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.
Let’s start with Minkowski spacetime in spherical coordinates:
This defines an inertial frame. Now suppose spacetime is filled with test particles moving radially, relative to the coordinate origin. Take coordinate speed , by analogy with the Schwarzschild and even Newtonian cases (choose one sign and stick with it). The 4-velocity is then:
which follows from normalisation . 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:
(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 and , is suggestive of a total differential which we would hope is the proper time . Unfortunately the expression is not a total differential, as seen by examining the coefficient of . But from inspection we can use an integrating factor: divide through by , simplify, and define the resulting expression as the differential of a new time coordinate :
(Incidentally, this easily integrates to plus a constant of integration.) While is not the proper time, its level sets coincide with the 3-space of the observers as shown next, which is sufficient for our embedding diagram. Since is by definition orthogonal to its local 3-space, the dual vector is also normal to this 3-space. But is parallel to , hence they are normal to the same 3-space, but any gradient is normal to the level sets , which proves the claim.
This is analogous to static observers in Schwarzschild spacetime. While the Schwarzschild -coordinate is not their proper time, setting still determines the same 3-space as these observers. Also we cannot replace the -coordinate with proper time while still retaining the coordinates , , and . The derivative for our fake black hole is also reminiscent of static observers in Schwarzschild spacetime.
Rearrange the earlier expression for and substitute into the line element to obtain:
plus the 2-sphere metric . These coordinates have no issue at , and while there is a coordinate singularity at 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 the coordinate is timelike everywhere. The 4-velocity in the new coordinates is . Integrating gives the travel time which is well behaved unlike Schwarzschild which diverges. The radial proper distance for our test particle observers is , which gets very small for compared to the inertial frame which measures radial distance everywhere.
A typical isometric embedding diagram for a spherically symmetric spacetime takes a slice of constant “time”, here , through the equator . This is matched isometrically with a surface in a 3-dimensional flat space. The flat space is taken to be Euclidean or Minkowski space, with the metric in cylindrical coordinates (the sign is unrelated to our previous sign choice). Our case requires the minus sign for Minkowski space since . It follows , which may be plotted in a scale-invariant way as against .
The particles must be accelerating, as their motion is not caused by gravity. In the new coordinates ingoing particles have 4-acceleration , outgoing particles have a different expression, but both have magnitude . Again these expressions are reminiscent of static observers in Schwarzschild spacetime. Each particle has a “Rindler” horizon at distance as measured in the instantaneous comoving frame, so in the original inertial frame this is contracted by the Lorentz factor and occurs at position (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 in the radial direction, and 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 and the shift . The extrinsic curvature (of the 3D spatial slices inside 4D Minkowski spacetime, not the 2D embedded slice) is times . This has trace or .
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 , 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.