Chapter 56Minkowski Spacetime — Relativity of Simultaneity and Proper Time

56.1 Transformation of Coordinate Differences and the Velocity Addition Law

Velocity is “change in position ÷ change in time,” so under the Lorentz transformation we must take into account that not only the position but also the change in time differs from one coordinate system to another. In Section 55.6 of the previous chapter, we differentiated the Lorentz transformation with respect to time and derived the velocity transformation law (Theorem 55.4), including the components perpendicular to the direction of motion. In this section we revisit that calculation in the form of “the transformation of the differences between the coordinates of two events.” The transformation of differences (Theorem 56.1) is the common starting point for simultaneity (Section 56.2), spacetime diagrams (Section 56.3), the invariant interval (Section 56.4), and proper time (Section 56.5). After that, as applications of the velocity transformation law, we examine the composition of velocities in the same direction and the aberration of light.

The setup and notation are the same as in Chapters 54 and 55. Frame S′ moves relative to frame S in the positive direction of the $x$ axis at a constant speed $v\ [\mathrm{m/s}]$ (the origins coincide at $t=t'=0$, and the axes are parallel to each other; Section 54.5), and the velocity of the particle (point mass) is written $\bm{u}=(u_x,u_y,u_z)=(\dd x/\dd t,\ \dd y/\dd t,\ \dd z/\dd t)$ in frame S and $\bm{u}'=(u'_x,u'_y,u'_z)=(\dd x'/\dd t',\ \dd y'/\dd t',\ \dd z'/\dd t')$ in frame S′ (this is distinct from the notation $u^\mu$ for the four-velocity; see Chapter 57).

56.1.1 The Lorentz transformation of coordinate differences

Let us write down the Lorentz transformation (Chapter 54) once more here.

This transformation is a first-degree expression (a linear transformation) in $x,t$. Therefore, the “differences” between the coordinates of two events $\mathrm{P}_1$ $(t_1,x_1,y_1,z_1)$ and $\mathrm{P}_2$ $(t_2,x_2,y_2,z_2)$ also transform by equations of the same form. Here an event is a happening specified by the spacetime coordinates of a single point—“when and where” it occurred (for example, “a flash of light was emitted at position $x_1$ at time $t_1$”).

56.1.2 Relation to the velocity transformation law

Suppose a particle is moving and its trajectory $x(t),y(t),z(t)$ is given in frame S. Let us apply Theorem 56.1 (in its infinitesimal form) to two neighboring events on the trajectory—the two points where the particle is located at time $t$ and at time $t+\dd t$. By the definition of velocity, the coordinate differences in frame S are $\dd x=u_x\dd t$, $\dd y=u_y\dd t$, $\dd z=u_z\dd t$. Substituting these, we obtain

The velocity in frame S′ is the change in position measured in S′ divided by the change in time measured in S′, so if we take the ratios in Eq. \eqref{eq:56-dxdt-prime} and cancel the common $\dd t$, exactly the same calculation as the derivation in Section 55.6.2 of the previous chapter gives the following result ($1/\gamma=\sqrt{1-v^2/c^2}$).

Conversely, since the velocity of S as seen from S′ is $-v$, the equations that give the velocity in frame S from the velocity in frame S′ are obtained by replacing $v\to-v$:

The detailed checks made along the way in the derivation—why the components $u_y,u_z$ perpendicular to the direction of motion also change, that for $v\ll c$ the result returns to the Galilean transformation (the high school addition of velocities), that the speed of light does not change, and that no composition of velocities exceeds $c$ (Theorem 55.5)—are given in Section 55.6 of the previous chapter. What is newly important in this section is the second equation of Eq. \eqref{eq:56-dxdt-prime}: the advance $\dd t'$ of the clock of frame S′ differs from the advance $\dd t$ of the clock of frame S by the factor $\gamma(1-vu_x/c^2)$ (this is the product of the time dilation factor $\gamma$ and the factor $1-vu_x/c^2$ for the shift in simultaneity; it is used again in Section 56.5 on proper time).

The behavior of the velocity transformation is shown in Figure 56.1. The Galilean transformation $u'_x=u_x-v$ is a straight line, but the curve of the Lorentz velocity transformation passes through $u'_x=\pm c$ at $u_x=\pm c$ and stays between $-c$ and $c$.

