Space–time intervals as light rectangles
N. David Mermin
Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853-2501
~Received 13 March 1998; accepted 11 May 1998!
Two inertial observers in relative motion must each see the other’s clock running at the same rate.
The representation of this symmetry of the Doppler effect in a two-dimensional space–time diagram
reveals an important geometrical fact: The squared interval between two events is proportional to
the area of the rectangle of photon lines with the events at diagonally opposite vertices. © 1998
American Association of Physics Teachers.
Recently I described in these pages1 an approach to the
space–time diagrams of Minkowski that leads naturally to a
simple geometric interpretation of the invariant interval between two events: The squared interval between two events,
represented as points in a two-dimensional space–time diagram, is proportional to the area of the unique rectangle of
four photon lines having those two points at diagonally opposite vertices. The proportionality constant ~of less interest
for many purposes! is given in terms of the invariant product
of two frame-dependent scale factors,2 l and m, used by
various observers to relate certain distances in the diagram to
the separations the observers assign to events in real space
and time ~see Fig. 1!.
I describe here a simpler route4 to these conclusions. For
this purpose it is best to define the interval between two
time-like separated events ~I return below to space-like or
light-like separated events! as the time between the events in
a frame in which they happen at the same place—i.e., as the
amount by which a uniformly moving clock, present at both
events, advances between the first and the second. Clearly
the squared time between any two events that both happen at
the same place in a single frame is proportional to the area of
any geometric figure of fixed shape that scales linearly with
the distance in the diagram between the events. The rectangle of photon lines is special because its area remains proportional to the interval between two events, even when the
events in different pairs are at the same place in different
frames. What must therefore be established is that if events
P1 and P2 happen at the same place in one frame ~Alice’s!,
and events Q1 and Q2 happen at the same place in another
~Bob’s!, then equality of the time between P1 and P2 in
Alice’s frame and the time between Q1 and Q2 in Bob’s is
the same as equality of the area of the two photon rectangles.
This follows directly from a gedanken experiment.
Let Alice be present at events P1 and P2 , with a clock
that reads 0 at the first event and T at the second, and let Bob
be present at Q1 and Q2 , with a clock that reads 0 and T at
those two events ~Fig. 2!. To compare the areas of the photon
rectangles determined by the two pairs of events, we may
translate Bob and his pair so that the points representing Q1
and P1 coincide ~Fig. 3!. The resulting figure can be interpreted as describing two clocks, in relative motion, which are
together when both read 0. I have added to the figure two
solid photon lines, which determine the time that Alice and
Bob each sees the other’s clock reading when their own
clock reads T. I have also extended those solid photon lines
~with dashed photon lines! into two rectangles, one of which
has Alice’s two events at opposite vertices, and the other of
which has Bob’s.
Note that Alice and Bob each bears the same relation to
the clock of the other: each is watching a clock moving away
at the same speed. The principles of relativity and the constancy of the velocity of light require that each must see the
moving clock running at a rate that differs from that of their
own by the same factor. In particular, Alice and Bob must
each see the other’s clock reading the same time t at the
moment their own clock reads T. It then follows directly
from some elementary proportions ~explained in the caption
of Fig. 3! that the two photon rectangles do indeed have the
same area. Therefore the squared interval between any two
time-like separated events is indeed proportional to the area
of the rectangle of photon lines having the events at diagonally opposite vertices.
To establish from this the invariance of the product of
scale factors and its relation to the proportionality constant,
note ~Fig. 4! that two copies of either of the rectangles of
Fig. 2 can be cut up and reassembled into a rhombus that has
the two events at adjacent vertices. The area of the rhombus
is equal to the product of the length mT of any side with the
distance lT between sides. The area of the original rectangle
is therefore V5 12lmT2, where l and m are the scale factors
Fig. 1. The scale factors l and m for a frame of reference with the indicated
lines of constant time and position. Events represented by points on the line
labeled 1 ns happen one nanosecond after events represented by points on
the line labeled 0 ns; events represented by points on the line labeled 1 f
happen one foot away ~Ref. 3! from events represented by points on the line
labeled 0 f. The scale factors ~in length of diagram per nanosecond or foot!
are the distances indicated in the diagram. Although l and m depend on
frame of reference, it turns out that their product lm, the area of the rhombus bounded by the four lines, does not. Evidently the scale factors for a
given frame of reference are related by l5m sin u, where u is the angle
between lines of constant time and position used in that frame.
