(c) 2009 Andisheh Mahdavi
Physics 725: Special and General Relativity
Describe the properties of the line element
= 
+
This would appear to be a combination wormhole + blackhole (a "black worm"). It would appear that the coordinate singularity has been moved from r=2M to 
. As a result, for b=0 we get the Schwarzschild geometry back; and for M=0 we get the wormhole metric back. For b>0, there is no essential singularity at r=0.
For 0<b<2M, something interesting happens. Unlike Schwarzschild, this solution is symmetric between positive and negative r. This means that the geometry looks exactly the same on either side of the black worm. However, for 0<b<2M, there is an event horizon at positive r; therefore, there is an event horizon and negative r as well. Once an observer falls into this kind of black worm, he is trapped forever. However, since there is no singularity at r=0, he will survive the process if he survives the tidal fields of the black worm (very likely if we're dealing with a low density black hole like the one at the center of our Galaxy). This type of black worm is therefore a truly eternal prison.
For b>2M there is no horizon. An observer at rest at r=R falls into the black worm, crosses into the parallel universe, and gets a free ride to a large distance r=-R from the black worm on the other side.
Let's now work out the exact radii of the various circular orbits.
Here's the metric corresponding to this line element:
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Here are the time and azimuthal Killing vectors:
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Let's define the four-velocity and and the u·u=-1 equation to begin with:
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We have two more equations from the Killing vectors; the inner product of each Killing ector with the four-velocity gives us a constant of motion:
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Now we can solve the three equations for the three components of u:
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We'll take the first solution offered, since 
has the correct (negative) sign there. Let's look at the effective potential, which is just given by the total minus the kinetic energy:
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For b=0 this gives us the Schwarzschild effective potential. Here's what it looks like for M=1 and different values of ℓ:
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The above are two contrasting examples ℓ=10 and ℓ=1. For the latter, there is no maximum (unstable circular orbit) or minimum (stable circular orbit); the only choice is to be captured by the black worm. For ℓ=10, scattering orbits are possible, and unstable circular orbits exist.
Let's find all the maxima and minima:
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There are a few interesting things going on here. First of all, there is a minimum at r=0. This means that once inside the black worm, if b>2M, you can have a stable circular orbit at r=0. If b<2M, there is an event horizon and this coordinate system is not valid inside the event horizon.
Other than this, stable circular orbits are at the final solution above, which is 
. Let's plot 
as a function of ℓ and b:
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Just like in Schwarzschild, it would seem that stable circular orbits are possible all the way until a radius given by l = 
. This radius is
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which is the same as the Schwarzschild when b=0. Interestingly, for b>6M, the innermost stable circular orbit becomes the one at r=0.
Now what about the unstable circular orbit? This is given by 
in the Schwarzschild metric, but what about for the black worm? Let's plot it as a function of ℓ and b, together with the location of the event horizon as a function of b, starting with the minimum value of ℓ:
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This is a little bit more complicated. The above plot shows two surfaces: the location of the event horizon, which is at 
, is the lower surface, and the higher surface is the location of the orbit. It seems that there is some sort of circular orbit at this radius, but what sort is it? Let's use the second derivative test to find out:
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It would seem that the second derivative flips sign along some locus of points in this space. When we have a negative second derivative, it shows that we have a maximum, or unstable circular orbit, just like we would in Schwarzschild. But when it's positive, we recover a stable circular orbit---and that turns out to be the same as the stable circular orbit at r=0.
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So, to summarize: there is a stable circular orbit at r=0. For 0<b<2M, this is the orbit of the imprisoned observer. For b>2M, this is simply an orbit at the throat of the wormhole.
Outside r=0, there is always a stable circular orbit. The innermost stable circular orbit outside r=0 is at 
. For b>6M the innermost stable orbit goes all the way down to the throat of the wormhole (r=0).
Finally, there are also unstable circular orbits for ℓ>b 
.