56.1.3 Composition of velocities in the same direction and the aberration of light

Let us look at two applications of the velocity transformation law. The first is the composition of motions in the $x$ direction only. Suppose S′ moves in the $x$ direction at speed $\beta_1c$ relative to S, and an object moves in the same $x$ direction at speed $\beta_2c$ relative to S′. Setting $v=\beta_1c$ and $u'_x=\beta_2c$ in the inverse transformation \eqref{eq:56-vel-inv}, we find that the speed $\beta c$ of the object in frame S is

($\beta_1,\beta_2$ are numbers between $-1$ and $1$). The term $\beta_1\beta_2$ in the denominator produces the deviation from the Galilean sum $\beta_1+\beta_2$. That $\abs{\beta}\lt1$ can be confirmed by factoring, $1-\beta=(1-\beta_1)(1-\beta_2)/(1+\beta_1\beta_2)$ and $1+\beta=(1+\beta_1)(1+\beta_2)/(1+\beta_1\beta_2)$, in which all the numerators and denominators are positive (the same calculation as Exercise 55.5 of the previous chapter). For example, for $\beta_1=\beta_2=0.5$ we have $\beta=1.0/1.25=0.8$, which does not reach the speed of light (for a numerical example, see Example 55.13 of the previous chapter). With the rapidity of Section 56.7, this composition law becomes the simple addition $\phi=\phi_1+\phi_2$.

The second is the transformation of the direction in which light travels, that is, the aberration of light. In the following example, we transform light traveling in an arbitrary direction by means of Eq. \eqref{eq:56-vel}.

56.2 The Relativity of Simultaneity

When we said in high school physics that “two events occurred at the same time,” we believed, almost without being aware of it, that this had the same meaning for everyone. For example, “the accident that occurred in Tokyo at 3 p.m. and the accident that occurred in Osaka at 3 p.m. are simultaneous.” The Galilean transformation $t'=t$ is precisely this belief written as a formula. However, the time equation of the Lorentz transformation, $t'=\gamma\,(t-vx/c^2)$, contains not only $t$ but also the position $x$. In this section we examine the consequence head-on. Two events that occur simultaneously in one coordinate system are not simultaneous in another—simultaneity is relative, determined separately for each coordinate system.

56.2.1 A shift in simultaneity seen from the Lorentz transformation

Let the differences between the coordinates of two events A and B be $\Delta t=t_B-t_A$ and $\Delta x=x_B-x_A$ in frame S, and $\Delta t'=t'_B-t'_A$ in frame S′. By Theorem 56.1 of the previous section (Eq. \eqref{eq:56-delta}),

holds. Even if the events are simultaneous in frame S ($\Delta t=0$), as long as the two points are apart ($\Delta x\neq0$), we have

This is the relativity of simultaneity.

Let us estimate the magnitude of Eq. \eqref{eq:56-simul-shift}. Since $c^2$ is enormous, that magnitude is almost zero at everyday speeds (Example 56.4). The shift in simultaneity is invisible merely because $v/c$ is small.

56.2.2 Einstein's train thought experiment

The relativity of simultaneity can be understood directly, from the principle of the constancy of the speed of light alone, through the following famous thought experiment (the train thought experiment that Einstein used in his popular exposition, slightly modified into the form often used in textbooks).

Consider a long train running along a straight track at a constant speed $v$. A light source at the center of the train emits a flash at some instant. The light travels toward the front end and the rear end of the train. To an observer riding on the train (frame S′), the light travels forward and backward at the same speed $c$, and both ends are at the same distance $L/2$ from the source ($L$ is the rest length of the train), so the light reaches the front end and the rear end simultaneously (panel (a) of Figure 56.2).

But to an observer standing beside the track (frame S), the situation is different. The speed of light is $c$ in every direction, even if the source is moving (the principle of the constancy of the speed of light). Meanwhile, the rear end of the train moves toward the oncoming light, and the front end moves away as if fleeing from the light. Therefore the light reaches the rear end first and the front end later (panel (b) of the figure). For the observer on the ground, the two events “the light reaches the rear end” and “the light reaches the front end” are not simultaneous.

It is meaningless to ask which observer's conclusion is “correct.” Each is a correct conclusion, obtained in that observer's own coordinate system with the speed of light $c$ and his or her own ruler and clock, and each is a manifestation of the fact that simultaneity differs from one coordinate system to another. Next, let us confirm this conclusion quantitatively using the Lorentz transformation.

56.2.3 When the order is reversed, and when it is not

The “before and after” of two events that are not simultaneous can also be reversed depending on the coordinate system (part (B) of Example 54.8 is such an example). Then, is causality, the principle that a cause occurs before its effect, safe? We examine this question in detail in Section 56.4. Here we give just one preview of the answer.

Conversely, if $\abs{\Delta x}\gt c\,\Delta t$ (the two events are so far apart that even light cannot connect them), there exist values of $\beta$ for which $\beta\Delta x/(c\Delta t)$ exceeds $1$, and then $\Delta t'\lt0$, so the order is reversed. The boundary between the two cases is the “light cone” of the following sections.

56.3 Spacetime Diagrams — World Lines, Light Cones, and the Coordinate Axes of S′

In high school physics, the motion of an object was represented by an $x$–$t$ graph (time $t$ on the horizontal axis, position $x$ on the vertical axis). This is the diagram in which the slope of the graph gives the velocity. Under the Lorentz transformation, time and space are mixed together as they transform. So if we use a diagram that treats time and position on an equal footing—the spacetime diagram (also called a Minkowski diagram)—then the relativity of simultaneity, and the light cone, causality, and proper time that we are about to study, become visible at a glance. The idea of this diagram was presented by Minkowski in 1908.

