| HPS 0410 | Einstein for Everyone |
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John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
So far all our discussion in special relativity has involved the motion of bodies in space over time. If you haven't already noticed, these motions can become rather complicated to visualize. Recall how tough it is to keep track of what the different ends of a moving rod are doing as a light signal bounces back and forth between them.
In 1907 the mathematician Hermann Minkowski explored a way of visualizing these processes that proved to be especially well suited to disentangling relativistic effects. This was their representation in spacetime. Quite puzzling relativistic effects could be comprehended with ease within the spacetime representation and work in the theory of relativity started to be transformed into work on the geometry of spacetime.
We build a spacetime by taking instantaneous snapshots of space at successive instants of time and stacking them up. It is easiest to imagine this if we start with a two dimensional space. The snapshots taken at different times are then stacked up to give us a three dimensional spacetime. In this spacetime, a small body at rest will be represented by a vertical line. To see why it is vertical, recall that it has to intersect each instantaneous space at the same spot. A vertical line will do this. If it is moving, it will intersect each instantaneous space at a different spot; a moving body is presented by a line inclined to the vertical.
A standard convention (that I will usually use) is to represent trajectories of light signals by lines at 45o to the vertical.
In the figure, a moving rod is represented by the
trajectories in spacetime of its ends. The zig-zag line is a light signal
bouncing back and forth between these two ends.
Here's another example. Take snapshots of the earth orbiting the sun in the three dimensional space around the sun in the course of a year, which will look like:
| Now we stack them up into the third dimension. |
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When we clean things up a
little, we have a spacetime. |
So far we have described how a two dimensional space is combined with the one extra dimension of time to generate a three dimensional spacetime, such as shown above in the figures. Our space is three dimensional. So when we add the extra dimension of time we generate a four dimensional spacetime.
There is no easy way to draw a picture of a four
dimensional spacetime. Visualizing it can be very
hard. But that does not make it mysterious. It is just another sort of
space that happens to transcend simple visualization. In physics, four
dimensions are actually quite modest. In statistical mechanics, we routinely
deal with phase spaces of 6 x number of molecules in a gas sample. For even
small samples of gas, that can come to 1025--a space with
10000000000000000000000000 dimensions. So we should not be too awed by a
mathematical space with only four dimensions!
That the speed of light is a constant is one of the most important facts about space and time in special relativity. That fact gets expressed geometrically in spacetime geometry through the existence of light cones, or, as it is sometimes said, the "light cone structure" of spacetime.
To see that structure, we imagine an event at which there is an explosion. Light will propagate out from it in an expanding spherical shell. In a two dimensional space, it will look like an expanding circle, as shown below.
| To see that structure, we imagine an event at which there is an explosion. Light will propagate out from it in an expanding spherical shell. In a two dimensional space, it will look like an expanding circle. |  |
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An animations makes the motion more visible. |
Now stack up these spatial snapshots to make a spacetime. The spacetime diagram that corresponds to it looks like a cone. As we proceed up the cone, we look in each instantaneous space to see how far the light has propagated. Each intersection of the cone with the space will be a circle.
In the figure, the expanding circle of light is represented by the top half of the cone. It is customary to draw in the bottom half of the cone, although it not part of the expansion of the light. In fact it represents the opposite. It depicts a circle of light collapsing in towards the original event at the apex of the cone. Here is that collapse, presented also as an animation.
A final animation now shows the association between the different stages of the collapsing and expanding light shell and the cross-sections of the light cone.
There is much potential for confusion in talking about
spacetimes. As a result a fairly precise vocabulary has been built up and it
is important to to use it correctly. Pay
attention to the following terms:
| Spacetime
When we add the extra dimension of time to a space, we produce a
spacetime. Minkowski spacetime There is nothing special about a spacetime. They can arise in classical physics. So if we mean a spacetime that also behaves the way special relativity demands, then we have a Minkowski spacetime. (Note for later: when we look at general relativity, we will meet spacetimes that are relativistic but not Minkowski spacetimes.) Event These are the individual points of a spacetime. They represent points in space at a particular time. Timelike Worldline This is the trajectory of a point moving less than the speed of light. These curves are contained within the light cone. They represent the trajectories of massive particles. Lightlike curve This is the trajectory of a point moving at the speed of light--a light signal. They lie on the surface of the light cone. Spacelike curve This is a curve that lies outside the light cone. If an object is to make this curve its trajectory, it would need to travel faster than light. Spacelike hypersurfaces These are the instantaneous spatial snapshots of spacetime. They are three dimensional in the case of a four dimensional spacetime. Past and future light cones All the lightlike curves through an event form the light cone at that event. The part of the cone to the future of that event is the future light cone. The part to the past is the past light cone. Light cone structure Since the speed of light is generally taken to be the fastest that causes can propagate their effects, once we know how the light cones are distributed in space we can say a great deal about what is possible and impossible causally in the spacetime. So this distribution is of great interest to us. It is called the light cone structure of the space. Timelike geodesic This term will be defined below. |
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It is quite demanding to try to visualize the relativity of simultaneity in the ordinary way of developing special relativity. It turns out to be very easy in the spacetime context. The judgments of simultaneity of each inertial observer correspond to slicing the spacetime up into a stack of spaces with each space formed from a set of simultaneous events. The relativity of simultaneity simply tells us that observers in relative motion slice the spacetime up differently. Here are three observers in relative motion in a spacetime.
