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The Light Cone Time past and time future What might have been and what has been Point to one end, which is always present.
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dennis garrett
Feb 9, 2020, 7:37:22 PM2/9/20
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The Light Cone
Time past and time future
What might have been and what has been
Point to one end, which is always present.
t. s. eliot, “Burnt Norton”
Well, the future for me is already a thing
of the past.
bob dylan, “Bye and Bye”
Absolute and Relative
Special relativity has shown us that time
and space are diff erent for diff erent observers.
A popular way in which one hears this expressed is with the phrase
“everything is relative.” But is that really so? For example, does the relativity of
simultaneity imply that the causal order of events is also relative? By changing
frames of reference, can we make the Second World War occur before Hitler’s
invasion of Poland? That is, can cause and eff ect be reversed by switching
frames of reference? The world would be a pretty peculiar place if that
were so.
We have seen that light has the same speed in all inertial frames. So the
invariance of the speed of light is certainly not “relative,” but is absolute in
special relativity. This fact implies that Einstein’s spacetime, unlike Newton’s
space and time, can be divided into regions, described by what is called the
“light cone.” In some of these regions, the order of events in time is the same
in all frames of reference, and in others the temporal order is relative. As we
will see, all pairs of events that are causally connected lie within a single region
of the fi rst type.
The Light Cone > 43
In fi gure 4.1, we present plots of the trajectory, that is, the variable ct versus
the position x, for the earth and the spaceship in the reference frame S(earth).
Such a trajectory is often called the “worldline” of an object. For ease and clarity
of plotting, we confi ne ourselves to trajectories lying entirely in the twodimensional
x and ct portion of spacetime; all of the points we consider have
y = z = 0. As we have done throughout, we choose the origin, x = 0, ct = 0, to be
the point where the earth and ship pass one another, and the moment at which
observers on both the earth and the ship choose to correspond to the origin of
time on their respective clocks.
The earth is at rest at x = 0 in this reference frame; its worldline lies along
the ct axis. Its position on the diagram at time t1 is x = 0, ct = ct1, ad we show
a segment with t stretching from some distance in the past, where t < 0, and
into the future, where t > 0. In order to plot the worldline of the ship we have
arbitrarily chosen, v = 0.8c, so that the position of the ship is given by x = 0.8ct.
The ship’s trajectory on the ct versus x plot will then be a straight line with
x
c t
x = c t
x = - c t
x = v t
past light cone past light cone
future light cone future light cone
worldline of spaceship
worldline of the earth
c t E
fig. 4.1. The worldline of a spaceship moving with respect
to the earth.
44 < Chapter 4
slope 0.8, relative to the vertical axis. All of our slopes are assumed to be measured
with respect to the vertical axis (this is because, unlike the diagrams you are probably
used to seeing, we have plotted ct along the vertical axis and x along the
horizontal axis).
We choose the variable ct rather than t for convenience so that both axes
have the same units of length. For an object moving with speed v, so that x = vt,
the t versus x curve is a straight line of slope v. The cases of interest will involve
values of v something like c / 2, and since c is a huge number in normal units,
the line in question would be almost vertical and indistinguishable from the
x axis. By taking the variable ct, the slope of the ct versus x curve becomes a
more manageable v / c. Using ct as the variable is equivalent to taking t as the
variable, but in units of light-seconds (the distance traveled by light in one
second, or about 300,000 kilometers), rather than seconds.
The two straight lines labeled x = ct and x = −ct, in fi gure 4.1 describe the
trajectories of light pulses moving in the positive and negative directions, respectively,
along the x axis and passing through the origin at t = 0. These lines
form what is called the “light cone.” This is the portion of the spacetime surface
x2 + y2 + z2 = (ct)2 lying in the ct versus x plane.
The light cone has a special signifi cance, because it plays the same role in
any inertial system. For example, as we know, if x = ct in S(earth), it is also true
that x' = ct' in the reference frame S'(ship). The invariant interval s2 between the
origin and any point on the light cone satisfi es the condition s2 = 0, and, as we
have discussed, s2 is left unchanged if one makes a Lorentz transformation to
a diff erent inertial frame.
The light cone divides the page into four quadrants. The bottom and top
portions of the light cone lie in the regions where t is, respectively, negative
and positive. That is, they correspond to the regions of spacetime that are,
respectively, before and after the time that we have called t = 0 when the spaceship
passes the earth. To observers on earth and on the spaceship at t = 0, these
regions are, respectively, in their past and future, and are called the past and
future light cones. At points inside the past and future light cones, x2 − (ct)2 < 0,
that is, s2, the invariant interval between those points and the origin is negative.
Such points are said to have a “timelike” separation from the origin. This is
because the “time” part of the interval is larger than the “space” part.
Let’s consider a particular event with time and space coordinates t1 and x1. If
someone on earth at t = 0 wants to aff ect this event, then he must either travel
or send a signal which travels at a speed u, where u is at least x1 / t1. Because
of the light barrier, we must have u / c = x1 / ct1 ≤ 1. That is, the slope of the ct1
The Light Cone > 45
versus the x1 curve cannot be greater than 1. Moreover, t1 must be greater than
zero, since we can only infl uence events in our future and not our past (we’re
not yet talking about time travel). These two conditions combine to describe
the future light cone, so the future light cone is just the set of events which can
be infl uenced by someone at the origin.
Let’s look at some examples. Suppose that at t = 0, Starfl eet Command on
earth receives information that space pirates are planning to attack three space
stations in exactly one year’s time. The three stations are located at x = 0.4
light-years, x = 1 light-year, and x = 1.2 light-years. (A light-year, recall, is the
distance light travels in one year.) This is before Starfl eet has developed warp
drive, so although they have spaceships with very powerful engines, they are
limited by the light barrier. What will happen?
Refer to fi gure 4.2. The closest station is inside the forward light cone. Assuming
that ships are available with top speeds greater than 0.4 c, one or more
ships can be dispatched to support the station, and the ships will arrive before
the marauding pirates. The second station is right on the edge of the light
cone. No aid can reach it in time, since material objects cannot attain the speed
of light. A signal can, however, be sent to the station using electromagnetic
x
c t
worldline of the earth
t = 0
c (1 year)
0.4 1 1.2 (light-years)
s < 0 s > 0 s = 0
E
future light cone future light cone
s > 0 spacelike
s = 0 lightlike
s < 0 timelike
fig. 4.2. “Space pirates.” The fi gure depicts the diff erence between
timelike, lightlike, and spacelike intervals.
46 < Chapter 4
waves, warning of the attack. (Unfortunately, it will not be a very timely warning,
since it will arrive just as the pirates appear.)
The most distant station, which is outside the light cone, is out of luck.
Nothing that Starfl eet can do will infl uence events at that station one year in
the future. Help cannot arrive before 1.2 years, and the station will have to fend
for itself in the meantime.
Now let’s consider the past light cone. The situation is similar, except the
past light cone is the region of spacetime that can infl uence, rather than be
infl uenced by, events on earth at t = 0. For example, worldlines of the earth and
the ship stretch out of the past light cone to reach the origin, and past events
on the ship, as well as the prior history of the earth, infl uence the earth at
t = 0.
The interior of the light cone, where s2 is negative, is thus the set of points
which are in causal contact with the spacetime origin and can aff ect or be affected
by what happens there. Since s2 is invariant under the Lorentz transformations,
the set of events in the interior of the light cone is the same in all
inertial frames. The temporal order of events in the interior of the light cone,
for example, whether an event is in the future or the past light cone of the event
at the origin, is also a Lorentz invariant. (We will show this in the following
paragraph.) Thus, given a pair of causally related events, observers in all inertial
frames will agree as to which is the cause and which is the eff ect.
To see this, note that under a Lorentz transformation to an inertial frame
moving with speed v relative to the frame S(earth), t' =
t –vx/c2
1–(v2 /c2 ) . Within
the forward light cone, as we have seen, all values of x satisfy x = ut, where
u / c < 1, and v /c < 1 because of the light barrier restriction on the speed of inertial
frames. Now let’s look at the numerator in t', which is t − vx / c2 = t − v(ut) / c2,
where we have substituted x = ut. We can factor the right-hand side of this last
equation to get t − v(ut) / c2 = t [1−(u / c)( v / c)]. Since u / c and v / c are both less
than 1, their product is also less than 1. The denominator of t' is also always a
positive expression. Thus t' involves t multiplied by a positive number, so that
t' has the same sign as t. As a result, if an event at the origin causes a later event
in one inertial frame, the eff ect will be seen to occur after the cause in every
inertial frame.
On the other hand, events outside the light cone, even though they occur
before t = 0, cannot infl uence the earth at t = 0, because the invariant interval
x2 + y2 + z2 − (ct)2 > 0. (Points separated by such an invariant interval are said to
have “spacelike” separation. This is because the “space” part of the interval is
larger than the “time” part.) Suppose a Starfl eet spy gained knowledge of the
The Light Cone > 47
pirates’ nefarious scheme two years in the past by overhearing some conversation
in a bar on the planet Tatooine (for the purists, we know, we’re mixing
Star Trek and Star Wars) located 4 light-years from earth. The information is not
going to do Starfl eet any good at t = 0, since it can’t reach them until t = 2 years,
which is 1 year after the pirate attack occurs. The temporal order of events in
the exterior of the light cone is not a Lorentz invariant and can be changed
by a Lorentz transformation. (The argument in the preceding paragraph fails
in this case because, for points outside the light cone, there is no guarantee
that u / c < 1. However, the temporal order is not critical in this case, because,
regardless of the sign of t, events outside the light cone can be of no help to
Starfl eet Command at t = 0.)