56.3.1 Drawing a spacetime diagram and world lines

In a spacetime diagram, we take position $x$ on the horizontal axis and $ct$ on the vertical axis (we either omit $y,z$ and draw only one-dimensional motion, or draw them in a separate figure). Compared with the high school $x$–$t$ graph, the vertical and horizontal axes are interchanged (by convention, time is put on the vertical axis, with upward as the future). Moreover, on the vertical axis we take not the time $t$ itself but $ct$, the time multiplied by the speed of light. The unit of $ct$ is $\mathrm{s\times m/s=m}$, that is, a length. There are the following reasons for this.

• To make the dimensions and scales of the two axes the same. If the vertical and horizontal axes are both lengths, the diagram can be drawn with the same scale on both, and the geometry of the figure (angles and slopes) becomes meaningful.
• Because the world line of light is at 45°. Light travels along $x=\pm ct$. If $ct$ and $x$ are drawn on the same scale, this is a straight line inclined at $\pm45^\circ$. The fact that the speed of light is the same $c$ in every coordinate system appears in the diagram in the form “the light line is at 45° in the spacetime diagram of any coordinate system.”
• Because the Lorentz transformation takes a symmetric form. If we write Eq. \eqref{eq:56-lorentz} as $x'=\gamma(x-\beta\,ct)$, $ct'=\gamma(ct-\beta x)$, then $ct$ and $x$ appear on the same footing (Section 54.7).

A point in a spacetime diagram is an event, which specifies “when and where.” As time passes, a particle traces out a single curve in the spacetime diagram.

The shape of the world line tells us the motion of the particle (Figure 56.3).

• The world line of a particle at rest ($x=$ constant) is a vertical straight line, parallel to the $ct$ axis. Only time advances; the position does not change.
• The world line of uniform rectilinear motion $x=x_0+vt$ is an inclined straight line. With position $x$ on the horizontal axis and $ct$ on the vertical axis of the spacetime diagram, we have $x=x_0+\beta\,ct$ ($\beta=v/c$), so for the angle $\theta$ that the line makes with the $ct$ axis, $\tan\theta=\dd x/\dd(ct)=\beta$. The faster the particle, the more the line leans away from the vertical. (In the high school $x$–$t$ graph the velocity was the slope, but here the vertical and horizontal axes are interchanged, so it is the $\tan$ of the inclination from the $ct$ axis that equals $v/c$.)
• For light, $\beta=1$, so $\theta=45^\circ$. A body with mass has $\abs{\beta}\lt1$, so its world line always stays inclined by less than $45^\circ$ from the $ct$ axis (Chapter 57).
• The world line of a motion in which the velocity changes (acceleration) is a curve. The slope of the tangent at each point of the curve, $\tan\theta=\dd x/\dd(ct)=v(t)/c$, represents the velocity at that instant.

56.3.2 The light cone

Consider the event at the origin O of the spacetime diagram—for example, a flash of light is emitted instantaneously at the point O. Light spreads in all directions of space at speed $c$, so in three-dimensional space, a time $t$ after the flash, a spherical surface of radius $ct$ has spread out. The coordinates $(x,y,z)$ of a point on this sphere satisfy $x^2+y^2+z^2=c^2t^2$ (the equation of the wavefront of light in Section 54.5). Rewriting this as a set of events to be drawn in the spacetime diagram, we obtain

If we omit $y,z$ and reduce to the one dimension of $x$ alone, we get $ct=\pm x$, that is, two straight lines at $45^\circ$ (the dashed orange lines in Figure 56.3). If we draw up to two space dimensions $(x,y)$, this becomes $ct=\pm\sqrt{x^2+y^2}$, that is, a cone whose axis is the $ct$ axis (Figure 56.3 (b)). We therefore call this surface the light cone.

If a particle passes through O and moves with speed $u\lt c$, its position at time $t$ satisfies $\sqrt{x^2+y^2+z^2}\lt ct$, that is, $c^2t^2-x^2-y^2-z^2\gt0$. Therefore, the world line of a particle passing through O always lies inside the light cone. With motion slower than light, one cannot go from O to an event outside the light cone. The light cone is the boundary that separates “the region that the event at O can influence (inside and on the future light cone)” from “the region by which O can be influenced (inside and on the past light cone).” We examine this in detail in Section 56.4, using the invariant interval.

56.3.3 The axes of frame S′ are tilted

Let us draw the coordinate axes $x'$ and $ct'$ of frame S′, which moves in the $x$ direction with speed $v$ relative to frame S, on the spacetime diagram of frame S. A coordinate axis is the set of points at which one of the coordinates is $0$.

