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>You wrote that the singularity is a point in space Because it is. Remember: space, not spacetime. Hence the remark in brackets in the original comment and my

>You wrote that the singularity is a point in space Because it is. Remember: space, not spacetime. Hence the remark in brackets in the original comment and my reminder to read precisely in the other one. And in Eddington Finkelstein it is most obvious that it is a point in space (i.e. it has spatial coordinate r=0 where r has the metric signature of a spatial dimension) that you can hit at various points in (global) time (and actually also in free falling observer time, but let's ignore that since it is not immediately obvious). You can literally trace incoming light rays crossing the event horizon and hitting the singularity at r=0 at a certain points in time in the diagram. This stuff is really not that weird once you choose less confusing coordinates. It only gets weird once you start asking what local observers can actually see, because from their perspective their relation to all other coordinates in spacetime gets really messy. That's probably where 95% of the confusion among laypeople comes from. But for that you can still resort to other coordinates which show it much better.

pdonis 23d ago
Sorry, you're just repeating the same wrong statement. I know you said "space", and I already explained that a spacelike line in spacetime cannot be a point in space. It can only be a moment of time. You are quite correct that, since the singularity is a line in spacetime, different incoming light rays (or free-falling observers, for that matter) can hit it at different points. Depending on how you choose your coordinates, you can set it up so that those points have different "time" coordinates. But that doesn't make the singularity a point in space. It means you're running up against relativity of simultaneity--whether or not different events on a spacelike line (or more generally a spacelike surface) happen at the same time depends on your choice of coordinates. You can, in fact, choose coordinates in which all events on the singularity happen at the same time (for a "time" coordinate that is genuinely timelike--see below). The standard Penrose chart does that, for example. You are also correct that a good choice of coordinates can make it easier to see certain properties of a spacetime geometry. But it can also make it harder to see other properties. In this case, your choice of Eddington-Finkelstein coordinates is making it harder for you to see why your claim that the singularity is a point in space is wrong, and why the things I said above are true. For example, inside the horizon, the Eddington-Finkelstein "time" coordinate that you are using is not timelike. It's spacelike. In other words, it's not actually a "time" coordinate (even though it's labeled as such). It is actually a "space" coordinate! You should be able to see this by observing that the singularity is a spacelike line, and in E-F coordinates it's a vertical line--i.e., the only coordinate that changes along it is the "time" coordinate. That means the "time" coordinate must actually be spacelike there. And, for extra confusion, the r coordinate in Eddington-Finkelstein coordinates is also spacelike, even inside the horizon (unlike in Schwarzschild coordinates, where it becomes timelike). So in this chart there is no coordinate that is timelike inside the horizon! That means any timelike curve inside the horizon must have more than one coordinate in this chart that changes along it (in the simplest case, a radial timelike curve, both the "time" and r coordinates must change along the curve).
- sigmoid10 21d ago
  Sorry, you are still arguing against things I never said or that you desperately want to misinterpret in a disingenuous way. If I was actually wrong about anything I said, all you had to do was write down the explicit metric and point out exactly where it disagrees with what I said. But if you did, you would immediately see that your argumentation falls apart.
  - pdonis 21d ago
    > you are still arguing against things I never said You said the singularity is a point in space. That's what I'm arguing against. The rest was an attempt to try to help you understand the correct physics. Evidently it was wasted effort. I won't do it again. > all you had to do was write down the explicit metric and point out exactly where it disagrees with what I said. Sure, it's the one in terms of t' and r in the Wikipedia article on Eddington Finkelstein coordinates. [1] > if you did, you would immediately see that your argumentation falls apart. No, I see that yours does. The article is using the timelike signature convention, so positive ds^2 is timelike and negative ds^2 is spacelike. Vertical lines in a spacetime diagram (like the one just a little bit below the metric, on the right, that shows the light cones) are intervals where only dt' is nonzero. It is obvious from the metric that for any r < 2M, i.e., anywhere inside the horizon, such intervals give a negative ds^2, since 1 - 2 GM / r is negative. So vertical lines, of which the singularity is one, are spacelike, and t' is a spacelike coordinate inside the horizon (just as r is). And a spacelike line cannot be a point in space. It can only be a moment of time. The fact that it is vertical on the diagram does not change that. [1] https://en.wikipedia.org/wiki/Eddington%E2%80%93Finkelstein_coordinates https://en.wikipedia.org/wiki/Eddington%E2%80%93Finkelstein_...

