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Black Holes, Black Holes Everywhere

Nowadays, one cannot watch a popular science tv show, read a popular science book, take an astrophysics class without hearing about black holes. The problem is that very few people discuss this topic appropriately. This is further evidenced that these same people also claim that the universe’s expansion is governed by the Friedmann equation as applied to a Friedmann-Lemaitre-Robertson-Walker (FLRW) universe.

The fact is that black holes despite what is widely claimed, are not astrophysical phenomena, they are a phenomena that arise from mathematical general relativity. That is, we postulate their existence from mathematical general relativity, in particular, Birkhoff’s theorem, which states the following (Hawking and Ellis, 1973):

Any $C^2$ solution of Einstein’s vacuum equations which is spherically symmetric in some open set V is locally equivalent to part of the maximally extended Schwarzschild solution in V.

In other words, if a spacetime contains a region which is spherically symmetric, asymptotically flat/static, and empty such that $T_{ab} = 0$, then the metric inn this region is described by the Schwarzschild metric:

$\boxed{ds^2 = -\left(1 - \frac{2M}{r}\right)dt^2 + \frac{dr^2}{1-\frac{2M}{r}} + r^2\left(d\theta^2 + \sin^2 \theta d\phi^2\right)}$

The concept of a black hole then occurs because of the $r = 0$ singularity that occurs in this metric.

The problem then arises in most discussions nowadays, because the very same astrophysicists that claim that black holes exist, also claim that the universe is expanding according to the Einstein field equations as applied to a FLRW metric, which are frequently written nowadays as:

The Raychaudhuri equation:

$\boxed{\dot{H} = -H^2 - \frac{1}{6} \left(\mu + 3p\right)}$,

(where $H$ is the Hubble parameter)

The Friedmann equation:

$\boxed{\mu = 3H^2 + \frac{1}{2} ^{3}R}$,

(where $\mu$ is the energy density of the dominant matter in the universe and $^{3}R$ is the Ricci 3-scalar of the particular FLRW model),

and

The Energy Conservation equation:

$\boxed{\dot{\mu} = -3H \left(\mu + p\right)}$.

The point is that one cannot have it both ways! One cannot claim on one hand that black holes exist in the universe, while also claiming that the universe is FLRW! Since, by Birkhoff’s theorem, external to the black hole source must be a spherically symmetric and static spacetime, for which a FLRW is not static nor asymptotically flat, because of a lack of global timelike Killing vector.

I therefore believe that models of the universe that incorporate both black holes and large-scale spatial homogeneity and isotropy should be much more widely introduced and discussed in the mainstream cosmology community. One such example are the Swiss-Cheese universe models. These models assume a FLRW spacetime with patches “cut out” in such a way to allow for Schwarzschild solutions to simultaneously exist. Swiss-Cheese universes actually have a tremendous amount of explanatory power. One of the mysteries of current cosmology is the origin of the existence of dark energy. The beautiful thing about Swiss-Cheese universes is that one is not required to postulate the existence of hypothetical dark energy to account for the accelerated expansion of the universe. This interesting article from New Scientist from a few years ago explains some of this.

Also, the original Swiss-Cheese universe model in its simplest foundational form was actually proposed by Einstein and Strauss in 1945.

The basic idea is as follows, and is based on Israel’s junction formalism (See Hervik and Gron’s book, and Israel’s original paper for further details. I will just describe the basic idea in what follows). Let us take a spacetime and partition it into two:

$\boxed{M = M^{+} \cup M^{-}}$

with a boundary

$\boxed{\Sigma \equiv \partial M^{+} \cap \partial M^{-}}$.

Now, within these regions we assume that the Einstein Field equations are satisfied, such that:

$\boxed{\left[R_{uv} - \frac{1}{2}R g_{uv}\right]^{\pm} = \kappa T_{uv}^{\pm}}$,

where we also induce a metric on $\Sigma$ as:

$\boxed{d\sigma^2 = h_{ij}dx^{i} dx^{j}}$.

The trick with Israel’s method is understanding is understanding how $\Sigma$ is embedded in $M^{\pm}$.  This can be quantified by the covariant derivative on some basis vector of $\Sigma$:

$\boxed{K_{uv}^{\pm} = \epsilon n_{a} \Gamma^{a}_{uv}}$.

