relativityfan said:
thank you for this link.
after having read this (ingoing and outgoing fluids) I do believe that even without the mass inflation, there is no white hole.
the structure of the "white hole" if we look at the metric should be exactly the same as the structure of the black hole, and matter could excape because it can be accelerated faster than the speed of light(like matter inside the event horizon) . So this would not be a white hole but a black hole, because the metric would be the same.
Does anyone disagree with this?
Yes, it is incorrect according to GR, which is a time-symmetric theory so that the time-reverse of any solution (like a black hole that forms at some time from a collapsing star, then exists eternally afterwards, the time-reverse of which would be a white hole that has existed eternally and then finally 'explodes' into an expanding star) must also be a valid GR solution. And if one wants to find the "maximally extended" spacetime for an eternal black hole (i.e. a spacetime where geodesics can be extended arbitrarily far in both directions of their proper time
unless they hit a singularity at some finite proper time), one must include a white hole interior region that's separate from a black hole interior region. The reason this is needed is that there are possible worldlines of particles outside the event horizon which in exterior Schwarzschild coordinates or ingoing Eddington-Finkelstein coordinates (see the bottom half of
this page for an intro. to different coordinate systems for a Schwarzschild black hole) have been rising
away from the event horizon for an infinite coordinate time--in the limit as coordinate time goes to negative infinity, the distance of these particles from the horizon approaches zero but their proper time approaches some finite value, so a "maximally extended" spacetime must include a region where they came from
before crossing the event horizon in the outward direction. On the page I linked to above, look at the orange worldlines to the right of the vertical red line representing the event horizon, which are moving away from the event horizon as time increases but which approach it in the limit as time goes to negative infinity:
Schwarzschild coordinates
Eddington-Finkelstein coordinates
By means of a coordinate transformation one can transform to Kruskal-Szekeres coordinates where these outgoing worldlines crossed the event horizon in the outward direction at finite coordinate time rather than at negative infinity (in this coordinate system the event horizon is split into two different lines at 45 degrees from vertical, one shown as pink and the other shown as dark red):
But this diagram does not illustrate a "maximally extended" spacetime because those outgoing worldlines just end at the event horizon (this diagram only shows the worldlines of outgoing
light rays which don't have a 'proper time' although one can define an 'affine parameter' for them that's similar to proper time, but in this diagram one could also draw on the worldlines of outgoing slower-than-light particles which would also cross the event horizon at finite coordinate time and at a finite value of their proper time, whereas once again 'maximally extended' implies that both proper time and other affine parameters should extend to arbitrary values unless the worldline runs into a singularity). To have a maximally extended spacetime you have to continue those outgoing worldlines by having them come from a "white hole interior region" at the bottom, and you also have to add a second "exterior region" (a sort of parallel universe) on the left:
Also, this statement of yours doesn't really make sense to me:
the structure of the "white hole" if we look at the metric should be exactly the same as the structure of the black hole, and matter could excape because it can be accelerated faster than the speed of light(like matter inside the event horizon) .
No matter moves "faster than the speed of light" in a local sense, whether inside or outside the event horizon. We can choose a coordinate system where light always moves at the same coordinate speed throughout the black hole spacetime, and where massive objects always travel slower than light--this is true of the coordinates used to draw a Penrose diagram and also of
Kruskal Szekeres coordinates which can be obtained by a transformation from Schwarzschild coordinates, I recommend reading the
nontechnical section of the wikipedia article on KS coordinates to get an idea of how things work in this coordinate system and how they relate to Schwarzschild coordinates. Note that region I on the diagram on that page corresponds to "our" region of spacetime outside the black hole, region II corresponds to the region in the interior of the black hole where infalling particles end up, and region IV on the bottom corresponds to the white hole interior region where outgoing particles must have come from (region III is the 'parallel universe' which is not entirely disconnected from our region since particles that have fallen into the black hole event horizon from region I may meet up with particles that have fallen in from region III).
Anyway, once you've reviewed the basic features of the Kruskal-Szekeres diagram (most of which are duplicated in the Penrose diagram), consider again what I said in an earlier post:
Penrose diagrams and Kruskal-Szekeres diagrams are designed with the property that all worldlines of light rays are at 45 degrees and all worldlines of slower-than-light objects have slopes closer to vertical than 45 degrees, just like a Minkowski diagram in SR, so from this perspective the reason it's impossible to enter a white hole horizon from the outside is just the same as the reason it's impossible to enter the past light cone of a given event from the outside in SR (because it's a surface moving inward at c so you can't catch up with it to cross it).
Do you see that in terms of these coordinate systems, as coordinate time progresses the white hole event horizon contracts inwards at the speed of light, until at the center of the diagram it becomes a black hole event horizon which expands outward at the speed of light? Can you see why this means that nothing can cross the white hole event horizon starting from the outside going in, while nothing can cross outward from the interior to the exterior of the black hole event horizon?