The result is shown in Figure 56.4. Both axes of frame S′ tilt toward the $45^\circ$ light line by the same angle $\theta$. The larger $\beta$ is, the more they tilt, as if closing a pair of scissors. In the limit $v\to c$ ($\beta\to1$), both axes coincide with the light line. This figure tells us the following.

• The coordinates $(x',ct')$ of an event in frame S′ are read off by drawing lines from that point parallel to the S′ axes and noting where they meet the axes (the green dotted lines in Figure 56.4; coordinates in frame S are read along horizontal and vertical lines, whereas coordinates in frame S′ are read “obliquely” along lines parallel to the axes). For example, the points with constant $x'$ lie on a straight line parallel to the $ct'$ axis.
• The points with constant $ct'$, that is, the events simultaneous in frame S′, lie on a straight line parallel to the $x'$ axis (a line rising to the right). Comparing this with the fact that events simultaneous in frame S lie on a horizontal line, we see that the “line of simultaneity” differs from one coordinate system to another (Section 56.3.4).

To fix the tick marks correctly, we need a quantity that does not depend on the coordinate system. In the previous chapter we saw that the Lorentz transformation leaves $c^2t^2-x^2$ unchanged ($c^2t'^2-x'^2=c^2t^2-x^2$) (Section 54.5). At the point $ct'=1$ ($x'=0$) on the $ct'$ axis of frame S′, $(ct')^2-x'^2=1$, so $(ct)^2-x^2=1$ holds in frame S as well. That is, the point $ct'=1$ lies on the hyperbola $(ct)^2-x^2=1$. Similarly, the point $x'=1$ on the $x'$ axis lies on the hyperbola $x^2-(ct)^2=1$. These are the red hyperbolas in Figure 56.4, and they give the tick marks of “the same length and the same time” in both coordinate systems.

56.3.4 Seeing the relativity of simultaneity on a spacetime diagram

On a spacetime diagram, “simultaneous” appears as “a line parallel to an axis.” Events that are simultaneous in frame S lie on a horizontal line ($ct$ constant), and events that are simultaneous in frame S′ lie on a line parallel to the $x'$ axis, rising to the right. The relativity of simultaneity is the fact that the same pair of events can lie on a horizontal line of frame S but not on a rising line of frame S′ (or vice versa). Let us draw the train thought experiment of the earlier Section 56.2.2 on a spacetime diagram (Figure 56.5).

Let frame S be the ground and frame S′ the train, with $v=0.6c$ ($\gamma=1.25$) and the rest length of the train $L=4$ (in arbitrary units of length). In frame S′, the light source is at $x'=0$, the rear end at $x'=-2$, and the front end at $x'=+2$. These world lines are all straight lines parallel to the $ct'$ axis, and in the diagram of frame S they become tilted parallel lines. Take the emission event E as the origin; the light travels to the left and right along $45^\circ$ lines. The event R, at which the light reaches the rear end, and the event F, at which it reaches the front end, lie in frame S′ on the same straight line $ct'=2$ parallel to the $x'$ axis, and so are simultaneous. Transforming these to frame S (using the inverse transformation $x=\gamma(x'+\beta ct')$, $ct=\gamma(ct'+\beta x')$), we obtain

(R lies on $x=-ct$ and F on $x=ct$, so both lie on light lines.) In frame S, R is at $ct=1$ and F is at $ct=4$, so they lie on different horizontal lines. F is much later. The time difference is $\Delta(ct)=3$, which agrees with $\gamma\beta L=1.25\times0.6\times4=3$ in Eq. \eqref{eq:56-train-dt}.

56.4 The Invariant Interval and Causality

On a spacetime diagram, when we changed the coordinate system, both the slopes of the coordinate axes and the coordinates of the points changed. However, there should be a quantity that does not change when the coordinate system is changed. For example, in high school we learned that the distance $\sqrt{\Delta x^2+\Delta y^2}$ between two points in a plane does not change when the coordinate axes are rotated. In spacetime, too, there is a quantity corresponding to such a “distance” that is unchanged by the Lorentz transformation. It is the invariant interval, which we introduce in this section. It is the extension to the general case of the invariance of $c^2t^2-x^2$ seen in the preceding chapters (Section 54.5).

56.4.1 Definition of the invariant interval and its invariance

From the time difference $\Delta t$ and the spatial differences $\Delta x,\Delta y,\Delta z$ of two events, we form the following quantity.

The sign is as in the $(+,-,-,-)$ convention of Section 54.5 (the time term is positive and the space terms are negative).

This fact is the spacetime version of the invariance of $\Delta x^2+\Delta y^2$ under a rotation of the two-dimensional plane. However, since the sign is different ($-$ instead of $+$), the “rotation” that effects the transformation preserves a hyperbola rather than a circle (Section 56.7). Also, since $\Delta x^2+\Delta y^2+\Delta z^2$ does not change when the orientation of the axes is changed (a spatial rotation), the invariant interval is also invariant under spatial rotations. Therefore, even if the spatial separation of two events is not along the $x$ axis, we can rotate the spatial axes to align it with that axis and then use the Lorentz transformation.