The Light Cone and Causality: A Summary
Because the light cone is so important for our future discussions, and since
this is a rather diffi cult section, it’s worth summarizing the ideas we’ve presented.
We recommend a careful study of the following discussion and
fi gure 4.3 (our treatment parallels that of Taylor and Wheeler, Spacetime Physics,
x
c t
x = c t
x = - c t
future light cone
past light cone
O
A
B
C
D
E
F
fig. 4.3. The light cone. The event O represents
the “present moment.” The figure shows what
events can aff ect, and be aff ected by, event O.
48 < Chapter 4
Sec. 6.3.1 Figure 4.3 shows a light cone associated with an arbitrary spacetime
event O (we have added one space dimension back in, to better illustrate the
“cone”).
Event A lies inside the future light cone of O, so O and A are separated by a
timelike interval, for example, s2 < 0. This means that a particle or signal traveling
slower than light, emitted at O at t = 0, can aff ect what is going to happen at
A. Event B lies on the future light cone of O, so O and B are separated by a lightlike
(“null”) interval, that is, s2 = 0. Therefore, a light signal emitted at O can aff ect
what is going to happen at B (in fact, the light ray arrives just as B occurs.) Event
C lies inside the past light cone of O. This means that O and C are separated by a
timelike interval, so a particle or slower-than-light signal emitted at event C can
aff ect what is happening at O. Similarly, event D lies on the past light cone of O, so
O and D are separated by a lightlike interval, and so a light signal emitted at D
can aff ect what is happening at O. The events E and F lie outside both the past and
future light cones of O, so each of these events are separated from O by a spacelike
interval, that is, s2 > 0. This means that for O to either aff ect, or be aff ected
by, events E and F would require faster-than-light signaling. (A worldline connecting
O with events E or F would have a slope of greater than 45°, and thus lie
outside the light cone.) Therefore, events E and F can have no causal infl uence
on O and vice versa. The time order of events A through D are invariant, that is,
the same in all frames of reference. The time order of events E and F is diff erent
in diff erent inertial frames. In some frames E and F will be seen as simultaneous;
in other frames E will be seen to occur before F, or vice versa.
There is a light cone structure, like that depicted in fi gure 4.3, associated
with every event in spacetime. The light cones defi ne the “causal structure” of
spacetime in that they determine which events can communicate with each
other.
Note to the reader: Do not be disheartened if you did not understand everything
in the last two chapters the fi rst time through. They are probably the
most demanding chapters in the book. You may need to read them more than
once to fully grasp the ideas. However, an understanding of the concepts introduced
here, particularly mastery of the notion of the “light cone,” will be crucial
to understanding the chapters on time travel and warp drives later on.
1. Taylor and Wheeler, Spacetime Physics, 2nd ed. (New York: W. H. Freeman and Co., 1992).
< 49 >
5
Forward Time Travel and
the Twin “Paradox”
It was the best of times, it was the worst of times.
charles dickens, A Tale of Two Cities
Baseball player: “What time is it?”
Yogi Berra: “You mean now?”
In the previous chapter, we saw that observers
in two different inertial frames
didn’t agree on whether their clocks were synchronized initially. When observers
in the frame of reference of the earth thought all their clocks read t = 0 at
the same time, those in the spaceship frame disagreed, and vice versa. In this
chapter we are going to see that observers in the two frames also disagree as
to whether their clocks are running at the same speed. We will see that special
relativity tells us that moving clocks appear to run slow. An observer who
sees clocks in the other frame as moving though space will think those clocks
are running slow compared to his own. Later in the chapter, we will see that
this prediction leads to one of the clearest experimental verifi cations of special
relativity and also to the conclusion that travel forward in time is possible.
Time Dilation and A Tale of Four Clocks
Recall that the two frames have coincident x and x' axes, with S'(ship) moving
in the positive x (and x' ) directions with speed v relative to S(earth). Recall also
that we placed clocks in the two frames at their respective origins and set them
to read t = t' = 0 at the moment they pass one another. Let us call these two
50 < Chapter 5
clocks C0 and C'0, respectively. Since C0 and C'0 are momentarily side by side
and simultaneously visible as they pass, observers in both frames will see them
in agreement at that point. Observers in the earth frame will see C'0 moving to
the right with speed v along with the reference frame to which it is attached.
Similarly, those in the ship will see C0 moving to the left with speed v. Refer to
fi gure 5.1 for the discussion in this section.
Now let’s introduce a third clock into the mix, located in the frame S(earth)
at the point x = x1. We’ll call this clock C1. Since C'0 is starting at the origin at
t = 0 in the unprimed frame, and traveling with speed v, it will pass C1 when
C1 reads t1, where x1 = vt1. There is no relativity needed here. This statement
involves three quantities all measured in the same reference frame, S(earth).
C
C C
C’ C’
C’
C
C’
C C
x = 0
x = 0 x = x
t’
- v
x = x = v t
C’
x’ = - x’
The view from S(earth) at t = t’ = 0
The view from S(earth) at t = t = x / v
x’ = 0
The view from S’(ship) at t’ = t’ = x’ / v
v
fig. 5.1. The time dilation eff ect.
Forward Time Travel and the Twin “Paradox” > 51
It’s just the familiar formula that distance traveled equals speed multiplied by
time, if all of these quantities are measured in the same reference frame.
Here, however, we do need some relativity. Now that we know when the
clocks pass each other in the frame S(earth), we would like to know what C'0
reads when it passes C1. That is to say, we have an event, C'0 passes C1, whose
coordinates in the frame S(earth) are t = t1, x = vt1. What is the time coordinate
t1' of that event as measured on the clock C'0, which is present at the event and
at rest in S'(ship)? To answer that question, we need to use the Lorentz transformation
equation t' =
t − (vx / c2 )
1− (v2 / c2 ) and put in the values for t1 and vt1 for
t and x, respectively. If we do this and factor out t1, we get t1 ' = t1
1− v2 / c2
1− v2 / c2
.
Since for any quantity q, q / q = q , just from the defi nition of the square
root, we arrive at the result
t1 '= t1 1− v2 /c( 2 ) .
Observers on earth agree that C'0 was set correctly at t = 0, because it agreed
with their clock C0 when the two passed each other. Now the time read on C'0 is
less than that read on C1, because the factor 1− (v2 / c2 ) is smaller than 1 unless
v = 0. Therefore, observers in the earth frame see the clock C'0, which for
them is a clock moving with speed v, running slow compared to their clocks
by a factor of 1− (v2 / c2 ). Special relativity thus leads to the remarkable conclusion
that moving clocks run slow by the factor 1− (v2 / c2 ), compared to clocks
at rest, where v is the speed of the clock. This phenomenon is called “time
dilation.”
There is a subtle point connected with this conclusion. Observers in both
the S(earth) and S'(ship) frames see the two clocks C1 and C'0 next to one another,
and both agree that the time as shown on C1 is greater than that shown
on C'0. Since C1 is a moving clock in S'(ship), why don’t observers in the ship
frame come to the conclusion that moving clocks run fast? Observers on the
ship agree that C0 read correctly at t' = 0. However, C1 is synchronized with C0
according to observers in S(earth). As we discussed in the last chapter, observers
in the two frames do not agree as to how to synchronize distant clocks.
Therefore, observers on the ship say you cannot draw any valid conclusions
from observations of C1 because it wasn’t set correctly to begin with.
The Lorentz transformations have been set up to guarantee that the prin52
< Chapter 5
ciples of relativity prevail. This means that observers in any inertial frame must
see moving clocks running slow, but they must determine this on the basis of
experiments that are valid in their own frame. To allow observers in S'(ship) to
do this, we must introduce a fourth clock, C'1, which plays the roles in S'(ship)
that C1 played in S(earth). That is, C'1 will be a clock at x' = –x'1; the minus sign
refl ects the fact that C0 will be moving in the negative x' direction relative to
S'(ship). Remember now that C'1 will be synchronized with C'0 according to observers
in the ship frame. If observers in S'(ship) compare the reading of what
they see as the moving clock, C0, with that of C'1 as they pass, they will fi nd that
C0, which was correct at t' = 0, is now reading slow.1
Note that the two events, clock C'0 passing clock C0, and C'0 passing clock C1,
occur at the same place in S'(ship). Thus the time between these two events can
be measured by a single clock, C'0, in this frame. The time between two events
that occur at the same place in some inertial frame, and thus can be measured by a
single clock, is called the “proper time.” In our example above, t' is therefore
the proper time. The name is somewhat misleading, as it seems to denote the
“correct” or “true” time. In fact, it implies neither of these. You can think of
proper time as the time measured by your wristwatch as you travel along your
worldline in spacetime.
To summarize, what we really mean by the phrase “moving clocks run
slow,” is that a clock moving at a constant velocity relative to an inertial frame
containing synchronized clocks will be found to run slow when timed by these
synchronized clocks. (An alternative, more geometric way of deriving the time
dilation formula, without using the Lorentz transformations directly, can be
obtained using a device known as a light clock, discussed in appendix 5.)
