Doubts about isotropy of space

[gionole]

I might have some other questions related to this topic, but I will ask them in further replies.

Isotropy of space is explained such as it should look the same in every direction. It's not enough to imagine ourselves to be in the center of the sphere, because definition says that to call space isotropic, wherever we are inside that space, it should look the same in every direction.

There's 2 definitions about space being isotropic and I believe both should be true to say that space is isotropic.
1. Visual - it looks the same in every direction
2. Behavioural - no preferred location in terms of movement.

Imagine there exists only an earth and vacuum around it and we're considering the space of sphere of 100m diameter that is put near the above of earth. When we consider this sphere, in terms of Visual definition, do we only apply the rule of "it looks the same in every direction" within the boundary of our space - we are only curious if the 100m diameter sphere is isotropic so I guess we only look around till boundary ? If so, then in terms of visual definition, space is isotropic. ofc, I know that this space of sphere is not isotropic because the behavioural definition breaks it - dropping a ball only goes to earth - so we got preferred location, but i am only asking about visual definition.

 

[phinds]

When we consider this sphere, in terms of Visual definition, do we only apply the rule of "it looks the same in every direction" within the boundary of our space

??? You are trying (unsuccessfully) to apply a concept that describes cosmological space to a trivial volume right next to the Earth.

 

[gionole]

??? You are trying (unsuccessfully) to apply a concept that describes cosmological space to a trivial volume right next to the Earth.

but that doesn't stop me from asking that question that I asked. I am asking in terms of visual definition, it looks the same in every direction applies to space boundaries that we consider or even outside if it ? or are you saying that "looks the same in every direction" is only valid for the whole universe space ? if so, then the usual definition of the isotropy of any space is that physics should behave the same in every direction then and not the definition of "looks the same in every direction". Thoughts ?

 

[Ibix]

Isotropic means that there is no preferred direction. It can include a privileged point - for example a solid sphere in an otherwise empty volume is isotropic around the sphere center, but not other points.

Physics usually also invokes homogeneity, which means no special point. The sphere in a vacuum is not homogeneous.

 

[gionole]

for example a solid sphere in an otherwise empty volume is isotropic around the sphere center, but not other points.

Then, do we say sphere is isotropic or not ? To call a space isotropic, it shouldn't matter where I stand, wherever I stand, either here or there, as long as I'm within the space, everything must be the same in every direction from here or there. is this definition incorrect ? if so, then whenever we say: space is isotropic, we should be adding: "around what point" in that space ? no ?

 

[Dale]

You appear to be mixing the concepts of isotropy and homogeneity. A sphere is isotropic, but not homogenous.

 

[gionole]

You appear to be mixing the concepts of isotropy and homogeneity. A sphere is isotropic, but not homogenous.

but in my previous reply, I asked a different thing. When you say sphere is isotropic, from which point do you assess the things(i.e start looking around in every direction and assume it's the same every direction) ? if it's a center, why the center ? if you consider the whole space isotropic, then it shouldn't matter where I stand, wherever I stand, either here or there, as long as I'm within the space, everything must be the same in every direction from here or there.

 

[Ibix]

Then, do we say sphere is isotropic or not ?

It's isotropic around its center, not about other points. As Dale says, you seem to be talking about homogeneity.

 

[Dale]

When you say sphere is isotropic, from which point do you assess the things

From the center of the sphere.

then it shouldn't matter where I stand, wherever I stand

That is homogeneity.

wherever I stand, …, everything must be the same in every direction from here or there.

That is homogeneity and isotropy together

 

[Ibix]

You can have a space that is homogeneous but not isotropic, I think. An example would be an empty space equipped with laws of physics such that free falling objects moving with respect to a particular frame start to rotate around the z-axis through their center of mass.

Everywhere it is the same, but there's a special direction.

 

[hmmm27]

You can have a space that is homogeneous but not isotropic, I think. An example would be an empty space equipped with laws of physics such that free falling objects moving with respect to a particular frame start to rotate around the z-axis through their center of mass.

Everywhere it is the same, but there's a special direction.

I'm not sure I get that. Could you explain your explanation ?

If the dimensionalities and orientation of the isotropy and homogeneity match that of the volume (or, at least of each other), then I don't see how you can have homogeneity without isotropy.

 

[Ibix]

I don't see how you can have homogeneity without isotropy.

A simpler example is the surface of an infinitely long cylinder. Every point is the same as every other (homogeneous), but in one direction a geodesic is a closed circle, but not in any other - so it's not isotropic.

 

[hmmm27]

A simpler example is the surface of an infinitely long cylinder. Every point is the same as every other (homogeneous), but in one direction a geodesic is a closed circle, but not in any other - so it's not isotropic.

A 2400km ride on a 120km dusty trail circular track is still 2400km of dusty trail. There's no room for "Hey, didn't we pass that bush an hour ago ?" because there are no bushes.

 

[Nugatory]

A 2400km ride on a 120km dusty trail circular track is still 2400km of dusty trail. There's no room for "Hey, didn't we pass that bush an hour ago ?" because there are no bushes.

Still plenty of ways of observing the anisotropy in this topology…. Point a laser beam in various directions, and there will be two in which you’ll be illuminating the back of your head.

 

[Ibix]

A 2400km ride on a 120km dusty trail circular track is still 2400km of dusty trail. There's no room for "Hey, didn't we pass that bush an hour ago ?" because there are no bushes.

