

B. Constructing Alice’s x-axis by a radar method 1

xQ ¼ 6 ðtr  te Þc: (6)
2
An inertial observer with a clock can assign rectangular
(t, x)-coordinates to all events on her worldline. Coordinate Since tr  te , the plus-sign corresponds to receiving the
t equals the reading on her clock at these events, with t ¼ 0 reflection from event Q on her forward side, so that xQ  0.
set at the origin event O. Coordinate x equals zero at these Note that there is a unique event P, with coordinates ðtQ ; 0Þ,
events because they are on her worldline. But, how can she on Alice’s worldline that she regards as simultaneous with
assign coordinates to distant events—those events not on her distant event Q.
worldline?
We introduce a radar method13–17,25–30 for assigning coor-
dinates. Then, we show how Alice constructs her “line of
constant time” (whose events are simultaneous according to
her) and how she uses a chain of her clock diamonds along
that line to measure spatial-displacements. We note that

Fig. 5. Alice applies the radar method. To measure event Q, she emits a
Fig. 4. Alice’s rectangular coordinate system. With origin event O, Alice’s
forward-directed light flash at time te ¼ 2 and receives its echo from event Q
time-axis is marked by a string of clock diamonds with their diagonals lined up
~ on her worldline. It will be shown that Alice’s x-axis is marked by a at time tr ¼ 8. Alice assigns event Q the coordinates tQ ¼ ð1=2Þðtr þ te Þ ¼ 5
along OT and xQ ¼ ð1=2Þðtr  te Þc ¼ 3c ¼ 3, in agreement with the assignment using
string of clock diamonds with their other diagonals lined up along OX ~ (parallel Alice’s clock diamonds from Fig. 4. Note that there is a unique event P,
~ ~ ~
to MN). As an example, since OQ ¼ 5 OT þ 3 OX, Alice assigns event Q rec- with coordinates ðtQ ; 0Þ, on Alice’s worldline that she regards as simultane-
tangular coordinates ðtQ ; xQ Þ ¼ ð5; 3Þ. Similarly, ðtW ; xW Þ ¼ ð5:25; 4:75Þ. ous with distant event Q.

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 Let us now use the radar method to locate events that
Alice regards as simultaneous with event O on her worldline;
that is, those events Xi with tXi ¼ tO ¼ 0 (see Fig. 6). For
such events, we have tei ¼ tri , so that xXi ¼ 6ð1=2Þð2tri Þc
¼ 6tri (since c ¼ 1). By choosing tri with spacing equal to
diagonal OT, we have xXi with spacing equal to diagonal
MN, the width of Alice’s clock diamond. This suggests that
we can mark Alice’s x-axis with a chain of her clock dia-
monds, arranged corner to corner, as in Fig. 6.
Alice’s clock diamond OMTN corresponds to radar meas-
urements of events M and N, with emissions at event O
(tO ¼ 0) and receptions at event T (tT ¼ 1). Events M and N
have (t, x)-coordinates (ð1=2Þ; ð1=2Þ) and (ð1=2Þ; ð1=2Þ),
respectively, and are therefore simultaneous according to
Alice. Thus, diagonal MN of Alice’s clock diamond can be
used to determine the “lines of constant time for Alice” and
the spacing of “tickmarks of space for Alice” (akin to diago-
nal OT marking the ticks of Alice’s clock). This construction
will be more fully appreciated when we apply it to the case
of a moving inertial observer Bob, drawn on Alice’s space-
time diagram.
Fig. 7. Alice’s describes event Q with radar coordinates ðtr ; te Þ ¼ ð8; 2Þ, rec-
C. Alice’s coordinate systems tangular coordinates ðtQ ; xQ Þ ¼ ð5; 3Þ, and light-cone coordinates ðuQ ; vQ Þ
For future reference, we establish, for an event Q, useful ¼ ð8; 2Þ.
relationships among the radar coordinates (tr, te), the rectan-
gular coordinates (tQ, xQ), and the light-cone coordinates number of timelike-diagonals plus the number of spacelike-
(uQ, vQ) that were briefly introduced in Sec. II (see Fig. 7). diagonals. In terms of radar coordinates, Eqs. (5) and (6)
The light-cone coordinates22,23 are expressed in terms of lead to
the rectangular coordinates by
u ¼ tr ; (9)
u  t þ x; (7)
v ¼ te : (10)
v  t  x; (8)
For the case of x < 0 (or, equivalently, u < v), we have
in our convention.24 Equation (7) tells us that in the grid of instead ðu; vÞ ¼ ðte ; tr Þ. Thus, an inertial observer’s light-
diamonds, the number of edges along the u-axis equals the cone coordinates for an event can be interpreted as that
observer’s radar-times needed to measure the event. For sim-
plicity, when dealing with radar coordinates, we will hence-
forth only consider cases in which x  0.

IV. BUILDING BOB’S LONGITUDINAL LIGHT

CLOCK
A. The calibration problem
Now consider another inertial observer, Bob, moving
in the x-direction with velocity b with respect to Alice (see
Fig. 8). Given b, we first use Alice’s ticks to draw Bob’s
worldline OV. The calibration problem is to locate on OV the
event F that marks the first tick on Bob’s clock after event
O. In accordance with the principles of special relativity,1
we expect that the first tick of Bob’s identically constructed
light clock will be drawn as Bob’s clock diamond, a causal
diamond with diagonal OF and edges parallel to the lightlike
lines in the grid.
Inspired by Bondi,13,14 we first locate event F indirectly
using a signal-sending experiment. Then, we develop the key
result that the area of Bob’s clock diamond is equal to that of
Alice’s clock diamond.6–12 (This result follows from the fact
Fig. 6. Alice’s positive x-axis constructed with the radar method. Note that that the Lorentz transformation has unit determinant, which
OX is parallel to the spacelike diagonal MN of Alice’s first clock diamond implies that events T and F lie on a hyperbola centered at O
OMTN. with asymptotes along O’s light cone.)

