What would happen if the force of gravitation suddenly starts varying as $1$$/$$r^3$ instead of $1/r^2$ ? Would the symmetry of universe now seen be disrupted?
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10$\begingroup$ If you're asking physics questions to satisfy your own curiosity at 16 years old, you don't need to apologize for how it sounds. Good luck on your continuing education. $\endgroup$– AdamRedwineOct 18, 2012 at 11:53
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2$\begingroup$ Not a direct answer but might be of interest. There are three cases of a central power law where all the orbits can be written down explicitly using elementary functions---inverse square, direct proportionality (Hooke's law) and the $\frac 1{r^3}$ of your question. In the latter case, the orbits are the so-called Cotes' spirals about which the internet is full of information. By the way, all three were known to Newton and are dealt with in his "Principia". $\endgroup$– mathuser4891Jun 18, 2013 at 9:58
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2 Answers
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It depends on what you mean by symmetry, But the force law $1/r^3$ does not have stable orbits. Small perturbations will de stabilize your orbits to fall inward or outward.
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6$\begingroup$ As addition to this answer: http://en.wikipedia.org/wiki/Bertrand%27s_theorem. $\endgroup$ Oct 18, 2012 at 11:45
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2$\begingroup$ Also the iron sphere theorem would not hold as it relies on area of a sphere increasing with $r^2$. Relativity gets around this by changing the geometry. $\endgroup$ Oct 18, 2012 at 15:54
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$\begingroup$ @NickKidman Wow! this and Bertrand's theorem are amazing! Thanks for the links. See also Dimension10's point 2 below and the Wiki page linked therein. $\endgroup$ Aug 29, 2013 at 5:04
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Such a change requires a 4+1-dimensional spacetime instead of a 3+1-dimensional one -- this would have several serious implications --
The Riemann curvature tensor gains new "parts" with interesting physical implications with each new spacetime dimension -- 1-dimensional manifolds have no curvature in this sense, 2-dimensional manifolds have a scalar curvature, 3-dimensional manifolds gain the full Ricci tensor, 4-dimensional manifolds get components corresponding to a new Weyl tensor and 5-dimensional geometry gets even more components, and general relativity in this spacetime is capable of explaining electromagnetism, too, so electromagnetism (along with the radion field) starts behaving as a part of gravity.
Apparently a 5-dimensional spacetime is unstable, according to wikipedia's "privileged character of 3+1-dimensional spacetime"[1] (now a transclusion of [2]).
The string theory landscape would be a bit smaller, since there are less dimensions to compactify.
The Ricci curvature in a vacuum on an Einstein Manifold would no longer be exactly $\Lambda g_{ab}$. There will be a coefficient of 2/3.
The magnetic field, among other things "cross product-ish", could not be written as a vector, unlike the electric field. This is because it would have 6 components whereas the spatial dimension is only 4. So, perhaps humans would become familiar with exterior algebras earlier than us who live in 3+1 dimensions. Either that or we would be trying to find out how magnetism works. Or we would just die out, for all the other reasons.
In string theory (see e.g. [3]), gravitational constants in successively higher dimensions are calculated as $G_{n+1}=l_sG_n$, where $l_s$ is the string length (the units must be different in order to accomodate the extra factor of $r$ in Newton's gravitational law). For distance scales greater than the string length, this causes gravity to be much weaker than in our number of dimensions, but stronger for length scales shorter than the string length. It's interesting how gravity's long-range ability peaks at 4 dimensions (it is a contact force below 4 dimensions).
See also some recent tests of the inverse square law at short length scales (to check for compactification -- [4].
answered Jun 17, 2013 at 10:12
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3$\begingroup$ Minor comment to the answer (v2): In the future please link to abstract pages rather than pdf files, e.g., arxiv.org/abs/hep-ph/0011014 $\endgroup$– Qmechanic ♦Jun 17, 2013 at 11:42
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$\begingroup$ The weak gravity is probably the killer. That would mean the Universe would be nothing but an expanding cloud of gas. So no 4D people to play Rubik's Tesseracts :( $\endgroup$ Feb 21, 2015 at 10:35