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bobrubino
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Which one would you use in order to map out a black hole and its connection to a white hole?
Neither.bobrubino said:Which one would you use in order to map out a black hole and its connection to a white hole?
Honest since it is an one-to-one mapping of a region of spacetime.Ibix said:But it's important to realise that the Euclidean representation is honest but not accurate.
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.cianfa72 said:Honest since it is an one-to-one mapping of a region of spacetime.
That is an excellent way to put itIbix said:it's important to realise that the Euclidean representation is honest but not accurate
KS diagrams (region I - IV) cover the entire spacetime manifold ?Ibix said:Kruskal diagrams are actually working tools, but the interpretation remains non-trivial.
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.cianfa72 said:KS diagrams (region I - IV) cover the entire spacetime manifold ?
However I believe it is a complete diagram of the underlying Schwarzschild spacetime manifold.Ibix said:I believe the diagram is attributed to Kruskal alone. So it's not a KS diagram.
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.cianfa72 said:I believe it is a complete diagram of the underlying Schwarzschild spacetime manifold.
The main difference between Minkowski spacetime and Euclidean spacetime is the nature of their metrics. Minkowski spacetime, used in the theory of relativity, has a metric with one time dimension and three spatial dimensions, characterized by the signature (-+++). In contrast, Euclidean spacetime has four spatial dimensions with a metric signature of (++++), meaning all dimensions are treated equally and there is no distinction between time and space.
Minkowski spacetime is crucial in the theory of relativity because it provides a geometric interpretation of spacetime where the effects of special relativity, such as time dilation and length contraction, can be naturally described. It allows for the unification of space and time into a single four-dimensional continuum, making it possible to describe the invariant interval between events and the behavior of objects moving at relativistic speeds.
Euclidean spacetime is not suitable for describing relativistic phenomena because it lacks the distinction between time and space that is essential for capturing the effects predicted by the theory of relativity. The Euclidean metric does not accommodate the invariant speed of light or the causal structure necessary to describe events in a relativistic framework.
In Minkowski spacetime, the concept of distance is defined by the spacetime interval, which can be positive, negative, or zero, depending on the relative positions of events in time and space. This interval is invariant under Lorentz transformations. In Euclidean spacetime, distance is always a positive quantity calculated using the Pythagorean theorem, treating all four dimensions symmetrically and without distinguishing between space and time.
The metric signature is fundamental in distinguishing between Minkowski and Euclidean spacetimes. Minkowski spacetime has a metric signature of (-+++), indicating one time dimension and three spatial dimensions, leading to the unique properties of time and causality in relativity. Euclidean spacetime, with a metric signature of (++++), treats all four dimensions as spatial, which is suitable for different mathematical and physical contexts but not for describing relativistic effects.