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 Fig. 8. Preparing to calibrate Bob’s identically constructed longitudinal light
clock in Alice’s frame of reference. Bob, with velocity 3/5 according to
Alice, emits light flashes at event O that reflect off his two light clock mir-
rors to be received by Bob at event F, 1 tick later on his clock. But where is
event F along Bob’s worldline OV? (How far away should Bob’s mirror
worldlines be for the reflections to be received at F?) The resulting parallelo-
gram OYFZ with diagonal OF would define Bob’s clock diamond. So, given
parallelogram OMTN and worldline OV, determine the event F on OV such
that tA B
T ¼ tF .
Fig. 9. Calibrating Bob’s longitudinal light clock. By counting the clock dia-
monds on her worldline from radar measurements, Alice determines
B. Constructing Bob’s clock diamonds k2 ¼ ðtr3  tr2 Þ=ðte3  te2 Þ ¼ 4, and thus k ¼ 2.
We now construct Bob’s clock diamonds using the
Doppler effect, framed in the context of television transmis- DtA B
r ¼ kDte ; (12)
sions sent and received by Alice and Bob.31 Following
Bondi,13,14 we will not use the Doppler formula. Using ¼ kðDtBr Þ ¼ kðkDtA
e Þ; (13)
Alice’s clock diamonds, we draw Bob’s worldline along a
sloped line corresponding to velocity b ¼ 3=5, chosen for so that it would take k2 of Alice’s hours to watch (in very
simplicity (see Fig. 9; where we have drawn Bob’s worldline slow-motion) her originally broadcasted one-hour program.
as OJ, with ðIJÞ=ðOIÞ ¼ 6=10 ¼ 3=5). By counting Alice’s clock diamonds off the spacetime dia-
We now begin the construction of Bob’s clock diamonds. gram in Fig. 9, one can determine, for b ¼ 3=5, the corre-
Alice sends to Bob two successive light flashes, the second sponding value of k2
flash sent one tick after the first. We can interpret these as
broadcasts marking the start and end of a one-hour television DtA tA A
r3  tr2 12  8
program produced by Alice. Due to the finiteness of the k2 ¼ r
A
¼ A A
¼ ¼ 4; (14)
Dte te3  te2 32
speed of light, Bob receives the first flash after a delay.
However, because Bob is receding from Alice, he receives so that k ¼ 2. The result is that there must two of Bob’s
the second flash after k of his ticks, where k is a positive pro- clock-ticks between events B2 and B3 on his worldline.
portionality constant to be determined. That is, Because the sides of the clock diamonds are traced out by
light flashes that are parallel to the lines of the rotated grid,
DtBr ¼ k DtA
e : (11) one is led to drawing two congruent causal diamonds along
Bob’s worldline between events B2 and B3, each a prototype
The above equation says that Bob’s measurement of the of Bob’s clock diamond. We have thus determined one
period between his receptions is equal to k times Alice’s “tick” for Bob’s light clock.
measurement of the period between her transmissions. It is
easy to see that k ¼ 1 for an observer at rest according to C. Constructing Bob’s x-axis and his rectangular coordi-
Alice, but k > 1 for an inertial observer who is receding from nate system
Alice. Thus, it would take k of Bob’s hours (assuming that
his clock had already been calibrated) to watch (in slow Using Bob’s clock diamonds, Bob’s rectangular coordi-
motion) a program that Alice broadcasts in one-hour. nate system is constructed in complete analogy with Alice’s.
To determine the value of k (and thus calibrate Bob’s Bob follows the radar method that Alice used to construct
clock), we arrange to have Bob immediately rebroadcast (or her x-axis (see Fig. 10 and compare with Fig. 6). In addition,
reflect) the received signals from Alice. By the principle of Bob uses his clock diamonds to measure displacements in
relativity, Alice must also receive the delayed broadcast spacetime (see Fig. 11 and compare with Fig. 3).
from Bob at the same slowed rate. Thus, it would take k of A striking feature of Bob’s rectangular coordinate system,
Alice’s hours to watch (in slow motion) a one-hour program shown in Fig. 12, is that Bob’s x-axis is not parallel with
broadcast by Bob. Therefore, Alice’s x-axis. Because an observer’s x-axis represents a set

348 Am. J. Phys., Vol. 84, No. 5, May 2016 Roberto B. Salgado 348

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 Fig. 10. Bob’s x-axis constructed with the radar method used by Alice in
Fig. 12. Bob’s rectangular coordinate system uses the diagonals of his clock
Fig. 6. Note that OG along Bob’s x-axis is parallel to the spacelike diagonal
diamonds to locate events (compare with Alice’s rectangular coordinate sys-
YZ of Bob’s clock diamond OYFZ.
tem in Fig. 4). His time-axis is marked by a string of clock diamonds with
timelike-diagonals along OF ~ on his worldline. His x-axis is marked by a
of events that are simultaneous with the event at t ¼ 0 for string of clock diamonds with their spacelike-diagonals along OG, ~ on a line
that observer, the spacetime diagram indicates that these of simultaneity according to him (parallel to YZ). From the diagram, since
observers will disagree on whether two distinct events (say, OW~ ¼ 3 OF ~ þ 2 OG, ~ Bob assigns event W rectangular coordinates ðtB ; xB Þ
W W
events O and G) are simultaneous. This is the relativity of ¼ ð3; 2Þ. Similarly, ðtBQ ; xBQ Þ ¼ ð4; 0Þ. Recall from Fig. 4 that ðtA A
W ;xW Þ ¼ ð5:25;
simultaneity. 4:75Þ and ðtA A
Q ;xQ Þ ¼ ð5;3Þ.
Note also that on Alice’s spacetime diagram Bob’s x-axis
is not perpendicular (in the familiar Euclidean sense) to his relativity (see Sec. VI). While we have tried to provide a
t-axis, as are her own axes.32 This indicates that the geome- good physical motivation through the use of physical meas-
try of spacetime is not Euclidean, but Minkowskian.2 Here, urements, we note some geometrical properties involving
the notion of perpendicularity—to be called Minkowski-per- causal diamonds that promote Lorentz invariance and lead to
pendicular—will be encoded by the statement: the diagonals more efficient calculations.
of a causal diamond are Minkowski-perpendicular to each Figure 13 redraws the construction of Bob’s clock dia-
other.33 mond from Fig. 9 with an emphasis on causal diamonds. By
counting the boxes on the rotated grid we observe:
V. THE AREA OF A CAUSAL DIAMOND
(1) The area of Bob’s clock diamond is equal to the area of
A. The area of causal diamond as an absolute Alice’s clock diamond. The edges of Bob’s diamond
With the radar construction of Bob’s clock diamond for a have been reshaped by factors k and 1/k.
given velocity (here, b ¼ 3=5) and the subsequent construc- (2) The causal diamond from events B2 to B3 has the same
tions of his time and space axes, one can immediately pro- shape (that is, the same aspect ratio) as one of Bob’s
ceed to the standard textbook examples from special clock diamonds.
(3) The area of the causal diamond from B2 to B3 (in units of
clock diamond areas) is equal to the square of the
proper-time interval along its timelike diagonal.
These statements can be checked for another inertial ob-
server, Carol, with b ¼ 4=5 and its associated factor k ¼ 3,
as shown in Fig. 14. We will make use of these observations
in the remainder of the paper.

B. The invariant unit hyperbola and perpendicularity

Using the equality of clock diamond areas, we can construct
other clock diamonds with lower corner at event O, which
reveals an invariant unit hyperbola centered at event O (see
Fig. 15 for the future-branch of the hyperbola). In Minkowski
spacetime, the hyperbola plays the role that the circle plays in
Fig. 11. Like Alice (in Fig. 3), Bob uses the diagonals of his clock diamonds to Euclidean space. In addition to being the curve of equal
measure (according to him) purely temporal and purely spatial displacements squared interval from event O, the hyperbola defines the
and the edges of his clock diamonds to measure lightlike displacements. notion of “perpendicular” in Minkowski spacetime geometry

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 event. This translates to the notion that, on a spacetime dia-
gram, an observer’s space-axis is Minkowski-perpendicular to
that observer’s time-axis (i.e., worldline).
The causal diamond encodes this notion of perpendicular-
ity. Figure 15 suggests that along a given direction from
event O, the tangent to the unit hyperbola is parallel to the
spacelike-diagonal of the corresponding causal diamond.
Thus, as mentioned earlier, the diagonals of the causal dia-
mond are Minkowski-perpendicular to each other.33 (Note
that the edges of clock diamond are not Minkowski-
perpendicular to each other. Thus, we use the terms
“diamond” and “parallelogram,” but not “rectangle.”)

