The apparent similarities between the Big Bang singularity and the singularity at the center of a black hole have intrigued physicists since the discovery of the Schwarzschild solution to the Einstein field equations. Both represent a breakdown of general relativity where the spacetime curvature becomes infinite and the concept of time loses meaning. The key difference is one of temporal orientation - the Big Bang singularity exists in the past for all timelike and null geodesics in our observable universe, while a black hole singularity exists in the future for all timelike and null geodesics that have crossed the event horizon.

However, the mathematics of general relativity is time-reversible, allowing for the theoretical existence of "white holes". A white hole is a time-reversed black hole where the singularity exists in the past and the event horizon can only be crossed in the outward direction. Infalling matter is eternally repelled, while outgoing null geodesics generate the event horizon. Thus, a white hole with a past singularity and an event horizon that cannot be entered from the outside begins to resemble our Big Bang and observable universe.

The major difference is that a white hole would seem to be a localized object embedded in a larger universe, while our observable universe has no known "exterior". The cosmic microwave background radiation is homogeneous and isotropic to one part in 100,000, suggesting that if an exterior universe exists, it must be nearly perfectly separated from our own.

However, for an observer inside the white hole, there would be no observational way to detect the external universe, as all outgoing geodesics end at the event horizon. So could our universe's Big Bang be a white hole singularity, with an unobservable "exterior" universe existing beyond our cosmological horizon? Some physicists, notably Raj Pathria in his 1972 paper on "Black Hole Cosmology", have proposed this idea.

The challenge is that the interior of a classic white hole, described by the maximally extended Schwarzschild metric, looks nothing like our universe. It is a vacuum solution that is utterly inhomogeneous and anisotropic, with the singularity occupying a single point in space and infinite tidal forces as one approaches the singularity along timelike or null geodesics.

In contrast, our universe appears extremely homogeneous and isotropic on large scales, and is filled with matter and radiation. The spacetime of our universe is well-described on cosmological scales by the Friedmann–Lemaître–Robertson–Walker (FLRW) metric, which assumes perfect spatial uniformity and isotropy. The FLRW metric corresponds to a family of solutions to the Einstein field equations parameterized by the scale factor a(t), which describes the expansion of space over time, and the curvature parameter k, which describes the geometry of spatial slices (open, flat, or closed).

However, theoretical work has shown that it is possible to embed an FLRW metric inside a black hole. The first example was the Oppenheimer-Snyder model of gravitational collapse, published in 1939. Oppenheimer and Snyder considered a simplified model of a pressureless "star" of uniform density, represented as a spherical ball of dust. As the dust ball collapses under its own gravity, the interior spacetime is described by an FLRW metric with a(t) decreasing towards zero. The exterior spacetime is patched to a Schwarzschild metric, representing a black hole. The boundary between the interior FLRW spacetime and the exterior Schwarzschild spacetime is the stellar surface, which collapses through the event horizon at a finite time as seen by exterior observers.

Reversing time in the Oppenheimer-Snyder model yields a white hole with an expanding FLRW interior, born from a past singularity. The stellar surface emerges from the white hole at a finite time in the past, and the interior spacetime expands forever according to the FLRW scale factor. With appropriate choice of parameters (spatially flat with a(t) proportional to t^(2/3), representing a matter-dominated universe), this interior spacetime could closely resemble our observable universe.

The Oppenheimer-Snyder model is highly simplified, assuming a perfect fluid with zero pressure, but it provides a proof-of-principle that an expanding FLRW spacetime can be embedded inside an eternal black/white hole. More realistic models of gravitational collapse with pressure and inhomogeneities lead to more complicated internal spacetimes, but the overall picture of a collapsing (or expanding) interior FLRW spacetime bounded by an event horizon remains valid.

Another key ingredient is Hawking radiation, derived by Stephen Hawking in 1974 by applying quantum field theory to the curved spacetime of a black hole. Hawking showed that black holes will emit thermal radiation from just outside their event horizons with a blackbody spectrum and a temperature inversely proportional to the black hole's mass. This causes black holes to slowly evaporate over time, with microscopic black holes evaporating almost instantly while astrophysical black holes take an unimaginable 10^100 years to evaporate.

In the 1990s, Hawking considered the case of an "eternal" black hole in thermal equilibrium with a surrounding radiation bath, constantly absorbing as much radiation as it emits. He showed that the quantum state of such a black hole is indistinguishable from a white hole, with Hawking radiation emission being equivalent to absorption by a white hole. This provides a mechanism for a black hole and a white hole to be the same object, with "Hawking radiation" being the time-reverse of infalling matter.

Putting the pieces together, we can construct a theoretical model for an "eternal" black hole with an expanding FLRW interior, representing a new universe being born inside the black hole. The model consists of three regions:

1. An exterior Schwarzschild spacetime, representing a black hole embedded in some "mother universe". The black hole is in thermal equilibrium, absorbing radiation from the mother universe at the same rate that it emits Hawking radiation. This equilibrium condition makes the black hole indistinguishable from a white hole.

2. An interior FLRW spacetime, representing a "baby universe" expanding from a past singularity. The scale factor a(t) would be chosen to match astronomical observations of our universe's expansion history, likely with an early inflationary phase followed by radiation domination and then matter domination. The spatial curvature would be nearly flat (|Ωk| < 0.001) to match cosmic microwave background constraints.

