HPS 0410 | Einstein for Everyone |

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John D. Norton

Department of History and Philosophy of Science

University of Pittsburgh

Once we have the notion of spacetime and the simple picture it brings of the relativity of simultaneity, we find that other processes and effects in special relativity become a great deal easier to understand. Here is a collection of a few of them.

One of the most intriguing entities in relativity theory are tachyons, hypothetical particles that travel faster than light. They are distinguished from "bradyons," particles that travel at less than the speed of light. While bradyons are familiar and include protons, electrons and neutrons, tachyons have never been observed.

Bradyons | Tachyons |

travel slower than light | travel faster than light |

ordinary matter | exotic matter (not found) |

Add energy and momentum and they speed up. |
Add energy and momentum and they slow down. |

c is the upper limit to their speeds | c is the lower limit to their
speeds |

For present purposes, the interesting fact is a curious property: for some observers they travel backwards in time. With the spacetime representation of the relativity of simultaneity it is now very easy to see how this comes about. The figures below show a tachyon being created and propagating into space; and how three different observers would judge the same tachyon.

Observer A judges it to be moving forward in time from its creation at the instant marked "now." It propagates from the "now" hypersurfaces of simultaneity towards the "later" hypersurfaces of simultaneity. |

Observer B moves in the direction of propagation of the tachyon. Observer B finds the tachyon to lie fully within one of B's hypersurfaces of simultaneity, the "now" hypersurface that contains the event of the tachyon's creation. Indeed the "now" hypersurface contains all the events in the tachyon's history. So the tachyon exists only "now" for observer B. That is, for B, the tachyon has infinite speed--it covers distance in no time--and it has disappeared to spatial infinity in the same instant "now." |

Finally observer C, who moves even faster in the same direction, judges the tachyon traveling into the past. It is created on the "now" hypersurface of simultaneity and propagates towards "earlier" hypersurfaces of simultaneity. It arrives at the earlier ones before it was created; that means it is traveling backwards in time. |

All three figures above are drawn with the tachyon
moving up the page. So it is easy to fall into the trap of imagining
that the figure with observer A shows what is really happening: that
the tachyon is really propagating forward in time and that the other
two figures represesent distorted reporting from observers B and C in
motion. That is not how it works. The principle of relativity assures
us that the reports of all of the observers are
equally good. C's reporting of the tachyon traveling back in
time is as good as A's reporting of the tachyon traveling forward in
time. That the figure showing observer A looks more natural is just an
accident of the way the figures have been drawn. We could equally well
draw the figures so that C's worldline is vertical. Then, as shown at
left, the natural reading would be to say that the tachyon propagates
backwards in time. None of the figures in any more correct than any other. The principle of relativity assures us that all the observers are equally good. Since they disagree on whether the tachyon propagates forward in time, the best we can say is that there is no observer independent fact of the direction of propagation, just as there is no fact as to which observer is really at rest. |

For some observers, tachyons travel backwards in time, that is, into the past. Does that mean that they can be used to affect the past, that is, to change the past? Does that mean that we can use them to create paradoxical situation?

The standard time travel paradox is the one in which a time traveler travels back in time and kills his or her grandfather; so that the time traveler is never born; so the time traveler doesn't travel back in time! This closed loop produces a contradiction. The time traveler both exists at some time and and does not exist at the same time.

Public domain http://en.wikipedia.org/wiki/File:Moordwillemzwijger2.jpg

It proves to be quite easy to conceive situations in which
tachyons are emitted and absorbed in such a way as to produce similar closed,
paradoxical cycles. The standard tachyonic paradox
employs spaceships run entirely by robots, programmed to behave in certain ways
according to whether or not a tachyonic signal has been received by them

In the figure below, the robot controlled spaceship A is programmed simply to self destruct if it receives a tachyonic signal; and if it still exists later to emit a tachyonic signal into the past.

The robot controlled spaceship B is programmed to switch into an "activated" mode upon receipt of a tachyonic signal; and to transmit a tachyonic signal later only if it is in the activated mode. In addition the motions of the spaceships and timing of the emissions are all carefully pre-programed so that the spaceships are in just the right positions for the sending of a signal the other will receive; and the receiving of a signal the other will send.