C. Subdivided grids and rational relative velocities and

k-factors
Figure 15 shows that our method can clearly display the
coordinate systems of numerous observers on the same dia-
gram. We briefly discuss the set of velocities that can be
handled using our method.
When the relative velocity b of an inertial observer is a
rational number, say b ¼ Dx=Dt where Dx and Dt are inte-
gers, the worldline of that observer can be constructed by
using a Minkowski right triangle formed by counting off Dt
of Alice’s temporally arranged diamonds, followed by count-
ing off Dx of Alice’s spatially arranged diamonds. Since the
Fig. 13. Calibrating Bob’s longitudinal light clock (Fig. 9), redrawn to squared interval of the hypotenuse is the integer Ds2 ¼ ðDtÞ2
emphasize causal diamonds. The causal diamond of B2B3 has height Dv ¼ ðDxÞ2 , the area of its causal diamond can be determined by
Dt and width Du ¼ k2 Dt. (Recall that Dt and Dx count diagonals and Du and
Dv count edges.) The width-to-height ratio Du=Dv ¼ k2 ¼ 4 suggests the counting grid boxes. To go further and count off an integer
shape of Bob’s clock diamonds. The area DuDv ¼ ðkDtÞ2 ¼ 4Dt2 suggests number Ds of clock diamonds along the hypotenuse, the
that there are 2 of Bob’s clock diamonds along the diagonal from B2 to B3. squared interval must be p a perfect-square. This occurs when
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Observe that the area of Bob’s clock diamond is equal to that of Alice’s. the Doppler k-factor k ¼ ð1 þ bÞ=ð1  bÞ is rational, with
value Ds=ðDt  DxÞ. These restrictions can be formulated in
(Minkowski34 uses the term “normal”), henceforth referred to terms of Pythagorean triples.35
as Minkowski-perpendicular: at any event on the hyperbola, its When the k-factors are ratios of small integers, it becomes
tangent is Minkowski-perpendicular to the radius vector to that easy to graphically construct the clock diamonds for these
observers, especially if the grid is suitably subdivided. As we
saw in Fig. 15, a 6  6 subdivision is useful because it can

Fig. 15. The clock diamonds and the unit hyperbola. By choosing our clock
diamond to have 6  6 subdivisions on the grid and by exploiting the equal-
ity of clock diamond areas, we construct some various Du-by-Dv clock dia-
monds from event O from the factors of 62. [The expressions for k and b in
terms of Du and Dv are based on Eqs. (1) and (4).] These clock diamonds
Fig. 14. Calibrating Carol’s longitudinal light clock (analogous to Fig. 13). reveal the invariant unit hyperbola DuDv ¼ 1 underlying our method. The
For velocity b ¼ 4=5, we find k ¼ 3. Here, we have chosen emission events inset diagram shows that the tangent to the hyperbola at P is parallel to the
e1 and e2 to avoid a larger diagram. spacelike-diagonal of OP’s clock diamond.

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 accommodate the k-factors 1, 3/2, 2, and 3, along with their earlier, this corresponds to k ¼ 2, which allows us to sequen-
reciprocals, which correspond to velocities 0, 65=13; 63=5, tially draw Bob’s clock diamonds to find that four such dia-
and 64=5 that are typically used because of the simple arith- monds can be constructed along OQ, as shown in the right
metic that results. panel.
While the restriction to rational k-factors may be desirable Alternatively, as illustrated in the middle panel, we can
for thinking in terms of clock diamonds along displacements, use the property that the proper-time elapsed along OQ is
we show that one can handle all rational velocities by think- equal to p theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
square root of the area of its causal diamond:
ing more invariantly in terms of causal diamonds and using ðOQÞ ¼ AreahOQi. Because we count the parea ffiffiffiffiffi to be 16,
the aspect ratio and area formulas [Eqs. (1) and (2)]. These the proper-time along OQ is computed to be 16 ¼ 4 ticks.
advanced methods will be established in Sec. VII. We can now draw 4 of Bob’s clock diamonds along OQ, as
For arbitrary velocities these methods still apply; however, shown in the right panel.
as with ordinary uses of graph paper only approximate quan- Note that while Bob declares the elapsed time between the
tities are obtained, with accuracy dependent on one’s skill events O and Q that he experiences (the proper time) is 4
with graphical tools. Fine subdivisions of the graph paper ticks, Alice declares the elapsed time between those events to
would certainly be helpful. Of course, one can advance to be 5 ticks according to her clock (using events O and P).
the use of algebraic formulas, whose meanings would have Since ðOPÞ > ðOQÞ, the elapsed time between O and Q
now been motivated by these special convenient velocities. observed by Alice is longer than the elapsed proper time, as
measured by Bob. This difference in measured times is the
VI. STANDARD EXAMPLES time dilation effect, with time-dilation factor c ¼ ðOPÞ=ðOQÞ
¼ 5=4. The completed diagram in the right panel of Fig. 16
We now present a series of standard problems in special encodes the key features of the problem, while suggesting the
relativity with our graphical methods. Some of these are con- reasoning behind its solution.
cise summaries of worksheets presented to introductory The Minkowski right triangle featured in the right panel of
undergraduate students, with emphasis on counting clock Fig. 16 satisfies the squared interval formula, which can be
diamonds along displacements. We also offer insights using regarded as the Minkowskian analogue of the Pythagorean
causal diamonds that are more appropriate for advanced theorem
students.
ðOQÞ2 ¼ ðOPÞ2  ðPQÞ2 : (15)
A. Time dilation
Question: After leaving inertial observer Alice at Using ratios b ¼ ðPQÞ=ðOPÞ and c ¼ ðOPÞ=ðOQÞ to
event O, another inertial observer Bob travels at express the right-hand side in terms of (OQ), we obtain the
velocity 3/5 according to Alice. Thus, according to familiar time-dilation factor
Alice, after 5 of her ticks have elapsed, Bob is 1
located 3 of her light-ticks away, at event Q. How c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi2 : (16)
much of Bob’s proper-time has elapsed between 1b
events O and Q, both of which are on his
worldline?
B. Symmetry of the inertial observers
To answer this question, we make use of Fig. 16. In the
left panel, Alice uses her clock diamonds to draw Bob’s (Refer to Fig. 17) In accordance with the principle of rela-
worldline using Minkowski right triangle OPQ, with tivity, for two events O and Q0 on Alice’s worldline, Bob
Minkowski-perpendicular legs OP and PQ chosen so that the will observe a longer time interval between those events than
ratio of their sizes is ðPQÞ=ðOPÞ ¼ b ¼ 3=5. As we found Alice will, with the same time-dilation factor c ¼ 5=4. As

Fig. 16. Time dilation. (Left) First, using Alice’s diamonds to construct two legs of a Minkowski right triangle so that b ¼ ðPQÞ=ðOPÞ ¼ 3=5, construct Bob’s
worldline. (Middle) Next, draw the causal diamondpof ffiffiffiffiffithe unknown interval OQ and compute the area of the causal diamond. Here, the area is 16 clock
diamond-areas. (Right) Divide the diagonal OQ into 16 ¼ 4 equal parts and then draw 4 congruent diamonds along OQ that are similar to the causal diamond
of OQ. These are Bob’s clock diamonds, marking the ticks of his light clock. This completed diagram encodes all of the relevant physical features of the prob-
lem. The time-dilation factor c is the ratio ðOPÞ=ðOQÞ ¼ 5=4.

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 shown in Fig. 17, Bob uses his clock diamonds and
Minkowski right triangle OP0 Q0 to construct the analogue of
Alice’s diagram.
In addition, it can be shown that triangles OIJ and OI 0 J 0
are congruent triangles in Minkowski spacetime, and that
they are related by a Lorentz transformation and a reflection.
The congruency is more apparent if the situation is drawn in
the “center of velocity” frame.

