Spacetime 47
is the same in all inertial frames even though px, py, pzand Eseparately may be dif-
ferent. This invariance was noted earlier in connection with Eq. (1.24); we note that
p^2 px^2 py^2 pz^2.
A more mathematically elaborate formulation brings together the electric and mag-
netic fields Eand Binto an invariant quantity called a tensor. This approach to
incorporating special relativity into physics has led both to a deeper understanding of
natural laws and to the discovery of new phenomena and relationships.Spacetime Intervals
The statements made at the end of Sec. 1.2 (P. 10) are easy to confirm using the idea
of spacetime. Figure 1.25 shows two events plotted on the axes xand ct. Event 1 oc-
curs at x0, t0 and event 2 occurs at x
x, t
t. The spacetime interval s
between them is defined by( s)^2 (c^ t)^2 (^ x)^2 (1.53)The virtue of this definition is that ( s)^2 , like the s^2 of Eq. 1.52, is invariant under
Lorentz transformations. If xand tare the differences in space and time between
two events measured in the Sframe and x and t are the same quantities meas-
ured in the S frame,( s)^2 (c^ t)(^2) ( x) (^2) (c t ) (^2) ( x ) 2
Therefore whatever conclusions we arrive at in the Sframe in which event 1 is at the
origin hold equally well in any other frame in relative motion at constant velocity.
Spacetime interval
between events
Figure 1.25The past and future light cones in spacetime of event 1.
FUTURE LIGHT CONE
PAST LIGHT CONE
Event 1
ct
∆x
c ∆t
Event 2
x=ct
x
x=−ct
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