Query: concluding scale of a dimension

[Varun Sethi]

Hi,

 

This is an excerpt from a paper I came across:

 

"In the case that the planetary orbits
are stable in more dimensions it is possible that the size of the extra dimensions can be
as large as our solar system, maybe even infinite. However, if planets cannot orbit stably
in a more dimensional world we will have to conclude that the extra dimensions have to be
smaller than astronomical scales."

 

I interpreted this as following:

 

If spatial dimensions are restricted to 2 (no 3rd dimension), length of 3rd spatial dimension is zero. Hence, no system can be a distance more than zero in 3rd dimension. Extending this to 3rd dimension being length L, no system can be a distance more than L in 3rd dimension.

 

However, following their mathematical analysis, which concludes

 

"When plotting this (potential energy) function for different dimensions it shows that the potential only has a
minimum value for (d−1) < 3. For higher dimensions than three there is either no extreme
value (for d = 4) or a maximum, which means that planetary orbits are not stable in higher
dimensions.",

 

is written

 

"They could exist, but the slightest knock would push the planets out of orbit.
The radial position of the planets could not be stable."

 

From last excerpt, I fail to understand, if, what L concluded, i.e., "no system can be a distance more than L in 3rd dimension", should imply system could exist but is unstable (P.E. function has no minimum), or if it could never exist. Also is the interpretation I arrived at correct?

 

Regards,

Varun