Start the cycle with the sending of a tachyonic signal by spaceship A. Tracing through the effects of that signal soon leads to the outcome that spaceship A self-destructs prior to sending the tachyonic signal. So the signal is not sent. And if the signal is not sent, tracing through the effects leads to the conclusion that spaceship A does not self-destruct.

A sends a tachyon. | B receives a tachyon. | B sends a tachyon. | A receives a tachyon. | A does NOT send a tachyon. | ||||

A does NOT send a tachyon. | B does NOT receives a tachyon. | B doesNOT send a tachyon. | A does NOT receive a tachyon. | A does send a tachyon. |

So A's tachyonic signal is sent. We have a contradiction:

A sends a tachyon |
if and only if |
A does not send a tachyon. |

If you know a little logic, you will find it easy from this to infer to a contradiction in the traditional form: (A's signal is sent AND A's signal is NOT sent.)

Since tachyons are candidates for serious science and not imaginings of science fiction, we cannot tolerate such an outright contradiction. Somehow it must be resolved. The most obvious resolution is the most severe. We could just suppose that these paradoxes show that there are no tachyons. That seems too severe to me since other weaker resolutions are possible.

The simplest resolution is just to suppose that the emission of tachyons is just not something that can be controlled by us. Just as the receipt of a signal is something that happens to us, the emission (or receipt) of a tachyon is again just something that happens to us. What makes this resolution plausible is that there is no absolute distinction between emission and receipt of a tachyon. What one observer counts as an emission another may count as a receipt. so we might expect the one rule over control to cover both emission and receipt.

We could look to other more fanciful resolutions. Perhaps tachyons exist but they don't interact with normal matter. Most people find that dubious. Since we are normal matter, that means we never interact with them and so we can never know they are there.

That inertial observers in relative motion will each judge the others' clocks to run slower is, by now, a quite familiar and readily understandable outcome of relativity theory. It does take a little while to get used to the idea, of course. When you first hear it, it seems strange and even paradoxical. How can each be correct in judging the other's clock to have slowed? What would happen if the two observers meet and compare their clocks? If relativity is right, each would have to read a time earlier than other; and surely that is impossible. Or is it?

We now know that these concerns are misplaced. The clocks cannot start out from the same place and then be re-united without one or both accelerating; and those accelerations so interfere with the analysis that no contradiction arises. When either accelerates, they cease to be inertial observers.

However an enduring literature has tried to generate some
sort of paradox from the effect of relativistic clock slowing. The most famous
of the these attempts is associated with a story of two twins. One stays on the earth--the "stay-at home-twin." The
stay-at-home twin's motion is inertial throughout. The other travels off
rapidly into space, journeys far and fast and then returns home. The traveling
twin must accelerate to complete this journey.

The story of what happens is readily told from the point of view of the stay-at-home twin. The traveling twin's clocks will slow due to the twin's rapid motion. That slowing encompasses all processes related to time. So the traveling twin's metabolism will slow as well. When the traveling twin returns to earth, the traveler will have aged significantly less. If the traveling twin had maintained a speed of 86.6% the speed of light, the internal clock would slow to 50% of the normal rate. Let us say the traveling twin returns, after what the stay-at-home twin finds to be 8 days. The traveling twin will experience merely the passing of 4 days.

All this can be depicted in a simple spacetime diagram. It shows the worldline of the
stay-at-home twin. The numbers 1, 2, ... , 8 are the days of time
elapsed on the stay-at-home twin's clock. It also shows the worldline
of the traveling twin, who moves away from earth, travels inertially
for four days of stay-at-home twin time; abruptly turns around; and
then takes another 4 days to return. Because the traveling twin is
moving so fast, the traveler's clock and the traveling twin's
metabolism run at half the speed of the stay-at-home twin. The numbers
1, 2, 3, 4 represent the days of time elapsed on the traveling twin's
clock and metabolism. The twins have set their clocks to zero at the start when the traveling twin leaves. When the traveling twin returns, the stay-at-home twin has aged 8 days, but the traveler has aged only 4 days. So far, the analysis is straightforward. Where the problems enter, is if we try to recount the story from the perspective of the traveling twin--and do it badly. The temptation is to say that, from the traveling twin's perspective, everything looks the same. The stay-at-home twin recedes, then turns around and comes back. So, we are tempted to ask, should the stay-at-home twin have aged less? Since the stay-at-home twin cannot have both aged more and less, do we have paradox--often called the "twin paradox"? We should not expect the stay-at-home twin to age less. The error made was to assume a symmetry in the two twins. The stay-at-home twin maintains inertial motion throughout the process. The traveling twin must at some point turn around and return. Even if very briefly, the traveling twin must accelerate. That acceleration makes a big difference and enables us to maintain different results for each twin. |

