Properties of body with spherical symmetry

[Soren4]

I'm studing Gauss law for gravitational field flux for a mass that has spherical symmetry.

Maybe it is an obvious question but what are exactly the propreties of a spherical simmetric body?

A body is said to have spherical symmetry if its density is function of the distance from the center only, and not of the angle coordinate. $$\rho=\rho(r)$$

Firstly does this imply that the body in question must be a sphere?

Secondly is it correct to interpret the definition as follows?

For any element of the body of mass [itex]dm[/itex] and volume [itex]dV[/itex] at a distance [itex]r[/itex] from the center of the body, there exists another identical element [itex]dm[/itex], [itex]dV[/itex] at the same distance[itex]r[/itex from the center of the body.

 

[rumborak]

First question of yours, I think the fact that a spherically symmetrical body has to be a sphere, is reasonably self-evident. Spherical symmetry means that you can rotate the object any way you like, and it will look indistinguishable from the non-rotated version. If you can find an object that fulfills that condition but is not a sphere, you might get a Nobel prize.

Your second one, no, I don't think the condition is strong enough. I could have a very lopsided object that still fulfills your condition but is not spherically symmetrical.

 

[Orodruin]

Firstly does this imply that the body in question must be a sphere?

No, but it is built out of homogeneous spherical shells.

For any element of the body of mass dm and volume dV at a distance r from the center of the body, there exists another identical element dmdmdm, dV at the same distance[itex]r[/itex] from the center of the body.

It means that at every point at the same distance from the center, the density is the same. It is difficult to tell whether that is exactly what you want to say with the above or not.

 

[Orodruin]

First question of yours, I think the fact that a spherically symmetrical body has to be a sphere, is reasonably self-evident. Spherical symmetry means that you can rotate the object any way you like, and it will look indistinguishable from the non-rotated version. If you can find an object that fulfills that condition but is not a sphere, you might get a Nobel prize.

This is not correct. A sphere is an object with one dimension less than the embedding space. If you are not thinking of this you are using a non-standard or colloquial definition of the word "sphere".

 

[rumborak]

This is not correct. A sphere is an object with one dimension less than the embedding space

What? I have never seen a definition of that kind, and it also doesn't make a lot of sense to me. A plane has one less dimension than the space it is embedded in. Is a plane a sphere?

 

[Orodruin]

What? I have never seen a definition of that kind, and it also doesn't make a lot of sense to me. A plane has one less dimension than the space it is embedded in. Is a plane a sphere?

No, you are misinterpreting things here. I just mentioned one of the properties of a sphere. If you want the interior part too you are talking about a ball. In three dimensions, a sphere of radius ##R## is a two-dimensional surface at a distance ##R## away from its center.

 

[rumborak]

I understand that when considered rigorously in mathematics, this distinction between sphere and ball is important, but at least from the OP wording it seemed he meant ball when saying sphere. Harping on this distinction seems to distract from the question he was asking, IMHO.

 

[Orodruin]

If you can find an object that fulfills that condition but is not a sphere, you might get a Nobel prize.

Harping on this distinction seems to distract from the question he was asking, IMHO.

I do not think it is a trivial matter when you make statements like the first one quoted. In particular not when the OP has received two seemingly incompatible answers - this should be explained.

 

[secur]

Orodruin, the distinction between sphere and ball appears only in math, in topics of solid geometry, topology, and so forth. In physics "sphere" is commonly used to include ball, "hollow sphere" or "spherical shell" being used when necessary. Only if you're a physicist working with math like algebraic topology, must you use the words that way. Note there's no such thing as "ballical symmetry"! Dirac, for instance, uses only the word "sphere", never "ball", in "Principles of QM".

rumborak, one object that fulfills the condition but wouldn't be called a "sphere" would be a uniform dust filling a 4-d spacetime, as in many cosmological models. You might even say a gas giant planet, while spherically symmetric, is "not a sphere" since there's no definable surface. But this is just nit-picking.

Relativistically there's a little problem with this definition: you can't in general define "center" such that all observers will agree.

It may be worth mentioning the related fact that in Quantum Mechanics a spherically symmetric state - i.e. with s.s. wave function - must have angular momentum zero.

 

[Soren4]

No, but it is built out of homogeneous spherical shells.
It means that at every point at the same distance from the center, the density is the same. It is difficult to tell whether that is exactly what you want to say with the above or not.

Thanks for the reply! When you say "homogeneous shell" what do you mean exactly? Is the volumetric density or area density of the shell constant?