The Twin “Paradox”
In this section we will discuss one of the most famous “paradoxes” of relativity,
the twin paradox. However, it should be noted at the outset that all of
these standard so-called paradoxes of relativity, including the twin paradox,
are really pseudo-paradoxes. That is, they only seem to be paradoxes because
the principles of relativity have been applied incorrectly. This distinguishes
1. If you want to verify this, you will need what are called the inverse Lorentz transformation
equations, which give t and x in terms of t' and x'. You can get these by taking the Lorentz transformation
equations given in chapter 3, interchange t and t' and x and x', and replace v [the velocity
of S'(ship) relative to S(earth)] with -v, since S(earth) will be moving to the left, in the negative x'
direction, as seen from S'(ship).
Forward Time Travel and the Twin “Paradox” > 53
them from the genuine logical consistency paradoxes which can occur in time
travel, such as the grandfather paradox, which we will discuss at length in later
chapters.
Let us introduce two twins, Jackie and Reggie, who are employed by a future
space agency. Jackie is a crew member on a manned fl yby of Alpha Centauri.
The trip will be made using a rocket that will fl y at constant speed to the star
4 light-years from earth, circle it, and return. (We ignore, very unrealistically,
the periods of acceleration and deceleration at the beginning and end of the
fl ight.) The rocket is capable of giving the spaceship a speed v, such that
1/ 1− (v2 / c2 ) = 10. A little arithmetic will convince you that this means v is
very nearly equal to c, the speed of light, so we will permit ourselves the luxury
of saying (a space engineer surely would not) that the 8-light-year round-trip
will require 8 years as seen by those on earth, though it would actually be a
little longer. Jackie and Reggie are accustomed to reading a book every week;
Reggie will read 416 books while Jackie is away, and Jackie stocks the spaceship
library appropriately (with e-readers, naturally, to save weight).
Happily, the trip goes off without a hitch, and, 8 years later, Reggie meets
the returning ship and the twins compare notes. Reggie is surprised to fi nd
that, for Jackie on the spaceship, only eight-tenths of a year have gone by, and
the forty-second book is only about fi nished. Similarly, Jackie is surprised to
fi nd that, while less than a year has gone by on the ship, there are the results of
two U.S. presidential elections to catch up with, and the campaign for a third
is, alas, already well underway.
In short, while 8 years have gone by for Reggie and the rest of the outside
world, less than a year has gone by for Jackie. This is just what we concluded in
chapter 2 would constitute time travel into the future, and just what happens
in the early pages of The Time Machine. Thus, we can say that Jackie has traveled
more than 7 years into the future. The only diff erence is that Wells envisioned
a time machine that remained stationary in space, while rapid travel through
space is the mechanism that produces relativistic forward time travel. One
could also achieve the same time dilation eff ect by traveling around a circular
path within a relatively limited region of space, rather than out and back as
with Jackie.
In the scenario we have discussed, there is no ambiguity as to which twin is
younger, and therefore no ambiguity as to whose clock was running slow. The
two twins are brought together again after the journey and can compare notes
in person. Everyone agrees that it was Jackie, due to the time dilation on the
moving spaceship, for whom time ran slowly.
54 < Chapter 5
But wait just a minute. The principle of relativity provides a sort of Declaration
of Independence for inertial frames. It proclaims in ringing terms, “all
inertial frames are created equal.” Jackie sees the earth move away and then return.
So one might conclude from this that Jackie would have read more books
than Reggie. If this argument were true, one would conclude that special relativity
indeed led to a paradox.
During the fi rst half of the last century there was a fair amount of controversy
engendered by this line of argument, with even some reputable physicists
suggesting that it struck at the logical foundation of special relativity. In fact,
there is no paradox, because there is a physical distinction between Jackie and
Reggie. Reggie has remained at rest in the reference frame of the earth. Apart
from corrections due to the earth’s rotation and orbital motion, which are
small because those velocities are very small compared to the speed of light,
the earth is an inertial frame, moving with constant velocity. It is the reference
frame that we have been denoting as S(earth). As an inertial frame, it is under
the protection of the principle of relativity’s grand proclamation of the equality
of all inertial frames. The same is true of the frame S'(ship), since prior to
the current discussion, we assumed that the ship was traveling with constant
velocity.
This is not true, however, of the reference frame of Jackie’s spaceship in the
twin paradox. That frame cannot move with constant velocity, because if the
twins are to be brought back together, the spaceship, traveling at relativistic
speed, must reverse its direction and thus undergo acceleration. It is not under
the protection of the principle of relativity’s guarantee of the equality of inertial
frames.
Invariant Interval and Proper Time
Consider the event, which we’ll call E for convenience, in which a clock C located
at x = 0 in a certain inertial frame, which we’ll call Se, reads t = T. Therefore
the invariant interval between E and the spacetime origin O (with coordinates
(0,0)) is s2 = − (ct)2 + (x)2 = − (cT)2
. The elapsed time on the clock, which was
present at both the spacetime origin O and E, is thus −s2 / c2 . (Recall that for
timelike intervals, s2 < 0, so –s2 > 0, and the quantity under the square root is
therefore positive.) The time of an event on a clock present at both the spacetime
origin and the event E, is the proper time of the event, as measured along
that clock’s particular worldline. However, proper time is not unique, in the sense
that it depends on worldline of the clock in question between the origin and E. Here
we have the special case that the clock is at rest in an inertial frame, and in that
Forward Time Travel and the Twin “Paradox” > 55
case we have the simple relation given above between the proper time on that
clock and the invariant interval (this gives us a new additional property of the
invariant interval, which we didn’t know before).
Now let’s say that, instead of a clock C remaining at rest, we consider a clock
C' that goes from the origin O with coordinates (0,0) to E with coordinates (0,cT)
by fi rst moving at constant velocity v to the intermediate spacetime event
A,with coordinates (x,ct) = vT
2
,
cT
2
⎛
⎝ ⎜
⎞
⎠ ⎟
. It then travels from event A to event E
along another path of constant speed v, but in the opposite direction. That
is, we fi rst give the clock a “kick” in the positive x direction, and then a kick
in the opposite direction. In the twin paradox, C' would correspond to a clock
on the spaceship, in the approximation in which we assume that Jackie’s
ship fl ies to Alpha Centauri at constant speed and then turns immediately
around and flies back, neglecting all the speeding up and slowing down
along the way. This is shown in fi gure 5.2. The heavy black lines represent
t = T/2
x
x
c t
= v T/2
t = 0
t = T
E
O
S(earth)
A
path 1
path 2
fig. 5.2. The twin paradox. Reggie’s worldline is the
straight line connecting events O and E . Jackie, the spaceship
twin, follows the “bent” worldline OAE . In this fi gure,
Jackie’s acceleration and deceleration periods are ignored.
56 < Chapter 5
the two legs of Jackie’s trip, outbound and then return. The dotted lines represent
the paths of light rays. The fact that the solid lines are so close to the
dotted lines indicates that Jackie’s spaceship is traveling very close to the speed
of light.
We already calculated the proper time elapsed along a straight worldline
from O to E, which would correspond to the time elapsed for the stay-at-home
twin, Reggie. That was simply T. Let us now calculate the proper time elapsed
along a “bent” worldline for Jackie. In this case, we can’t fi nd the invariant interval,
and hence, the elapsed proper time on the clock C' at one stroke, because
the directions of travel along the two segments of the path are diff erent. But
since the clock moves at the same constant speed along each side, we can use
the invariant interval for each side to fi nd the elapsed time for each segment of
the trip, and since elapsed time has no direction, they can be added to get the
total elapsed time.
The invariance of the spacetime interval can be expressed as
s2 = –(ct' )2 + (x' )2 = –(ct)2 + (x)2
where t' will denote the proper time along the “bent” worldline. This would be
Jackie’s “wristwatch time.” Let us fi rst calculate the proper time elapsed along
the fi rst leg of Jackie’s trip, from O to A. We’ll call this part of the trip path 1
and call the spacetime interval along this path s2
1. In Jackie’s frame, all events
occur at the same place, namely, x' = 0. Therefore the spacetime interval, in
terms of her coordinates, is just s2
1 = –(ct'1)2, where t'1 is the elapsed proper time
for Jackie along path 1.
To get the spacetime interval along path 1 in terms of Reggie’s coordinates,
notice that the coordinates of event A in S(earth) are x = vT / 2, ct = cT / 2. From
the invariance of the spacetime interval, all observers must agree on the value
of the interval along a given path. Therefore, our earlier equation becomes
s2
1 = –(ct'1)2 = –(cT / 2)2 + (vT / 2)2.
Multiplying both sides by –1 and factoring out c2 and (T / 2)2 gives
c2(t'1)2 = c2T2
4
1− v2
c2
⎛
⎝ ⎜
⎞
⎠ ⎟
.
If we now cancel the c2, and take the square root of both sides, we get
Forward Time Travel and the Twin “Paradox” > 57
t'1 = T
2
1− v2
c2 .
Since the bent path is symmetrical, it’s not too hard to convince yourself
that the spacetime interval along path 2 will be equal to that along path 1, that
is, s2
1 = s2
2. An identical calculation to the one just performed would then show
that the elapsed proper time along path 2, t2', is the same as that along path 1.