The projections of the Einstein tensor is then given by Gauss’ theorem and the Codazzi equation:

$\boxed{\left[E_{uv}n^{u}n^{v}\right]^{\pm} = -\frac{1}{2}\epsilon ^{3}R + \frac{1}{2}\left(K^2 - K_{ab}K^{ab}\right)^{\pm}}$,

$\boxed{\left[E_{uv}h^{u}_{a} n^{v}\right]^{\pm} = -\left(^{3}\nabla_{u}K^{u}_{a} - ^{3}\nabla_{a}K\right)^{\pm}}$,

$\boxed{\left[E_{uv}h^{u}_{a}h^{v}_{b}\right]^{\pm} = ^{(3)}E_{ab} + \epsilon n^{u} \nabla_{u} \left(K_{ab} - h_{ab}K\right)^{\pm} - 3 \left[\epsilon K_{ab}K\right]^{\pm} + 2 \epsilon \left[K^{u}_{a} K_{ub}\right]^{\pm} + \frac{1}{2}\epsilon h_{ab} \left(K^2 + K^{uv}K_{uv}\right)^{\pm}}$

Defining the operation $[T] \equiv T^{+} - T^{-}$, the Einstein field equations are given by the Lanczos equation:

$\boxed{\left[K_{ij}\right] - h_{ij} \left[K\right] = \epsilon \kappa S_{ij}}$,

where $S_{ij}$ results from defining an energy-momentum tensor across the boundary, and computing

$\boxed{S_{ij} = \lim_{\tau \to 0} \int^{\tau/2}_{-\tau/2} T_{ij} dy}$.

The remaining dynamical equations are then given by

$\boxed{^{3}\nabla_{j}S^{j}_{i} + \left[T_{in}\right] = 0}$,

and

$\boxed{S_{ij} \left\{K^{ij}\right\} + \left[T_{nn}\right] = 0}$,

with the constraints:

$\boxed{^{3}R - \left\{K\right\}^2 + \left\{K_{ij}\right\} \left\{K^{ij}\right\} = -\frac{\kappa^2}{4} \left(S_{ij}S^{ij} - \frac{1}{2}S^2\right) - 2 \kappa \left\{T_{nn}\right\}}$.

$\boxed{\left\{^{3} \nabla_{j}K^{j}_{i} \right\} - \left\{^{3}\nabla_{i}K\right\} = -\kappa \left\{T_{in}\right\}}$.

Therefore:

1. If black holes exist, then by Birkhoff’s theorem, the spacetime external to the black hole source must be spherically symmetric and static, and cannot represent our universe.
2. Perhaps, a more viable model for our universe is then a spatially inhomogeneous universe on the level of Lemaitre-Tolman-Bondi, Swiss-Cheese, the set of $G_{2}$ cosmologies, etc… The advantage of these models, particular in the case of Swiss-Cheese universes is that one does not need to postulate a hypothetical dark energy to explain the accelerated expansion of the universe, this naturally comes out out of such models.

Under a more general inhomogeneous cosmology, the Einstein field equations now take the form:

Raychauhduri’s Equation:

$\boxed{\dot{H} = -H^2 + \frac{1}{3} \left(h^{a}_{b} \nabla_{a}\dot{u}^{b} + \dot{u}_{a}\dot{u}^{a} - 2\sigma^2 + 2 \omega^2\right) - \frac{1}{6}\left(\mu + 3p\right)}$

Shear Propagation Equation:

$\boxed{h^{a}_{c}h^{b}_{d} \dot{\sigma}^{cd} = -2H\sigma^{ab} + h^{(a}_ch^{b)}_{d}\nabla^{c}\dot{u}^{d} + \dot{u}^{a}\dot{u}^{b} - \sigma^{a}_{c} \sigma^{bc} - \omega^{a}\omega^{b} - \frac{1}{3}\left(h^{c}_{d}\nabla_{c}\dot{u}^{d} + \dot{u}_{c}\dot{u}^{c} - 2\sigma^2 - \omega^2\right)h^{ab} - \left(E^{ab} - \frac{1}{2}\pi^{ab}\right)}$

Vorticity Propagation Equation:

