Mimicking a black hole in flat spacetime

Difficulty level:   ★ ★ ★

For a Schwarzschild-Droste black hole, the curvature of 3-dimensional space is often depicted as a funnel shape (Flamm 1916 ). As I emphasise in forthcoming papers, this assumes the static slicing of spacetime, whereas other slicings yield different embedding diagrams. This leads to the question, could we slice flat spacetime in such a way that we get a similar funnel, or mimic other properties of a black hole? While this cannot of course change the fact the 4-dimensional spacetime is flat, the point is there is much flexibility in defining the 3-space, because it depends only on the chosen slicing or observers.

Embedding diagram for a fake black hole, representing an unusual spatial slice of Minkowski spacetime
Spoiler: Yes you can! This is an embedding diagram for our “fake black hole”, representing an unusual spatial slice of Minkowski spacetime. This looks more like a spinning top than a funnel.

Let’s start with Minkowski spacetime in spherical coordinates:

    \[ds^2 = -dt^2 + dr^2 + r^2(d\theta^2+\sin^2\theta\,d\phi^2)\]

This defines an inertial frame. Now suppose spacetime is filled with test particles moving radially, relative to the coordinate origin. Take coordinate speed dr/d\tau = \pm\sqrt{2M/r}, by analogy with the Schwarzschild and even Newtonian cases (choose one sign and stick with it). The 4-velocity is then:

    \[u^\mu = \bigg(\sqrt{1+\frac{2M}{r}},\pm\sqrt\frac{2M}{r},0,0\bigg)\]

which follows from normalisation \mathbf u\cdot\mathbf u=-1. Next we define a new time coordinate. A natural first attempt is to try the proper time of the particles. This may be obtained via local Lorentz boosts, or equivalently by a neat trick of lowering the index on the 4-velocity vector then taking its negative:

    \[-\mathbf u^\flat = \sqrt{1+\frac{2M}{r}}dt \mp\sqrt\frac{2M}{r}dr\]

(I explain this approach in a forthcoming paper, but it is inspired by Martel & Poisson 2001  and ultimately based on Frobenius’ theorem: see the variant for 1-forms described in de Felice & Clarke §2.12.) Expressing the dual velocity this way, as an explicit sum of the coordinate dual basis vectors dt and dr, is suggestive of a total differential which we would hope is the proper time d\tau. Unfortunately the expression is not a total differential, as seen by examining the coefficient of dt. But from inspection we can use an integrating factor: divide through by \sqrt{1+2M/r}, simplify, and define the resulting expression as the differential of a new time coordinate T:

    \[dT := dt \mp\frac{1}{\sqrt{1+r/2M}}dr\]

(Incidentally, this easily integrates to T = t \mp 4M\sqrt{1+r/2M} plus a constant of integration.) While T is not the proper time, its level sets T=\textrm{const} coincide with the 3-space of the observers as shown next, which is sufficient for our embedding diagram. Since \mathbf u is by definition orthogonal to its local 3-space, the dual vector \mathbf u^\flat is also normal to this 3-space. But dT is parallel to \mathbf u^\flat, hence they are normal to the same 3-space, but any gradient dT is normal to the level sets T=\textrm{const}, which proves the claim.

This is analogous to static observers in Schwarzschild spacetime. While the Schwarzschild t-coordinate is not their proper time, setting t=\textrm{const} still determines the same 3-space as these observers. Also we cannot replace the t-coordinate with proper time while still retaining the coordinates r, \theta, and \phi. The derivative dT/d\tau = 1/\sqrt{1+2M/r} for our fake black hole is also reminiscent of static observers in Schwarzschild spacetime.