Hence, the total proper time along the bent path is given by t' = t'1 + t'2, and the
elapsed proper time for Jackie for the entire trip is
t' = T 1− v2
c2 .
Therefore, the unambiguous result is that t' < T, which means less time has
elapsed for Jackie than has for Reggie. So Jackie is the younger of the twins
when they reunite.2
You might be worried about the fact that we ignored the periods of acceleration
and deceleration during the trip. To show that this is not crucial to the
argument, let’s look at fi gure 5.3, where we have “rounded off the corners” of
Jackie’s trajectory to include these eff ects. We could, if we wished, break up the
curved path into a lot of tiny (approximately) straight-line segments. Then we
could work out the proper time along each straight-line segment, as we did in
the previous example, and add them up. Our result would still be that Jackie is
younger than Reggie when they reunite. This also dispels the commonly cited
fallacy that because acceleration is involved, special relativity is not applicable
and one must use general relativity to resolve the paradox.
In both fi gures 5.2 and 5.3, the “bent” line path between O and E is actually
shorter, in terms of elapsed proper time, than the vertical straight-line path between
the same two events! But, you say, it certainly doesn’t look that way in the
fi gure. This is because we are forced to illustrate the geometry of spacetime in special
relativity (you need to remember the minus sign in the interval!) on a piece of
paper which has the geometry of Euclidean space. It may help you to recall that, in
our earlier spacetime diagrams, lines inclined at 45° (representing the paths of
light rays) actually have zero length in spacetime, that is, s2 = 0. In fi gure 5.2, the
two legs of Jackie’s trip lie very close to 45° lines, and hence, together have much
shorter length (in terms of proper time) than the vertical straight-line path.
2. A good fi ctional portrayal of a forward time travel scenario is given in Poul Anderson’s novel
Tau Zero, Gollancz SF collector’s edition (London: Gollancz, 1970).
58 < Chapter 5
Practical Considerations and Experiments
Special relativity clearly allows the theoretical possibility of traveling forward in
time. In this section we will examine briefl y why such trips are not a very realistic
possibility for human beings or other macroscopic objects, as well as the evidence
that they are rather commonplace in the world of elementary particles.
Suppose you really cannot wait to see what kind of electronic miracles await
us 20 years in the future, and you’re only patient enough to spend 2 years getting
there. In order to make the trip, you need a space capsule large enough
to accommodate you that is capable of achieving a speed v through space, such
that 1− (v2 /c2 ) = 1/10 . Then the clocks on the ship, including your own biological
clock, run at about one-tenth the rate of clocks outside. Since the energy
of an object is mc2 / 1− (v2 / c2 ), you would need to increase the total energy of
your space capsule by about 9 mc2 to bring its speed up to v. How much energy
t = T/2
x
c t
t = 0
t = T
The curved path between O and E
is shorter than the straight path!
x = v T/2
O
E
fig. 5.3. A worldline for a spaceship observer including
periods of acceleration and deceleration. These make no difference
to the main argument. As a result of the geometry
of spacetime, the curved worldline connecting events and
is actually shorter, in terms of proper time elapsed, than the
straight worldline connecting the same two events!
Forward Time Travel and the Twin “Paradox” > 59
this is depends, of course, on m. Let’s try a value of about 1,000 kilograms for
m. That’s about the mass of a car, and your capsule would no doubt have to be
much larger. But even for a mass of 1,000 kilograms, it turns out that mc2 is
about equal to the entire annual electrical power output of the United States!
Thus, all the generators in the country would have to devote their full time
for a year to supplying power for your projected trip. There are other serious
technical problems as well. But the energy requirement by itself is enough to
demonstrate that time travel into the future using relativistic time dilation is
not going to be practical anytime in the near future, if ever.
One experiment has been done in which a macroscopic object was sent into
the future. The object was an atomic clock, and it was fl own around the world
on a commercial jetliner. The typical speed of such planes is around 500 miles
per hour, or around 1⁄7 of a mile per second, which gives a value of v2 / c2 of less
than 10–12. Nevertheless, the experimental group reported that the clock on
the plane lost about 1 nanosecond (one-billionth of a second) during the trip,
compared to a corresponding clock that had remained behind on the ground.
That was about the limit of the accuracy of the experiment. A supporter of
special relativity would not feel terribly secure if the experimental evidence in
support of time dilation hung only by this rather slender nanosecond. (Actually,
the result of this experiment is a test of both Einstein’s special theory of
relativity and his general theory of relativity, that is, his theory of gravity. The
calculations done by the experimenters to make their prediction depended on
both the aircraft’s speed [special relativity] and the height of the aircraft above
the earth’s surface [general relativity].)
Fortunately, there is a wealth of other evidence from the world of elementary
particles. Physicists at high-energy labs routinely observe these small masses
achieve relativistic speeds; we also observe such speeds for cosmic ray particles
incident on earth from outer space. Many of these particles are radioactively
unstable and decay with a well-established time interval called a “half-life.”
That is, half the particles decay, on the average, after one half-life, half of the
remainder after the next, and so on, and the rate of decay can be observed by
detecting the decay products with various detectors such as Geiger counters
or photomultiplier tubes. As a result, a sample of a number of such particles
provides a clock. It is routinely observed that particles produced with higher
energy—and thus with speeds close to the speed of light, and hence, a smaller
value of 1− v2 / c2 —decay more slowly. That is, they have a longer half-life,
as seen in the laboratory, than similar particles that decay at rest. In general the
observations are consistent with the relativistic prediction that the time read
60 < Chapter 5
on a moving clock, which is inversely proportional to the half-life in the case of
decaying particles, is proportional to 1− v2 / c2 , or equivalently, to mc2 / E.
One experiment of particular interest involved a circulating beam of particles
called muons. These particles decay radioactively with a half-life of about
a microsecond. The main purpose of the experiment was to compare magnetic
properties of muons and electrons. In the process, it was confi rmed with rather
high precision, that the lifetime of muons in motion was equal to the known
muon half-life at rest multiplied by the predicted factor of 1/ 1− v2 / c2 . In
contrast to most such experiments, which involve linear beams of particles,
this one involved a circulating beam with the circulating particles returning
periodically to their starting point. Thus, it modeled the twin paradox. To no
one’s surprise the circulating muons, playing the role of the traveling twin,
underwent time dilation, compared to muons remaining at rest.
The predictions of special relativity are tested literally thousands of times a
day in high-energy physics accelerators all around the world. In fact, the “nuts
and bolts” engineers who design these accelerators must take into account the
eff ects of special relativity, such as the increase in energy with velocity. Otherwise,
their machines would not function.
A Final Look at Forward Time Travel through The Door into Summer
We’ll conclude this chapter with a quick look at another possible mechanism
for forward time travel—one that does not involve relativity nor primarily even
physics, but rather, biology and medicine. The look is inspired by Robert Heinlein’s
book, The Door into Summer. If we had the time, we would be able to meet
one of the most engaging groups of characters, both human and feline, in science
fi ction. However, we must forego such pleasures and attend to business.
In the book the protagonist travels forward, then back, and then forward
again in time. Not surprisingly, Heinlein does not provide a detailed mechanism
for backward time travel, resorting instead to a glorifi ed “black box.” But
he does provide a mechanism for the forward time travel parts of the journey,
namely, “cold” or cryogenic sleep. The characters’ bodies are cooled to liquid
helium temperatures, after which it is hypothesized that all aging processes
stop. That is, the biological clocks of those stored are slowed—essentially
stopped—with respect to the fl ow of time in the outside world until they are
brought out of storage at some prearranged future time.
When Allen proposed this to his time travel classes as being a form of time
travel, his students tended to rebel and think he was cheating. It clearly wasn’t
Forward Time Travel and the Twin “Paradox” > 61
what they were used to thinking of as time travel, perhaps because the travelers
were all too clearly there throughout the process, rather than in an invisible
time machine (which, in fact, should have been visible if Wells had gotten the
physics right). Actually, Heinlein’s scheme is exactly the sort of thing we said
in chapter 2 would constitute forward time travel, namely, time going by very
slowly for a time traveler inside a time machine relative to the rate at which it
was going by outside.
In this area it seems likely the necessary physics has already been done.
Although low-temperature physicists continue to make advances toward the
unreachable goal of absolute zero, they are already so close that the progress
comes in small fractions of a degree that seem unlikely to be relevant to the
cold sleep problem. One guesses that, if this sort of forward time travel can be
done at all, it can be done at the very low temperatures already attainable.
We are neither MDs nor trained biologists and have no wisdom to off er on
the likelihood, or even the plausibility, of cryogenic time travel ever becoming
a reality. It’s not clear to us whether practitioners of the relevant disciplines
are in a position at this stage to off er any wisdom either. However, given the
technological problems confronting the relativistic version, which we’ve only
touched on, it’s not impossible to imagine that forward time travel will turn
out to be a fi eld for the biologists, not the physicists.
In the meantime, if you haven’t read it and you come across a copy of The
Door into Summer, get it; it’s a fun read.
Time past and time future
What might have been and what has been
Point to one end, which is always present.
t. s. eliot, “Burnt Norton”
Well, the future for me is already a thing
of the past.
bob dylan, “Bye and Bye”
Absolute and Relative
Special relativity has shown us that time
and space are diff erent for diff erent observers.
