Wormhole proper radial distance

In summary, The authors of the article "Wormholes in spacetime and their use for interstellar travel a tool for teaching general relativity" state that in the general case, the radial coordinate r is ill behaved near the throat, but the proper radial distance l(r) must be well behaved everywhere. This requires that l(r) is finite throughout spacetime, which also implies that 1-b(r)/r must be greater than or equal to 0 throughout spacetime. This is important because if this quantity is zero, the proper radial distance would not be finite, but it does not necessarily mean that the derivative of the distance is infinite.
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PLuz
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I've been reading the excellent article from Morris and Thorne: "Wormholes in spacetime and their use for interstellar travel a tool for teaching general relativity" and there's something I don't quite get it. In the ninth page the authors state:
"[...] in the general case, the radial coordinate [itex]r[/itex] is ill behaved near the throat; but the proper radial distance [tex]l(r)=\pm \int_{b_0}^{r}\frac{dr'}{(1-b(r')/r')^{\frac{1}{2}}}[/tex] must be well behaved everywhere;i.e., we must require that [itex]l(r)[/itex] is finite throughout spacetime, which also implies that [tex]1-\frac{b(r)}{r}\geqslant 0[/tex] throughout spacetime."

After this very long transcript. My actual question is: why the greater or equal in the last expression? If the quantity is zero the proper radial distance isn't finite...
 
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PLuz said:
My actual question is: why the greater and equal in the last expression? If the quantity is zero the proper radial distance isn't finite...
No, it would mean the derivative of the proper radial distance dl(r)/dr is infinite. There are plenty examples of functions that remain finite even though their derivative is infinite.
 
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1. What is a wormhole proper radial distance?

A wormhole proper radial distance is the measure of the shortest distance between the entry mouth and exit mouth of a wormhole. It is also known as the throat radius and is a crucial factor in determining the size and stability of a wormhole.

2. How is wormhole proper radial distance calculated?

Wormhole proper radial distance is calculated using the Schwarzschild radius formula, which takes into account the mass and energy of the object creating the wormhole. It can also be calculated using the Morris-Thorne metric, which considers the shape and curvature of the wormhole.

3. What is the significance of wormhole proper radial distance?

The proper radial distance of a wormhole determines its size and stability. A smaller proper radial distance indicates a more stable and traversable wormhole, while a larger distance may lead to instability and collapse.

4. Can wormhole proper radial distance be manipulated?

There is currently no known way to manipulate the proper radial distance of a wormhole. However, some theories suggest that advanced technologies or exotic matter may be able to alter the properties of a wormhole, including its proper radial distance.

5. How does wormhole proper radial distance affect time travel?

The proper radial distance of a wormhole has a direct impact on the potential for time travel. A traversable wormhole with a proper radial distance of less than 1.5 times its length could allow for time travel, while a distance greater than 1.5 may not permit it.

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