I know that the cosmological principle states that the universe is both isotropic and homogeneous, but I was wondering if these both had to exist at the same time? Is there a way that the universe could be isotropic, but not homogenous or the other way around and what would that look like?
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$\begingroup$ Does this help? physics.stackexchange.com/q/24881/226902 Studies of the large-scale distribution of matter in the visible universe and analysis of the CMB confirm that this assumption is justified. $\endgroup$– QuilloCommented Apr 26, 2023 at 23:03
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$\begingroup$ @Quillo The theory underlying the same studies is also projecting that in the far future the observable universe will "shrink" to a local group of galaxies that are gravitationally bound and that will not look either homogenous or isotropic. The OP has a very good question there, I believe. The cosmological principle might be a good approximation for the current era measured at the current instrumental limits but it's probably not a good assumption in general. $\endgroup$– FlatterMannCommented Apr 26, 2023 at 23:20
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2 Answers
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I know that the cosmological principle states that the universe is both isotropic and homogeneous, but I was wondering if these both had to exist at the same time?
No, they don't necessarily have to, in general.
For example, a uniform electric field in the $\hat x$ direction is a homogeneous vector field. But it is not isotropic.
Such an electric field is homogeneous because it obeys: $$ \vec E(\vec r) = \hat x E(\vec r) = \hat x E(\vec r + \vec d) = \vec E(\vec r + \vec d)\;, $$ for any $\vec d$.
But an active rotation, say by $\pi/2$ about the $\hat z$ axis, changes the components of $\vec E$. $$ R(\pi/2;\hat z)\vec E(\vec r) = E(\vec r)R(\pi/2;\hat z)\hat x = E(\vec r)\hat y \neq \vec E(\vec r)\;. $$
Is there a way that the universe could be isotropic, but not homogenous or the other way around and what would that look like?
Yes, things can be isotropic but not homogeneous.
For example, suppose you are considering isotropy with respect to the origin, $\vec r_0 = (0,0,0)$, of some coordinate system.
Then any function $f(\vec r)$ that only depends non-trivially on the magnitude of $\vec r$ is isotropic, but not homogeneous (because points with different $r$ values generally have different $f(r)$ values).
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Example of a field that is isotropic but not homogeneous:
- The electric field of a proton. For an observer on the proton, they observe the same electric field in every direction. No other observer observes this electric field, so this field is not homogeneous.
Example of a field that is homogeneous but not isotropic:
- The electric field of an infinitely large charged sheet. This results in a uniform electric field that points in one direction. Every observer above the charged sheet sees the same field, but the field is not isotropic, because there's a clear preferred direction ("away").