A popular way in which one hears this expressed is with the phrase
“everything is relative.” But is that really so? For example, does the relativity of
simultaneity imply that the causal order of events is also relative? By changing
frames of reference, can we make the Second World War occur before Hitler’s
invasion of Poland? That is, can cause and eff ect be reversed by switching
frames of reference? The world would be a pretty peculiar place if that
were so.
We have seen that light has the same speed in all inertial frames. So the
invariance of the speed of light is certainly not “relative,” but is absolute in
special relativity. This fact implies that Einstein’s spacetime, unlike Newton’s
space and time, can be divided into regions, described by what is called the
“light cone.” In some of these regions, the order of events in time is the same
in all frames of reference, and in others the temporal order is relative. As we
will see, all pairs of events that are causally connected lie within a single region
of the fi rst type.
The Light Cone > 43
In fi gure 4.1, we present plots of the trajectory, that is, the variable ct versus
the position x, for the earth and the spaceship in the reference frame S(earth).
Such a trajectory is often called the “worldline” of an object. For ease and clarity
of plotting, we confi ne ourselves to trajectories lying entirely in the twodimensional
x and ct portion of spacetime; all of the points we consider have
y = z = 0. As we have done throughout, we choose the origin, x = 0, ct = 0, to be
the point where the earth and ship pass one another, and the moment at which
observers on both the earth and the ship choose to correspond to the origin of
time on their respective clocks.
The earth is at rest at x = 0 in this reference frame; its worldline lies along
the ct axis. Its position on the diagram at time t1 is x = 0, ct = ct1, ad we show
a segment with t stretching from some distance in the past, where t < 0, and
into the future, where t > 0. In order to plot the worldline of the ship we have
arbitrarily chosen, v = 0.8c, so that the position of the ship is given by x = 0.8ct.
The ship’s trajectory on the ct versus x plot will then be a straight line with
x
c t
x = c t
x = - c t
x = v t
past light cone past light cone
future light cone future light cone
worldline of spaceship
worldline of the earth
c t E
fig. 4.1. The worldline of a spaceship moving with respect
to the earth.
44 < Chapter 4
slope 0.8, relative to the vertical axis. All of our slopes are assumed to be measured
with respect to the vertical axis (this is because, unlike the diagrams you are probably
used to seeing, we have plotted ct along the vertical axis and x along the
horizontal axis).
We choose the variable ct rather than t for convenience so that both axes
have the same units of length. For an object moving with speed v, so that x = vt,
the t versus x curve is a straight line of slope v. The cases of interest will involve
values of v something like c / 2, and since c is a huge number in normal units,
the line in question would be almost vertical and indistinguishable from the
x axis. By taking the variable ct, the slope of the ct versus x curve becomes a
more manageable v / c. Using ct as the variable is equivalent to taking t as the
variable, but in units of light-seconds (the distance traveled by light in one
second, or about 300,000 kilometers), rather than seconds.
The two straight lines labeled x = ct and x = −ct, in fi gure 4.1 describe the
trajectories of light pulses moving in the positive and negative directions, respectively,
along the x axis and passing through the origin at t = 0. These lines
form what is called the “light cone.” This is the portion of the spacetime surface
x2 + y2 + z2 = (ct)2 lying in the ct versus x plane.
The light cone has a special signifi cance, because it plays the same role in
any inertial system. For example, as we know, if x = ct in S(earth), it is also true
that x' = ct' in the reference frame S'(ship). The invariant interval s2 between the
origin and any point on the light cone satisfi es the condition s2 = 0, and, as we
have discussed, s2 is left unchanged if one makes a Lorentz transformation to
a diff erent inertial frame.
The light cone divides the page into four quadrants. The bottom and top
portions of the light cone lie in the regions where t is, respectively, negative
and positive. That is, they correspond to the regions of spacetime that are,
respectively, before and after the time that we have called t = 0 when the spaceship
passes the earth. To observers on earth and on the spaceship at t = 0, these
regions are, respectively, in their past and future, and are called the past and
future light cones. At points inside the past and future light cones, x2 − (ct)2 < 0,
that is, s2, the invariant interval between those points and the origin is negative.
Such points are said to have a “timelike” separation from the origin. This is
because the “time” part of the interval is larger than the “space” part.
Let’s consider a particular event with time and space coordinates t1 and x1. If
someone on earth at t = 0 wants to aff ect this event, then he must either travel
or send a signal which travels at a speed u, where u is at least x1 / t1. Because
of the light barrier, we must have u / c = x1 / ct1 ≤ 1. That is, the slope of the ct1
The Light Cone > 45
versus the x1 curve cannot be greater than 1. Moreover, t1 must be greater than
zero, since we can only infl uence events in our future and not our past (we’re
not yet talking about time travel). These two conditions combine to describe
the future light cone, so the future light cone is just the set of events which can
be infl uenced by someone at the origin.
Let’s look at some examples. Suppose that at t = 0, Starfl eet Command on
earth receives information that space pirates are planning to attack three space
stations in exactly one year’s time. The three stations are located at x = 0.4
light-years, x = 1 light-year, and x = 1.2 light-years. (A light-year, recall, is the
distance light travels in one year.) This is before Starfl eet has developed warp
drive, so although they have spaceships with very powerful engines, they are
limited by the light barrier. What will happen?
Refer to fi gure 4.2. The closest station is inside the forward light cone. Assuming
that ships are available with top speeds greater than 0.4 c, one or more
ships can be dispatched to support the station, and the ships will arrive before
the marauding pirates. The second station is right on the edge of the light
cone. No aid can reach it in time, since material objects cannot attain the speed
of light. A signal can, however, be sent to the station using electromagnetic
x
c t
worldline of the earth
t = 0
c (1 year)
0.4 1 1.2 (light-years)
s < 0 s > 0 s = 0
E
future light cone future light cone
s > 0 spacelike
s = 0 lightlike
s < 0 timelike
fig. 4.2. “Space pirates.” The fi gure depicts the diff erence between
timelike, lightlike, and spacelike intervals.
46 < Chapter 4
waves, warning of the attack. (Unfortunately, it will not be a very timely warning,
since it will arrive just as the pirates appear.)
The most distant station, which is outside the light cone, is out of luck.
Nothing that Starfl eet can do will infl uence events at that station one year in
the future. Help cannot arrive before 1.2 years, and the station will have to fend
for itself in the meantime.
Now let’s consider the past light cone. The situation is similar, except the
past light cone is the region of spacetime that can infl uence, rather than be
infl uenced by, events on earth at t = 0. For example, worldlines of the earth and
the ship stretch out of the past light cone to reach the origin, and past events
on the ship, as well as the prior history of the earth, infl uence the earth at
t = 0.
The interior of the light cone, where s2 is negative, is thus the set of points
which are in causal contact with the spacetime origin and can aff ect or be affected
by what happens there. Since s2 is invariant under the Lorentz transformations,
the set of events in the interior of the light cone is the same in all
inertial frames. The temporal order of events in the interior of the light cone,
for example, whether an event is in the future or the past light cone of the event
at the origin, is also a Lorentz invariant. (We will show this in the following
paragraph.) Thus, given a pair of causally related events, observers in all inertial
frames will agree as to which is the cause and which is the eff ect.
To see this, note that under a Lorentz transformation to an inertial frame
moving with speed v relative to the frame S(earth), t' =
t –vx/c2
1–(v2 /c2 ) . Within
the forward light cone, as we have seen, all values of x satisfy x = ut, where
u / c < 1, and v /c < 1 because of the light barrier restriction on the speed of inertial
frames. Now let’s look at the numerator in t', which is t − vx / c2 = t − v(ut) / c2,
where we have substituted x = ut. We can factor the right-hand side of this last
equation to get t − v(ut) / c2 = t [1−(u / c)( v / c)]. Since u / c and v / c are both less
than 1, their product is also less than 1. The denominator of t' is also always a
positive expression. Thus t' involves t multiplied by a positive number, so that
t' has the same sign as t. As a result, if an event at the origin causes a later event
in one inertial frame, the eff ect will be seen to occur after the cause in every
inertial frame.
On the other hand, events outside the light cone, even though they occur
before t = 0, cannot infl uence the earth at t = 0, because the invariant interval
x2 + y2 + z2 − (ct)2 > 0. (Points separated by such an invariant interval are said to
have “spacelike” separation. This is because the “space” part of the interval is
larger than the “time” part.) Suppose a Starfl eet spy gained knowledge of the
The Light Cone > 47
pirates’ nefarious scheme two years in the past by overhearing some conversation
in a bar on the planet Tatooine (for the purists, we know, we’re mixing
Star Trek and Star Wars) located 4 light-years from earth. The information is not
going to do Starfl eet any good at t = 0, since it can’t reach them until t = 2 years,
which is 1 year after the pirate attack occurs. The temporal order of events in
the exterior of the light cone is not a Lorentz invariant and can be changed
by a Lorentz transformation. (The argument in the preceding paragraph fails
in this case because, for points outside the light cone, there is no guarantee
that u / c < 1. However, the temporal order is not critical in this case, because,
regardless of the sign of t, events outside the light cone can be of no help to
Starfl eet Command at t = 0.)