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First we have an observer whose worldline runs vertically up the page. |
| The next observer moves to the right with respect to the first. |  |
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The third observer moves to the left with respect
to the first. |
Notice how differently they slice up the spacetime into spaces of simultaneous events. That difference simply is the relativity of simultaneity. It is expressed in the tilting of the hypersurfaces of simultaneity as we move the judgments of simultaneity of events from inertial observer to inertial observer.
| Two points to watch
when you are drawing this tilting of hypersurfaces.
First, setting an observer into motion to the right will tilt the observer's world line to the right; and the hypersurface of simultaneity will also tilt up on the right side to meet it. Second, if one follows the usual convention of drawing light lines at 45o, then the angle of the observer's worldline to the vertical will be the same as the angle of the hypersurface of simultaneity to the horizontal. |
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The tilting of the hypersurfaces gives us a simple picture of how inertial observers in relative motion assign times to events.
| An inertial observer carries a clock that marks
the time of events along the observer's worldline as "1," "2," "3,"
... That settles the time of events only on the worldline for the
observer. What time should be assigned to events not on the observer's worldline?
The observer's hypersurfaces of simultaneity answer. Consider the hypersurface that passes through the event of the clock showing "1." All these events are simultaneous in the judgement of the observer. Therefore all these events are assigned time "1. The same applies for the remaining hypersurfaces that pass through the events of the clock ticking "2" and "3." All the events on those hypersurfaces are assigned times "2" and "3," respectively. |
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The same analysis
obtains for a new inertial observer who moves relative to the
original observer. The new observer's clock assigns times to events
on the observer's world line. The observer's hypersurfaces of
simultaneity are then used to propagate the times throughout the
spacetimes. Clearly the original and new observer will differ on the times each assigns to the same event in almost every case. Is there a sense in which one is assigning times correctly and the other not? There cannot be. The prinicple of relativity requires each observer's frame to be equivalent. If the procedure is good in one inertial frame, then it is equally good in all. This reminds us once again that there is no frame independent notion of simultaneity in a Minkowski spacetime. |
Just why is it that hypersurfaces of simultaneity tilt when we change the state of motion of the observer or reference frame? A simple construction shows how it comes about.
| Imagine that some inertial observer wants to
determine which events in spacetime are simultaneous with some event
O on the observer's world line. The simple way to do it is with light
signals. Following Einstein's original idea of
1905, the observer sends out light signals, reflects them off
positions in space and then notes when they return. In the figure opposite, there is a light signal leaving the observer's worldline, reflecting at event A and then arriving back at the observer's worldline. Since the event O is exactly midway in time between the departure and arrival events, the observer judges event A to be simultaneous with O. Also it is obvious that light signals reflected at A' and A must arrive back at the observer at the same time, since they departed the observer at the same time. It is just symmetry. |
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This same reasoning applies to all the remaining events shown in the figure: B', C', B and C. In each case, there are arrival and departure events at the observer's worldline for light signals that are reflected at B', C', B and C. The event O is midway between the arrival and departure events in each case. Therefore, the observer judges each of B', C', B and C to be simultaneous with O. The totality of these events will form a flat plane perpendicular to the observer's worldline.
Now let us consider a second case in which a new inertial observer moves relative to the original inertial observer. The new observer's worldline will be drawn as a tilted line. Which events will that observer judge as simultaneous with an event O on that observer's worldline? Although the worldline is now tilted, the same procedure just described can also be used to pick out the events simultaneous with O. Indeed the principle of relativity requires this, for otherwise we would have some intrinsic difference between the first inertial frame and the second; only in the first could this procedure be used.
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The construction proceeds as before and it is
drawn for you at left. As long as we adhere to the light postulate and draw our lightlike curves
at 45 degrees to the vertical, we will end up plotting out events A',
B', C', A, B and C that lie on a tilted hypersurface. This happens because the departure and arrival events for the light have been displaced to the left and right respectively. We now need to locate the bends in the lightlike curves--the reflection events--in such a way that, as before, the light signals from A and A' return at the same arrival event (and so on for B and B' and for C and C'). That can only happen if we displace the reflection events into the tilted hypersurface shown. If you are having any trouble seeing this, the simplest remedy is to draw the figure for yourself, being careful to keep all light signals propagating along lines that are at 45 degrees to the vertical. |
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. |
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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 45o to the vertical. |
| 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. |
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Just what is so special about a Minkowski spacetime? One might think that is it the idea of representing space and time together in a four dimensional geometry, where the four dimensionality of the geometry outstrips our immediate powers of visualization. It is certainly the case that the four dimensionality if both interesting and hard to visualize. But there is nothing inherently relativistic about it. One can take all the physics of Newton and re-express it in four dimensional terms.