1077 Am. J. Phys. 66 ~12!, December 1998 © 1998 American Association of Physics Teachers 1077
Fig. 2. ~a! The world line of a clock present at two events P1 and P2 that are
at the same place and a time T apart in Alice’s frame. ~b! The same as ~a! for
two other events Q1 and Q2 that are at the same place and are the same time
T apart in Bob’s frame. The distance in the diagram mBT between the points
representing the events Q1 and Q2 differs from the distance in the diagram
mAT between the points representing the events P1 and P2 , because Alice
and Bob use different scale factors m to relate separation in time to distance
in the diagram between events that happen at the same place. However the
areas of the two rectangles formed by photon trajectories emerging from the
events are the same. This is established in Fig. 3.
for the frame in which the events happen at the same place.
Since a pair of events at the same place and separated by a
time T in Alice’s frame gives rise to a photon rectangle with
the same area V as does a pair of events at the same place
and separated by the same time T in Bob’s, it follows that the
product of scale factors lm52V/T2 must be the same for
Alice and Bob: lAmA5lBmB .
If two events are space-like rather than time-like separated, then the interval can be defined as the distance between them in a frame in which they happen at the same
time. Given two space-like separated events, one can reflect
them in a photon line to get two other events which are
Fig. 3. The two parts of fig. 1 have been translated to describe a situation in
which both clocks read 0 at the same place and time. At the moment each
clock reads T, Alice and Bob ~who are with their clocks! each looks at the
other’s clock and sees it reading t. For either clock, the ratio of t and T is just
the ratio of the distances in the diagram mt and mT from the events in which
the clock had those readings, back to the event at which it read 0. ~One uses
mA for Alice’s clock and mB for Bob’s.! It is evident that this ratio ~as read
from the trajectory of Alice’s clock! is the same as the ratio of a to A or ~as
read from the trajectory of Bob’s clock! the same as the ratio of b to B. But
if a/A5b/B then Ba5bA—i.e., the rectangle of photon lines with Bob’s
clock reading 0 and T at opposite vertices has the same area as the corresponding rectangle for Alice’s clock.
1078 Am. J. Phys., Vol. 66, No. 12, December 1998
Fig. 4. The area of the photon rectangle in ~a! is half the area of the rhombus
in ~b!. But the area of the rhombus is the length mT of a side times the
distance lT between sides. So the area of the rectangle is 21lmT2.
time-like separated ~Fig. 5!. The time ~in nanoseconds! between the time-like separated events in the frame in which
they happen at the same place is the same as the distance ~in
feet!3 between the space-like separated events in the frame in
which they happen at the same time. The square of that time,
however, is 2/lm times the area of the photon rectangle determined by the time-like separated events. Since that rectangle goes under reflection into the photon rectangle determined by the two space-like separated events, and since the
area of a rectangle is invariant under reflection, the same
geometric interpretation of the squared interval holds for
space-like separated events: It is proportional ~with the same
proportionality constant 2/lm! to the area of the rectangle of
photon lines having those events at diagonally opposite vertices.
For light-like separated events ~events that can be joined
by a light trajectory! the photon rectangle degenerates to a
single line, as it should since the interval between such
events is zero.
Fig. 5. The black dots above the heavy dashed light line represent a pair of
time-like separated events, a time T apart in the frame in which they happen
at the same place. The length of the line connecting them is mT and the area
of the photon rectangle on whose vertices they lie is 21lmT2 ~where l and m
are the scale factors for that frame!. This entire structure appears mirrored
below the heavy dashed photon line. Now it represents two space-like separated events, a distance D apart in the frame in which they happen at the
same time. Since the lengths mT and mD are the same and the areas of the
rectangles are the same, the area of the photon rectangle on whose vertices
the space-like separated events lie is 21lmD2.
N. David Mermin 1078
Fig. 6. The two large black dots are two events. The two thin solid lines are
lines of constant time and constant position in a frame in which the events
happen a time T and a distance D apart. The difference between the areas of
the thin-sided right triangles formed by those lines and the thin photon lines
is 21ba2 21( fb)( fa). This can be re-expressed as 21@(11 f )b#@(12 f )a#,
which the figure reveals to be the area of the thick-sided right triangle
formed by the thick solid line joining the events and the thick photon lines.
