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well I didn't want to get to the details since this is math forum but not exactly.Do you consider what's called Dark Energy, or the Cosmological Constant to be an intrinsic property?

That's causing the space that is our universe to expand.

cosmological constant historically was introduced to achieve the static universe. then it became the mysterious origin of expansion. it is called dark energy since it is in the form of energy density and it is basically related to the right hand side of the field equations. what I'm looking for is far from today's explanations. but it's just an idea and I want to see if there is some intrinsic properties that could explain expansion in geometry.

And I have to be precise about your question. the cosmological constant is not going to be considered intrinsic property. I want it to vanish and then replace it with a geometrical process.

This eventually is going to be explained in a certain Lagrangian equation. in that equation cosmological constant will vanish and other terms will have to explain its effect.

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That equivalence I am looking for is for this. if the expansion in a certain object is happening intrinsically then that equivalence is possible to achieve.

No, not in math. In math the situation is exactly as I explained it. For each time $t$ we have the circle $x^2 + y^2 = t^2$. We can think of it as a collection of circles parameterized by $t$, or we can think of it as the "same" circle growing over time. It's the same either way.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.

It's just like the idea of a block universe. We can consider the state $S_t$ of the universe at time $t$, as $t$ flows from the past to the future. Or we can think of it happening all at once. Input a value for $t$, look up the state $S_t$. But there's no flow of time, there's just all the values of $t$ considered all at once.

You are asking a physics question. There could be angels pushing outward on the circle, or a force of some kind, or a curvature or distortion of space of some kind. These are matters of physics. If you have a physical idea you can use math to model it. But math doesn't know "why" the circle is expanding, any more than when we write $f(x) = x^2$, "why" the function $f$ always squares its input. That's just what it does. If you want to say there's an invisible daemon inside the function using a pocket calculator to square each input, you can do that. But that story or model or explanation is extraneous to the math.

Here's another idea. You have a point that runs around the circle. At time $t$ the point is at $(\cos t, \sin t)$ in the plane. Then as time goes from the deep past to the deep future, the point keeps going around the circle at a uniform rate of speed. You can say it's on a frictionless circular track and someone gave it an initial push, if you like. Or maybe the ground is made of hard rocky dirt and you have a mule pulling the point around the circle. Maybe the point being pulled has a mass, and the mule has to be fed a certain amount of food to pull a certain amount of mass through a given distance. Those physical considerations are whatever you want to make up or stipulate. There's no inherent mathematical reason a circle is sitting there being a circle.

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Can you tell me what kind of textbooks can I read about expanding objects in geometry? any reference about the issue?No, not in math. In math the situation is exactly as I explained it. For each time $t$ we have the circle $x^2 + y^2 = t^2$. We can think of it as a collection of circles parameterized by $t$, or we can think of it as the "same" circle growing over time. It's the same either way.

It's just like the idea of a block universe. We can consider the state $S_t$ of the universe at time $t$, as $t$ flows from the past to the future. Or we can think of it happening all at once. Input a value for $t$, look up the state $S_t$. But there's no flow of time, there's just all the values of $t$ considered all at once.

You are asking a physics question. There could be angels pushing outward on the circle, or a force of some kind, or a curvature or distortion of space of some kind. These are matters of physics. If you have a physical idea you can use math to model it. But math doesn't know "why" the circle is expanding, any more than when we write $f(x) = x^2$, "why" the function $f$ always squares its input. That's just what it does. If you want to say there's an invisible daemon inside the function using a pocket calculator to square each input, you can do that. But that story or model or explanation is extraneous to the math.

Here's another idea. You have a point that runs around the circle. At time $t$ the point is at $(\cos t, \sin t)$ in the plane. Then as time goes from the deep past to the deep future, the point keeps going around the circle at a uniform rate of speed. You can say it's on a frictionless circular track and someone gave it an initial push, if you like. Or maybe the ground is made of hard rocky dirt and you have a mule pulling the point around the circle. Maybe the point being pulled has a mass, and the mule has to be fed a certain amount of food to pull a certain amount of mass through a given distance. Those physical considerations are whatever you want to make up or stipulate. There's no inherent mathematical reason a circle is sitting there being a circle.

I don't really know any physics textbooks. Any text on multivariable calculus would give you the tools to analyze motion in three-space. But I gather you're more interested in the physics aspect.Can you tell me what kind of textbooks can I read about expanding objects in geometry? any reference about the issue?

It's not restricted to calculus. It is a matter of geometry. For example, consider a cone. Assume that the height dimension which stretches from apex to flat circle base, is the time dimension. now the 2-D circle is a space moving along the time dimension. as it moves in time it expands.but this expansion is not anything but a geometrical effect. how? because I say the equation of the cone itself is the limit for the 2-D circle along the time dimension. If we say the equation of cone is the limit then the expansion is entirely a geometrical effect. it has nothing to do with any sort of energy. how can I say all of this in mathematical form?I don't really know any physics textbooks. Any text on multivariable calculus would give you the tools to analyze motion in three-space. But I gather you're more interested in the physics aspect.

I don't follow you at all.It's not restricted to calculus. It is a matter of geometry. For example, consider a cone. Assume that the height dimension which stretches from apex to flat circle base, is the time dimension. now the 2-D circle is a space moving along the time dimension. as it moves in time it expands.but this expansion is not anything but a geometrical effect. how? because I say the equation of the cone itself is the limit for the 2-D circle along the time dimension. If we say the equation of cone is the limit then the expansion is entirely a geometrical effect. it has nothing to do with any sort of energy. how can I say all of this in mathematical form?

Take the simpler case of the graph of the function $f(x) = x^2$. The graph is a parabola opening upward with its vertex at the origin.

What makes it have that particular shape? What forces are causing the graph to take the particular shape that it has? The answer is that there aren't any forces. It's just a particular mathematical function. With your cone, there's nothing making the cone be a cone. It's a cone, a mathematical object. There are no causes. A cone is just a set of points that satisfies an equation.

Exactly. I'm not looking for forces or anything in the cone. No, I agree with you. It is the way it is, because of math and because we defined it that way. that is my point when I say I want intrinsic property. because with the cone I don't have to bring any forces in. It's just pure math and we have to say it is the way it is. That is what I'm looking for. I'm not asking for forces. I want situations like this cone.I don't follow you at all.

Take the simpler case of the graph of the function $f(x) = x^2$. The graph is a parabola opening upward with its vertex at the origin.

What makes it have that particular shape? What forces are causing the graph to take the particular shape that it has? The answer is that there aren't any forces. It's just a particular mathematical function. With your cone, there's nothing making the cone be a cone. It's a cone, a mathematical object. There are no causes. A cone is just a set of points that satisfies an equation.

Here's the punch line: I honestly don't know how to write all I have said into mathematical form. Don't get confused. tell me how to explain the cone scenario itself in mathematical form. please write it in math for me.

Now the cone is one example that I thought of, that satisfies my need. which is explaining expansion with intrinsic properties in geometry. what other ways is there to explain expansion (not universe, don't go there. just think about expansion in geometry) in situations like this? which expansion happens just because of the definition or some property of space or the object.

en.wikipedia.org

Another idea I had was that perhaps you would be interested in vector fields. So at each point of space you have a vector representing the force at that point, for example the velocity and direction of the wind, or gravity, etc. Again, general relativity or differential geometry. Or basic multivariable calculus.

I don't think I can help with your idea of expansion happening because of the definition of some property of space. I don't understand it. A cone's a cone.

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