The fellow 120km ahead is strangely familiar, though.

There's a more precise definition in terms of pull backs of vector fields in Carroll's lecture notes (chapter 8, Cosmology, IIRC). He'll probably refer you to Wald for detail.

 

[Dale]

A 2400km ride on a 120km dusty trail circular track is still 2400km of dusty trail. There's no room for "Hey, didn't we pass that bush an hour ago ?" because there are no bushes.

This objection doesn’t work. If you are supposing a track then you can suppose marks on the track. The topology described is a correct example of a manifold with global anisotropy.

There are also spacetimes with homogeneity but with local anisotropy. The classic example is the Goedel spacetime

https://en.m.wikipedia.org/wiki/Gödel_metric

 

[Baluncore]

Isotropy of space is explained such as it should look the same in every direction.

Is there any evidence that suggests space might be anisotropic?

Space is linear to EM radiation, in all directions. We do not see sum and difference frequencies of EM waves passing through shared space.

Gravity seems the same in all directions, so we get orderly orbits, and spherical objects, that become elliptical when spinning.

How could we observe an anisotropy?

 

[vanhees71]

According to GR locally for any free-falling observer space is the usual Euclidean space. That's Einstein's realization of the equivalence principle, i.e., that at any spacetime point there's a local reference frame, where the laws of special relativity hold, i.e., coordinates, where at this point $g_{\mu \nu}=\eta_{\mu \nu}$.

Whether the universe is globally homogeneous and isotropic even on the coarse-grained large spacetime level is of course an open question. It's the basic assumption of modern cosmology, the "cosmological principle", though, and the cosmological standard model ($\Lambda\text{CDM}$ model) works pretty well. But just recently there's again trouble with the "Hubble tension", and one type of theoretical explanation is that the cosmological principle might not be true.

See, e.g.,

https://iopscience.iop.org/article/10.1088/1361-6382/ac086d (open access)

 

[gionole]

On the attached image(left image), it's said that it is homogeneous and non-isotropic. I am wondering why that is. To be homogeneous, the density property must be $f(r) = f(r+r')$ everywhere, doesn't that mean it's isotropic as well - $f(r) = f(|r|)$ ? or homogenous means as long as $f(r) = f(r+r')$ is valid in one direction only it's enough to call it homogeneous ?, but if so, in that one direction, it should be valid till the end of the material right ? if so, there're a little bit of spaces in between bricks, so if I take $r'$ to be quite small, then it could occur that $f(r)$ corresponds to density where the brick is and $f(r+r')$ corresponds to that empty space between 2 bricks and won't be equal.

 

[A.T.]

On the attached image(left image), it's said that it is homogeneous ....

Said by whom in what context?

https://en.wikipedia.org/wiki/Homogeneity_(physics)#Context

The definition of homogeneous strongly depends on the context used. For example, a composite material is made up of different individual materials, known as "constituents" of the material, but may be defined as a homogeneous material when assigned a function.

 

[gionole]

@A.T. you're right, never mind. I got a better question and I think this sums it up for me.

From the following answer, we note the following:

isotropic space means that all mechanical properties are the same in all directions. Mechanical properties refer to anything you can think of: density, electric charges, force, anything...

Question 1: should also the things we see be the same in every direction ? like if you're seeing a chair in front of you, you must be looking the chair to your right direction. If no, why does the isotropic definition says that density must be the same as well ? doesn't density include "things" ?

Question 2: it's said that in inertial frame, space is isotropic, but I thought isotropic meant around you, you must be looking at the same things, separated by the same distance, same force, anything you can think of. If i am in a train that's moving with constant speed(yep, it's inertial frame), but how is it isotropic since i'm definitely seeing different things in train in every direction. Does the isotropic have different definition when we apply it to the concept of frames ?

 

[A.T.]

Question 1: should also the things we see be the same in every direction ? like if you're seeing a chair in front of you, you must be looking the chair to your right direction. If no, why does the isotropic definition says that density must be the same as well ? doesn't density include "things" ?

Question 2: it's said that in inertial frame, space is isotropic, but I thought isotropic meant around you, you must be looking at the same things, separated by the same distance, same force, anything you can think of. If i am in a train that's moving with constant speed(yep, it's inertial frame), but how is it isotropic since i'm definitely seeing different things in train in every direction. Does the isotropic have different definition when we apply it to the concept of frames ?

You seem to be confusing space itself with things that exist in space. The statements about space being isotropic and homogeneous mean that if you rotate/translate the entire setup (things and observer) he will still make the same observations.

 

[gionole]

@A.T. The following answer confuses me. It says:

It basically means that all mechanical properties are the same in all directions. Mechanical properties refer to anything you can think of: density, electric charges, anything...

This leads to an easy consequence: if all directions have EXACTLY the same properties: they are indistinguishable. You just cannot tell whether you're looking to the North or to the East. Anywhere you look, you're seeing the same amount of things, separated by the same distance, with exactly the same force, whatever.

Why does it say: "Anywhere you look, you're seeing the same amount of things, separated by the same distance" ?

 

[A.T.]

@A.T. The following answer confuses me. It says:

Why does it say: "Anywhere you look, you're seeing the same amount of things, separated by the same distance" ?

That would be isotropy of matter distribution, which is different than isotropy of space itself. As already said, these terms need context to be meaningful.