# Evolution of a sphere in expanding and non-expanding spaces

#### johnconner

I'm not very damiliar with differential geometry but I wanted to know in the most general sense when can we say that the evolution of an expanding sphere in a static space is equivalent to the evolution of a static sphere in an expanding space? In other words, what are the necessary assumptions for the spaces and two spheres so that the spheres could be considered equivalent in terms of dynamical evolution in two different spaces? One assumption I know, which is we consider both spheres are non-rigid, so if the space expands the sphere expands with it.

#### Maschke

If I'm understanding you correctly, it's all about changing the metric.

Let's do this in one dimension for simplicity. Take the line segment on the real axis from 0 to 1; that is, the unit interval [0,1]. The distance is given by the usual Euclidian distance formula $d(x, y) = |x - y|$.

Now suppose God or gravity or whatever comes along and changes the distance formula to $d(x, y) = 2 |x - y|$.

Or suppose God stretches the universe to $[0,2]$.

There is no difference in these two scenarios. If you lived in the space you couldn't tell the difference between the space expanding or your measuring rod shrinking.

Is that an answer to the question you asked? I'm not sure. You can take space to be the entire real line and your sphere to be the unit interval.

#### johnconner

If I'm understanding you correctly, it's all about changing the metric.

Let's do this in one dimension for simplicity. Take the line segment on the real axis from 0 to 1; that is, the unit interval [0,1]. The distance is given by the usual Euclidian distance formula $d(x, y) = |x - y|$.

Now suppose God or gravity or whatever comes along and changes the distance formula to $d(x, y) = 2 |x - y|$.

Or suppose God stretches the universe to $[0,2]$.

There is no difference in these two scenarios. If you lived in the space you couldn't tell the difference between the space expanding or your measuring rod shrinking.

Is that an answer to the question you asked? I'm not sure. You can take space to be the entire real line and your sphere to be the unit interval.
Thank you so much for answering. Of course that is not my answer but at least you helped me with the basic part.
what I'm looking for is actually a lot deeper, and although I'm not familiar with all of it but what I need to do is to prove that this could be explained in mathematics.
your assumptions are that everything is non-rigid so nothing would actually understand the expansion in any way. what I want is not exactly the situation in our world and regarding relativity but it's close and it's about a thought experiment that is trying to explain something in physics. but the part that I asked is completely in realms of mathematics.

so the scenario is actually this and correct me if I'm wrong. When I said spaces I should've been more precise. They both are space-time but the expanding one's spatial dimensions are expanding. their metrics are different. one is expanding and one is not. and because expansion doesn't affect objects that are rigid (well basically objects which are bound with gravity or EM) then objects or in better words frames can detect the expansion. now it is different. if I leave a non-rigid sphere in the expanding space, the sphere would expand over time (solely as a consequence of expansion of the space) and in the static space scenario the sphere is expanding itself (there's a force acting on it so it expands).

So yes. You are right. the metrics are different. My problem is that is there any mapping between these two spaces? can I introduce a mapping between two spaces with different metric so I could finally write an equation which is going to be like this:
The evolution of the sphere in expanding space (which is expanding because of expansion of the space) = The evolution of the sphere in static space (which is expanding on its own; for example there's forces that are expanding it despite the space being static)

I want know how to introduce the assumptions correctly, so I would be able to write this equation. which properties of spaces should be similar and which properties should be different? their metric are different but should they be homeomorphic to each other? how should say all of this?
how can I prove that I can write that equation above? someone told me you can't write that equation at all. because you can't have two spaces with different metrics that have a mapping between them.

#### johnconner

For example what kind of functions can I introduce to these spaces? or could they be metric spaces? or more generally speaking, what kind of topological spaces are able to satisfy that equation? what kind of symmetries they require?
If I had a specific question I would ask but what I am looking for is the most general case of spaces that could satisfy the equation above.

#### Maschke

I wonder if you can make your idea more clear by giving an example.

In general there are many functions between any two mathematical sets or spaces. Do you have certain properties in mind for your map? Continuous or differentiable perhaps, or metric-preserving, or angle-preserving, or something along those lines?