C. The clock effect

Question: After meeting Alice at event O, Bob
travels away at velocity 3/5, instantaneously
turning around at a distant event Q and returning
with velocity –3/5. At the reunion event Z, if Alice
had aged 10 ticks since the separation event, how
much did Bob age?
Referring to Fig. 18, we determine Bob’s clock diamonds
by treating his non-inertial trip as piecewise-inertial seg-
ments. By constructing the causal diamonds from O to Q and
ffiffiffiffiffi Q to Z, we determine from the diagram that there
thenpfrom
are 16 ¼ 4 ticks along OQ, followed by another 4 ticks
along QZ, for a total of 8 ticks. This does not equal the 10
ticks logged by Alice’s inertial trip from O to Z. This route-
dependence of elapsed proper time between two events in
spacetime is the clock effect. Fig. 18. Clock effect. Alice ages 10 ticks along her inertial worldline from O
Observe that time dilation implies that Area<OP> > to Z, whereas Bob (traveling to and from event Q with speed 3/5) ages 8
Area<OQ> and Area<PZ> > Area<QZ>, where P is the ticks along his non-inertial worldline.
event on her worldline OZ that Alice regards as simultaneous
with a distant event Q. Thus, the trip with the longest proper alternative approach by Marzke and Wheeler26,27 and by
time from O to Z is Alice’s inertial trip through P. Schild39 in which Bob effectively switches to a third inertial-
In passing, we note that since the kink in Bob’s non- clock he meets at the turn-around event Q.
inertial worldline is associated with an acceleration, there are
issues involving the rigidity of a real light clock carried by
Bob and changes in reference frame. This, of course, affects D. Length contraction
the reliability of using of a real light clock and radar methods Question: Bob is traveling at a velocity 3/5 with
in our idealized situation. We refer the reader to articles on respect to Alice, and carries a ladder that is 5 ticks
the clock effect that discuss this point, some29,30,36–38 of long. How long is that ladder according to Alice?
which analyze a clock undergoing acceleration. We adopt an
Referring to Fig. 19, Alice uses her clock diamonds to
draw Bob’s worldline and his clock diamonds. Then, Bob
uses his clock diamonds to measure from event O a spatial
displacement of 5 ticks (his ladder’s rest-length) to event Y.
Through event Y, Bob then constructs the worldline of his
ladder’s far end as a parallel to his worldline OQ.
Alice determines the length of Bob’s ladder by finding the
spatial distance between his ladder worldlines using events
O and X, which Alice regards as simultaneous. Since OX is
the spacelike diagonal of the causal diamond between O and
X, the length of the ladder according to Alice is the square-
root of the magnitude of p the area of ffithe causal diamond
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
between O and X: ðOXÞ ¼ jAreahOXij ¼ 4 ticks.
Because ðOXÞ < ðOYÞ, Alice’s observed length of the lad-
der is shorter than its rest-length. This is the length contrac-
tion effect, with length-contraction factor ðOXÞ=ðOYÞ
¼ 4=5. Since it can be shown that triangles OPQ and OYX
are similar triangles in Minkowskian geometry, we have
ðOXÞ=ðOYÞ ¼ ðOQÞ=ðOPÞ ¼ 1=c. That is, the length-
contraction factor is equal to the reciprocal of the time-
dilation factor c.
Fig. 17. Symmetry of time-dilation, extending Fig. 16. Because the small
diagram we have been using becomes cluttered when displaying the symme- By symmetry, Bob determines the length of Alice’s identi-
try, we have included additional pairs of events {I, J} and fI0 ; J 0 g. (This dia- cal ladder (5 ticks long according to her, with its far end
gram also suggests how to draw the situation on Bob’s spacetime diagram.) along a worldline parallel to OP, through events Y 0 and X0 )

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 clock diamonds to locate event E. Then, Alice uses her clock
diamonds to find ðtA A
E ; xE Þ ¼ ð7; 9Þ. This agrees with the
result from the inverse40 Lorentz coordinate transformation
formulas
     
A B A B 5 3
tE ¼ c tE þ bBob xE ¼ 2 þ 6 ¼ 7; (17)
4 5
     
A B 5 3
xAE ¼ c x B
E þ b t
Bob E ¼ 6 þ 2 ¼ 9; (18)
4 5

where c ¼ 5=4 is the time-dilation factor for velocity

bABob ¼ 3=5.
For completeness, recall the graphical method coordinate
assignments of events W and Q from Fig. 12. Given
ðtBW ;xBW Þ ¼ ð3;2Þ, we had determined ðtA A
W ;xW Þ ¼ ð5:25;4:75Þ.
That assignment agrees with the results from the transforma-
tion formulas given in Eqs. (17) and (18). Similarly, we verify
that ðtBQ ;xBQ Þ ¼ ð4;0Þ and ðtA A
Q ;xQ Þ ¼ ð5;3Þ also agree with the
Fig. 19. Length contraction: Using Minkowski right triangle OYX with hy- formulas.
potenuse OX, Alice determines that Bob’s ladder is shorter than its rest This inverse40 Lorentz coordinate transformation can be
length: LA B
ladderB ¼ ðOXÞ < ðOYÞ ¼ LladderB . By symmetry, using triangle derived by drawing on Fig. 20 a rectangle with its edges par-
OX0 Y 0 , where X0 Y 0 represents the far end of Alice’s ladder, Bob would deter-
mine that Alice’s ladder is shorter than its rest length: LBladderA ¼ ðOX0 Þ
allel to Alice’s axes and its corners at T and T 0 (see Refs. 14
< ðOY 0 Þ ¼ LA
and 17). We offer a proof in Sec. VII C using the k-factor
ladderA .
reshaping of the clock diamonds.
to be shorter than its rest-length by the same length-
contraction factor. Using triangle OY 0 X0 , we find ðOX0 Þ= F. Velocity transformation
ðOY 0 Þ ¼ 4=5. Question: Alice, Bob, and Carol met briefly at
event O. Bob, with velocity 3/5 according to Alice,
E. Lorentz coordinate transformation observes Carol with velocity 5/13. What is the
velocity of Carol according to Alice?
Question: Bob, with velocity 3/5 according to
Alice, assigns event E coordinates ðtBE ; xBE Þ Referring to Fig. 21, Alice uses Minkowski right triangle
¼ ð2; 6Þ. What coordinates would Alice assign to OPQ and her clock diamonds to draw Bob’s worldline with
E? (Both Alice and Bob regard their meeting event velocity 3/5 and his clock diamonds. Next, using Minkowski
O as the origin of their coordinates.) right triangle OXY, Bob uses his clock diamonds to construct
Carol’s worldline along OY, with OX and XY chosen so that
Referring to Fig. 20, Alice uses her clock diamonds to ðXYÞ=ðOXÞ ¼ 5=13. Using Minkowski right triangle OTY,
draw Bob’s worldline and his clock diamonds. Bob uses his

Fig. 20. Comparison of coordinates without using the Lorentz coordinate

transformation. The causal diamond with spacelike diagonal OE has width
Du ¼ 16 and height Dv ¼ 2 according to Alice. Similarly, this causal dia-
mond has width Du ¼ 8 and height Dv ¼ 4 according to Bob. Both deter- Fig. 21. Velocity transformation: bA A B A B
Carol ¼ ðbBob þ bCarol Þ=ð1 þ bBob bCarol Þ
mine the square interval of OE to be AreahOEi ¼ DuDv ¼ 32 ¼ ð72  92 Þ ¼ ðð3=5Þ þð5=13ÞÞ=½1þ ð3=5Þð5=13Þ ¼ 16=20 ¼ 4=5. Note three Minkowski
¼ ð22  62 Þ. right triangles here: OTY, OPQ, and OXY.

