Active transformation use-case in homogeneity

[gionole]

I'm sorry that I have asked so many questions about this subject and the endless discussion that I caused. This was not my intention at all. There is nothing I'd love more than to close this chapter of confusion.

I'd appreciate if you could read my exact questions and only try to answer them and if you don't understand the questions, you can say so and I will make it clearer if possible. This is really important as I believe, deviating from these questions is what's causing my confusion.

After your help, I now understand that mathematically, active and passive transformations are the same, but this post will only be about active transformation and its usage in the subject of homogeneity. I've seen that people use active transformation as mathematical tool for the homogeneity detection and here is what I'm curious about.

Imagine you got a huge area that includes only 3 objects - earth, sun and ball - Let's call all of this to be "space" - i.e area that includes its objects in it.

Now, what one can do is move ONLY the ball somewhere else and observe its behaviours. If it's the same behaviour as it had before moving, then space can be said to be homogeneous. Nice and easy.

My question though is this: When would it be required to move two objects instead of one ? (i.e move the earth and ball both) - It seems to me that moving only the ball(i.e one object) is enough to show mathematically whether space can be said to be homogeneous or not. Just to repeat the question again - in terms of homogeneity, when would moving 2 objects be much more insightful than moving only one ? and please, let's use my example that I mentioned above.

I don't want to ask any other related questions yet on this not to confuse the discussion yet. I would understand as well if you don't want to participate in the discussion. Thank you everyone for your help so far.

 

[BvU]

If it's the same behaviour as it had before moving, then space can be said to be homogeneous

What's that, "behaviour"? If I release a ball close to the moon it will fall towards the moon. When I move it close to earth, it will fall towards earth. Different behaviour or not ?

$\$

 

[gionole]

By behaviour I mean, that it exactly moves the same way(no difference at all - there's nothing you can distinguish in its movement). i.e the same physics.

 

[BvU]

I suppose you can distinguish that it moves the other way?

 

[gionole]

In that case, if universe is said to be isotropic on a scale of 150mpc, then i get that averagely, in every direction, taking 150mpc volume contains the same density of matter and force averagely but if you do the experiment letting go of a ball in one direction, then in another, you will still be able to distinguish directions, because in some directions, closer to the ball, there will be more mass and more gravity so in that sense, how is it said to be isotropic on a scale of 150mpc ?

I believe since we're talking 150mpc scale, doing a ball experiment close to me is wrong and I should let go of a ball pretty far from me in every direction ? maybe 90mpc away from me in every direction ? would the ball behaviour would be the same in every direction then?

 

[Hill]

In that case, if universe is said to be isotropic on a scale of 150mpc, then i get that averagely, in every direction, taking 150mpc volume contains the same density of matter and force averagely but if you do the experiment letting go of a ball in one direction, then in another, you will still be able to distinguish directions, because in some directions, closer to the ball, there will be more mass and more gravity so in that sense, how is it said to be isotropic on a scale of 150mpc ?

I believe since we're talking 150mpc scale, doing a ball experiment close to me is wrong and I should let go of a ball pretty far from me in every direction ? maybe 90mpc away from me in every direction ? would the ball behaviour would be the same in every direction then?

If every cube of 150 Mpc size contains the same amount of mass-energy, then you can in a thought experiment smooth the content of every such cube so that the mass-energy density everywhere inside it equals to its average density. After doing so, your test particle will not distinguish between directions.

 

[Ibix]

how is it said to be isotropic on a scale of 150mpc ?

The surface of a table can be flat, so a marble placed anywhere on it won't roll in any direction. That's homogeneous and isotropic, at least in the horizontal directions. But zoom in on the table and you'll see it's probably got tiny scratches and, at the very least, it's rough at the atomic scale. A marble placed on it almost certainly does roll a micron or two - but a population of marbles at random locations will all roll in different directions, and none of them will roll far. So homogeneous and isotropic is a reasonable model for most purposes.

Similarly, the matter distribution in the universe has a lot of non-uniformity in it. But the non-uniformity is at (relatively) small scales, and at very large scales it is well approximated by a model that is homogeneous and isotropic.

 

[gionole]

If every cube of 150 Mpc size contains the same amount of mass-energy, then you can in a thought experiment smooth the content of every such cube so that the mass-energy density everywhere inside it equals to its average density. After doing so, your test particle will not distinguish between directions.

Thanks very much. If you do smooth the content, sure, but by itself, without our interference, universe still doesn't seem isotropic at large scale because as I said, particle will behave differently in different directions even though every 150mpc cube in every direction has the same average mass-energy density. Don't you agree ?

 

[gionole]

The surface of a table can be flat, so a marble placed anywhere on it won't roll in any direction. That's homogeneous and isotropic, at least in the horizontal directions. But zoom in on the table and you'll see it's probably got tiny scratches and, at the very least, it's rough at the atomic scale. A marble placed on it almost certainly does roll a micron or two - but a population of marbles at random locations will all roll in different directions, and none of them will roll far. So homogeneous and isotropic is a reasonable model for most purposes.

Similarly, the matter distribution in the universe has a lot of non-uniformity in it. But the non-uniformity is at (relatively) small scales, and at very large scales it is well approximated by a model that is homogeneous and isotropic.

I agree so far, but by itself, without our interference, universe still doesn't seem isotropic at large scale as particle will behave differently in different directions. Note the word: "by itself", because as @Hill said, if you smooth the content, sure, but the whole idea is to do no interfere and check it as it is currently by itself.

 

[Ibix]

Sure.

Note that this is the matter distribution you're talking about, not the laws of physics.

 

[gionole]

Sure.

Note that this is the matter distribution you're talking about, not the laws of physics.

By the matter distribution, universe is isotropic at large scale I agree, but in terms of laws of physics, I don't think so. Do you ?

 

[Hill]

Thanks very much. If you do smooth the content, sure, but by itself, without our interference, universe still doesn't seem isotropic at large scale because as I said, particle will behave differently in different directions even though every 150mpc cube in every direction has the same average mass-energy density. Don't you agree ?

To probe that scale, the "particle" needs to have a size of galaxy cluster or so. Such particles do not behave differently in different directions.

 

[gionole]

To probe that scale, the "particle" needs to have a size of galaxy cluster or so. Such particles do not behave differently in different directions.

Yes, in that sense, true. I guess that's because things that are closer to such big particle in one direction and not in other direction won't have big impact on it and can be neglected.

 

[Ibix]

By the matter distribution, universe is isotropic at large scale I agree, but in terms of laws of physics, I don't think so. Do you ?

You seem to be saying the exact opposite to what I said, so no.