Are you saying that the space expands but there are some objects (which you call bound by gravity, but of course that's not right because everything in the universe is bound by gravity to everything else) that don't expand? You could certainly write down a piecewise metric.

I had a hard time understanding your question, perhaps you can clarify it.

johnconner

#### johnconner

I wonder if you can make your idea more clear by giving an example.

In general there are many functions between any two mathematical sets or spaces. Do you have certain properties in mind for your map? Continuous or differentiable perhaps, or metric-preserving, or angle-preserving, or something along those lines?

Are you saying that the space expands but there are some objects (which you call bound by gravity, but of course that's not right because everything in the universe is bound by gravity to everything else) that don't expand? You could certainly write down a piecewise metric.

I had a hard time understanding your question, perhaps you can clarify it.
Yes. the big problem is that I don't know where to start. If I knew I would find some textbooks and read about it.

Let me tell you what is it exactly that I want. I was thinking about an arbitrary space-time and our universe's space-time. I want to introduce a mapping between these two so that I could say if a non-rigid sphere in our space-time expands due to cosmological expansion, then I can introduce another non-rigid sphere in the arbitrary space-time which its evolution is equal to the evolution of the sphere in our space-time. I want the equivalence so that I could take one sphere of our space-time to an arbitrary one so I could study the geometrical effects without disturbance.
If the evolution of the two expanding spheres are equal (they're both expanding but for different reasons), then I can study the geometrical effects of expansion in the arbitrary space-time.

now as I think, there's probably an affine transformation between two spaces. but I still don't know how to write the transformation.
for studying the issue I have a friend who's an expert but I have to come up with the basics so that I could go to him.

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#### Maschke

Sounds like you want to study general relativity.

But honestly your exposition is very difficult to understand. I'd still like a specific example. Do it in the plane in two dimensions with circles instead of spheres, and try to describe what you have in mind using high school analytic geometry. You have some circle with a radius and the radius changes with respect to time according to some function; and meanwhile the metric of the plane keeps changing over time according to some other function. You can work this out once you write down your ideas clearly.

#### johnconner

Sounds like you want to study general relativity.

But honestly your exposition is very difficult to understand. I'd still like a specific example. Do it in the plane in two dimensions with circles instead of spheres, and try to describe what you have in mind using high school analytic geometry. You have some circle with a radius and the radius changes with respect to time according to some function; and meanwhile the metric of the plane keeps changing over time according to some other function. You can work this out once you write down your ideas clearly.
Ok how about this? it's actually another question altogether but the answer will help me in the first question. what kind of geometrical objects have intrinsic expansion? I mean that is there any object in geometry that we could say it has intrinsic expansion? or we can attribute expansion so that it would be considered intrinsic for that object.

#### Maschke

I don't understand what you mean by intrinsic expansion.

For example using high school analytic geometry I can imagine the unit circle centered at the origin with radius 1, consisting of the points $(x,y)$ in the plane that satisfy $x^2 + y^2 = 1$.

So that's, I supposes we could call it a static circle.

Now let's introduce a time variable $t$. Let's say for example that we have a circle centered at the origin of radius $t$ at every time $t$. So at 2 seconds the radius is 2, and so forth.

Then the equation at time $t$ is $x^2 + y^2 = t^2$. We have in effect a different circle for each moment of time. Or "the same" circle getting bigger. Which is the true visualization? As we get older are we the same, or are we just a succession of different people, one person at each instant, but related by some continuity conditions?

Is this what you mean?

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topsquark

#### johnconner

Sort of. In physics we have to come up with a good explanation as why this sphere is expanding or why the space is expanding; I want to know if there's certain mathematical properties that can explain expansion as a geometrical phenomenon, not as a consequence of a force or energy. kind of like what Einstein did to gravity. with the difference that in relativity Einstein considers gravity as a result of presence of matter. I'm thinking expansion to be intrinsic. without any relation to matter. so there has to be some kind of intrinsic properties of geometry or topology for the space so it expands over time. That's why I want to know if there's any geometrical objects that are expanding solely because of intrinsic geometrical properties.

If you want an example for this it would be: curvature in space-time results in acceleration. that acceleration is a geometrical phenomenon. curvature is the intrinsic property of space-time that is has the key role for gravity here.

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