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 Alice uses her clock diamonds to determine the velocity of
Carol’s worldline OY. From the graph paper, we determine
bA
Carol ¼ 16=20 ¼ 4=5, which agrees with the result obtained
by the velocity transformation formula
3 5
bA B þ
Bob þ bCarol 5 13 4
bA
Carol ¼ ¼  ¼ : (19)
1 þ bA B
Bob bCarol
3 5 5
1þ
5 13

To derive this formula, one can start with bA A A

Carol ¼ xY =tY
and substitute the analogues of Eqs. (18) and (17); since
xBY ¼ bBCarol tBY , the formula follows directly (see Refs. 18 and
19). We offer a proof in Sec. VII C using the k-factor reshap-
ing of the clock diamonds.
Note that because OY is along Carol’s worldline and its
causal diamond (of dimensions Du ¼ 36 and Dv ¼ 4) has an
area ð36Þð4Þ ¼ ð12Þ2 that is a perfect-square, it is easy to
construct Carol’s clock diamonds along her worldline. Along
OY, there will be 12 of her clock diamonds reshaped by the
pffiffiffiffiffiffiffiffiffiffi Fig. 22. An elastic collision (with P ~1;i þ P
~2;i ¼ P
~1;f þ P
~2;f ) drawn on
factor k ¼ 36=4 ¼ 3. (Carol’s clock diamond was drawn
Alice’s energy-momentum diagram. The particles have rest-masses m1 ¼ 12
in the calibration of Fig. 14 and was included in the hyper- and m2 ¼ 8, initial-velocities b1;i ¼ 5=13 and b2;i ¼ 15=17, and final-
bola in Fig. 15.) velocities b1;f ¼ 4=5 and b2;f ¼ 3=5. We have drawn a “mass-diamond”
for the total energy-momentum vector of the system, P ~COM . The aspect ratio
G. Collisions in energy-momentum space corresponds to the square of the Doppler factor for the center-of-momentum
frame. The area of this mass-diamond, (40)(20), corresponds to the square of
In addition to kinematical examples in spacetime, the the invariant-mass of this system of two particles.
method described in this paper can be applied to collision
problems in energy-momentum space. Analogous to the
causal diamond of a timelike displacement, let us consider To avoid this restriction, we establish some important
the “mass-diamond” of a timelike energy-momentum vector, properties involving causal diamonds that will allow us to
whose area is equal to the square of the associated rest-mass. calculate more efficiently and invariantly. This effectively
The magnitudes of energy-momentum vectors can now be shifts emphasis away from clock diamonds along those seg-
described in terms of unit-mass diamonds (analogous to ments to clock diamonds along an observer’s worldline used
clock diamonds).41 for making radar-measurements. Although this method is
In the energy-momentum diagram of Fig. 22, we describe more abstract—though arguably more practical in the real
an elastic collision42 of two particles with rest-masses m1 ¼ 12 world—it allows us to handle all rational velocities with our
and m2 ¼ 8, initial-velocities b1;i ¼ 5=13 and b2;i ¼ 15=17, rotated-graph-paper method.
and final-velocities b1;f ¼ 4=5 and b2;f ¼ 3=5.
From the system’s mass-diamond (of size 40  20 with
unit-mass diamonds in Alice’s frame), we have [Eq. (1)] the A. The aspect ratio formula and the Doppler k-factor
aspect ratio k2 ¼ 40=20 ¼ 2 so that [Eq. (4)] the velocity of As we saw in the construction of Bob’s clock diamond
the center-of-momentum frame is bCOM ¼ ð2  1Þ=ð2 þ 1Þ (see Fig. 9), the shape of the causal diamond is related to the
¼ 1=3. The magnitude of the total energy-momentum vec- Doppler effect. We illuminate this relation by deriving the
tor—the pinvariant-mass offfi this system of particles—is
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi aspect ratio formula,10 Eq. (1), and the Doppler formula, Eq.
PCOM ¼ ð40Þð20Þ ¼ 800, the square-root of the area of (3). To clarify the role of measurements, we distinguish the
the mass diamond of P ~COM . Alternatively, one can construct two k-factors relating the two observers. In Eqs. (11) and Eq.
~COM as its hypotenuse, B A
a Minkowski right triangle with P (12), we could have written kAlice and kBob for the k-factors
whose legs [by counting or by using the analogues of Eqs. determined by Bob and Alice, respectively, as the receiver.
(7) and (8)] are Etot ¼ ð40 þ 20Þ=2 ¼ 30 and ptot ¼ ð40 But since the principle of relativity requires that these values
20Þ=2 ¼ 10, so that bCOM ¼ ptot =Etot ¼ 10=30 ¼ ð1=3Þ are equal, we simply used k.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi Referring back to Fig. 13, we express k2 from Eq. (14) in
and PCOM ¼ E2tot  p2tot ¼ ð30Þ2  ð10Þ2 ¼ 800. terms of Alice’s measurements of the width Du and height
Dv of the causal diamond of B2B3 by using the light-cone
VII. CALCULATING WITH CAUSAL DIAMONDS coordinates from Eqs. (9) and (10). Thus, we obtain the as-
pect ratio formula [Eq. (1)]
Counting clock diamonds along line segments of interest
is akin to reading measurements off a clock or a ruler along  2
those segments. As mentioned, this method is limited to DuA DtA
r k2 DtAe A
¼ ¼ ¼ k Bob (20)
rational velocities associated with rational k-factors (or, DvA DtA
e DtAe
equivalently, triangles associated with Pythagorean triples
and squared intervals that are perfect squares). from this special case.43 Using Eqs. (7) and (8), we have

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 rffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 A
Du Dt þ Dx 1 þ Dx=Dt AreahB2 B3 i ¼ DuDv ¼ ½ðkBob Þ Dte DtA
e ; (28)
A
kBob ¼ ¼ ¼
Dv Dt  Dx 1  Dx=Dt A 2
¼ ðkBob DtA
e Þ ; (29)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ bBob ¼ ðDtBB3 B2 Þ2 ¼ ðtBB3  tBB2 Þ2 ; (30)
¼ ; (21)
1  bBob
which is independent of k and, thus, the choice of external
which shows that the k-factor is the familiar Doppler factor. measuring observer. Thus, AreahB2 B3 i is equal to the square
Although one could solve Eq. (21) for bBob to obtain the of the proper-time along its timelike diagonal B2B3.
inverse relation [Eq. (4)], we instead solve for t and x in Eqs. Moreover, the area of Alice’s clock diamond (representing
(7) and (8) and use the slope-form of the velocity, followed one tick on her clock) is equal to the area of Bob’s clock dia-
by the aspect ratio formula [Eq. (1) or Eq. (20)]. We find mond (representing one tick on his identical clock).
By using Eq. (13) in a different way, Eq. (30) can be
Dx ðDu  DvÞ=2 Du=Dv  1 written
bBob ¼ ¼ ¼
Dt ðDu þ DvÞ=2 Du=Dv þ 1 !
  1
A
2 A A A B B
k 1 AreahB2 B3 i ¼ Du Dv ¼ kBob DtB3 B2 A
DtB3 B2 ;
¼ Bob
2 : (22) kBob
A
kBob þ 1
(31)

which suggests that different observers (e.g., Alice) will use

their clock diamonds to decompose the area of that causal di-
B. The area formula and the absolute squared interval amond differently, depending on their k-factors with respect
As we saw in the construction of Bob’s clock diamond to Bob. We will relate this to a Lorentz coordinate transfor-
(Fig. 9), the area of the causal diamond is related to the mation in Sec. VII C.
squared interval. We illuminate this relation by deriving the Applying Eqs. (25) and (30) to the Minkowski right trian-
area formula, Eq. (2), that was introduced in Sec. II. gle in the time dilation diagram (Fig. 16), we can now estab-
Referring once again to Fig. 13, and using Eqs. (7) and (8) lish the Minkowskian analogue of the Pythagorean theorem,
to use rectangular coordinates (but suppressing the subscripts introduced as Eq. (15)
B2 and B3 unless needed), we find that the area of the causal
AreahOQi ¼ ðtBQ  tBO Þ2 ¼ ðtA A 2 A A 2
Q  tO Þ  ðxQ  xO Þ ; (32)
diamond of B2B3 can be expressed as
A 2 A 2
AreahB2 B3 i ¼ DuA DvA ¼ ðDt þ DxÞðDt  DxÞ; (23) ¼ ðtA A
P  tO Þ  ðxQ  xP Þ ; (33)

¼ ðDtÞ2  ðDxÞ2 ; (24) ðOQÞ2 ¼ ðOPÞ2  ðPQÞ2 : (34)