The Light Cone and Causality: A Summary
Because the light cone is so important for our future discussions, and since
this is a rather diffi cult section, it’s worth summarizing the ideas we’ve presented.
We recommend a careful study of the following discussion and
fi gure 4.3 (our treatment parallels that of Taylor and Wheeler, Spacetime Physics,
x
c t
x = c t
x = - c t
future light cone
past light cone
O
A
B
C
D
E
F
fig. 4.3. The light cone. The event O represents
the “present moment.” The figure shows what
events can aff ect, and be aff ected by, event O.
48 < Chapter 4
Sec. 6.3.1 Figure 4.3 shows a light cone associated with an arbitrary spacetime
event O (we have added one space dimension back in, to better illustrate the
“cone”).
Event A lies inside the future light cone of O, so O and A are separated by a
timelike interval, for example, s2 < 0. This means that a particle or signal traveling
slower than light, emitted at O at t = 0, can aff ect what is going to happen at
A. Event B lies on the future light cone of O, so O and B are separated by a lightlike
(“null”) interval, that is, s2 = 0. Therefore, a light signal emitted at O can aff ect
what is going to happen at B (in fact, the light ray arrives just as B occurs.) Event
C lies inside the past light cone of O. This means that O and C are separated by a
timelike interval, so a particle or slower-than-light signal emitted at event C can
aff ect what is happening at O. Similarly, event D lies on the past light cone of O, so
O and D are separated by a lightlike interval, and so a light signal emitted at D
can aff ect what is happening at O. The events E and F lie outside both the past and
future light cones of O, so each of these events are separated from O by a spacelike
interval, that is, s2 > 0. This means that for O to either aff ect, or be aff ected
by, events E and F would require faster-than-light signaling. (A worldline connecting
O with events E or F would have a slope of greater than 45°, and thus lie
outside the light cone.) Therefore, events E and F can have no causal infl uence
on O and vice versa. The time order of events A through D are invariant, that is,
the same in all frames of reference. The time order of events E and F is diff erent
in diff erent inertial frames. In some frames E and F will be seen as simultaneous;
in other frames E will be seen to occur before F, or vice versa.
There is a light cone structure, like that depicted in fi gure 4.3, associated
with every event in spacetime. The light cones defi ne the “causal structure” of
spacetime in that they determine which events can communicate with each
other.
Note to the reader: Do not be disheartened if you did not understand everything
in the last two chapters the fi rst time through. They are probably the
most demanding chapters in the book. You may need to read them more than
once to fully grasp the ideas. However, an understanding of the concepts introduced
here, particularly mastery of the notion of the “light cone,” will be crucial
to understanding the chapters on time travel and warp drives later on.
1. Taylor and Wheeler, Spacetime Physics, 2nd ed. (New York: W. H. Freeman and Co., 1992).
< 49 >
5
Forward Time Travel and
the Twin “Paradox”
It was the best of times, it was the worst of times.
charles dickens, A Tale of Two Cities
Baseball player: “What time is it?”
Yogi Berra: “You mean now?”
In the previous chapter, we saw that observers
in two different inertial frames
didn’t agree on whether their clocks were synchronized initially. When observers
in the frame of reference of the earth thought all their clocks read t = 0 at
the same time, those in the spaceship frame disagreed, and vice versa. In this
chapter we are going to see that observers in the two frames also disagree as
to whether their clocks are running at the same speed. We will see that special
relativity tells us that moving clocks appear to run slow. An observer who
sees clocks in the other frame as moving though space will think those clocks
are running slow compared to his own. Later in the chapter, we will see that
this prediction leads to one of the clearest experimental verifi cations of special
relativity and also to the conclusion that travel forward in time is possible.
Time Dilation and A Tale of Four Clocks
Recall that the two frames have coincident x and x' axes, with S'(ship) moving
in the positive x (and x' ) directions with speed v relative to S(earth). Recall also
that we placed clocks in the two frames at their respective origins and set them
to read t = t' = 0 at the moment they pass one another. Let us call these two
50 < Chapter 5
clocks C0 and C'0, respectively. Since C0 and C'0 are momentarily side by side
and simultaneously visible as they pass, observers in both frames will see them
in agreement at that point. Observers in the earth frame will see C'0 moving to
the right with speed v along with the reference frame to which it is attached.
Similarly, those in the ship will see C0 moving to the left with speed v. Refer to
fi gure 5.1 for the discussion in this section.
Now let’s introduce a third clock into the mix, located in the frame S(earth)
at the point x = x1. We’ll call this clock C1. Since C'0 is starting at the origin at
t = 0 in the unprimed frame, and traveling with speed v, it will pass C1 when
C1 reads t1, where x1 = vt1. There is no relativity needed here. This statement
involves three quantities all measured in the same reference frame, S(earth).
C
C C
C’ C’
C’
C
C’
C C
x = 0
x = 0 x = x
t’
- v
x = x = v t
C’
x’ = - x’
The view from S(earth) at t = t’ = 0
The view from S(earth) at t = t = x / v
x’ = 0
The view from S’(ship) at t’ = t’ = x’ / v
v
fig. 5.1. The time dilation eff ect.
Forward Time Travel and the Twin “Paradox” > 51
It’s just the familiar formula that distance traveled equals speed multiplied by
time, if all of these quantities are measured in the same reference frame.
Here, however, we do need some relativity. Now that we know when the
clocks pass each other in the frame S(earth), we would like to know what C'0
reads when it passes C1. That is to say, we have an event, C'0 passes C1, whose
coordinates in the frame S(earth) are t = t1, x = vt1. What is the time coordinate
t1' of that event as measured on the clock C'0, which is present at the event and
at rest in S'(ship)? To answer that question, we need to use the Lorentz transformation
equation t' =
t − (vx / c2 )
1− (v2 / c2 ) and put in the values for t1 and vt1 for
t and x, respectively. If we do this and factor out t1, we get t1 ' = t1
1− v2 / c2
1− v2 / c2
.
Since for any quantity q, q / q = q , just from the defi nition of the square
root, we arrive at the result
t1 '= t1 1− v2 /c( 2 ) .
Observers on earth agree that C'0 was set correctly at t = 0, because it agreed
with their clock C0 when the two passed each other. Now the time read on C'0 is
less than that read on C1, because the factor 1− (v2 / c2 ) is smaller than 1 unless
v = 0. Therefore, observers in the earth frame see the clock C'0, which for
them is a clock moving with speed v, running slow compared to their clocks
by a factor of 1− (v2 / c2 ). Special relativity thus leads to the remarkable conclusion
that moving clocks run slow by the factor 1− (v2 / c2 ), compared to clocks
at rest, where v is the speed of the clock. This phenomenon is called “time
dilation.”
There is a subtle point connected with this conclusion. Observers in both
the S(earth) and S'(ship) frames see the two clocks C1 and C'0 next to one another,
and both agree that the time as shown on C1 is greater than that shown
on C'0. Since C1 is a moving clock in S'(ship), why don’t observers in the ship
frame come to the conclusion that moving clocks run fast? Observers on the
ship agree that C0 read correctly at t' = 0. However, C1 is synchronized with C0
according to observers in S(earth). As we discussed in the last chapter, observers
in the two frames do not agree as to how to synchronize distant clocks.
Therefore, observers on the ship say you cannot draw any valid conclusions
from observations of C1 because it wasn’t set correctly to begin with.
The Lorentz transformations have been set up to guarantee that the prin52
< Chapter 5
ciples of relativity prevail. This means that observers in any inertial frame must
see moving clocks running slow, but they must determine this on the basis of
experiments that are valid in their own frame. To allow observers in S'(ship) to
do this, we must introduce a fourth clock, C'1, which plays the roles in S'(ship)
that C1 played in S(earth). That is, C'1 will be a clock at x' = –x'1; the minus sign
refl ects the fact that C0 will be moving in the negative x' direction relative to
S'(ship). Remember now that C'1 will be synchronized with C'0 according to observers
in the ship frame. If observers in S'(ship) compare the reading of what
they see as the moving clock, C0, with that of C'1 as they pass, they will fi nd that
C0, which was correct at t' = 0, is now reading slow.1
Note that the two events, clock C'0 passing clock C0, and C'0 passing clock C1,
occur at the same place in S'(ship). Thus the time between these two events can
be measured by a single clock, C'0, in this frame. The time between two events
that occur at the same place in some inertial frame, and thus can be measured by a
single clock, is called the “proper time.” In our example above, t' is therefore
the proper time. The name is somewhat misleading, as it seems to denote the
“correct” or “true” time. In fact, it implies neither of these. You can think of
proper time as the time measured by your wristwatch as you travel along your
worldline in spacetime.
To summarize, what we really mean by the phrase “moving clocks run
slow,” is that a clock moving at a constant velocity relative to an inertial frame
containing synchronized clocks will be found to run slow when timed by these
synchronized clocks. (An alternative, more geometric way of deriving the time
dilation formula, without using the Lorentz transformations directly, can be
obtained using a device known as a light clock, discussed in appendix 5.)