3. A thin shell representing the interface between the exterior and interior regions, analogous to the stellar surface in the Oppenheimer-Snyder model. This shell emerges from an initial singularity and expands outwards, asymptotically approaching the event horizon. Exterior observers would see the shell hover just above the event horizon, slowly growing and accumulating mass-energy from infalling Hawking radiation. Interior observers would see the shell receding at nearly the speed of light, defining an "edge" to their observable universe.

From the perspective of observers in the interior FLRW spacetime, their universe would appear to begin with a Big Bang singularity and then expand forever according to the FLRW scale factor, with a nearly flat geometry and a mix of matter and radiation matching astronomical observations. The shell separating the interior and exterior regions would be forever out of causal contact, as it recedes at nearly the speed of light. There would be no observational evidence of the exterior "mother universe".

This model provides a mechanism for generating "baby universes" with FLRW metrics inside of eternal black holes, which could repeat ad infinitum to create a "multiverse" without requiring an explicit assumption of inflationary bubble nucleation or quantum fluctuations. Each black hole in a given universe could spawn a new universe in its interior, with slightly different physical constants or initial conditions.

This scenario was proposed by physicist Lee Smolin in his 1992 paper on "cosmological natural selection". Smolin suggested that, if the fundamental constants of physics could vary between universes, then a form of natural selection would favor universes that maximize their number of black holes, since those universes would have more "offspring". Over time, the multiverse would come to be dominated by universes "fine-tuned" for black hole production.

This could potentially explain the apparent fine-tuning of our universe for the emergence of structure and complexity, as stars and galaxies are necessary precursors to black hole formation. If the strength of gravity were slightly weaker, or the cosmological constant were slightly larger, matter would not be able to collapse into black holes before the accelerating expansion of the universe pulls everything apart. If the proton-to-electron mass ratio or the fine-structure constant were slightly different, stars and black holes might be impossible.

Cosmological natural selection provides an alternative to the anthropic principle for explaining the apparent fine-tuning of our universe, without invoking an inflationary multiverse or quantum fluctuations. It is perhaps the most developed version of the idea that our universe could exist inside a black hole.

However, the scenario remains highly speculative and there are many open questions and potential problems. It is unclear whether the physical constants could actually vary between universes, or if there is a large enough "landscape" of possible values to allow for selection effects. Most theories of quantum gravity, such as string theory, predict a unique set of fundamental constants.

Also, for the daughter universes to inherit some of the properties of their parent universes (as required for evolution by natural selection), there would need to be a subtle correlation between the initial conditions of the baby universe and the late-time state of the mother universe. In other words, the initial singularity of the baby universe (analogous to the Big Bang) would need to "remember" the black hole it formed from, including the physical constants of the mother universe. This requires violation of the "no-hair theorem" for black holes, which states that a black hole is completely characterized by its mass, charge, and spin, with no other information about its past observable from the outside.

Some physicists, such as Nikodem Poplawski, have proposed that violations of the no-hair theorem could occur due to quantum gravity effects, allowing information about the parent universe to be transmitted to the baby universe. However, this remains speculative and there is no widely accepted mechanism for such information transmission.

Furthermore, astronomical observations have constrained primordial black holes (black holes formed in the early universe, soon after the Big Bang) to make up less than 10^-7 of the total mass of the universe. If primordial black holes were abundant, they would have noticeable effects on Big Bang nucleosynthesis and the cosmic microwave background. This suggests that black hole formation is a relatively rare event in our universe, which is in tension with the idea that our universe is optimized for black hole production.

Finally, the idea that our universe exists inside a black hole raises philosophical questions about the nature of time and causality. In general relativity, the singularity at the center of a black hole represents a "future boundary" of spacetime, where timelike geodesics terminate. This means that the formation of the singularity (and any "baby universe" inside it) cannot be the cause of the black hole's formation, as that would require information to propagate "backwards in time" from the singularity to the collapsing star or dust cloud.

Instead, the causal order is reversed: the formation of the black hole is the cause of the singularity and any baby universes inside it. This leads to a "chicken-and-egg" paradox where each universe is both the cause and effect of its parent/child universes. Some physicists, such as Sean Carroll, have argued that this undermines the explanatory power of cosmological natural selection, as it requires a disconnect between the local causal structure of spacetime and the global causal structure of the multiverse.

In conclusion, the hypothesis that our universe exists inside a black hole is a fascinating but highly speculative idea that pushes the boundaries of general relativity and quantum mechanics. It provides a potential mechanism for generating an infinite "multiverse" with varying physical constants, which could explain the apparent fine-tuning of our universe for complexity and black hole production.

However, the scenario faces significant theoretical and observational challenges, and raises deep questions about the nature of time, causality, and information in quantum gravity. Testing the idea experimentally may be impossible, as any "mother universe" would be forever hidden behind an event horizon, and any "baby universes" inside black holes would be causally disconnected from our own. For now, the idea remains an intriguing thought experiment and a theoretical "sandbox" for exploring the frontiers of cosmology and fundamental physics. But without direct observational support, it is likely to remain on the fringes of mainstream scientific research.