A 2 A 2
¼ ðtA A
B3  tB2 Þ  ðxB3  xB2 Þ ; (25)
C. Velocity and coordinate (Lorentz) transformations
which is the squared interval as measured by Alice between !
For clarity in this section, let vectors ~ u Alice ¼ ON and
the timelike-related events B2 and B3. This is the area for- !
mula given in Eq. (2). v Alice ¼ OM represent the edges of Alice’s clock diamond,
~
!
We can provide an operational interpretation of the area where ^t Alice ¼ OT ¼ ~ u Alice þ~v Alice (see Fig. 23). Similarly,
formula in terms of radar-coordinates. Using Eqs. (9) and ! ! !
let ~ u Bob ¼ OZ and ~ v Bob ¼ OY , where ^t Bob ¼ OF ¼ ~ u Bob
(10), let us write þ~ v Bob . Now, referring to Fig. 24, recall that, compared
to Alice’s clock diamonds, the widths of Bob’s clock dia-
AreahB2 B3 i ¼ DuB2 B3 DvB2 B3 ¼ DtA A
r Dte ; (26) A
monds are stretched by a factor of kBob and the heights
¼ ðtA A A A are compressed by the same factor (in order to preserve the
r3  tr2 Þðte3  te2 Þ; (27)
area)
which, with Eq. (24), is essentially the Robb25 formula for the A
u Bob ¼ kBob
~ ~
u Alice ; (35)
squared interval as the product of two radar time-intervals44
as measured by Alice. As mentioned earlier, the interpretation
is elegant43 for this case when xB2 and xB3 are non-negative.
By extending the Robb formula to arbitrary pairs of corners of
the causal diamond, one finds that the areas of causal dia-
monds encode the metrical information of Minkowski space-
time. In fact, the signature of the Minkowski metric is
encoded in the required sequence of radar measurement
events for a displacement in spacetime, as illustrated by
Geroch.15 We leave the details to the reader.
To establish the invariance of the squared interval, use Eq.
(13) (which used the principle of relativity) in Eq. (26) to
obtain a special case of Eq. (25) Fig. 23. Clock diamonds edges described by vectors (based on Figs. 3 and 11).

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 Eqs. (7) and (8) to introduce rectangular coordinates, we
obtain the Lorentz coordinate transformations
1  A A 1  B


uE 6vE ¼ k tE þ xBE 6k1 tBE  xBE ;
2 2   
k6k1 B k7k1 B
¼ tE þ xE : (45)
2 2
pffiffiffiffiffiffiffiffiffiffiffiffiffi
Using Eq. (3), we p have ðk þ k1 Þ=2 ¼ 1= 1  b2 ¼ c
ffiffiffiffiffiffiffiffiffiffiffiffiffi
and ðk  k1 Þ=2 ¼ b= 1  b2 ¼ bc. With the upper signs
in Eq. (45), the left-hand side is tA E and we obtain the ana-
logue of Eq. (17) applied to event E. With the lower signs,
Fig. 24. The velocity and coordinate (Lorentz) transformation formulas can the left-hand side is xA E and we obtain the analogue of Eq.
be obtained from the area-preserving reshaping of clock diamonds (based on (18) applied to event E.
Fig. 15).
VIII. FINAL REMARKS
A 1
v Bob ¼ ðkBob
~ Þ ~
v Alice : (36) We have shown how calculations in special relativity are
facilitated by visualizing ticks of a clock as clock diamonds
This is an active Lorentz transformation, expressed in a drawn on a sheet of rotated graph paper. When the velocities
basis of its lightlike eigenvectors, with eigenvalues k and between observers have rational Doppler k-factors, the arith-
1=k. To recover the standard rectangular form, there is a der- metic and graphical constructions become simple. Our
ivation similar to what is done for the Lorentz coordinate
approach allows us to place emphasis first on the physical
transformation [Eq. (45)].
interpretation and geometrical modeling of situations in spe-
We first derive the velocity transformation formula. For a
cial relativity. Thus, the standard textbook examples involv-
third observer Carol (see Fig. 24), we have the analogous
ing these velocities can be discussed without the traditional
relations for the u-direction
relativistic formulas. More generally, when the relative
~ A
u Carol ¼ kCarol ~
u Alice ; (37) velocities are rational (even when k is not rational), we can
construct causal diamonds, whose area can be counted and
~ B
u Carol ¼ kCarol ~
u Bob : (38) whose shape and size can be operationally interpreted in
terms of radar measurements.
Equating Eqs. (37) and (38), and inserting Eq. (35), we Some of these methods can be extended to light clocks for
obtain uniformly accelerated observers. Such an extension will be
treated elsewhere. In addition, we are exploring the possible
A B A connection between causal diamond area and the notion of
kCarol u Alice ¼ kCarol
~ ðkBob u Alice Þ;
~ (39)
quadrance in rational trigonometry.45
A B A
kCarol ¼ kCarol kBob ; (40)
ACKNOWLEDGMENTS
a multiplicative relation13 that embodies the velocity trans-
formation formula, Eq. (19). [Proof: Plug this expression The author wishes to thank his students in AST 110L and
into bA A 2 A 2 PHY 216 at Mount Holyoke College, PHYS 63 at Bowdoin
Carol ¼ ½ðkCarol Þ  1=½ðkCarol Þ þ 1, an analogue of
College, PHYS 141 at Lawrence University, and PHY 453 at
Eq. (22), and then use the analogue of Eq. (21) twice to
the University of Wisconsin, La Crosse who tried out various
obtain Eq. (19).]
We next derive the Lorentz coordinate transformation for- worksheets and homework problems based on this work. Their
mula. Referring to Fig. 20, consider the causal diamond of performance and feedback helped to improve the presentation
OE. Because each observer counts with her or his clock dia- of the material. The author also thanks Tom Moore, who
monds to decompose the same displacement OE, ~ we require suggested the term “light clock diamond” which the author
shortened to “clock diamond,” as well as Tevian Dray, Stephen
uA
E~u Alice ¼ uBE ~
u Bob ; (41) Naculich, and the anonymous referees for useful comments on
the drafts of this manuscript. Finally, the author thanks
vA
E~v Alice ¼ vBE~
v Bob : (42) Wolfgang Christian for creating the Special Relativity on
Rotated Graph Paper Model46 with the Easy Java Simulations
From Eqs. (35) and (36), it follows that Alice counts k of modeling tool.
her clock diamonds for each of Bob’s clock diamonds along
the u-direction and 1=k of her clock diamonds for each of
APPENDIX A: EQUALITY OF CLOCK DIAMOND
Bob’s clock diamonds along the v-direction
AREAS: SHORT PROOF
uA B
E ¼ kuE ; (43)
We provide a proof of the equality of clock diamond areas
vA ¼ k1 vBE : (44) (elaborating on a geometric argument from Mermin10) that
E
concisely highlights the role of physical laws. We show that
(Equations (43) and (44) correspond to Eqs. (12) and (11), the construction of event F using the invariance of the clock
respectively, in Fig. 9 where Bob’s clock is being cali- diamond area follows from the Doppler effect and from the
brated.) By forming half of the sum and difference, and using similarity of triangles in Minkowski spacetime.

