# Charting manifolds

#### mikeeey

Hi
How can a person chart a manifold if he does not know how the manifold looks like ?
E.g. The 2-sphere manifold can have 2 charts and symmetric charts with the chart goes like this ( theta from zero to pi , psy from minus pi to pi ) but the problem for unknown manifold , e.g. In general relativity (4-manifold) when solving GR equations , (consider the vacuum solutions by Schwarzschild ) how to take a chart here to do analysis ? And further more he considers [ spherically symmetric spacetime manifold ] , does this mean that the spatial manifold is 3-sphere , so he can chart by considering ( r larger than zero , theta from 0 to pi , psy from minus pi to pi ) , but who says that the manifold is spherical S^3 ?

Thanks

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#### HallsofIvy

Homework Helper
He can't- the definition of a "manifold" require that there already be a chart on the point set. You can then convert from that chart to another consistent chart but not just "create" a chart. How you determine what "consistent" means for a given manifold depends on the chart already given.

#### mikeeey

But u can't imagine a 3- spatial manifold !! How can one chart it ?! As i said in GR , u have the equations , in order to solve u need to know how the manifold looks like to u use charting map and base u r equations on the chart !

#### lavinia

Gold Member
But u can't imagine a 3- spatial manifold !! How can one chart it ?! As i said in GR , u have the equations , in order to solve u need to know how the manifold looks like to u use charting map and base u r equations on the chart !
You just draw local charts until the manifolds is completely covered. The charts will overlap but that is what you want.

#### mikeeey

There is no such thing called drawing in higher dimensional manifold, u can not visualized it to draw it ! , its only inserting some prodicting chart ! , E.g. Consider the black hole by schwarzschild solution , how the manifold would be taken to be ?! In wikipedia the manifold would considered to be spherically symmetric 3-spatial manifold !!

#### lavinia

Gold Member
There is no such thing called drawing in higher dimensional manifold, u can not visualized it to draw it ! , its only inserting some prodicting chart ! , E.g. Consider the black hole by schwarzschild solution , how the manifold would be taken to be ?! In wikipedia the manifold would considered to be spherically symmetric 3-spatial manifold !!
Generally local charts are easy to construct mathematically. If you want to talk about Space-Time then there are many methods for someone living in it. Your question seems misposed. What are you really asking?

#### mikeeey

Sorry i meant how to visualize the manifold in order to draw the chart !! By taking the coordinates limits , a manifold can be covered by a single chart like earth single atlas or by multi charts

#### lavinia

Gold Member
Still not sure what you mean.

Suppose you were the first map maker on Earth and you wanted to chart your local neighborhood. You have no idea you are on a round planet. What would you do.?

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#### mikeeey

This is the problem ,its the same as GR issue , it's like how to solve einstein's field equations , how to set the coordinate chart without knowing how the lorentzian manifold would look like , the only way to solve then by using assumptions for some famous topological spaces like the n-dimensional sphere and the torus !
About what u said , i don't know ! , what would u do ?!

Thanks Lavinia