Charting Manifolds: Tips & Techniques

[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

 

[HallsofIvy]

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]

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]

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]

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.?

 

[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

 

[lavinia]

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

Well the Ancients drew local maps of their environs without knowing anything about the whole earth.