Rearrange the earlier expression for dT and substitute into the line element to obtain:

    \[ds^2 = -dT^2 \mp \frac{2}{\sqrt{1+r/2M}}dT\,dr + \bigg(1+\frac{2M}{r}\bigg)^{-1}dr^2\]

plus the 2-sphere metric r^2(d\theta^2+\sin^2\theta\,d\phi^2). These coordinates have no issue at r=2M, and while there is a coordinate singularity at r=0 the metric was degenerate there even in our initial spherical coordinates. The Riemann tensor is zero, as it must be since this is still flat spacetime. Since g^{TT}=-(1+2M/r)^{-1} the coordinate T is timelike everywhere. The 4-velocity in the new coordinates is u^\mu=(1/\sqrt{1+2M/r},\pm\sqrt{2M/r},0,0). Integrating dT/dr = u^T/u^r gives the travel time T(r) which is well behaved unlike Schwarzschild t(r) which diverges. The radial proper distance for our test particle observers is (1+2M/r)^{-1/2}dr, which gets very small for r\ll 2M compared to the inertial frame which measures radial distance dr everywhere.

A typical isometric embedding diagram for a spherically symmetric spacetime takes a slice of constant “time”, here T=\textrm{const}, through the equator \theta=\pi/2. This is matched isometrically with a surface z=z(r) in a 3-dimensional flat space. The flat space is taken to be Euclidean or Minkowski space, with the metric dr^2+r^2d\phi^2\pm dz^2 in cylindrical coordinates (the sign is unrelated to our previous sign choice). Our case requires the minus sign for Minkowski space since g_{rr}<1. It follows z = 4M\sqrt{1+r/2M}, which may be plotted in a scale-invariant way as z/M against r/M.

Embedding diagram for a fake black hole, an unusual choice of spatial slice in Minkowski spacetime
The same embedding diagram from a lower viewpoint and with further comments. The diagram is the same for both ingoing and outgoing particles / observers. This surface extends to the origin, unlike Flamm’s paraboloid for Schwarzschild space which corresponds to static observers and hence is only defined outside r=2M.

The particles must be accelerating, as their motion is not caused by gravity. In the new coordinates ingoing particles have 4-acceleration (0,-M/r^2,0,0), outgoing particles have a different expression, but both have magnitude a=(1+2M/r)^{-1/2}M/r^2. Again these expressions are reminiscent of static observers in Schwarzschild spacetime. Each particle has a “Rindler” horizon at distance 1/a as measured in the instantaneous comoving frame, so in the original inertial frame this is contracted by the Lorentz factor \gamma=\sqrt{1+2M/r} and occurs at position r\mp r^2/M (simultaneous in the instantaneous comoving frame).

The kinematic decomposition of the particle worldlines yields zero vorticity, which is fortunate because by Frobenius’ theorem this is the condition for the local 3-spaces to all patch together consistently. The expansion tensor, expressed in the frame of the particles (different frames for ingoing and outgoing), is \mp\sqrt{M/2r^3} in the radial direction, and \pm\sqrt{2M/r^3} in the tangential directions. The shear is twice this amount in the radial direction, and half this amount in the tangential directions.

In the new coordinates the lapse is \sqrt{1+2M/r} and the shift (\mp 2M/r\cdot\sqrt{1+r/2M},0,0). The extrinsic curvature (of the 3D spatial slices inside 4D Minkowski spacetime, not the 2D embedded slice) is \pm\sqrt{2M/r^3} times -dr^2/2(1+2M/r)+r^2(d\theta^2+\sin^2\theta\,d\phi^2). This has trace K = \pm\sqrt{M/2r^3}(1/(1+2M/r)-4) or \pm\sqrt{M/2r^3}(3+8M/r)/(1+2M/r).

Finally, Flamm’s paraboloid is an iconic image, and I defend visualisations and metaphors in general as helpful and intuitive. But one should understand the limitations, in contrast to Painlevé 1921  for example who found a slicing of Schwarzschild spacetime into Euclidean 3-spaces \mathbb R^3, but drew some overly zealous conclusions from this (thanks to Andrew Hamilton for discussion on this point). Admittedly the static slicing in Schwarzschild spacetime is a natural choice, while my “fake black hole” slicing is contrived. But still, the reproduction of a funnel-shaped embedding in flat spacetime shows the need for caution in interpreting Flamm’s paraboloid as gravity.