If it's a common myth that a black hole contains a singularity, what does a black hole actually (likely) contain?

Question

It's a common myth (especially in popsci) that a black hole contains a singularity. However, I cannot find an explanation for what we think a black hole actually does contain. The best I've seen is "in order to answer this question, we have to unify GR and quantum mechanics." So there are no attempts to answer this question despite our incomplete picture of physics? Let's say Schwarzschild black hole to start with.

Answer 1

Same answer as for the electric Coulomb field. Physicists start with the free Lagrangian

$$T = \frac{p^2}{2m} $$

for determination of test particle trajectories in space time. Adding the symmetry of translations in momentum space (Galilei invariance) yields

$$T = \frac{(\vec p - \frac{e}{c} \vec A)^2}{2m} $$

with no effect on trajectories (straight lines at constant speed $\vec v = \nabla_p T$, even if momentum p is not constant), if $\vec A $ is a constant and even more, a derivative $\vec A = \nabla f$ for any smooth function $f$

Replace space by 4d space-time, then the Coulomb field is a vacuum solution of Maxwells equation in empty flat space-time

$$ (\Phi, \vec A ) = ( \frac{1}{|\vec r|}, 0 ) $$

that is a cylindrical symmetric function around the world line $r=0$

What is inside at r=0? A pole-singularity of the field. No one ever has found an answer.

Quantum theory had to be introduced to explain the nature of charge. There are no point charges, smooth densities only.

In quantum theory, spatial infinity is very far away, the Coulomb field outside a sphere of atomic dimensions degenerates to the winding number of the common electric field of all charged particles on the radial cylinder at infinity with time axis as center.

The winding number is a topological invariant. It has to be kept constant on any space slice in time during movement of the charge singularities, because nobody can change any data at spacelike infinity instantaneously (Einstein causality). Funny: Topologically, the charge is a boundary condition at infinity. The singularity at $r=0$ is an unavoidable consequence of the field equation and boundary conditions.

Mathematically its a fundamental distributional solution, that is the kernel for integrating over smooth densities (superposition principle for a linear PDE)

Einsteins step in 1916 after introducing SR and making SR locally in momentum space by coordinate transforms:

Gauge the space-time scalar product for a free test particle, that yields the invariant mass m, Einsteins true, originally personal invention in 1905 besides the photon quantum.

$$m^2 = g_{ik}(p_i-\frac{e}{c} A_i)(p_k-\frac{e}{c} A_k)$$

to

$$m^2 = g(x)_{ik}(p_i-\frac{e}{c} A(x)_i)\ (p_k-\frac{e}{c} A(x)_k)$$

and Hilbert showed him how to formulate the condition, that the local metric is pure gauge and what is the source term of curvature yielding the gravitational field as a metric in vacuum space-time $V=\frac{m}{r}$ at spacelike infinite large cylinders $r$ around the time axis.

In electomagnetism, the exterior derivative of the vector potential $\vec A$ yields the force fields $E,B$ as a 2x2 antisymmetric tensor $F$

$$ F = dA = dx^i \wedge \partial_i ( A_k dx^k) $$

$F=0$ eg is the condition for a pure gauge field, the second mixed derivatives $\partial_i A_k = \partial_k A_i$ being equal locally everywhere.

The charge representing stress fields $D,H$ are the Hodge dual of $F$ by exchange of the 2-forms $dx^i \wedge dx^k$ with the dual counter parts $dx^l \wedge dx^m$ in the 4-volume element. So as a bottom line, charge-current densities are given by the Hodge Laplacian of the vector fields

$$ \rho dt + \vec j \cdot \vec dx = *d(*dA) $$

where the Hodge dual $*f$ always denotes the exchange of the differential in the 4-volume element $dV = dct \wedge dx^1 \wedge \dots dx^3$ of space-time.

Analoguously, Hilbert and Einstein used the recently introduced covariant derivative $\nabla= \vec \partial + \Gamma$ of the Italians to derive the set of equations, that differentiate pure gauge transformation of the quadratic metric form $g(x)$ in cotangential space from a metric in curved space-time.

From the beginning, the pure gauge metric tensors were always present in Hamiltonian mechanics of test particles by coordinate transforms.