56.4.2 Timelike, spacelike, and lightlike

Since the sign of the invariant interval does not depend on the coordinate system, we can use it to classify the relation between two events. Consider the interval $s^2=c^2t^2-x^2-y^2-z^2$ between the event O (the origin) and another event P.

This is also a statement about position relative to the light cone (Definition 56.2). Taking the event O as the origin, timelike events lie inside the light cone (on the future side and the past side), spacelike events lie outside the light cone, and lightlike events lie on the light cone. Since the sign of the invariant interval does not depend on the coordinate system, a pair of events that is “timelike” in one coordinate system is “timelike” in all coordinate systems, and the same holds for spacelike and lightlike pairs. Changing the coordinate system does not change whether two events lie inside or outside the light cone.

56.4.3 The light cone and causality

As a basic premise of physics, a cause occurs before its effect (causality). Whether this holds in every coordinate system becomes a problem in the theory of relativity, because simultaneity is relative and even the order of events can be reversed.

The answer is given by the light cone. For an event O to be the cause of another event P, some influence (a force, a signal, or matter) must propagate from O to P. The speed at which matter and signals propagate is at most the speed of light $c$ (that the speed of a massive particle cannot exceed $c$ is shown in Chapter 57, and that the speed of electromagnetic waves is $c$ was seen in the preceding chapters). Then, for P to be influenced by O, P must lie inside or on the future light cone of O ($\Delta s^2\ge0$ and $\Delta t\gt0$).

• When P is timelike (or lightlike) and later than O, P can be influenced by O (O can be a cause of P). The order is the same in all coordinate systems: this follows from Theorem 56.5 (1) in the timelike case, and from Theorem 56.3 when the lightlike case is included. That is, causality is preserved in all coordinate systems. The inside of the future light cone of O is called the absolute future, and the inside of the past light cone the absolute past.
• When P is a spacelike event, a signal faster than light would have to be sent to connect O and P, so no causal relation is possible. Although the order is reversed depending on the coordinate system, there is no causal relation, so which event comes first has no physical meaning. This region is called elsewhere (the spacelike region).

56.5 Proper Time

In the previous chapter we learned that a moving clock runs slow. There, for a clock moving at a constant speed $v$, we saw that the time $\Delta t$ of an observer at rest and the time $\Delta\tau$ ticked off by the moving clock are related by $\Delta t=\gamma\Delta\tau$ (Section 55.2). In this section, using the invariant interval (Definition 56.3), we generalize this relation so that it can also be used for motion in which the speed varies. It is the time measured by “riding along” on a moving object—the proper time.

56.5.1 The definition of proper time

Consider the world line of a single particle in a spacetime diagram. The invariant interval between two neighboring events on the world line—the events at times $t$ and $t+\dd t$—is, as seen in frame S, $\dd s^2=c^2\dd t^2-\dd x^2-\dd y^2-\dd z^2$. During $\dd t$ the particle moves by $\dd\bm{r}=\bm{u}\dd t$, so, writing the speed as $u=\abs{\bm{u}}$, we have

Since $u\lt c$, $\dd s^2\gt0$; that is, neighboring events on the particle's world line are separated by a timelike interval.

On the other hand, let $\mathrm{S}_0$ denote the inertial frame in which this particle appears (at that instant) to be at rest (the instantaneous rest frame). In $\mathrm{S}_0$ the particle does not move, so $\dd\bm{r}_0=\bm{0}$, and if we write $\dd\tau$ for the time advanced by a clock attached to the particle, then $\dd s^2=c^2\dd\tau^2$. Since the invariant interval has the same value whichever coordinate system it is computed in (Theorem 56.4), the relation

holds.

The proper time of Definition 55.2 in the previous chapter (the time between two events occurring at the same place, in the frame in which the clock is at rest) is a special case of this definition, namely the case of motion at a constant speed. The form of Definition 56.5 can be used as it is even for motion in which the speed varies.

If the particle moves at a constant speed $v$, then $\gamma$ is constant, so integrating $\dd t=\gamma\dd\tau$ gives $\Delta t=\gamma\Delta\tau$, that is, the formula for time dilation of the previous chapter. This can be verified directly from the Lorentz transformation. Let $\Delta\tau=\Delta t'$ be the time difference in frame S′ between two events of a clock at rest in frame S′ ($\Delta x'=0$ in S′); from the inverse transformation $t=\gamma(t'+vx'/c^2)$, the time difference in frame S is $\Delta t=\gamma(\Delta t'+v\Delta x'/c^2)=\gamma\Delta t'=\gamma\Delta\tau$.