The big difference between Newtonian and relativistic spacetimes lies in how they are sliced up into three dimensional spaces. That slicing is done by picking out sets of simultaneous events to form three dimensional spaces.
In Newtonian spacetimes, there is only one way to do this, so a Newtonian spacetime unstacks into a unique set of spaces. In this sense, space and time remain distinct even if we represent the physics in a spacetime.
In a relativistic (i.e. Minkowski) spacetime, the relativity of simultaneity tells us that there are many ways to do this; there is no unique, preferred unstacking. In this sense, space and time get fused together and this fusion is the real novelty of the spacetime approach in relativity theory.
This novelty is surely what Hermann Minkowski had in mind when he wrote in the introduction to his famous lecture "Space and Time" of 1908
"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics and therein lies their strength. They are radical. Henceforth space by itself and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."
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 an 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. |  |
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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. |  |
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.
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; 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. So A's tachyonic signal is sent. We have a contradiction: A's tachyonic signal is sent if and only if it 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.
One of the most enduring topics of discussion in relativity
theory concerns one of the simplest effects that arises in special
relativity. We know that a rapidly moving clock slows. This effect can be
incorporated into 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 his 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, he will have
aged significantly less. If the traveling twin had maintained a speed of say
99.5% the speed of light, his internal clock would slow to 10% of the normal
rate. Let us say the traveling twin returns, after what the stay at home twin
says is one year. The traveling twin will experience merely the passing of
365/10 = 36.5 days.
All this is clear. An enormous literature has emerged, however, from failed attempts to redescribe clearly the process from the perspective of the traveling twin. They typically yield contradictory results. Out of these confusions has emerged the idea that the effect is somehow paradoxical. So it is often mistakenly called the "twin paradox."
Now relativity theory does allow us to construct a perfectly good account of how this process appears from the traveling twin's point of view. What makes it a little messy is that the traveling twin's motion is accelerated. That means that at different times in the journey, the traveling twin will invoke different judgements of simultaneity. However if one incorporates them carefully into the analysis the traveling twin will determine that he must age less. |
The key thing is that the traveling twin must change his state of inertial motion when it comes to returning home. Up to that turn around, the traveling twin will use his judgments of simultaneity to arrive at the conclusion that the stay at home twin has aged less. However, after turning around and changing his state of inertial motion, the traveling twin will slice up the spacetime into a quite different set of hypersurfaces of simultaneity. That results in a dramatic shift in his judgment of which event in the stay at home twin's life is simultaneous with the traveling twin's turning around. That shift leaves the traveling twin judging that the stay at home twin has actually aged much more than the traveling twin. That judgment persists throughout the traveling twin's return journey. It is a helpful exercise to draw the spacetime diagram that displays this big change in the traveling twin's judgments of simultaneity. Try it! |
The traps people fall into go something like this.
Question: If the stay at home twin sees the other
twin slow, does not the principle of relativity require that the traveling
twin see the same thing of the other 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 him only. There is no problem in the traveling twin deciding that
he is moving, 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. |
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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.
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The definition is exactly
like that of the geodesic of Euclidean space, except that we
have replaced shortest spatial length by greatest
proper time. It tells us that we proceed from event A to event B with
greatest elapsed time if we follow an inertial trajectory. |
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.
We can also use spacetime diagrams to give us a much simpler, geometric picture of how it is possible that two moving observers can each judge the other's clocks to have slowed. The set-up employs just half of the twin effect above. We imagine that each twin sets off at exactly the same speed but in opposite directions.
Each twin now judges the rate of the other's clock. (How this is done is a detail we don't fuss with. They might use light signals, for example, and correct for the time of flight of the signals to figure what the other twin's clock was reading at each instant.)
The figure shows the essential result. Each twin will use different judgments of the simultaneity of distant events to determine the reading on the other's clock. (The clock readings are the numbers next to each twin's worldline.) Those differing judgments of simultaneity are represented by the hypersurfaces of simultaneity of each twin. And it is clear that each will say that the reading of a later event on their clock (e.g. a "4") coincides with an earlier time reading (e.g. a "3") on the other twin's clock. Thus each infers the other twin's clock is running slow. Of course, an observer on earth would judge that the rates of both twin's clocks to be the same.
Copyright John D. Norton. January 2001, September 2002; July 2006; February 3, 2007.