Recent content by Onyx

Would it also be sufficient to instead make sure that the derivatives of the coordinate functions (metric components) are the same at the point of the boundary?

Although clearly a line on the surface of the solid that crosses from the cylinder to the hemisphere could not be described by just one equation; it would be described by two different equations, one for each part.

If it were a hemisphere being glued to the top of a cylinder, would the derivative condition then be met?

I see. So if you wanted to plot a geodesic that passes through the boundary, it would not be a smooth function, because the metric tensor is different for each region.

Well I'm assuming that if two particular manifolds can be joined continuously, there would be some metric tensor describing the resulting manifold. After the gluing, how would one find such a metric, if not already known?

It seems like knowing ##\ell## requires already knowing ##g_{ab}##.

So there is a whole different notation for the junction conditions?

I say combined because two manifolds are glued together across the shell to make a new one. If it is a continuous junction, there should be a continuous metric for the whole thing.

A Relativist's Toolkit (2004) lists the Israel junction conditions as: ##1. [h_{ab}]## ##2. S_{ab}=[K_{ab}]-[K]h_{ab}## Where ##S_{ab}## is the stress-energy tensor of the shell only, and ##[K_{ab}]## and ##[K]## are ##K_{ab}^--K_{ab}^+## and ##K^--K^+## respectively. My understanding is that...

This seems right if considering a significantly rotating disk, but unless I'm mistaken most galaxies don't rotate very fast in proportion to their disk radius. I found another metric describing a stationary and static ellipse, as described in this paper. However, I don't understand why the...

How can I create a metric describing the space outside a large disk, like an elliptical galaxy? In cylindrical coordinates, ##\phi## would be the angle restricted the the plane, as ##\rho## would be the radius restricted to the plane. I think that if ##z## is suppressed to create an embedding...

Am I correct in assuming that it is just like the OS collapse case, where there has to be two different metrics for two different manifolds? Also, if not a bag of gold, what is this person describing (not a reliable source)?

Is that because the gluing process is so complicated?

What is the metric for a bag-of-gold spacetime?

But how would a traveller get from ##l## to ##-l## without passing through the throat?