Relativity theory is able to give a consistent
account of times elapsed for both twins. The clearest account shows the
judgments of simultaneity made by each twin. First, the diagram opposite shows the judgments of simultaneity of the stay-at-home twin. As time passes on the clock of the stay-at-home twin, we can trace out the corresponding times on the clock of the traveling twin. The two clocks are both set to 0 at the outset. Then, the traveler's clock starts to lag. After 2 days have elapsed for the stay-at-home twin, we follow the hypersurface of simultaneity from the worldline of the stay-at-home twin to the traveling twin to find that just one day has elapsed for the traveler. Repeating for the remaining times, we see that for each time elapsed on the stay-at-home twin's clock--2, 4, 6, 8 days--half the time has elapsed on the traveling twin's clock--1, 2, 3, 4. |

Now consider the judgments of simultaneity of the traveling twin, as shown in
the spacetime diagram opposite. Since the traveling twin is moving very
rapidly, the traveler's hypersurfaces of simultaneity are quite
tilted. |

Careful attention to the differing judgments of simultaneity of the two twins shows that there is nothing paradoxical in the twin effect. The brief moment of acceleration of the traveling twin completely alters the traveler's judgments of simultaneity and this alteration is key to seeing how relativity provides a consistent account of the effect.

Nevertheless, many still get confused by the twin effect. The
traps they fall into go something like this.

**Question**: If the stay-at-home twin judges the other
twin's clock to slow, does not the principle of relativity require that the
traveling twin see the same thing for the stay-at-home twin? Otherwise, could
we not use the difference to detect the absolute motion of the traveling
twin?

**Answer**: The principle of relativity applies to inertial motion. Only the
stay-at-home twin moves inertially. So the principle of relativity is applied
to that twin only. There is no problem in the traveling twin deciding that he
is moving from the relative slowing of his clock, since what he is really
inferring is that he is accelerating.

**Question**: OK--forget the principle of relativity. Is
there not a symmetry in the situation. Each sees the other moving so if one
sees the other's clock slow, should not both?

**Answer**: There is not a perfect symmetry in the two twins. One moves
inertially; the other accelerates. So there is no basis for expecting
symmetrical effects and we do not get them.

The reason I have gone into such detail on the story of the
twin effect is that it turns out to be especially simple to understand when we
relate it to the geometry of a Minkowski spacetime.

The result that will interest us is one of the most fundamental results of
Euclidean geometry, that is, of the ordinary
geometry of our space. If one has two points in space, which of all possible
curves is the straight line the connects them? The answer is that the
straightest is the shortest.

This shortest curve is
called a "geodesic." That the straight lines are the shortest is
a very familiar fact of experience. If I need to go from one side a
large hall to another quickly, I choose the straight path since that is
the shortest. The figure shows a straight line as the shortest curve
connecting two events A and B. |

There is an analogous notion in
Minkowski spacetime (and in all relativistic spacetimes). Think of all
the timelike trajectories that might represent the motion of some physical
system. How do we distinguish those that are inertial? In the spacetime
diagrams, they are drawn as straight lines since they are straight in several
senses. The one that matters to us here is that they are geodesics analogous to
the geodesics of Euclidean geometry.

In a Euclidean space, every curve has a length. If we drive a
car on some trip, the length of the road we traverse is measured by the car's
odometer. In spacetime there is a similar notion. As you or I traverse some
timelike worldline in spacetime, we carry an instrument that measures the
curve's "length" in spacetime, analogous to the car's odometer. That instrument
is our wristwatch or any other clock we carry with us. The length of a timelike curve in spacetime is just the time
elapsed as read by a co-moving clock.

So now we can say which of all timelike trajectories
connecting two event A and B in a Minkowski spacetime is the inertial
trajectory. It is just the timelike geodesic, where: **Timelike geodesic**:
The timelike curve connecting two events of *greatest* proper time.

