# Spacetime curvature around a black hole

I recently got an understanding of general relativity and what it means for spacetime to be bend, and what 4d spacetime really is (https://www.youtube.com/watch?v=sryrZwYguRQ).

I understand that due to the equivalence theorem in GR, and since the magnitude of the four-velocity is always the speed of light, as time slows down near an object, movement in the x, y, and z.

In addition, this animation was good at giving some visuals, explaining how the geodesics apply here. https://www.youtube.com/watch?v=DdC0QN6f3G4.

My question is, in a black hole, what would happen to the curvature of the spacetime in the video link directly above.

When I looked at the answer to this question, Spacetime around a Black Hole, it has these pictures that aren't very helpful. This notion of spacetime some how being ripped, doesn't provide me an intuitive (or mathematical understanding) of what is going on. A mathematical explanation of what is going on would be nice (as long as it isn't incredibly complex).

• It's incredibly complex.........because the visuals of it are not to be relied upon and we can use only math (not pictures) to really describe it properly. I am kinda kidding, it depends on what math you know already, if you look at Amazon and read the Table of contents of Relativity by Hartle, you will get an idea of what you need to know.
– user163104
Commented Jul 19, 2017 at 22:29
• It's not really clear to me what you want to know here - are you looking for someone to write down the solution for the metric of a Schwarzschild blackhole? Without watching the videos you linked, it's not clear what sort of understanding you are after, and questions on our site should be self-contained. Commented Jul 19, 2017 at 22:54
• I don't want to sound discouraging or anything, but you definitely did not understand curved spacetime with a couple of YouTube videos. It takes people years of serious study. Commented Jul 19, 2017 at 22:57
• @user262328 what it means mathematically for a black hole to "rip spacetime" is that our best mathematical understanding of spacetime stops looking normal at the horizon. Spacetime is not damaged in some way by black holes, but our understanding of spacetime is incomplete. Commented Jul 20, 2017 at 16:02
• I recommend reading this: en.wikipedia.org/wiki/Eddington–Finkelstein_coordinates Your question, if still unanswered, will hopefully get clearer then.
– J.G.
Commented Jul 21, 2017 at 23:12

## 2 Answers

General relativity is a physical model. We expect this model to fail in the center of a black hole since the theory tells us that the curvature there diverges which we don't believe. Most probably the solution to this singularity-problem of classical GR is a quantized renormalizable theory of gravity. But maybe not (It could be a bookshelf there as well ;) ). People are working on that. Important to know is just that GR is believed not to be the end of the story.

The math to understand GR is basically Differential geometry. Being familiar with the basic concepts is probably enough to see that GR has singularities (see e.g. in Bernard Schutz - A First Course in General Relativity. This book can also be found as a pdf using google).

Edit: This is the answer why people draw the spactime of a black hole as "ripped" in a diagram. Its due to the singularity at the center of the black hole not the event horizon. At the event horizon there is no singularity and also no "ripping".

The core of the trick is that when you are parallel transporting vectors in a curved surface, the final direction of the vector depends on the path you take. So, if you start out with two timelike vectors, and let one of them take a path toward the black hole and the other one staying outside the black hole, the one that falls in will have its path curve in such a way that its time and space coordinates get "mixed up" relative to the vector staying outside of the black hole. At the event horizon, the radial direction of the first vector will actually be the "foward in time vector".

What the curvature of the Schwarzschild spacetime does is simply to mix up space and time directions in such a way that one person's "falling into the black hole" is another's "going forward in time".