56.5.2 When the speed varies: integrating the proper time

When the speed $u(t)$ of a particle varies with time, we add up (integrate) the infinitesimal proper times $\dd\tau=\dd t/\gamma(t)$ of the successive instants along the world line. The time ticked off by the particle's clock between two events (at times $t_1$ and $t_2$) is

(University Mathematics, Section 5.2). This is the general defining formula of the proper time. Since it is the sum, along the world line, of the invariant interval $\dd s$ divided by $c$, namely $\tau=\frac1c\int\dd s$, it gives the same value whether it is computed in frame S or in any other inertial frame.

56.5.3 A straight world line has the maximum proper time

There are many possible motions of a particle joining the same two events A and B. For example, there is motion from A to B at a constant speed (the world line is a straight line), and motion with a detour or with acceleration along the way (a broken line or a curve). From Eq. \eqref{eq:56-tau-int} we can see the following.

This is the heart of the twin paradox in the next section. Contrary to the intuition of Euclidean geometry that “a straight road is the shortest,” in spacetime “a straight world line has the maximum proper time,” and along a world line with a detour the clock advances only a little.

56.6 The Twin Paradox

There is a question that everyone who studies special relativity inevitably runs into. A moving clock runs slow. So suppose that, of a pair of twin brothers, the younger brother makes a round trip to a distant star in a high-speed rocket and comes back. From the viewpoint of the older brother who stayed on Earth, the younger brother's clock is slow, so the younger brother is younger than the older brother. From the viewpoint of the younger brother, however, it was the older brother who was moving (the Earth receded and then came back), so the older brother's clock is slow, and the older brother should be the younger one. Which of them ends up younger? This is the twin paradox. In this section we use the proper time of Section 56.5 to answer this question with concrete numbers.

56.6.1 Proper time of the round trip

We consider the following setup. Seen from the Earth (frame S), the younger brother B leaves in a rocket at a constant speed $v=0.8c$ for a star $4$ light-years away ($1$ light-year $=$ the distance light travels in $1$ year), turns around immediately on arriving at the star, and returns to the Earth at the same speed. The older brother A stays on Earth. (Making the turnaround instantaneous is an idealization. Even if the actual acceleration takes time, the effect is small as long as it is short compared with the whole trip.)

By Earth clocks, a one-way trip takes $D/v=4\ \text{light-years}/0.8c=5$ years, so the round trip takes $T=10$ years. A's proper time is $10$ years (because he is at rest on the Earth). With $\gamma=1/\sqrt{1-0.8^2}=5/3$ ($1/\gamma=0.6$), integrating $\dd\tau=\dd t/\gamma$ of Eq. \eqref{eq:56-dtau} over each leg gives B's proper time:

Let us check this with the invariant interval. For the two events of one leg (departure and turnaround), the differences in the coordinates are $c\Delta t=5$ light-years and $\Delta x=4$ light-years, so $\Delta s=\sqrt{5^2-4^2}=3$ light-years ($\Delta\tau=\Delta s/c=3$ years). The outbound and return legs together make $6$ years. Thus, when the brothers meet again, A has aged $10$ years and B only $6$ years. The younger brother is four years younger (Figure 56.7).

The result agrees with Theorem 56.6. The world line of A, who stays on Earth, is a straight line (inertial motion), and its proper time is the maximum, $10$ years. The world line of B, who turns around, is a broken line, and its proper time is smaller.

56.6.2 Why is it asymmetric?

The argument “from the older brother's viewpoint the younger brother's clock is slow, and from the younger brother's viewpoint the older brother's clock is slow” is correct, but only during a single interval. For example, during the outbound leg, seen from the inertial frame $\mathrm{S}_1$ in which B's rocket is at rest, the Earth's clock runs at $1/\gamma=0.6$ times the normal rate. During a one-way trip of $3$ years (by B's clock), A's clock advances only $0.6\times3=1.8$ years. The same holds for the return leg: $3$ years by B's time, and A's clock advances $1.8$ years. Together that is $3.6$ years, which does not agree at all with the actual $10$ years. Where did the missing $10-3.6=6.4$ years come from?

The answer is that at the moment B turns around, B's “now” (the set of simultaneous events) changes drastically (the relativity of simultaneity of Section 56.2). In the outbound inertial frame $\mathrm{S}_1$, let Q be the event on the Earth ($x=0$) that is simultaneous with the event T at which B turns around (in frame S, $x=4$ light-years and $ct=5$ years), and let us find its time $t_{\mathrm Q}$ using Eq. \eqref{eq:56-simul}. From the condition that T and Q are simultaneous in $\mathrm{S}_1$, the time in frame S is $t_{\mathrm Q}=t_{\mathrm T}-vx_{\mathrm T}/c^2=5-0.8\times4=1.8$ years (by $\Delta t=v\Delta x/c^2$, the time difference in frame S is $3.2$ years). In the return inertial frame $\mathrm{S}_2$ (velocity $-0.8c$) the sign is reversed, and $t_{\mathrm Q}=5+0.8\times4=8.2$ years. Across the turnaround, the Earth time corresponding to B's “now” jumps all at once by $6.4$ years, from $1.8$ years to $8.2$ years (the red dotted lines in Figure 56.7). The missing $6.4$ years is the amount of this jump.