$\boxed{h^{a}_{b}\dot{\omega}^{b} = -2H\omega^{a} + \sigma^{a}_{b}\omega^{b} - \frac{1}{2}\eta^{abcd}\left(\nabla_{b} \omega_{c} + 2\dot{u}_{b}\omega_{c}\right)u_{d} + q^{a}}$

Constraint Equations:

$\boxed{h^{a}_{c} h^{c}_{d} \nabla_{b} \sigma^{cd} - 2h^{a}_{b}\nabla^{b}H - \eta^{abcd}\left(\nabla_{b}\omega_{c} + 2 \dot{u}_{b} \omega_{c}\right)u_{d} + q^{a} = 0}$,

$\boxed{h^{a}_{b} \nabla_{a}\omega^{b} - \dot{u}_{a}\omega^{a} = 0}$,

$\boxed{H_{ab} - 2\dot{u}_{(a}\omega_{b)} - h^{c}_{(a}h^{d}_{b)}\nabla_{c} \omega_{d} + \frac{1}{3} \left(2\dot{u}_{c}\omega_{c} + h^{c}_{d} \nabla_{c}\omega^{d}\right)h_{ab} - h^{c}_{(a}h^{d}_{b)} \eta_{cefg}\left(\nabla^{e}\sigma^{f}_{d}\right)u^{g}=0}$.

Matter Evolution Equations through the Bianchi identities:

$\boxed{\dot{\mu} = -3H\left(\mu + p\right) - h^{a}_{b}\nabla_{a}q^{b} - 2\dot{u}_{a}q^{a} - \sigma^{a}_{b}\pi^{b}_{a}}$,

$\boxed{h^{a}_{b}\dot{q}^{b} = -4Hq^{a} - h^{a}_{b}\nabla^{b}p - \left(\mu + p\right)\dot{u}^{a} - h^{a}_{c}h^{b}_{d}\nabla_{b} \pi^{cd} - \dot{u}_{b}\pi^{ab} - \sigma^{a}_{b}q^{b} + \eta^{abcd}\omega_{b}q_{c}u_{d}}$.

One also has evolution equations for the Weyl curvature tensors $E_{ab}$ and $H_{ab}$, these can be found in Ellis’ Cargese Lectures.

Despite the fact that these modifications are absolutely necessary if one is to take seriously the notion that our universe has black holes in it, most astronomers and indeed most astrophysics courses continue to use the simpler versions assuming that the universe is spatially homogeneous and isotropic, which contradicts by definition the notion of black holes existing in our universe.

By Dr. Ikjyot Singh Kohli

Sikh, Theoretical and Mathematical Physicist, main research in the structure and dynamics of Einstein's field equations.

2 replies on “Black Holes, Black Holes Everywhere”

AutisticPhysicistsays:

Oh my god, that stress-energy tensor defined across the boundary trick is amazing. I love that. I wish I knew GR formalism better. It seemed like there were separate field equations for both regions and another field equation for the boundary using the boundary stress-energy tensor. What is the physical meaning of that border stress-energy tensor, is it uniquely determined by the stress energy tensor of the two spacetime regions? I’m not sure Black Holes are just a mathematical singularity. Like with the singularities in any theory it’s up to physicists to find out what nature is telling us with those singularities. I mean a black hole has a physical meaning too right? Like the electron degeneracy pressure, baryon degeneracy pressures that normally fight the gravity pushing the star inwards actually contribute to the gravitational force because pressure is momentum flux in GR. My physical understanding of why GR creates these singularities is because the pressures that keep a star from collapsing actually generate more gravity by contributing more momentum=energy=mass to the field equation further curving the spacetime. In Newtonian gravity we only care about mass, so the star’s thermal pressure, and other internal forces would just outweigh gravity. But in GR those pressures actually create more gravity and under certain circumstances the gravity will win against all these pressures. I though black holes made perfect physical sense in addition to their mathematical existence.

AutisticPhysicistsays:

There are also qft attempts to explain expansion acceleration. Most theories try to get the ground state of the vacuum and compare it to cosmological constants. No dice for most of them, they just get intractable infinities. No quantum gravity today! But some of the lattice theories that cut off space time at the Plank length seem promising in garnering a quantum vacuum ground state that provides expansion fuel.