356 Am. J. Phys., Vol. 84, No. 5, May 2016 Roberto B. Salgado 356

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 In Fig. 25, we draw the main features of the calibration Similarly, because OYS (shaded) and OMT are similar and
problem (see also Fig. 8). Start with the causal diamond of OT the ray SF is lightlike, the ratio of time-intervals measured
(parallelogram OMTN) and Bob’s worldline OV, and note their by Alice is
intersection events: the meeting event O and the event E on
T’s past-light-cone. Regard triangle OET as a Doppler effect tA
T vT vT
relating a moving source Bob with emission period tBE and a ¼ ¼ : (A4)
tA
S
vS vF
stationary receiver Alice with reception period
Now form the ratio ðOFÞ=ðOTÞ two ways. Using Eq. (A1)
tA A B
T ¼ kBob tE ; (A1) (Doppler) and Eq. (A3) (similar triangles), we obtain
A
where kBob is a proportionality factor determined by Alice tBF tB 1 uF
that depends on their relative speed b. [This equation is A
¼ AF B ¼ A : (A5)
tT kBob tE kBob uT
essentially Eq. (12).]
For any event F on Bob’s worldline, construct the causal
diamond of OF and the intersection events O and S on Alternatively, using Eq. (A2) (Doppler) and Eq. (A4)
Alice’s worldline. Regard triangle OSF as a Doppler effect (similar triangles) we obtain
involving a stationary source Alice with emission period tAS
and a moving receiver Bob with reception period B
tBF kAlice tA
S B vF
A
¼ A
¼ kAlice : (A6)
tT tT vT
tBF ¼ B
kAlice tA
S; (A2)
By multiplying Eqs. (A5) and (A6), we can express the
B
where kAlice is the analogous factor determined by Bob. [This square of the ratio of time-intervals on the left-hand-side in
equation is essentially Eq. (11).] terms of the ratio of areas of the corresponding causal
Following Mermin,10 describe the edges of the causal dia- diamonds
monds with Alice’s light-cone coordinates [Eqs. (7) and (8)].
Because triangles ONE (shaded on Fig. 25) and OZF are sim- !2
B B
tBF kAlice uF vF kAlice AreahOFi
ilar and the ray ET is lightlike (so uE ¼ uT), the ratio of time- ¼ ¼ A : (A7)
intervals measured by Bob is tA
T
A
kBob uT vT kBob AreahOTi

tBF uF uF Since the principle of relativity requires the k-factors to be

¼ ¼ : (A3)
tBE uE uT equal, we now have

!2
tBF uF vF AreahOFi
¼ ¼ : (A8)
tA
T
uT vT AreahOTi

This means that, in units of the area of Alice’s clock

diamond AreahOTi, the area of the causal diamond of
OF, AreahOFi; is equal to the squared interval of segment
OF. To locate the event F that marks Bob’s first tick,
we require tBF =tA
T ¼ 1. Thus, event F is located at the oppo-
site corner of a causal diamond with diagonal along
OV and with area equal to that of Alice’s clock diamond
(along OT).

APPENDIX B: THE AREA OF A CAUSAL DIAMOND,

ALGEBRAICALLY
In this section, we begin with a brief discussion of some
subtleties concerning the use of “area” and our choice of
units. We then provide an algebraic foundation of our
approach.
The non-Euclidean geometry of Minkowski spacetime
prevents us from using our Euclidean intuition in comparing
lengths of segments along different directions. However,
since both Euclidean space and Minkowski spacetime satisfy
Fig. 25. Calibrating Bob’s light clock (based on Fig. 8). Interpreting triangles Euclid’s parallel postulate, they are examples of so-called
OSF and OET as Doppler effects, the reception periods are tBF ¼ k tA S and
affine geometries.47 In affine geometry, one can compare
tA B
T ¼ k tE , with the same k-factor (in accordance with the principle of relativ-
lengths of segments along the same line, not by assigning a
ity). By similar triangles, we form the ratios tBF =tBE ¼ uF =uT and number to a segment (which requires the specification of a
tA A B A
T =tS ¼ vT =vF . When tF ¼ tT , it follows that uF vF ¼ uT vT . This means the
dot-product or metric), but by expressing the length of a seg-
required parallelograms OYFZ and OMTN have equal areas, and that events T ment as a multiple of the length of another segment on that
2
and F lie on the hyperbola uv ¼ ðtAT Þ with asymptotes along O’s light cone. line.

