B Dilating or expanding a closed ball in Riemannian geometry

[johnconner]

Hello. If a closed ball is expanding in time would we say it's expanding or dilating in Riemannian geometry? better saying is I don't know which is which? and what is the function that explains the changes of coordinates of an arbitrary point on the sphere of the ball?

 

[andrewkirk]

Expansion is a concept that does not even require topology, let alone geometry, to define it. All we need is set theory. An object is expanding if and only if for any two times t1 and t2 such that t1 < t2, the set of points covered by the object at time t2 properly contains the set of points covered by the object at time t1. So the geometry does not affect whether the object is judged to be expanding.

 

[johnconner]

Expansion is a concept that does not even require topology, let alone geometry, to define it. All we need is set theory. An object is expanding if and only if for any two times t1 and t2 such that t1 < t2, the set of points covered by the object at time t2 properly contains the set of points covered by the object at time t1. So the geometry does not affect whether the object is judged to be expanding.

thank you. then what is dilation? when we can use that? if we consider time as the fourth dimension, then can we say expansion in time is a dilation?

 

[Ibix]

Is this off the back of your thread in cosmology? If so, I rather suspect you're still misusing terminology, so the answers you get may well not be to the questions you think you are asking.

 

[PeroK]

Expansion is a concept that does not even require topology, let alone geometry, to define it. All we need is set theory. An object is expanding if and only if for any two times t1 and t2 such that t1 < t2, the set of points covered by the object at time t2 properly contains the set of points covered by the object at time t1. So the geometry does not affect whether the object is judged to be expanding.

Okay, but this was posted under "Differential Geometry", which assumes we are talking about some sort of metric expansion.

 

[Orodruin]

Okay, but this was posted under "Differential Geometry", which assumes we are talking about some sort of metric expansion.

Does it? Metric expansion is typically used in cosmology, but it then refers to a particular family of spatial submanifolds of a Lorentzian manifold. The OP seems to imply talking about Riemannian geometry. To me it just seems as if the OP has not written a well defined question and may think he is asking something but in reality that something is going to be interpreted differently by different people. There is a need of context from OP (ie, I agree with #4).

 

[PeroK]

Does it? Metric expansion is typically used in cosmology, but it then refers to a particular family of spatial submanifolds of a Lorentzian manifold. The OP seems to imply talking about Riemannian geometry. To me it just seems as if the OP has not written a well defined question and may think he is asking something but in reality that something is going to be interpreted differently by different people. There is a need of context from OP (ie, I agree with #4).

Cosmology doesn't have a monopoly on the term "metric"! In any case, "ball" or "sphere" are only defined in sets with a metric.

 

[Orodruin]

Cosmology doesn't have a monopoly on the term "metric"! In any case, "ball" or "sphere" are only defined in sets with a metric.

This is not the point. The point is that the question is ambiguous.