Eternal Black Holes

What is the definition of an eternal black hole? Studying white holes and the term appears in relation to this field of research.

• Related by OP: physics.stackexchange.com/q/161201 Jan 25, 2015 at 19:27
• It refers to a theoretical idealisation of a (usually stationary) black hole that has existed forever, rather than being formed from the collapse of some matter or similar. This leads to some strange consequences (white holes, Einstein-Rosen bridges, ...) so one must be a little careful when interpreting the physics from these models. Jan 25, 2015 at 19:32
• how could u form a black hole from nothing collapsing then? Jan 25, 2015 at 19:43
• See e.g. p.2 in arXiv:hep-th/0106112. Jan 25, 2015 at 21:34
• Albert, the eternal black hole is a non-physical solution; it is the static, spherically symmetric vacuum solution to Einstein's gravitational field equations. There is no matter to collapse to a black hole in this solution. The event horizon is an eternal, unchanging feature of the spacetime. Jan 25, 2015 at 21:48

When students are taught GR one of the metrics they learn about first is the Schwarzschild metric. We tend to say, rather casually, that the Schwarzschild metric describes a non-rotating black hole. However the Schwarzschild metric is time independant, so it describes a non-rotating black hole that has existed for an infinite time and will exist an infinite time into the future. In other words it describes an eternal black hole.

Real black holes obviously can't have formed more than 13.8 billion years ago, because that's how old the universe is. So the Schwarzschild metric is only an approximate description of real black holes, though we would expect the approximation to be exceedingly good for a black hole of any reasonable age.

Anyhow, the Schwarzschild metric is written using the Schwarzschild coordinates time $t$, radial distance $r$, and the angles $\theta$ and $\phi$. And written this way it looks fairly unremarkable. But the trouble is that the Schwarzschild coordinates are singular at the event horizon and badly behaved inside it ($r$ becomes timelike and $t$ spacelike). So ever since Schwarzschild first wrote down the metric general relativists have looked for a better set of coordinates to use. One very useful set of coordinates is the Kruskal-Szekeres coordinates because these remain non-singular except at the central singularity, plus the $u$ coordinate is spacelike everywhere and the $v$ coordinate is timelike everywhere. So the coordinates remain well behave everywhere.

But there's something a bit odd about the KS spacelike $u$ coordinate. In the Schwarzschild coordinates the $r \ge 0$ because it's just a radial distance and obviously that can't be less than zero. But the KS $u$ coordinate can be less than zero, and if we draw a spacetime diagram of our block hole using the KS coordinates we get:

The top red line is the singularity and the dashed line is the event horizon.

The dashed lines split the spacetime diagram into four parts, helpfully labelled 1 - 4, and parts 1 and 2 are the regions of space time described by the Schwarzschild coordinates. But by using the KS coordinates we see there are two new regions of spacetime that Schwarzschild didn't know about. Region 3 is a region of spacetime outside the event horizon and linked to our patch of spacetime by a non-traversible wormhole (specifically an Einstein Rosen bridge). Region 4 is inside the event horizon of a past white hole.

So if we draw a spacetime diagram using the full range of the Kruskal-Szekeres coordinates (known as maximally extended coordinates) we find that our apparently boring black hole must include a white hole, a wormhole and a region of spacetime that is normal but forever inaccessible to us.

And this is why you'll find the concepts of white holes and eternal black holes to be linked. Because they are both present when we draw the spacetime diagram of the eternal black hole.

But ...

The physical relevance of regions 3 and 4 is debatable. Because our mathematical model predicts them doesn't mean they have to exist. It may just mean we've pushed the model beyond the bounds of its physical applicability. And of course real black holes aren't eternal, and aren't described by the spacetime diagram above anyway.

• I'll try to find a diagram to complement yours that shows qualitatively the (spherically symmetric) collapsing surface of a star. In that case, only the geometry outside the surface is Schwarzschild eliminating some of region 2 and all of regions 3 and 4. Jan 25, 2015 at 21:53

Real black holes form through stellar collapse (and possibly a few other mechanisms) and from the point of view of an observer observing the black hole from a safe distance away they would be able to easily talk about a time before the black hole existed.

When the first black hole solutions were created they used the idealization that black holes, from the point of the faraway observer, had always existed. After all, for example, when describing planetary orbits using Newtonian physics we don't often need to worry about how solar system came to form and so don't include it in our models.

