|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.
|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
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.|
|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 is 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.
To have a light cone, we do not need light to be present. The cones map out the trajectories light would take if light were to be present. Since it is just the possibilities that are mapped out, not necessarily the trajectories of actual light. Spacetime still has a light cone structure in the dark!
To describe the light cone just now, we picked an event in spactime and imagined all the possible trajectories that light could take in propagating through that event. We could have picked any event in the spacetime. We would have found lightcones at every one of them. That means that spacetime is completely filled with light cones. There is one at every event.
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:
When we add the extra dimension of time to a space, we produce a
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 later.
Knowing the light cone structure of a spactime tells us what connects with what. Its importance is comarable to what is represented on ordinary maps of countries. Here is a map of ancient Greece:
Greece at the beginning of the Peloponnesian war. p. 17 in W. S. Shepherd, Historical Atlas, New York: Henry Holt & Co., 1911.
We read from the map that, if we are in Greece, we can travel North and eventually arrive in Thrace. However we cannot get to Crete; or at least we cannot get to Crete by land. We would need to cross a sea.
|The light cone structure catalogs analogous "what connects with what" information for a spacetime. It reveals its most basic geography.
To see how this works, pick any event "O" in the spacetime. The future light cone at O contains all the events in the spacetime that can be reached from O by future directed timelike or lightlike curves. If we make the usual assumption that all causal processes propagate at or less than the speed of light, we conclude that these are all the events that we can causally affect from O.
More simply, if you are at O, the events in the forward light cone are just those events that you can reach with a physical signal, such as an ordinary particle or a light flash that you may emit.
|For that same event O, the past light cone contains all the events in spacetime from which one can reach event O by future directed timelike or lightlike curves.
Assuming again that all causal processes propagate at less than or equal to the speed of light, we conclude that this past light cone contains all the events that can causally affect event O.
More simply, if you are at any event in the past light cone of O, you can always send to O a physical signal consisting of an ordinary particle or a light flash.
|The remaining region of the spacetime is outside both past and future light cones. It is a new sort of region that does not appear in pre-relativistic spacetimes. It is an "elsewhere" region.
It collects all events that cannot be connected to event O by timelike or lightlike curves. Its events can only be connected to O by spacelike curves. That is, its events are "spacelike separated" from O.
If we assume that no causal processes propagate faster than light, these events are causally disconnected from O. If we are at O we cannot causally affect or be causally affected by an occurrence at an event in this "elsewhere" region. Correspondingly, we cannot exchange signals between the event at O and any spacelike separated event in this region.
There is no corresponding region in a pre-relativistic spacetime. In Newtonian theory, it is assumed that there are propagations that are arbitrarily fast and even instantaneous. An example of an instantaneus propagation is changes in the Newtonian gravitational field. If the sun were to disappear, we would know instantly on earth, according to Newtonian theory, for the sun would no longer exert a gravitational pull on us.
Copyright John D. Norton. January 2001, September 2002; July 2006; February 3, 2007; January 23, September 24, 2008; February 3, 2012.