The definition is exactly
like that of the geodesic of Euclidean space, except that we
have replaced |

But that fact is just the result of the
twin effect! The stay-at-home twin travels to some event in his future
along an inertial trajectory. The traveling twin follows an accelerated
trajectory along which less proper time elapses.

So we see that the twin effect is as fundamental to the geometry of a Minkowski spacetime as is the simple idea in ordinary geometry that a straight line is the shortest distance between two points.

Special relativity requires us to believe something that at first seems unbelievable: that two inertial observers in relative motion will judge the same speed for the same light signal. We know now that the relativity of simultaneity solves the problem. The two can judge the same speed for light since, through the relativity of simultaneity, they set the clocks used to measure the speed of light differently.

Visualizing just how the relativity of simultaneity enables the light postulate to hold for all inertial observers is not easy as long as we try to picture things in ordinary space. It does become dramatically simpler once we depict them in spacetime and use the simple geometric picture of the relativity of simultaneity that it affords.

To see how this comes about, let us first sharpen the problem by describing the difficulty in a quite concrete case. Once we see how the relativity of simultaneity resolves this one case, others are obviously analogous. Imagine that we have an inertially moving rod and a light signal that bounces back and forth between its two ends.

For a observer at rest on the rod, the light signal will take the same time for the forward and return journey.

Now imagine that we redescribe the process from the perspective of an observer on a nearby planet, who judges the rod to be in uniform motion along its length. That planet observer would judge the light to need more time to traverse the rod in the direction of the rod's motion and less in the direction opposite to rod's motion. | Remember how it goes? When the light signal moves in the direction of the rod, it chases after a fleeing end and needs more time to catch it. When the light signal moves in the opposite direction, its destination is an end that approaches; so it needs less time to reach it. Note that the speed of light is assumed to be the same in both directions. |

If the light postulate is to hold for both inertial observers, somehow both have to be right. The forward the return journal should take the same time for the rod observer; and they should take different times for the planet observer.

Let's now look at the spacetime diagrams for this process.

First here's a spacetime diagram that depicts the rod observer's judgments. In particular,
the hypersurfaces of simultaneity reflect the rod observer's judgments
of simultaneity of events.
The equal spacing of the hypersurfaces reflects the rod observer's judgment that equal times are needed for the forward and return journey of the light signal. More precisely, the events in question are the arrivals of the light signal at either end of the rod. The hypersurfaces reflect how the rod observer associates these events with events simultaneous with them on the rod observer's own world line. The rod observer will then use a single clock carried with the observer to judge the equality of times elapsed between these latter events. |

Here's the spacetime diagram that depicts the planet observer's judgments. In particular, the
hypersurfaces of simultaneity reflect the judgments of simultaneity of
events by the planet observer.
It is clear that a greater time is needed for the light to traverse the rod when the light propagates in the direction of the rod's motion; and less time is needed for the return trip. The speed of light in both directions remains the same. That is
captured by the fact that the lightlike curves in both directions are
at 45 |

How can both views cohere? That becomes apparent
immediately if we now depict how the planet observer judges the rod
observer's hypersurfaces of simultaneity to be spread over the
spacetime.
The planet observer notices that the rod observer's judgments of simultaneity differ from the planet observer's. The difference lies in a tilting of the rod observer's hypersurfaces of simultaneity. Indeed that tilting is precisely what is needed to restore the equality of times for the forward and return trips of the light signal. |

In comparing the last two figures,
the key element to notice is that the rod and bouncing light signal remain the same. All that
changes is the way that the two observers slice up the spacetime into
hypersurfaces of simultaneity. The planet observer's slicing leads to the judgment that the light's forward journey takes longer. The rod observer's slicing leads to the judgment that the light's forward and return journeys take the same time. The animation at left shows the two figures overlayed and that the rod and light signal world lines are the same in both. |

- How the relativity of simultaneity is used to infer that tachyons travel backwards in time for some observers.
- What the twin effect is and why it isn't paradoxical.
- What a timelike geodesic is and how it relates to the twin effect.
- How the relativity of simultaneity enables all inertial observers to judge that light propagates at the same speed c.

Copyright John D. Norton. January 2001, September 2002; July 2006; February 3, 2007; January 23, September 24, 2008; January 21, 2010; February 1, 2012.