Isotropy and homogeneity

In summary: Is this the case in your 3-sphere example?In summary, the conversation discusses an exercise that asks to show why a connected 3-space cannot be isotropic about 2 distinct points without being homogeneous. The exercise is questioned by one person who suggests a counterexample using the 3-dimensional sphere, where the matter distribution can be radially symmetric around two antipodal points. However, the matter density does not have to be homogeneous and can increase with distance from these points. Another person suggests trying to come up with an explicit expression for a scalar field on the 3-sphere to prove the exercise right or the counterexample wrong.
  • #1
lark
163
0
http://camoo.freeshell.org/27.16wrong.pdf" [Broken]

Mistake by the author?

Laura

Latex source below for quoting purposes but the .pdf may've been edited since then.

Exercise 27.16 asks you to show why a connected 3-space can't be
isotropic about 2 distinct points without being homogeneous.

Counterexample, though. Suppose the space is $S^3$, the
3-dimensional sphere. You could think of it as the equation for
$x^2+y^2+z^2+w^2=1$.

Then let the 2 separate points be antipodal points on the sphere. For
example $x=(1,0,0,0)$ and $-x=(-1,0,0,0)$.

You could have a matter distribution that was radially symmetric
around both of these points, because a rotation around x is also a
rotation around $-x$! But it doesn't have to be homogeneous. The
matter density could go up with distance from x or $-x$, up to the
"equator" $y^2+z^2+w^2=1$.

Am I missing something, or is this exercise just wrong?
\end{document}
 
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  • #2
However if the universe were isotropic about 3 separate points, then it has to be homogeneous!

Laura
 
  • #3
lark said:
However if the universe were isotropic about 3 separate points, then it has to be homogeneous!

Laura

Do you know a proof for this?

Christian
 
  • #4
One trivial way is to constrain the space to be flat, but that's not what your getting at here :wink:. I'm still trying to visualize your example but failing. I can see a 2-sphere being isotropic around two poles and not homogeneous, but I'm having trouble with the 3-sphere. Maybe it would be fruitful to try to come up with an explicit expression for a scalar field on the 3-sphere that would satisfy isotropy around two points without being homogeneous and see if you can do it. That might either prove you right or the problem right. I am not sure if a rotation about (-1,0,0,0) is the same as a rotation about (1,0,0,0).

In the two sphere picture the rotations are the same because you are basically spinning a globe on its axis so the rotation isn't a general rotation about a point but one around an axis.
 

1. What is isotropy and homogeneity?

Isotropy and homogeneity refer to the properties of a physical system or environment. Isotropy means that the system is the same in all directions, while homogeneity means that the system is the same at all points in space.

2. How are isotropy and homogeneity related?

Isotropy and homogeneity are closely related concepts. In fact, a system cannot be isotropic without also being homogeneous. This means that if a system is the same in all directions, it must also be the same at all points in space.

3. Why are isotropy and homogeneity important in science?

Isotropy and homogeneity are important because they allow us to make predictions and draw conclusions about a system based on limited information. If a system is isotropic and homogeneous, we can assume that the same laws and principles apply at all points in the system, making it easier to study and understand.

4. How do scientists determine if a system is isotropic and homogeneous?

Scientists use various methods to determine if a system is isotropic and homogeneous. This can include experiments, observations, and mathematical models. By analyzing data and looking for patterns, scientists can determine if a system exhibits these properties.

5. What are some examples of isotropic and homogeneous systems?

Some examples of isotropic and homogeneous systems include the universe, a perfect gas, and a perfectly mixed solution. These systems are the same at all points and in all directions, making it easier for scientists to study and understand them.

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