Let the maximal extension of the Schwarzschild spacetime be given. It admits as coordinates the Kruskal-Szekeres coordinates $(T,X,\theta,\phi)$ with $$T^2-X^2<1$$

since the singularity occurs at $T^2-X^2=1$. This spacetime is divided into four regions:

  • Region I: this is the exterior region. One can define in this region the usual $t,r$ coordinates by $$r=2M\left(1+W_0\left(\dfrac{X^2-T^2}{e}\right)\right),\quad t=4M\tanh^{-1}\dfrac{T}{X}$$

  • Region II: this is the black hole region. One can also define the above two coordinates here, but now they are $$r=2M\left(1+W_0\left(\dfrac{X^2-T^2}{e}\right)\right),\quad t=4M\tanh^{-1}\dfrac{X}{T}$$

  • Region III: this is the parallel exterior region, on which we have $$r=2M\left(1+W_0\left(\dfrac{X^2-T^2}{e}\right)\right),\quad t=4M\tanh^{-1}\dfrac{T}{X}$$

  • Region IV: this is the white hole region, on which we have $$r=2M\left(1+W_0\left(\dfrac{X^2-T^2}{e}\right)\right),\quad t=4M\tanh^{-1}\dfrac{X}{T}$$

Now, regions I and II together comprise the usual Schwarzschild spacetime. On the other hand, we have also regions III and IV. I remember Wald says in his book these regions are unphysical.

Is that really the case? There is no physical interpretation for the regions III and IV? Specially the parallel exterior region, isn't there any known physical interpretation of what it might be physically or how it might really exist in a given situation?

Is the maximal extension of Schwarzschild spacetime physically meaningful or is it just mathematically meaningful?


2 Answers 2


Is that really the case? There is no physical interpretation for the regions III and IV? Specially the parallel exterior region, isn't there any known physical interpretation of what it might be physically or how it might really exist in a given situation?

Yes, that's really the case. Those regions don't exist for a black hole that forms by gravitational collapse. For a black hole that forms by gravitational collapse, the Penrose diagram looks like this:

penrose diagram of an astrophysical black hole

Related: Can black holes form in a finite amount of time?


While for the description of an astrophysical black hole the maximally extended Schwarzschild spacetime is indeed unphysical/inapplicable, it is possible to gain a degree of physical interpretation and/or intuition regarding regions III and IV if we consider this maximally extended spacetime to be a limit of a continuum of T-invariant spacetimes that on the other end have an explosion of white hole — collapse into black hole.

First, let us clarify how the “white hole” could appear. Consider the spherically symmetric collapse of dust-like matter into a black hole. We can choose initial data at a certain space-like slice (let it be the section $t=0$) in a way that the spacetime would be symmetric under time reversal. Then, while for positive times we would have collapse of the dust cloud toward the black hole, at negative time we would have the dust cloud expanding from under the past horizon, in other words we would have a white hole exploding in a cloud of dust. This spacetime is geodesically complete, worldline of every matter particle starts at the white hole singularity and ends in the black hole singularity after a finite proper time.

To construct the explicit version of such a spacetime we could glue together two regions, the empty region with a Schwarzschild metric and a region with matter which is a patch of closed isotropic homogeneous FLRW universe with dust-like matter. The regions are glued along the spherically symmetric and invariant under time reversal boundary composed of worldlines of dust particles. The junction conditions for this boundary are: 1) the metrics induced on it by both regions are the same and 2) there is no finite surface densities of stress-energy on it. The second condition simply means that the worldlines of boundary particles are both comoving geodesics of FLRW spacetime and radial geodesics with a turning point at $t=0$ of the Schwarzschild metric. Birkhoff theorem and GR version of the shell theorem imply that it is enough to just match proper time of FLRW universe from Big Bang to Big Crunch to proper time from white hole singularity to black hole singularity of the Schwarzschild metric (this would gives us correct FLRW solution), and to cut out from this FLRW universe patch bounded at the moment of maximal expansion by a sphere with radius equal to maximum value of radial variable $r$ on the Schwarzschild side.

We could illustrate this construction by a series of images taken from the book

  • Frolov, V., & Novikov, I. (2012). Black hole physics: basic concepts and new developments (Vol. 96). Springer Science & Business Media, Google Books.

(this family of solutions was obtained by I. Novikov in 1963).

Expanding and collapsing dust sphere

To the left, there is a spacetime diagram in Lemaître-like coordinates (so that $R=\mathrm{const}$ corresponds to radial timelike geodesic). Note, that along the line $R=\mathrm{const}$ the radial variable $r$ starts from zero at both top and bottom singularities and achieves maximal value at the turning point $T=0$ (which is also $t=0$). Doubly dashed region is a FLRW patch, Schwarzschild region is to the right. Big Bang singularity of the FLRW metric joins with the past singularity of a white hole, while Big Crunch joins with the future singularity of a black hole. The right part of the image is an embedding diagram for the spatial section of a spacetime at the turning moment $t=0$, when expansion of a dust sphere changes to contraction (with one of the angular coordinates suppressed). The dust matter part of the geometry is precisely part of a $S_3$ sphere while outside region is an embedding of spatial Schwarzschild geometry.

Radial Schwarzchild geodesics are characterized by the value of specific energy $\epsilon=E/mc^2$, with $\epsilon<1$ since we are interested only in bound motion. By varying this parameter we would obtain a family of similar solutions. Values $\epsilon=1-\text{small number}$ correspond to dust cloud expanding to a large radius before collapsing, smaller positive value of $\epsilon$ corresponds to smaller radius of maximal dust cloud expansion.

When $\epsilon=0$, the maximal expansion radius is equal to Schwarzschild radius of the black hole:

E&C dust sphere with the maximum expansion radius = rg

We have precisely half of FLRW solution glued together with precisely half of maximally extended Schwarzschild manifold, (full region I, right halves of regions II and IV). Dust from Big Bang singularity only touches the horizon without emerging from it, and then falls back. The spatial embedding diagram for $t=0$ has half of a sphere glued together with spatial Schwarzshild along the horizon.

But we also know that inside the horizon there are geodesics with negative specific energies. Such solutions would have following diagram:

Dust universe cloud is inside the horizon

We see that there appears a piece of the parallel world region III of the Schwarzschild manifold. And this region together with the FLRW patch (which is now comprises more than a half of a manifold) could be interpreted as a separate world, causally separated from the world of region I. This world is a universe of closed cosmology in which (once the universe expanded large enough) a void appears without dust matter but with a black hole of equivalent mass. The embedding diagram of spatial section $t=0$ exhibits a characteristic throat feature, the Einstein–Rosen bridge.

As the parameter $\epsilon$ becomes closer and closer to $-1$ the time interval between Big Bang and Big Crunch of the parallel world become longer and longer, while the size of dustless void becomes larger and larger. So taking the limit and replacing closed cosmology with asymptotically flat spacetime we arrive at the maximally extended Schwarzschild manifold:

enter image description here

This family of solution is, of course, rather artificial/contrived and could not be considered realistic. But still similar manifolds could have some physical meaning in quantum theory in light of ER=EPR conjectures.

For example, in the paper by van Raamsdonk, arXiv:1005.3035, maximally extended Schwarzshild-AdS spacetime appears as a gravity dual to a state of pair of entangled CFT .

image from van Raamsdonk's paper


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