As working with diagonal symmetric matrices is easy on the one hand, and paradigmatically universal by the diagonalization theorem, its very easy to advance the Maxwell paradigma to the metric case by use of orthogonal coordinates and their tangent basis space (partial derivatives $\partial_x$, velocities $\partial_t x(t)$) and cotangent space (exterior differentials of the coordinates $dx$ as tangent hypersurfaces, momentum variables $p$ ).

A metric is pure gauge, if it is the square of the Jacobi matrix of a coordinate transform from cartesian coordinates in Minkowsks space:

$$ J(y) = \nabla_y\ x(y),\ \ g(y) = J(y)^T \cdot J(y) , \ \ dV = \det(J) \wedge_i dy^i $$

Playing the same game with the Jacobian as vector potentials $$e_i(y) J(y)^i_k \ dy^i $$ that maps tangent vectors in two coordinate systems, its exterior derivative, the covariant derivative $\nabla$ to generate the connection coefficients (virtual forces proportional to products of velocities) (for a syntactically accurate indexing the summations confer Wheeler et al. Gravitation)

$$(\Gamma(x) = \mathbb d J(x) = dx^i \wedge \nabla_i J = dx^i \wedge \partial_i J + \Gamma(x)^i_j J_i) $$ (notational dilemma denoting the Einstein summation symbolically)

the derivative of curvature potentials

$$ R = \mathbb d *J = d*J + \Gamma^2 \wedge J $$ (2 $\Gamma$'s for two indices)

In order to generate the fundamental, rotational solution in vacuum with $1/r$ decay at spacelike infinity, as an operator in the field equations, the Einstein tensor emerges as the trace of the Riemann curvature tensor, called the Ricci tensor, corrected for its own trace, the curvature scalar times the metric, set equal to the 4-momentum transport tensor

$$\left(( R_{ik}) = (R)^s_i)_{sk} - \frac{1}{2} (R) g_{ik} = T_{ik}\right) * dx^i\otimes dx^k$$ .

The Einstein vacuum equations with boundary values $1/r$ at spacelike infinity in a spacelike spherical, space-time cyindrical, static and spherical symmetric metric gave Schwarzschild after 2 hours paperwork in 1916

$$g = f(r) \ dt\otimes dt + \frac{1}{f(r)} \ dr\otimes dr + r^2\ (d\theta \otimes d\theta + \sin^2 \theta \ d\phi \otimes d\phi) $$ with $$f(r) = (1-\frac{2m}{r})$$

The radial variable is not the metric radius, it is the instantaneuos circumference of a sphere $2\pi r$ or surface $\frac{4 \pi}{3}\ r^2$ at constant time.

In Schwarzschild coordinates, this formula yields the fact, that a cylinder parallel to the time axis with spherical static spatial spherical surface $\frac{4 \pi}{3}\ r^2$ has a tangent vector of length 0 in time direction. Inside this cylinder, that is by $dt\wedge dr = 1$ a light cone.

Any other metric with spatial spherical symmetry and an energy momentum tensor respecting this or any other spatial boundary condition, and is respecting the equations of motion of a static mechanical model can be used to continue smoothly the metric tensor to complete the manifold.

As for the Maxwell-Dirac-Feynman case of the electromagnetic quantum gauge theory, the metric field at infinity yields a topological fixed charge. But it does not say any word about the microscopic, mass and charge generating model of the fundamental solution of the field equations at $r=0$.

That is, there is a sphere of no surface open to speculations how to fill it with a very local mechanical model.

Historically there are known two solutions continuing smoothly into the inner of the 2m-cone:

The continuation of the vacuum solution everywhere yields the double, hyperboloid singularities, where all timelike geodesics end or emerge from in time.

This is a quite a different aspect of gravity compared to the spatial singularity along the time axis for the Coulomb field.

Unfortunately, the vacuum completion of the manifold by spacelike geodesics generates a secondary, mirror universe inside, with another black horizon and an own, different spatial infinity, too.

The second is Schwarzschild's interior solution, that provides a mass distribution inside a star with surface greater than ${frac {4 \pi}{3} \ (2 m)^2 $.