What does this spacetime look like?
How can we draw a picture of this spacetime? Forget time, let's just try to
capture the space part. One possible picture is a disk missing a wedge of angle
8 Pi G M , with an added rule that the inhabitants of the spacetime cross the
missing wedge as if the two edges were joined smoothly together. In this manner,
we guarantee that any inhabitant will measure the distance of 2 Pi (1 - 4 G M) R
for a circular path of radius R around the central mass M. Here it is below,
with the missing wedge set to Pi/2
In[48]:=
Show[Graphics[Disk[{0,0},3,{-Pi/4, Pi/4}]],
AspectRatio -> Automatic, Axes -> True,
PlotRange -> {{-3,3},{-3,3}}]
Out[49]=
-Graphics-
What would happen if we literally sewed those two edges together? It would look
like a cone. The way to make this cone mathematically is to embed the cone in
regular three-dimensional flat space and decompose the direction that was called
"R" into two new directions, called a and z, so that R^2 = a^2 + z^2.
Let's require that the coordinate "a" measure the circumference of a
closed path around the mass M, so that 2 Pi a = 2 Pi (1-4GM) R.
So we know that a = (1 - 4GM) R.
If we tr to parametrize z = R Cos[fi0] and a = R Sin[fi0] for some angular
parameter fi0, then Sin[fi0] = (1 - 4 G M). But fi0 is also half of the apex
angle of the cone determined by z[a] = a Cot[fi0] in a cylindrical coordinate
system with coordinates {a, theta, z}. This is shown in the plot below where the
apex angle is fi0 = ArcSin[3/4]:
In[50]:=
Plot3D[-ArcSin[3./4] Sqrt[x^2 + y^2],{x,-3,3},{y,-3,3}]
Out[51]=
-SurfaceGraphics-
Here's another way to plot the same surface, to make it look more like a cone:
In[52]:=
ParametricPlot3D[{a Cos[theta], a Sin[theta],
-ArcSin[3./4] a},{a,0,3},{theta,0,2 Pi}]
Out[53]=
-Graphics3D-
This spacetime was first described and generalized by three physicists named
Deser, Jackiw and 't Hooft. Their calculation was more complicated and
generalized to several masses, and a point spin. Look up their paper and others
on the subject of three-dimensional gravity in the Spires high energy physics
database at the Stanford Linear Accelerator Center (SLAC).
Up to What is the spacetime geometry?
|