pdonis 23d ago

Sorry, you're just repeating the same wrong statement. I know you said "space", and I already explained that a spacelike line in spacetime cannot be a point in space. It can only be a moment of time. You are quite correct that, since the singularity is a line in spacetime, different incoming light rays (or free-falling observers, for that matter) can hit it at different points. Depending on how you choose your coordinates, you can set it up so that those points have different "time" coordinates. But that doesn't make the singularity a point in space. It means you're running up against relativity of simultaneity--whether or not different events on a spacelike line (or more generally a spacelike surface) happen at the same time depends on your choice of coordinates. You can, in fact, choose coordinates in which all events on the singularity happen at the same time (for a "time" coordinate that is genuinely timelike--see below). The standard Penrose chart does that, for example. You are also correct that a good choice of coordinates can make it easier to see certain properties of a spacetime geometry. But it can also make it harder to see other properties. In this case, your choice of Eddington-Finkelstein coordinates is making it harder for you to see why your claim that the singularity is a point in space is wrong, and why the things I said above are true. For example, inside the horizon, the Eddington-Finkelstein "time" coordinate that you are using is not timelike. It's spacelike. In other words, it's not actually a "time" coordinate (even though it's labeled as such). It is actually a "space" coordinate! You should be able to see this by observing that the singularity is a spacelike line, and in E-F coordinates it's a vertical line--i.e., the only coordinate that changes along it is the "time" coordinate. That means the "time" coordinate must actually be spacelike there. And, for extra confusion, the r coordinate in Eddington-Finkelstein coordinates is also spacelike, even inside the horizon (unlike in Schwarzschild coordinates, where it becomes timelike). So in this chart there is no coordinate that is timelike inside the horizon! That means any timelike curve inside the horizon must have more than one coordinate in this chart that changes along it (in the simplest case, a radial timelike curve, both the "time" and r coordinates must change along the curve).

sigmoid10 21d ago
Sorry, you are still arguing against things I never said or that you desperately want to misinterpret in a disingenuous way. If I was actually wrong about anything I said, all you had to do was write down the explicit metric and point out exactly where it disagrees with what I said. But if you did, you would immediately see that your argumentation falls apart.
- pdonis 21d ago
  > you are still arguing against things I never said You said the singularity is a point in space. That's what I'm arguing against. The rest was an attempt to try to help you understand the correct physics. Evidently it was wasted effort. I won't do it again. > all you had to do was write down the explicit metric and point out exactly where it disagrees with what I said. Sure, it's the one in terms of t' and r in the Wikipedia article on Eddington Finkelstein coordinates. [1] > if you did, you would immediately see that your argumentation falls apart. No, I see that yours does. The article is using the timelike signature convention, so positive ds^2 is timelike and negative ds^2 is spacelike. Vertical lines in a spacetime diagram (like the one just a little bit below the metric, on the right, that shows the light cones) are intervals where only dt' is nonzero. It is obvious from the metric that for any r < 2M, i.e., anywhere inside the horizon, such intervals give a negative ds^2, since 1 - 2 GM / r is negative. So vertical lines, of which the singularity is one, are spacelike, and t' is a spacelike coordinate inside the horizon (just as r is). And a spacelike line cannot be a point in space. It can only be a moment of time. The fact that it is vertical on the diagram does not change that. [1] https://en.wikipedia.org/wiki/Eddington%E2%80%93Finkelstein_coordinates https://en.wikipedia.org/wiki/Eddington%E2%80%93Finkelstein_...