Mimicking a black hole in flat spacetime

For a Schwarzschild-Droste black hole, the curvature of 3-dimensional space is often depicted as a funnel shape (Flamm 1916 research paper (astronomy) ). 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 research paper (astronomy) and ultimately based on Frobenius’ theorem: see the variant for 1-forms described in de Felice & Clarke textbook §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 research paper 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.

Leave a Reply Cancel reply