The Twin “Paradox”
In this section we will discuss one of the most famous “paradoxes” of relativity,
the twin paradox. However, it should be noted at the outset that all of
these standard so-called paradoxes of relativity, including the twin paradox,
are really pseudo-paradoxes. That is, they only seem to be paradoxes because
the principles of relativity have been applied incorrectly. This distinguishes
1. If you want to verify this, you will need what are called the inverse Lorentz transformation
equations, which give t and x in terms of t' and x'. You can get these by taking the Lorentz transformation
equations given in chapter 3, interchange t and t' and x and x', and replace v [the velocity
of S'(ship) relative to S(earth)] with -v, since S(earth) will be moving to the left, in the negative x'
direction, as seen from S'(ship).
Forward Time Travel and the Twin “Paradox” > 53
them from the genuine logical consistency paradoxes which can occur in time
travel, such as the grandfather paradox, which we will discuss at length in later
chapters.
Let us introduce two twins, Jackie and Reggie, who are employed by a future
space agency. Jackie is a crew member on a manned fl yby of Alpha Centauri.
The trip will be made using a rocket that will fl y at constant speed to the star
4 light-years from earth, circle it, and return. (We ignore, very unrealistically,
the periods of acceleration and deceleration at the beginning and end of the
fl ight.) The rocket is capable of giving the spaceship a speed v, such that
1/ 1− (v2 / c2 ) = 10. A little arithmetic will convince you that this means v is
very nearly equal to c, the speed of light, so we will permit ourselves the luxury
of saying (a space engineer surely would not) that the 8-light-year round-trip
will require 8 years as seen by those on earth, though it would actually be a
little longer. Jackie and Reggie are accustomed to reading a book every week;
Reggie will read 416 books while Jackie is away, and Jackie stocks the spaceship
library appropriately (with e-readers, naturally, to save weight).
Happily, the trip goes off without a hitch, and, 8 years later, Reggie meets
the returning ship and the twins compare notes. Reggie is surprised to fi nd
that, for Jackie on the spaceship, only eight-tenths of a year have gone by, and
the forty-second book is only about fi nished. Similarly, Jackie is surprised to
fi nd that, while less than a year has gone by on the ship, there are the results of
two U.S. presidential elections to catch up with, and the campaign for a third
is, alas, already well underway.
In short, while 8 years have gone by for Reggie and the rest of the outside
world, less than a year has gone by for Jackie. This is just what we concluded in
chapter 2 would constitute time travel into the future, and just what happens
in the early pages of The Time Machine. Thus, we can say that Jackie has traveled
more than 7 years into the future. The only diff erence is that Wells envisioned
a time machine that remained stationary in space, while rapid travel through
space is the mechanism that produces relativistic forward time travel. One
could also achieve the same time dilation eff ect by traveling around a circular
path within a relatively limited region of space, rather than out and back as
with Jackie.
In the scenario we have discussed, there is no ambiguity as to which twin is
younger, and therefore no ambiguity as to whose clock was running slow. The
two twins are brought together again after the journey and can compare notes
in person. Everyone agrees that it was Jackie, due to the time dilation on the
moving spaceship, for whom time ran slowly.
54 < Chapter 5
But wait just a minute. The principle of relativity provides a sort of Declaration
of Independence for inertial frames. It proclaims in ringing terms, “all
inertial frames are created equal.” Jackie sees the earth move away and then return.
So one might conclude from this that Jackie would have read more books
than Reggie. If this argument were true, one would conclude that special relativity
indeed led to a paradox.
During the fi rst half of the last century there was a fair amount of controversy
engendered by this line of argument, with even some reputable physicists
suggesting that it struck at the logical foundation of special relativity. In fact,
there is no paradox, because there is a physical distinction between Jackie and
Reggie. Reggie has remained at rest in the reference frame of the earth. Apart
from corrections due to the earth’s rotation and orbital motion, which are
small because those velocities are very small compared to the speed of light,
the earth is an inertial frame, moving with constant velocity. It is the reference
frame that we have been denoting as S(earth). As an inertial frame, it is under
the protection of the principle of relativity’s grand proclamation of the equality
of all inertial frames. The same is true of the frame S'(ship), since prior to
the current discussion, we assumed that the ship was traveling with constant
velocity.
This is not true, however, of the reference frame of Jackie’s spaceship in the
twin paradox. That frame cannot move with constant velocity, because if the
twins are to be brought back together, the spaceship, traveling at relativistic
speed, must reverse its direction and thus undergo acceleration. It is not under
the protection of the principle of relativity’s guarantee of the equality of inertial
frames.
Invariant Interval and Proper Time
Consider the event, which we’ll call E for convenience, in which a clock C located
at x = 0 in a certain inertial frame, which we’ll call Se, reads t = T. Therefore
the invariant interval between E and the spacetime origin O (with coordinates
(0,0)) is s2 = − (ct)2 + (x)2 = − (cT)2
. The elapsed time on the clock, which was
present at both the spacetime origin O and E, is thus −s2 / c2 . (Recall that for
timelike intervals, s2 < 0, so –s2 > 0, and the quantity under the square root is
therefore positive.) The time of an event on a clock present at both the spacetime
origin and the event E, is the proper time of the event, as measured along
that clock’s particular worldline. However, proper time is not unique, in the sense
that it depends on worldline of the clock in question between the origin and E. Here
we have the special case that the clock is at rest in an inertial frame, and in that
Forward Time Travel and the Twin “Paradox” > 55
case we have the simple relation given above between the proper time on that
clock and the invariant interval (this gives us a new additional property of the
invariant interval, which we didn’t know before).
Now let’s say that, instead of a clock C remaining at rest, we consider a clock
C' that goes from the origin O with coordinates (0,0) to E with coordinates (0,cT)
by fi rst moving at constant velocity v to the intermediate spacetime event
A,with coordinates (x,ct) = vT
2
,
cT
2
⎛
⎝ ⎜
⎞
⎠ ⎟
. It then travels from event A to event E
along another path of constant speed v, but in the opposite direction. That
is, we fi rst give the clock a “kick” in the positive x direction, and then a kick
in the opposite direction. In the twin paradox, C' would correspond to a clock
on the spaceship, in the approximation in which we assume that Jackie’s
ship fl ies to Alpha Centauri at constant speed and then turns immediately
around and flies back, neglecting all the speeding up and slowing down
along the way. This is shown in fi gure 5.2. The heavy black lines represent
t = T/2
x
x
c t
= v T/2
t = 0
t = T
E
O
S(earth)
A
path 1
path 2
fig. 5.2. The twin paradox. Reggie’s worldline is the
straight line connecting events O and E . Jackie, the spaceship
twin, follows the “bent” worldline OAE . In this fi gure,
Jackie’s acceleration and deceleration periods are ignored.
56 < Chapter 5
the two legs of Jackie’s trip, outbound and then return. The dotted lines represent
the paths of light rays. The fact that the solid lines are so close to the
dotted lines indicates that Jackie’s spaceship is traveling very close to the speed
of light.
We already calculated the proper time elapsed along a straight worldline
from O to E, which would correspond to the time elapsed for the stay-at-home
twin, Reggie. That was simply T. Let us now calculate the proper time elapsed
along a “bent” worldline for Jackie. In this case, we can’t fi nd the invariant interval,
and hence, the elapsed proper time on the clock C' at one stroke, because
the directions of travel along the two segments of the path are diff erent. But
since the clock moves at the same constant speed along each side, we can use
the invariant interval for each side to fi nd the elapsed time for each segment of
the trip, and since elapsed time has no direction, they can be added to get the
total elapsed time.
The invariance of the spacetime interval can be expressed as
s2 = –(ct' )2 + (x' )2 = –(ct)2 + (x)2
where t' will denote the proper time along the “bent” worldline. This would be
Jackie’s “wristwatch time.” Let us fi rst calculate the proper time elapsed along
the fi rst leg of Jackie’s trip, from O to A. We’ll call this part of the trip path 1
and call the spacetime interval along this path s2
1. In Jackie’s frame, all events
occur at the same place, namely, x' = 0. Therefore the spacetime interval, in
terms of her coordinates, is just s2
1 = –(ct'1)2, where t'1 is the elapsed proper time
for Jackie along path 1.
To get the spacetime interval along path 1 in terms of Reggie’s coordinates,
notice that the coordinates of event A in S(earth) are x = vT / 2, ct = cT / 2. From
the invariance of the spacetime interval, all observers must agree on the value
of the interval along a given path. Therefore, our earlier equation becomes
s2
1 = –(ct'1)2 = –(cT / 2)2 + (vT / 2)2.
Multiplying both sides by –1 and factoring out c2 and (T / 2)2 gives
c2(t'1)2 = c2T2
4
1− v2
c2
⎛
⎝ ⎜
⎞
⎠ ⎟
.
If we now cancel the c2, and take the square root of both sides, we get
Forward Time Travel and the Twin “Paradox” > 57
t'1 = T
2
1− v2
c2 .
Since the bent path is symmetrical, it’s not too hard to convince yourself
that the spacetime interval along path 2 will be equal to that along path 1, that
is, s2
1 = s2
2. An identical calculation to the one just performed would then show
that the elapsed proper time along path 2, t2', is the same as that along path 1.
Hence, the total proper time along the bent path is given by t' = t'1 + t'2, and the
elapsed proper time for Jackie for the entire trip is
t' = T 1− v2
c2 .
