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John
D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
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In the main text, we saw that it is possible to take a spacetime that has infinite spatial and temporal extent and represent it in a diagram in which the infinities of time, of space and of light propagation appear at finite locations on the diagram. The analogy drawn was to a perspective picture. It uses a perspective transformation to take an infinite space and represent infinite parts of it in a finite diagram.
The perspective transformation was just an analogy to
something familiar. What is the transformation that is used to create
these conformal diagrams? It is--unsurprisingly--a "conformal
transformation." In more precise technical terms, it is a map on
a spacetime that can:
• change the proper spatial and proper temporal separation of events; but
always in a way that
• preserves lightlike connections between events.
That is, if two events are connected by a lightlike curve, they remain
connected that way, no matter how the temporal and spatial separations of
events have changed.
Let us try to get a picture how such a conformal transformation can come about. We are interested specifically in those with the special capacity of being able to bring in the infinities of time and space to finite locations. We will take the easy and familiar case of a Minkowski spacetime. The general ideas, however, apply to fancier spacetimes.
First, we know that we want the transformation to contract things spatially, so that we can bring in events at spatial infinity. To see how such a contraction behaves, consider the timelike geodesics of an inertial frame of reference in a Minkowski spacetime. Here is how a spatial contraction acts on them:
The timelike geodesics are here at equal spatial separations. The contraction reduces the space between them. It is the effect that we want.
So far so good. But there is a problem. The figure also shows lightlike curves. In our figure, we want them to remain as lines drawn at 45 degrees to the vertical. However the squeezing in the spatial direction has the effect of rotating them towards the vertical. They are no longer lines at 45 degrees to the vertical.
We also want to contract things temporally. We want to bring the infinite past and the infinite future to finite locations in the diagram. Now consider spacelike geodesics that belong to the same slicing of spacetime into hypersurfaces of simultaneity. Here is how a temporal contraction acts on them:
The hypersurfaces of simultaneity represented by these
spacelike geodesics are squeezed together temporally. That is the effect
we want. However we have the same problem
with the lightlike curves. They are supposed to remain at 45 degrees to
the vertical. However they are now rotated further away from the vertical.
The two transformations above each work as we wish individually for time and space, but they individually fail to give what we want for lightlike curves. The remedy should be clear. Both transformations should be applied together. If they are coordinated so that the temporal contraction exactly matches the spatial contraction in magnitude, then the lightlike curves will remain at 45 degrees to the vertical.
These coordinated contractions have the property we want:
they preserve the lightlike curves at 45
degrees to the vertical. They implement a "conformal transformation."
The conformal transformations above are a special case. They contract by the same amount everywhere. We, however, need contractions that become greater and greater as we move closer to the infinities of interest.
Take again the case of spatial contractions. If we pick some spatial midpoint, we do not need the spatial contractions to be large in the vicinity of the midpoint. However as we move off spatially towards spatial infinity, the spatial contractions must become larger and larger. Indeed they must become so without limit so that their combined effect is to bring spatial infinity to some finite point on the diagram.
How easy is to have increasing contractions that can bring infinity in to some finite point? It is quite easy. Here is one way. Let us assume that the timelike geodesics of the figure about are separated by unit distances.Then the total distance covered by all of them in one direction from the midpoint is just the sum of infinitely many unit distances:
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + ... = ∞
Here's a simple contraction scheme:
first distance 1 is contracted to 1/2;
second distance 1 is contracted to 1/4;
third distance 1 is contracted to 1/8;
fourth distance 1 is contracted to 1/16;
etc.
After these contractions, the total distance covered to what was infinity is just
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 1
An infinite spatial distance has been contracted to a unit spatial distance.
It is not so easy to represent this sort of contraction in an animation. Here's one attempt.
Since the contractions become more extreme the farther out we go, timelike geodesics from spatially far away are brought into the finite realm of the diagram. The timelike geodesics that were originally distributed all the way out to spatial infinity in two directions are now bunched together at the two sides of the diagram.
This last transformation contracted only space. It was not a conformal transformation, but merely one that contracted space so that points all the way to infinity are brought to finite points in the diagram.
We recover the full conformal transformation sought if we combine this last spatial transformation with a similar temporal transformation that brings the infinite future and the infinite past into finite points in the diagram. As before we coordinate the two transformations so that lightlike curves remain at 45 degrees to the vertical.
The combined effect looks like this:
This transformation takes the infinite set of timelike geodesics and spacelike geodesics of a Minkowski spacetime and compresses them into the conformation diagrams that we have been using in the main text.