# Got a question

Is there any way to construct a fundamentally different physics system for the fourth dimension where gravity diminishes with an inverse square law instead of inverse cube law?

This was originally alkalines question

dextercioby
Homework Helper
Hold on.What "inverse cube law" are u referring to...?

Daniel.

3D = inverse square law
4D+ = inverse cube law

thus planets cannot form in 4D+ universes

dextercioby
Homework Helper
Is this (4D) a result of GTR...?What metric did they use...?Care to make a referece...?

Daniel.

dextercioby
Homework Helper
Interesting.I don't recall coming across the referred calculations in a GR book.Perhaps i didn't know where to look for.Let's hope that here,in GenPhys one of our experts in GR will provide more info on the calculations themselves (my interest) and the interpretation (yours).

Daniel.

pervect
Staff Emeritus
I'd expect gravity to drop off as an inverse cube in 4-d space based on the Newtonian limit. Basically, just Gauss's law - the force*area goes to a constant at large distances.

Something like MOND could change that, I suppose. I'm only vaguely familiar with MOND, so I looked it up. Based on the following

http://motls.blogspot.com/2004/10/mond-and-holography.html

The MOND theories, roughly speaking, say that the Newtonian laws are modified in such a way that the inverse square law (1/r^2) for gravity is continously replaced by the inverse distance law (1/r) for all objects whose acceleration is smaller than the critical value a_0.

It would appear to be possible for some sort of MOND-like theory to change the 1/r^3 force of gravity to something that allows stable orbits (anything slower than 1/r^3 will do) in a 4D universe. But the critical accleration value would have to be very large to allow stable planetary orbits (MOND for instance switches from 1/r^2 to 1/r only when distances are on a cosmological scale - Newtonian gravity works fine for planetary orbits. )

This would probably present some problems, becuase the critical acceleration is closely related to 1/H in current versions of MOND.

But if you're looking for something vaguely plausible (but highly unlikely), some sort of MOND-like gravity might work.

Thanks, what do you mean by large acceleration values?

And...so it is possible

pervect
Staff Emeritus
Gold Barz said:
Thanks, what do you mean by large acceleration values?

MOND in 3space+1time has a critical acceleration in which gravity changes from a 1/r^2 force to a 1/r force. (See the link I quoted). I was speculating that there is some 4space+1time version of MOND in which gravity changes from a 1/r^3 force to a 1/r^2 force below the "critical" acceleration. One problem with this idea is that it would be hard to make this work for planetary orbits with any value of the critical acceleration parameter that's similar to all to that in our universe.

I would stop well short of claiming that it's definitely possible to do what you want - just that I had this wild idea based on some promising but non-standard theories of gravity that might sort-of do what you want.

But the problem with the whole inverse cube law thing is that they got the inverse cube law just by adding an additional dimension but applying the same geometry as before, wouldnt you have to do more than that? because 3D geometry and 4D geometry are different from each other

pervect
Staff Emeritus
Basically, the inverse cube law for gravity in 4-d space comes from the gravitational analog to Gauss's law.

See for instance http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html
for more on Gauss's law - which is a law about electric charges.

While Gauss's law is a law about electric charges, Newtonian gravity also satisfies Gauss's law. General Relativity is more complicated than Newtonian gravity, but reduces to Newtonian gravity in the weak field limit. Thus we have GR satisfying Gauss's law in the weak field / low velocity etc. limit where it is the same as Newtonian gravity.

So in order to do what you want, you need gravity that does not satisfy Gauss's law, which means it can't be Newtonian in the weak field limit - the real problem isn't with Einstein, it's with Newton.

So theres no way around it?

plus wouldnt the consenquences of adding an additional spatial dimension be emerging constants and effects we do not know of and cannot know

pervect
Staff Emeritus
You need something that modifies Newtonian gravity to get around your problem. MOND, which I mentioned, stands for "Modified Newtonian Dynamics". So it does modify Newton. It turns out that MOND even modifies things in the right direction. (Apparently, anyway - that's something I had to look up).

But in the form that's applicable to our universe, MOND is not nearly a strong enough modification to affect planetary orbits, it only shows up (in theory) over a much larger scale. And it's still quite a controversial idea - GR is still the standard theory of gravity.

Whether or not a form of MOND can be made to do what you want is somethign that I'm not familiar enough with the theory to answer.

Aside from modifying Gauss's law, it might be possible to make garavity anisotropic. Think of a gravity-like force that for some reason acted only in a specific plane in 3d space, and you'll get the basic idea. The 4-d equivalent would be a force that acted only act in a 3-d hyperplane. This idea is also pretty far out.

So there is a way around the inverse cube law in a 4-D universe?

I mean in a 4-D universe SHOULD there be the inverse cube law? but adding an additional spatial dimension changes things and probably change the extrapolation of inverse square law to the inverse cube law, im guessing the laws would either change up or be tweaked