However it was recognized that these idealized solutions had features that more realistic solutions would not have and so they came to be know as "eternal black holes". One of these features is that when you maximally extend the Schwarzschild solution by time reversing the rays moving radially outward from the black hole you encounter a white hole region (as well as another asymptotically flat region).

The way I personally think about a white hole is that they exist in the infinite past of the faraway observer observing the black hole. However a note of caution is that these is only the most roughest and ready of pedagogic explanations and much better understanding can be got by looking at (and understanding) the Penrose diagram of the maximally extended Schwarzschild solution.

• Apologies for my ignorance, but how could we say that from the point of view of the observer that the black hole could exist forever, when the observer witnessed (and acknowledged) both the creation and the subsequent time before its creation? Jan 25, 2015 at 20:13
• It's an idealized situation so from the point of view of the faraway observer they never witness any time before the creation of the black hole, even though this is not a realistic way to describe an actual black hole. Jan 25, 2015 at 20:16
• So the observer only witnesses the hole's formation, and nothing else? Jan 25, 2015 at 20:23
• No they never witness the formation of the black hole as from their point of view it is eternal (without temporal beginning or end). Jan 25, 2015 at 20:27
• ah so an eternal black hole is just an idealization? Jan 25, 2015 at 20:29

I think it helps to define quite a few terms.

We have spacetime, it's where we keep all our stuff both the places where they happen and the times when they happened. For instance the point $(10,0)$ could be a distance ten away from our origin, and a time of zero from our origin. And the point $(10,1)$ could be a distance ten away from our origin, and a time of one unit from our origin, same place, different time, so different point in spacetime. Conversely, the point $(9,1)$ could be a distance nine away from our origin, and a time of one unit from our origin, same time, but different place, so different point in spacetime.

OK, then the interesting thing is that we can assign the coordinates to keep all these events separate, but change how we measure the distances and times to take into account the curvature. So now we have some $(x,y,z,t)$ labels, but we have to do math to find out how far or long they are from each other.

So now we have spacetime, and we can define the next idea, an event horizon. An event horizon is a boundary from one region of spacetime and another such that you can only cross one way. Since you always move towards the future and always at the speed of light, if a surface expanded at the speed of light then once you cross it you are stuck on that side. This is because it is a boundary in spacetime, not just a surface in space.

A black hole is called a black hole because of the event horizon, it means that we never see whats on the inside (so to us the inside is black, in that we never see it). But inside the black hole there is something totally different, or there might be, its all theoretical .. by definition. Since we can't see it.

The thing inside might be a singularity. For a normal (nonrotating) black hole, the hypotehtical singularity might fill the whole inside in such a way that if you cross the event horizon you will hit the singularity no matter what, it's in your future and in the way and every direction you take to the future after crossing the event horizon hit it.

let's draw this. Set up a 2d space. Set the line $y=x$ for $x>0$ to be the event horizon. Set lines $y=sx$ for $x>0$ with $s$ in $-1<s<1$ as lines of constant time outside the horizon ($s=-1$ would be like $t=-\infty$ and $s=+1$ would be like $t=+\infty$. Note that the event horizon looks like a surface of $t=+\infty$ and that's not an accident, that's because the time is time measured by the outside, and time on the outside never sees you cross, and never seeing is like seeing something that happened at $t=+\infty$ so you can see it get close to happening but you literally never see it happen. The outside never sees the event horizon.

And curves like $x^2-y^2=s^2$ with $x>0$ to be surfaces of constant distance $s$ from the surface of the event horizon. This is an eternal black hole in that you can start from outside the black hole at any time,no matter how far back, and the event horizon is already there when you get to it, and then later so is the singularity. Let's put in that singularity.

The curve $y=\sqrt{1+x^2}$ is the singularity. And we can set up the metric so that particles can go upwards always, and sloping towards the left is going into the black hole and sloping right (as you go up) is away.

So once you cross the line $y=x$ even if you slope as far right as you can (45 degrees slope is moving at the speed of light away from the center) then you hit the singularity. It is unavoidable. And if you are outside you can slope left (the most you can slope is 45 degrees to the left as you go up) and cross the event horizon.