In short, A stays in a single inertial frame from beginning to end, but B switches from the outbound inertial frame to the return inertial frame. Therefore A and B are not on an equal footing. The argument “from B's viewpoint, A's clock is slow” can be used only while B is in a single inertial frame.

The same result is obtained even if we do not use acceleration at all. Consider the following. Set the clock of the outbound rocket (speed $0.8c$), which passes by the Earth, to $0$ at the instant it passes. Near the star, this rocket passes the return rocket (speed $-0.8c$), which is heading for the Earth. As they pass, the outbound rocket reports its clock reading of $3$ years ($=5/\gamma$), and the return rocket sets its own clock to that reading. When the return rocket arrives at the Earth, its reading is $3+3=6$ years, and by then the older brother on Earth has aged $10$ years. Each of the two rockets stays in a single inertial frame, and no one accelerates. Even so, the total of the clock readings is $6$ years, which does not agree with the older brother's $10$ years. What makes the difference is not the acceleration itself but the bend in the world line, “switching from the outbound inertial frame to the return inertial frame.”

56.7 Rapidity — The Lorentz Transformation Is a Hyperbolic Rotation

In high school mathematics, we learned that the transformation $x'=x\cos\theta+y\sin\theta$, $y'=-x\sin\theta+y\cos\theta$, which rotates the coordinate axes through an angle $\theta$, leaves $x^2+y^2$ (the square of the distance from the origin) unchanged, and that when two rotations are performed in succession, the angles add ($\theta_1+\theta_2$). The Lorentz transformation can also be regarded as a kind of “rotation.” However, it is a transformation that preserves a hyperbola rather than a circle, and hyperbolic functions ($\cosh,\sinh,\tanh$, which we studied in University Mathematics (AD 02), Chapter 2) appear in place of trigonometric functions. What provides this viewpoint is the rapidity. Velocities could not be added (Section 56.1), but with rapidity they can.

56.7.1 Definition of rapidity

The speed ratio $\beta=v/c$ lies only in the range $-1\lt\beta\lt1$. The function that stretches this finite range onto all numbers from $-\infty$ to $+\infty$ is the inverse function of the hyperbolic tangent $\tanh$ (University Mathematics, Section 2.5).

The equality on the right is obtained by solving $\tanh\phi=\beta$ for $\phi$. Substitute $\tanh\phi=(e^{\phi}-e^{-\phi})/(e^{\phi}+e^{-\phi})$, clear the denominator, and collect the terms in $e^{\phi}$ and the terms in $e^{-\phi}$ on the two sides:

(In the last step we multiplied both sides by $e^{\phi}$ and divided by $1-\beta$.) Taking the natural logarithm of both sides gives $\phi=\frac12\ln\frac{1+\beta}{1-\beta}$. This is the inverse hyperbolic function $\tanh^{-1}\beta$ (University Mathematics, Section 2.6). For $\beta\ll1$ we have $\tanh\phi\approx\phi$, so $\phi\approx\beta$. That is, for slow motion the rapidity is almost equal to “the speed with the speed of light taken as 1.”

Figure 56.8 is drawn only up to $\phi=2.6$, but $\beta$ practically sticks to $1$ once $\phi$ exceeds about $2$. $\beta=0.96$ ($\phi=1.95$) is already close to $1$, and for $\beta=0.999$ ($\phi=3.8$) the difference can be seen only by going outside the figure. The two values of $\beta$ are both close to $1$, but measured by rapidity the values of $\phi$ differ greatly. As a yardstick of “how close to the speed of light,” the rapidity expresses finer differences than $\beta$ does.

56.7.2 The Lorentz transformation as a hyperbolic rotation

Using Formula 56.2, we rewrite the Lorentz transformation $ct'=\gamma(ct-\beta x)$, $x'=\gamma(x-\beta ct)$. Substituting $\gamma=\cosh\phi$ and $\gamma\beta=\sinh\phi$, we obtain

(this is the matrix $\Lambda(\beta)$ of Section 54.7). We write this matrix as $\Lambda(\phi)$. A Lorentz transformation of this kind, which sets the coordinate system in motion at a constant velocity (“changing over” to a frame moving in the direction of the velocity), is called a boost, to distinguish it from a rotation of space. Let us place it beside the rotation formula we learned in high school.

The forms are strikingly similar. The differences are that $\cos,\sin$ become $\cosh,\sinh$ and that some signs change. And the quantities that are preserved differ. A rotation preserves $x^2+y^2$ by virtue of $\cos^2\theta+\sin^2\theta=1$, whereas the Lorentz transformation preserves $(ct)^2-x^2$ by virtue of $\cosh^2\phi-\sinh^2\phi=1$. Indeed,

(the cross terms $\mp2\sinh\phi\cosh\phi\,ct\,x$ cancel each other). This is another expression of Theorem 56.4. Just as the circle $x^2+y^2=1$ is a curve that is invariant under rotations, the hyperbola $x^2-(ct)^2=1$ is a curve that is invariant under the Lorentz transformation (the curve that gives the tick marks in Figure 56.4).

