3.4 Curves and Arc Lengths
27
3.4 Curves and Arc Lengths
The lengths of lines and curves in the spacetime of special relativity have some
peculiar and interesting properties. Let us study the time elapsed for travelers aboard
rocket ships having diverse trajectories, curves in spacetime, using the geometric
view that we have developed. The proper time interval for such a traveler is equal to
the square root of the line element divided by c, as in (3.2),
cdτ = ds =
c 2 dt 2 − d x 2 = cdt
1 − (d x/cdt) 2 =
1 − β 2 cdt,
(3.30)
where the space and time intervals are measured in some inertial system such as our
lab. Notice that this has meaning only so long as the velocity of the rocket ship is
less than c, for otherwise the proper time becomes imaginary. That is the trajectory
must always have a slope of over 45°. The time elapsed for a traveler is thus simply
the integral of the line element along the trajectory, or the arc length of the curve
between initial and final points in spacetime,
cτ = s =
f
i
1 − β 2 cdt.
(3.31)
It is obvious from the integrand in (3.31) that this arc length is largest when the
velocity of the rocket remains small along the trajectory. In particular the longest
curve in spacetime for the roundtrips shown in Fig. 3.4 is the straight line along the
time axis; any other curve is shorter, and as the curve approaches the 45° lines (light
cone) its length approaches zero!
A straight line of this type is the longest distance between 2 points in spacetime,
whereas it is the shortest distance between two points in Euclidean space. The minus
sign in the line element (3.2) produces this profoundly different behavior. A physical
consequence of this is that someone who leaves earth and travels at high velocity, say
to a nearby star, and returns to earth will be younger than indicated by an earthbound
clock. The infamous “twin paradox” is based on this peculiar behavior. See Exercise
3.6.
Fig. 3.4 The longest arc length (elapsed travel time) is the straight line along the time axis; the arc
length along the 45° lines is zero
27
3.4 Curves and Arc Lengths
The lengths of lines and curves in the spacetime of special relativity have some
peculiar and interesting properties. Let us study the time elapsed for travelers aboard
rocket ships having diverse trajectories, curves in spacetime, using the geometric
view that we have developed. The proper time interval for such a traveler is equal to
the square root of the line element divided by c, as in (3.2),
cdτ = ds =
c 2 dt 2 − d x 2 = cdt
1 − (d x/cdt) 2 =
1 − β 2 cdt,
(3.30)
where the space and time intervals are measured in some inertial system such as our
lab. Notice that this has meaning only so long as the velocity of the rocket ship is
less than c, for otherwise the proper time becomes imaginary. That is the trajectory
must always have a slope of over 45°. The time elapsed for a traveler is thus simply
the integral of the line element along the trajectory, or the arc length of the curve
between initial and final points in spacetime,
cτ = s =
f
i
1 − β 2 cdt.
(3.31)
It is obvious from the integrand in (3.31) that this arc length is largest when the
velocity of the rocket remains small along the trajectory. In particular the longest
curve in spacetime for the roundtrips shown in Fig. 3.4 is the straight line along the
time axis; any other curve is shorter, and as the curve approaches the 45° lines (light
cone) its length approaches zero!
A straight line of this type is the longest distance between 2 points in spacetime,
whereas it is the shortest distance between two points in Euclidean space. The minus
sign in the line element (3.2) produces this profoundly different behavior. A physical
consequence of this is that someone who leaves earth and travels at high velocity, say
to a nearby star, and returns to earth will be younger than indicated by an earthbound
clock. The infamous “twin paradox” is based on this peculiar behavior. See Exercise
3.6.
Fig. 3.4 The longest arc length (elapsed travel time) is the straight line along the time axis; the arc
length along the 45° lines is zero