So that's a black hole that never formed but was just always there, the event horizon marks the inside versus the outside, once in you never escape and inside there is a singularity that can't be avoided once you cross in. This is not realistic, just a simple example of a possible universe that would satisfy the theory of GR, it's not our universe.

The part covered by the lines with $y=sx$ with $x>0$ and $s$ in $-1<s<+1$ cover all the possibilities outside the event horizon for all times. But there are future pointing curves that go from outside to inside the event horizon.

So we have spacetime, event horizons, and a black-hole singularity that lives inside. Are there other things inside? One interesting possibility that is consistent with GR is that there is a whole other universe that shares the inside of the black hole, that if someone there and someone here both went in they could meet each other. Since once either goes in they won't get out, the two different outsides can't ever see each other unless they go inside, and once they go inside they can't send a signal to either universe. So let's make this second universe.

We didn't use $x<0$ for the outside, so let's use it. A line like $y=sx$ for $x<0$ is a line of constant outside time for the other universe, and there is a second event horizon, $y=-x$, and now from either outside you can cross one fo the two event horizons and then you'll reach the same inside, and hit the same singularity $y=\sqrt{1+x^2}$. The curves like $x^2-y^2=s^2$ with $x<0$ are now to be surfaces of constant distance $s$ from the surface of the second event horizon in the second universe in the second outside. Both universes would think of themselves at the primary ones and the other one as weird that can only be reached.

OK, so we have spacetime. We have an event horizon, and an outside. And a whole different outside with a whole different event horizon, and inside the two event horizons we have the inside of the black hole, and it has an unavoidable singularity. it is unavoidable, because we set up our metric to require the $y$ always get larger, and that you can angle at most 45 degrees (either right or left).

So basically when $y>|x|$ you are inside the black hole. When $x>|y|$ you are outside the black hole in one universe, and when $x<|y|$ you are outside the black hole in another universe. Eternal black holes could be very weird (we don't know for sure, we've never been inside, and couldn't report back even if we did). But it was weird that when you get inside you see things you never saw before. But is there another way in beside coming from on of these black holes?

Again, the math of GR allows yet another option. This is where the white hole will come in. The inside of the black hole never affects the outside(s). If you imagined running time backwards, then the outside never "reverse-affects" the inside. And we have a region for something like that. Inside the black hole (where $y>|x|$) you hit the singularity $y=+\sqrt{1+x^2}$ no matter what you do. What if we put a singularity at $y=-\sqrt{1+x^2}$, then inside the black hole $(0,1/2)$ is a fine event, someone from either universe could get there if they moved fast enough and started early enough. But someone from $(0,-1/2)$ could get there too. So mathematically it is possible to have a third place people can come from that end up in the inside of the black hole.

This third place is very weird it is like the inside of a black hole, but with things running backwards. In the sense that you can leave from it to the two outside and you have a singularity in your past. And every point in this region has a singularity in it's past.

Like us, if there was a singularity in the big bang, then it's in our past for everything in our universe, not just from some parts of it. If you go from this region, you have a singularity in your past and you could go to either outside or to the inside of the black hole. Since you can get to those outsides, this region is not inside the black hole. Just as the outside can treat the inside of the black hole as beyond the $t=+\infty$ surface since they can never see it. The white hole region is beyond (earlier than) $t=-\infty$ surface.

The white hole is from before the outside. If you pick any outside time (line $y=sx$ with $s$ in $-1<s<+1$), then the region of the white hole is inaccessible because its too far in the past, just like you can't see inside the black hole because it is inaccessible for being too far in the future (for the outside). A point in the outside has already been fully affected by every part of the white hole and trying to move to the inside takes you to the black hole instead of the white hole. It's like the cemetary, you can go there, but you aren't going to talk to someone that is gone because they just aren't there anymore.

That's the point of assigning coordinates to these event, so we can see what can affect what and take into account that rulers and clocks one person uses might be relative and different than what someone somewhere or somewhen else would expect.

So the white hole can effect both outsides and the black hole, and either outside can get to the inside of the black hole but then will never get out. But if you describe your entire outside at some fixed outside time $t$ which corresponds to a line $y=sx$ with $-1<s<+1$ then every part of the black hole has already had it's last chance to affect the outside, in that since it is over. And that line $y=sx$ with $-1<s<+1$ still has the chance to get to any point inside the black hole, so the black hole is still 100% in the future.