Summarized in a table, the correspondence is as follows (Figure 56.9).

56.7.3 Additivity of rapidity

If a rotation of the plane through an angle $\theta_1$ is followed by a further rotation through $\theta_2$, the result is a rotation through the angle $\theta_1+\theta_2$. The same holds for the Lorentz transformation.

This result agrees with the velocity addition law \eqref{eq:56-beta-add} that we derived directly from the transformation law of velocities in Section 56.1. We have thus obtained the same result in two ways (from the transformation law of velocities, and from the addition of rapidities). The additivity of rapidity expresses a property of the Lorentz transformation: velocities cannot be added the way they are under the Galilean transformation, but instead rapidities can be added.

56.8 Summary and Exercises

56.8.1 Summary

• Transformation of coordinate differences (Section 56.1): Since the Lorentz transformation is linear, the coordinate differences $\Delta t,\Delta x$ of two events transform in the same form: $\Delta x'=\gamma(\Delta x-v\Delta t)$, $\Delta t'=\gamma(\Delta t-v\Delta x/c^2)$, $\Delta y'=\Delta y$, $\Delta z'=\Delta z$.
• Velocity addition law (Section 56.1; derived in Section 55.6 of the previous chapter): $u'_x=\dfrac{u_x-v}{1-vu_x/c^2}$, $u'_y=\dfrac{u_y\sqrt{1-v^2/c^2}}{1-vu_x/c^2}$, and $u'_z$ likewise. If $u_x=c$ then $u'_x=c$ (constancy of the speed of light), and if $v,u\ll c$ the law reduces to Galilean addition. No speed exceeds $c$ (Theorem 55.5 of the previous chapter). For composition in the same direction, $\beta=(\beta_1+\beta_2)/(1+\beta_1\beta_2)$.
• Relativity of simultaneity (Section 56.2): Even if $\Delta t=0$, if $\Delta x\neq0$ then $\Delta t'=-\gamma v\Delta x/c^2\neq0$. In the train thought experiment, the events of the light reaching the two ends are simultaneous in the train but not simultaneous on the ground (the rear end comes first, with a time difference $\gamma Lv/c^2$). The temporal order of two events with $\abs{\Delta x}\le c\Delta t$ is the same in every coordinate system.
• Spacetime diagram (Section 56.3): The horizontal axis is $x$ and the vertical axis is $ct$. A world line is the trajectory of a particle: rest is vertical, a constant speed $v$ is a line inclined from the $ct$ axis with $\tan\theta=v/c$, and light is at $45^\circ$. The light cone is $c^2t^2-x^2-y^2-z^2=0$. Of the axes of frame S′, the $ct'$ axis is $x=\beta ct$ and the $x'$ axis is $ct=\beta x$; both are tilted toward the light line by the same angle $\theta$. The tick marks are fixed by the hyperbolas $(ct)^2-x^2=\pm1$.
• Invariant interval (Section 56.4): $\Delta s^2=c^2\Delta t^2-\Delta x^2-\Delta y^2-\Delta z^2$ is invariant under the Lorentz transformation. It is timelike if $\gt0$, spacelike if $\lt0$, and lightlike if $=0$. For two timelike-separated events there is a coordinate system in which they occur at the same place, and their temporal order is invariant. For two spacelike-separated events there is a coordinate system in which they are simultaneous, and their temporal order can be reversed. A causal relation is possible only when the separation is timelike or lightlike, so causality is preserved.
• Proper time (Section 56.5): $\dd\tau=\dd s/c=\dd t/\gamma=\dd t\sqrt{1-u^2/c^2}$. For motion with varying speed, $\tau=\int\dd t/\gamma(t)$. Proper time does not depend on the coordinate system. The proper time along a straight world line (inertial motion) is the maximum.
• Twin paradox (Section 56.6): If the younger brother makes a round trip, he is the younger of the two when they meet again (for $0.8c$ and $4$ light-years, the older brother ages $10$ years and the younger brother $6$ years). The cause of the asymmetry is that the younger brother switches inertial frames. Because his line of simultaneity jumps before and after the turnaround, the numbers also agree without contradiction.
• Rapidity (Section 56.7): $\tanh\phi=\beta$, $\gamma=\cosh\phi$, $\gamma\beta=\sinh\phi$. The Lorentz transformation is the hyperbolic rotation $\begin{pmatrix}\cosh\phi&-\sinh\phi\\-\sinh\phi&\cosh\phi\end{pmatrix}$, which preserves $(ct)^2-x^2$. Composing boosts in the same direction adds the rapidities: $\phi=\phi_1+\phi_2$.

56.8.2 Exercises

56.8.3 References