Therefore, the unambiguous result is that t' < T, which means less time has
elapsed for Jackie than has for Reggie. So Jackie is the younger of the twins
when they reunite.2
You might be worried about the fact that we ignored the periods of acceleration
and deceleration during the trip. To show that this is not crucial to the
argument, let’s look at fi gure 5.3, where we have “rounded off the corners” of
Jackie’s trajectory to include these eff ects. We could, if we wished, break up the
curved path into a lot of tiny (approximately) straight-line segments. Then we
could work out the proper time along each straight-line segment, as we did in
the previous example, and add them up. Our result would still be that Jackie is
younger than Reggie when they reunite. This also dispels the commonly cited
fallacy that because acceleration is involved, special relativity is not applicable
and one must use general relativity to resolve the paradox.
In both fi gures 5.2 and 5.3, the “bent” line path between O and E is actually
shorter, in terms of elapsed proper time, than the vertical straight-line path between
the same two events! But, you say, it certainly doesn’t look that way in the
fi gure. This is because we are forced to illustrate the geometry of spacetime in special
relativity (you need to remember the minus sign in the interval!) on a piece of
paper which has the geometry of Euclidean space. It may help you to recall that, in
our earlier spacetime diagrams, lines inclined at 45° (representing the paths of
light rays) actually have zero length in spacetime, that is, s2 = 0. In fi gure 5.2, the
two legs of Jackie’s trip lie very close to 45° lines, and hence, together have much
shorter length (in terms of proper time) than the vertical straight-line path.
2. A good fi ctional portrayal of a forward time travel scenario is given in Poul Anderson’s novel
Tau Zero, Gollancz SF collector’s edition (London: Gollancz, 1970).
58 < Chapter 5
Practical Considerations and Experiments
Special relativity clearly allows the theoretical possibility of traveling forward in
time. In this section we will examine briefl y why such trips are not a very realistic
possibility for human beings or other macroscopic objects, as well as the evidence
that they are rather commonplace in the world of elementary particles.
Suppose you really cannot wait to see what kind of electronic miracles await
us 20 years in the future, and you’re only patient enough to spend 2 years getting
there. In order to make the trip, you need a space capsule large enough
to accommodate you that is capable of achieving a speed v through space, such
that 1− (v2 /c2 ) = 1/10 . Then the clocks on the ship, including your own biological
clock, run at about one-tenth the rate of clocks outside. Since the energy
of an object is mc2 / 1− (v2 / c2 ), you would need to increase the total energy of
your space capsule by about 9 mc2 to bring its speed up to v. How much energy
t = T/2
x
c t
t = 0
t = T
The curved path between O and E
is shorter than the straight path!
x = v T/2
O
E
fig. 5.3. A worldline for a spaceship observer including
periods of acceleration and deceleration. These make no difference
to the main argument. As a result of the geometry
of spacetime, the curved worldline connecting events and
is actually shorter, in terms of proper time elapsed, than the
straight worldline connecting the same two events!
Forward Time Travel and the Twin “Paradox” > 59
this is depends, of course, on m. Let’s try a value of about 1,000 kilograms for
m. That’s about the mass of a car, and your capsule would no doubt have to be
much larger. But even for a mass of 1,000 kilograms, it turns out that mc2 is
about equal to the entire annual electrical power output of the United States!
Thus, all the generators in the country would have to devote their full time
for a year to supplying power for your projected trip. There are other serious
technical problems as well. But the energy requirement by itself is enough to
demonstrate that time travel into the future using relativistic time dilation is
not going to be practical anytime in the near future, if ever.
One experiment has been done in which a macroscopic object was sent into
the future. The object was an atomic clock, and it was fl own around the world
on a commercial jetliner. The typical speed of such planes is around 500 miles
per hour, or around 1⁄7 of a mile per second, which gives a value of v2 / c2 of less
than 10–12. Nevertheless, the experimental group reported that the clock on
the plane lost about 1 nanosecond (one-billionth of a second) during the trip,
compared to a corresponding clock that had remained behind on the ground.
That was about the limit of the accuracy of the experiment. A supporter of
special relativity would not feel terribly secure if the experimental evidence in
support of time dilation hung only by this rather slender nanosecond. (Actually,
the result of this experiment is a test of both Einstein’s special theory of
relativity and his general theory of relativity, that is, his theory of gravity. The
calculations done by the experimenters to make their prediction depended on
both the aircraft’s speed [special relativity] and the height of the aircraft above
the earth’s surface [general relativity].)
Fortunately, there is a wealth of other evidence from the world of elementary
particles. Physicists at high-energy labs routinely observe these small masses
achieve relativistic speeds; we also observe such speeds for cosmic ray particles
incident on earth from outer space. Many of these particles are radioactively
unstable and decay with a well-established time interval called a “half-life.”
That is, half the particles decay, on the average, after one half-life, half of the
remainder after the next, and so on, and the rate of decay can be observed by
detecting the decay products with various detectors such as Geiger counters
or photomultiplier tubes. As a result, a sample of a number of such particles
provides a clock. It is routinely observed that particles produced with higher
energy—and thus with speeds close to the speed of light, and hence, a smaller
value of 1− v2 / c2 —decay more slowly. That is, they have a longer half-life,
as seen in the laboratory, than similar particles that decay at rest. In general the
observations are consistent with the relativistic prediction that the time read
60 < Chapter 5
on a moving clock, which is inversely proportional to the half-life in the case of
decaying particles, is proportional to 1− v2 / c2 , or equivalently, to mc2 / E.
One experiment of particular interest involved a circulating beam of particles
called muons. These particles decay radioactively with a half-life of about
a microsecond. The main purpose of the experiment was to compare magnetic
properties of muons and electrons. In the process, it was confi rmed with rather
high precision, that the lifetime of muons in motion was equal to the known
muon half-life at rest multiplied by the predicted factor of 1/ 1− v2 / c2 . In
contrast to most such experiments, which involve linear beams of particles,
this one involved a circulating beam with the circulating particles returning
periodically to their starting point. Thus, it modeled the twin paradox. To no
one’s surprise the circulating muons, playing the role of the traveling twin,
underwent time dilation, compared to muons remaining at rest.
The predictions of special relativity are tested literally thousands of times a
day in high-energy physics accelerators all around the world. In fact, the “nuts
and bolts” engineers who design these accelerators must take into account the
eff ects of special relativity, such as the increase in energy with velocity. Otherwise,
their machines would not function.
A Final Look at Forward Time Travel through The Door into Summer
We’ll conclude this chapter with a quick look at another possible mechanism
for forward time travel—one that does not involve relativity nor primarily even
physics, but rather, biology and medicine. The look is inspired by Robert Heinlein’s
book, The Door into Summer. If we had the time, we would be able to meet
one of the most engaging groups of characters, both human and feline, in science
fi ction. However, we must forego such pleasures and attend to business.
In the book the protagonist travels forward, then back, and then forward
again in time. Not surprisingly, Heinlein does not provide a detailed mechanism
for backward time travel, resorting instead to a glorifi ed “black box.” But
he does provide a mechanism for the forward time travel parts of the journey,
namely, “cold” or cryogenic sleep. The characters’ bodies are cooled to liquid
helium temperatures, after which it is hypothesized that all aging processes
stop. That is, the biological clocks of those stored are slowed—essentially
stopped—with respect to the fl ow of time in the outside world until they are
brought out of storage at some prearranged future time.
When Allen proposed this to his time travel classes as being a form of time
travel, his students tended to rebel and think he was cheating. It clearly wasn’t
Forward Time Travel and the Twin “Paradox” > 61
what they were used to thinking of as time travel, perhaps because the travelers
were all too clearly there throughout the process, rather than in an invisible
time machine (which, in fact, should have been visible if Wells had gotten the
physics right). Actually, Heinlein’s scheme is exactly the sort of thing we said
in chapter 2 would constitute forward time travel, namely, time going by very
slowly for a time traveler inside a time machine relative to the rate at which it
was going by outside.
In this area it seems likely the necessary physics has already been done.
Although low-temperature physicists continue to make advances toward the
unreachable goal of absolute zero, they are already so close that the progress
comes in small fractions of a degree that seem unlikely to be relevant to the
cold sleep problem. One guesses that, if this sort of forward time travel can be
done at all, it can be done at the very low temperatures already attainable.
We are neither MDs nor trained biologists and have no wisdom to off er on
the likelihood, or even the plausibility, of cryogenic time travel ever becoming
a reality. It’s not clear to us whether practitioners of the relevant disciplines
are in a position at this stage to off er any wisdom either. However, given the
technological problems confronting the relativistic version, which we’ve only
touched on, it’s not impossible to imagine that forward time travel will turn
out to be a fi eld for the biologists, not the physicists.
In the meantime, if you haven’t read it and you come across a copy of The
Door into Summer, get it; it’s a fun read.
dennis garrett
Feb 22, 2020, 10:22:33 PM2/22/20
to
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> and bolts” engineers who design these accelerators ...
Denny Garrett
Mar 10, 2020, 6:29:26 PM3/10/20
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On Sunday, February 9, 2020 at 4:37:22 PM UTC-8, dennis garrett wrote:
On Sunday, February 9, 2020 at 4:37:22 PM UTC-8, dennis garrett wrote:
newgirlf...@gmail.com
Feb 12, 2021, 8:11:48 PM2/12/21
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Dennis Garrett
Oct 27, 2022, 9:19:31 PM10/27/22
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Dennis Garrett
Dec 13, 2023, 12:31:36 AM12/13/23
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