But the area of the thick-sided triangle is 41mlI2, the area of the large
thin-sided triangle is 41mlT2, and the area of the small thin-sided triangle is
1
4mlD2. Therefore I25T22D2.
Figure 6 provides the connection with the coordinate-
based definition of interval. It shows ~using triangular halves
of the relevant light rectangles, so as not to clutter the figure!
that if two time-like separated events are a time T and a
distance D apart in a frame in which they do not happen at
the same place, then the value I2 of the squared interval is
indeed equal to uT22D2u. ~The same figure, reflected in a
45° photon line, demonstrates the same conclusion for two
space-like separated events.!
What about the aspect ratio of the photon rectangle determined by a pair of events? Unlike its area, this depends on
the flexibility observers have, in setting up a diagram, to
choose the angles between lines of constant time and position. The aspect ratio can, however, be simply related to the
velocity v, in the frame in which lines of constant time and
Fig. 7. This demonstrates that if v is the velocity, in the frame using vertical
and horizontal lines of constant position and time, of the frame in which two
events happen at the same place, then the aspect ratio of the photon rectangle determined by the events is (11v)/(12v).
Fig. 8. It follows that the longitudinal Doppler shift is given by
A(12v)/(11v). Take the unit of area in the diagram to be that of a photon
rectangle associated with events separated by an interval of 1 ns. Then a unit
square of photon lines determines the 1-ns scale on a vertical line of constant position ~shown on the left vertical line!. The unit photon rectangle
associated with successive nanosecond readings of a clock moving with
velocity v in that frame then has sides A(12v)/(11v) and
A(11v)/(12v) ~since its area is 1 and its aspect ratio is (11v)/(12v).
Consequently the clock is seen by somebody on a vertical line of constant
position as running slowly ~left vertical line! or fast ~right vertical line! by
these factors. ~The unit square is drawn with thicker photon lines, as are the
segments of photon lines making up the unit rectangle. The pairs of white
circles containing 0’s and 1’s represent events in the history of two actual
clocks at which they read 0 and 1 ns. The grey circles represent events in
which the clock with the nonvertical world-line is seen to be reading 0 or 1
by people whose world-lines are the two vertical lines.!
position are orthogonal, of the frame in which the events
happen at the same place ~or at the same time!. Figure 7
shows that the aspect ratio is (12v)/(11v). Finally, Fig. 8
extracts from this the usual expression for the longitudinal
Doppler shift.
ACKNOWLEDGMENT
This work was supported by the National Science Foundation, Grant No. PHY9722065.
1N. David Mermin, ‘‘An introduction to space–time diagrams,’’ Am. J.
Phys. 65, 476–486 ~1997!. The points made in the present note can be
understood without consulting this paper. It is, however, important for
instilling the proper attitude toward space–time diagrams.
2Multiplication by the scale factor l converts the time between two events
to the ~perpendicular! distance between the lines of constant time on which
lie the points that represent those events. ~And it converts the actual spatial
distance between two events to the distance in the diagram between the
lines of constant position on which the points lie!. The scale factor m for a
given frame converts the time between two events that happen at the same
place in that frame to the distance in the diagram between the points that
represent the events. ~And it converts the actual spatial distance between
two events that happen at the same time to the distance in the diagram
between the points.! The relation between the two scale factors belonging
to a given frame is just l5m sin u, where u is the angle in the diagram
between its lines of constant time and constant position. The units of the
scale factors could be, for example, centimeters of diagram per nanosec
1079 Am. J. Phys., Vol. 66, No. 12, December 1998 N. David Mermin 1079
ond of time ~or per foot—i.e., light-nanosecond—of distance!. but on something that transcends frame-dependent conventions: how fast
3As used here the ‘‘foot’’ is the light nanosecond, 29.979 245 8 cm. See, each sees the other’s clock run. It is also simpler directly to establish the
for example, N. David Mermin, ‘‘Light Feet,’’ The New Yorker, May 16, representation of the squared interval as the area of a rectangle of photon
1994, p. 10. lines, from which the invariance of the product of scale factors follows
4The argument here improves on that in Secs. IV and V of Ref. 1 by effortlessly, rather than deriving these results in the opposite order, as in
focusing not on how fast Alice and Bob each says the other’s clock runs, Ref. 1.
1080 Am. J. Phys., Vol. 66, No. 12, December 1998 N. David Mermin 1080