Minkowski Spacetime vs Euclidean Spacetime

Which one would you use in order to map out a black hole and its connection to a white hole?

bobrubino said:

Which one would you use in order to map out a black hole and its connection to a white hole?

Neither.

Neither.

First of all, there is no such thing as Euclidean spacetime so that’s out. Second, Minkowski spacetime is a flat affine spacetime and doesn’t contain anything like a black hole. What you are looking for is Schwarzschild spacetime.

thanks

You can draw a thing called a Kruskal diagram on a Euclidean plane, which is a map of the maximally extended Schwarzschild spacetime, which is probably what you are talking about (the wiki article on Kruskal-Szekeres coordinates is pretty good). But it's important to realise that the Euclidean representation is honest but not accurate. No representation of spacetime on a Euclidean plane can really be accurate because there's no minus sign in Pythagoras' theorem and there always is one in the equivalent thing in locally-Minkowski spacetimes. No matter if you try to hide it by using imaginary coordinates - the effects of it are still there.

Ibix said:

But it's important to realise that the Euclidean representation is honest but not accurate.

Honest since it is an one-to-one mapping of a region of spacetime.

cianfa72 said:

Honest since it is an one-to-one mapping of a region of spacetime.

Well, I just meant "honest" in contrast to the "marble on a dip in a sheet", which is hopelessly misleading for almost anything. Kruskal diagrams are actually working tools, but the interpretation remains non-trivial.

Ibix said:

it's important to realise that the Euclidean representation is honest but not accurate

That is an excellent way to put it

Ibix said:

Kruskal diagrams are actually working tools, but the interpretation remains non-trivial.

KS diagrams (region I - IV) cover the entire spacetime manifold ?

cianfa72 said:

KS diagrams (region I - IV) cover the entire spacetime manifold ?

Ideally they cover the entirety of two dimensions of it, yes. An actual diagram only covers a finite region unless you know where to buy an infinite sized piece of paper. Penrose diagrams cover the whole of the same two dimensions on a finite piece of paper, at the expense of yet more coordinate transforms.

Edit: although the coordinates are called Kruskal-Szekeres I believe the diagram is attributed to Kruskal alone. So it's not a KS diagram.

Ibix said:

I believe the diagram is attributed to Kruskal alone. So it's not a KS diagram.

However I believe it is a complete diagram of the underlying Schwarzschild spacetime manifold.

cianfa72 said:

I believe it is a complete diagram of the underlying Schwarzschild spacetime manifold.

It is a complete diagram of a 2D subspace of the maximal analytic extension of the Schwarzschild spacetime manifold. The 2D subspace is the one that is orthogonal to the 2-sphere subspace of the manifold that is induced by spherical symmetry. So every point on the diagram that is within the manifold (i.e., within the boundaries given by the two singularities) represents a 2-sphere.