357 Am. J. Phys., Vol. 84, No. 5, May 2016 Roberto B. Salgado 357

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 2
Similarly, one can compare areas of regions on the same H. Minkowski, “Space and time” (1909) in The Principle of Relativity, by
plane. By describing the area of causal diamonds in units of H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl (Dover
Publications, New York, 1923).
the numerically unspecified area of Alice’s clock diamond, 3
G. W. Gibbons and S. N. Solodukhin, “The geometry of small causal dia-
our counting of diamond-areas applies equally well for monds,” Phys. Lett. B 649, 317–324 (2007).
Euclidean and Minkowskian geometries. 4
See “Alexandrov interval or neighborhood” in, e.g., R. Penrose,
Next, as noted at the end of Sec. III B, because we are Techniques of Differential Topology in Relativity (SIAM, Philadelphia,
counting diamonds, the coordinates t and x will be dimen- 1972), pp. 29–33.
5
sionless. This raises a subtle issue with units, which we now See “optical parallelogram” in, e.g., A. A. Robb, A Theory of Time and
address. Space (Cambridge U.P., Cambridge, England, 1914).
6
S. Daubin, “A geometrical introduction to special relativity,” Am. J. Phys.
In the construction of Alice’s clock diamond, recall that 30, 818–824 (1962).
her timelike diagonal OT and spacelike diagonal MN each 7
D. Bohm, The Special Theory of Relativity (Benjamin, New York, 1965).
have size 1 tick. The area of Alice’s diamond is ð1=2Þ tick2 8
Y. S. Kim and M. E. Noz, “Dirac’s light-cone coordinate system,” Am. J.
since it is half of the area of the rectangle of height OT and Phys. 50, 721–724 (1982).
9
width MN. Since this factor of 1=2 will complicate our I. M. Yaglom, A Simple Non-Euclidean Geometry and its Physical Basis
counting calculations, we have chosen the units of diamond- (Springer Verlag, Berlin, 1978).
10
N. D. Mermin, “Space-time intervals as light rectangles,” Am. J. Phys. 66,
area to be “the area of Alice’s clock diamond,” or simply 1077–1080 (1998).
“diamond,” rather than tick2 . Recall that when counting 11
D. Brill and T. Jacobson, “Spacetime and Euclidean geometry,” Gen.
clock diamonds along a segment, we are really counting the Relativ. Gravitation 38, 643–651 (2006).
12
number of diagonals or edges that span that segment, as sug- R. B. Salgado, “Visualizing proper-time in Special Relativity,” Phys.
gested in Figs. 3 and 11. Teach. (Indian Physical Society), 46, 132–143 (2004); available at
Finally, we state without proof some useful vectorial rela- arXiv:physics/0505134v1 [physics.ed-ph]
13
H. Bondi, Relativity and Common Sense (Dover, New York, 1962).
tions that will help us to write an expression for the area of a 14
! G. F. R. Ellis and R. M. Williams, Flat and Curved Space-Times (Oxford,
causal diamond. Referring back to Fig. 23, let ^t ¼ OT ; New York, 1988).
! ! ! 15
R. P. Geroch, General Relativity from A to B (University of Chicago Press,
x^ ¼ OX ; ~ u ¼ ON , and ~ v ¼ OM , and note the relations Chicago, 1978).
^t ¼ ~ u þ~v; x^ ¼ ~u ~ v, u^ ¼ ð^t þ x^Þ=2, and ^v ¼ ð^t  x^Þ=2. 16
J. L. Synge, Relativity: The Special Theory (North-Holland, Amsterdam,
~ ^
Given A ¼ At t þ Ax x^ ¼ Au~ u þ Av~ v, we have At ¼ ðAu 1956).
17
þAv Þ=2; Ax ¼ ðAu  Av Þ=2; Au ¼ At þ Ax , and Av ¼ At  Ax . T. A. Moore, Six Ideas That Shaped Physics, Unit R, 2nd ed. (McGraw-
For any two-dimensional vector space (without any need Hill, New York, 2002).
18
E. F. Taylor and J. A. Wheeler, Spacetime Physics (W.H. Freeman, New
for a dot-product), we can write A ~ B ~¼ ðAt Bx  Ax Bt Þð^t  x^Þ York, 1966).
¼ ðAu Bv  Av Bu Þð~ u ~vÞ, where we associate ð~ u ~vÞ with the 19
E. F. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed. (W.H.
signed-area of Alice’s clock diamond—the causal diamond Freeman, New York, 1992).
20
of OT. From our definitions of the basis vectors, we have T. Dray, The Geometry of Special Relativity (CRC Press, Boca Raton, FL,
~ B~ ¼ ðAt Bx  Ax Bt Þ 2012).
ð^t  x^Þ ¼ 2ð~ u ~ vÞ, and thus A 21
T. A. Moore (private communication) suggested “light clock diamonds”
½2ð^ u  ^v Þ ¼ ðAu Bv  Av Bu Þð~
u ~ vÞ. (2011).
We now specialize to the Minkowski dot-product. In our 22
P. A. M. Dirac, “Forms of relativistic dynamics,” Rev. Mod. Phys. 21,
signature convention, we have ^t  ^t ¼ 1; x^  x^ ¼ 1, and ^t  x^ 392–399 (1949).
23
¼ 0. This implies that ~ u ~u ¼ 0; ~ v ~
v ¼ 0, and ~u ~
v ¼ 1=2. L. Parker and G. M. Schmieg, “A useful form of the Minkowski diagram,”
~ B~ ¼ At Bt  Ax Bx ¼ ðAu Bv þ Av Bu Þ=2. Am. J. Phys. 38, 1298–1302 (1970). pffiffiffi
Thus, A 24
Alternate conventions for signs and factors of 1= 2 would complicate the
We can now relate the squared interval of a vector Q ~ counting calculations.
(Ds2Q~ ¼ Q~  Q)~ with the signed area of the causal diamond 25
A. A. Robb, Optical Geometry of Motion (Heffer, Cambridge, 1911).
26
R. F. Marzke and J. A. Wheeler, “Gravitation as geometry–I: The geome-
whose diagonal is Q. ~ Let U~ and V ~ be the pair of lightlike vec- try of space-time and the geometrodynamical standard meter,” in
tors, along ~ v, respectively, that form the edges of that
u and ~ Gravitation and Relativity, edited by H.-Y. Chui and W. F. Hoffman
~¼ U ~þV ~. We calculate the area of this (Benjamin, New York, 1964), pp. 40–64.
diamond such that Q 27
C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W.H.
causal diamond (in units of clock diamonds) using both rec- Freeman, New York, 1973).
tangular and light-cone coordinates. Since Q ~ ¼ Qt ^t þ Qx x^, 28
J. L. Anderson and R. Gautreau, “Operational approach to space and time
~ ~
we have U ¼ ðQt þ Qx Þð^t þ x^Þ=2 and V ¼ ðQt  Qx Þð^t  x^Þ=2. measurements in flat space,” Am. J. Phys. 37, 178–189 (1969).
29
C. E. Dolby and S. F. Gull, “On radar time and the twin ‘paradox’,” Am. J.
So, AreahOQi ~ ¼ ðU ~V ~Þ ¼ 2ðUt Vx  Ux Vt Þ ¼ 2½ð1=2Þ Phys. 69, 1257–1261 (2001).
ðQt þ Qx Þ ð1=2ÞðQx  Qt Þ  ð1=2ÞðQt þ Qx Þð1=2ÞðQt  Qx Þ 30
A. Eagle, “A note on Dolby and Gull on radar time and the twin ‘para-
¼ Q2  Q2 ¼ Ds2 . In light-cone coordinates, we have Q ~ dox’,” Am. J. Phys. 73, 976–979 (2005).
t x ~
Q 31
N. Calder, “Einstein’s Universe,” film produced by the BBC (1979).
u þ Qv~
¼ Qu ~ ~ ¼ Qu~
v with U ~ ¼ Qv~
u and V ~
v. Then, AreahOQi 32
Indeed, this is one property that makes Minkowski spacetime diagrams
~ ~ 2
¼ ðU  V Þ ¼ Uu Vv  Uv Vu ¼ Qu Qv ¼ DsQ~. This is an alge- difficult to interpret. Note, however, that Galilean spacetime diagrams
(position-time graphs) also have this property.
braic proof of the area formula introduced as Eq. (2) and 33
A parallelogram with edges a~ u and b~ v has diagonals ða~ u þ b~
vÞ
developed as Eq. (24). and ða~ u  b~vÞ, with dot-product ða2~ u ~u  b2~
v ~vÞ. Using the dot-
product of Minkowski spacetime, the diagonals of a causal diamond are
a)
perpendicular to each other since the edges are lightlike (~ u ~
u ¼ 0 and
Electronic mail: rsalgado@uwlax.edu ~v ~
v ¼ 0).
34
This is equivalent to Minkowski’s definition: the radius vector drawn to a
point on the hyperbola is “normal” (perpendicular) to the tangent vector at
1
A. Einstein, “On the electrodynamics of moving bodies” (1905) in The that point (see Ref. 2, p. 85).
Principle of Relativity, by A. Einstein, H. A. Lorentz, H. Minkowski, and 35
When ðDxÞ2 þ ðDsÞ2 ¼ ðDtÞ2 for nonnegative integers Dx; Ds, and Dt,
H. Weyl (Dover, New York, 1923). with Dx < Dt and Ds Dt for future-timelike displacements, then

358 Am. J. Phys., Vol. 84, No. 5, May 2016 Roberto B. Salgado 358

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81.194.22.198 On: Tue, 31 May 2016 15:31:26
 41
ðDx; Ds; DtÞ form a Pythagorean triple. For triples generated by Taylor (Ref. 19 on p. 198.) had used “handles” on the energy-momentum
Dx ¼ kðm2  n2 Þ, Ds ¼ kð2mnÞ, and Dt ¼ kðm2 þ n2 Þ, with positive inte- vectors due to W. A. Shurcliff.
42
gers k, m, and n (with m  n), we find k ¼ m=n. Alternatively, when Ds ¼ Based on Fig. 7.6 (p. 207) in Ref. 19 and problem R10.S3 (p. 189) in Ref. 17.
43
kðm2  n2 Þ and Dx ¼ kð2mnÞ, we find k ¼ ðm þ nÞ=ðm  nÞ. The relations Du ¼ DtA A
r and Dv ¼ Dte hold only when the timelike diago-
36
L. Marder, Time and the Space-Traveller (University of Pennsylvania nal of the causal diamond is on the forward (x  0) side of the observer.
Press, Philadelphia, 1971). However, if, for example, xB3 > 0 but xB2 < 0, then we have the less-
37
F. V. Kowalski, “Accelerating light clocks,” Phys. Rev. A 53, 3761–3766 elegant relations Du ¼ tA A A A
r3  te2 and Dv ¼ te3  tr2 . Hence, we restrict to
(1996). the case x  0 for simplicity.
38 44
M. Pauri and M. Vallisneri, “Marzke-Wheeler coordinates for accele- See Ref. 15, pp. 91–96, Ref. 16, p. 26, and Ref. 25, p. 31.
45
rated observers in special relativity,” Found. Phys. Lett. 13, 401–425 N. J. Wildberger, “Chromogeometry and relativistic conics,” KoG, 13,
(2000). 43–50 (2009).
39 46
A. Schild, “The clock paradox in relativity theory,” Am. Math. Monthly W. Christian, computer program, Special Relativity on Rotated Graph
66, 1–18 (1959). Paper Model, Ver. 1.0 (2011), <http://www.compadre.org/Repository/
40
The inverse transformations express the coordinates of the “lab frame” document/ServeFile.cfm?ID=11540&DocID=2464>.
47
(here, Alice) in terms of those of the “moving frame” (Bob). The standard P. Bamberg and S. Sternberg, A Course in Mathematics for Students of
transformations (with the minus signs) are obtained by solving these equa- Physics (Cambridge U. P., Cambridge, England, 1991), Vol. 1, pp 1–2,
tions for tBE and xBE . See p. 107 in Ref. 17. and 35.

359 Am. J. Phys., Vol. 84, No. 5, May 2016 Roberto B. Salgado 359

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