Black holes

 Misleading to show the black hole as a funnel or as a rubber sheet

   You’ve probably seen the above picture of a black hole depicted as a kind of funnel, or alternatively as a rubber sheet depressed by a heavy round mass. Far away from the funnel, or the depression in the rubber sheet, the surface is supposed to be flat and its variants have appeared in countless magazines, newspapers, popular books, and even on the cover of a textbook. In many science museums, visitors are invited to toss a small ball onto the surface of an actual funnel shaped construction. If you toss the ball with sufficient speed in an angular direction, it will orbit around the central funnel, slowly spiraling into the dark “bottomless” pit in the center. And of course, if you toss the ball in the radial direction, it will fall right in, “sucked in by the irresistible force” of the black hole, often thought of as a “source of evil” in the visitor’s mind. You know of course that this display depicts the sun equally well. This museum display entertains the visitors and educates them to some extent, but D.Marolf has pointed out that it is misleading at best. For sure, it has seriously confused some students.

                                          (1)

This popular picture and the display that goes with it are obtained by setting t equal to some constant and  θ = π/2 in the Schwarzschild metric (1) to obtain

                 (2)

which is then embedded in 3-dimensional Euclidean space E3. (The museum staff could hardly do otherwise.) Using the usual cylindrical coordinates (z, ρ, ϕ) for E3, we specify the embedding by writing z = f (r), ρ = r, ϕ = ϕ. You can work out f (r) if you want, but it is not necessary here. In science museums, they don’t use the actual f (r), but instead, use an f (r) such that f (r)→constant for large r and f '(a)=−∞for some small value of a. So you see why we don’t need to draw a picture in this case.

Marolf’s point is that this picture represents a slice in time and is not directly connected to the gravitational attraction of the black hole. (The actual force “sucking” the ball into the funnel is of course supplied externally, by the earth.) In fact, there are spacetimes with the same t − ϕ slice as (2) but with totally different gravitational fields as in the Schwarzschild case.

To obtain a more appropriate representation of the black hole, we should take a slice of (1) with θ and ϕ both constant (in contrast to the funnel picture based on a slice with t and θ constant) and then embed the slice (3) in (2 + 1)-dimensional Minkowski spacetime M2,1. The resulting picture contains two flanges.

Embedding in Minkowski spacetime

The motion of any observer which maintains zero angular momentum must take place in such a plane whether the observer falls through the horizon or remains outside. Since dθ = 0 = dϕ on this plane, the metric on our surface is given by the first two terms of (1):

           (3)

after doing the embedding we get the diagram like this:

Here, the vertical (T) direction is timelike and the horizontal (X and Y ) directions are spacelike. Thus, time again runs up and down while space runs across the diagram. Units have been chosen in which the speed of light (c) is one, so that light rays travel at 45 degrees to the T axis.

Some Features of the embedding diagram

(1) The diagram is a smooth surface, with little to distinguish one point from another. In particular, it is not immediately clear which points lie on the horizon r = 2MG. This is an excellent way to show students that the horizon is not essentially different from any other part of the spacetime.

(2) Moving along the flanges in the +Y direction allows one to move a large proper distance in a spacelike direction. Thus, these flanges must correspond to the asymptotic regions far away from the black hole.

(3) One of the most important uses of this sort of diagram is that it does allow one to see the gravitational ‘attraction’ of the black hole. Specifically, it allows one to see the worldlines followed by freely falling observers, and to see that they curve toward the middle of the diagram, away from the asymptotic regions. The point is that, in General Relativity, freely falling observers follow ‘geodesics,’ the straightest possible lines on a curved surface. Let us consider a freely falling observer moving up one of the flanges. Because our surface bends away from the ends of the flanges and toward the center, our observer will follow this curve and also move closer to the center of the surface.

Wordline shows the trajectory of the particle in spacetime and above is worldline of an observer who falls freely from rest at r equals some big value, starting at T=0.

(4) Note that the two light rays given by Y = 0, X = ±T represented as two straight dark lines. Lie completely in our surface. The reader will immediately see that these light rays do not move along the flanges at all (since they stay at Y = 0) and thus neither of these light rays actually move away from the black hole. Instead, the light rays are trapped near the black hole forever.

The lines Y = 0, X = ±T represents the null geodesics at the surface, where null geodesics are the path followed by the light rays.

Summary: In this we have seen why one must consider r-t slice instead of t − ϕ slice, for the t − ϕ slice is just a static time slice and no variation is shown with time in such cases and the r-t slice of the black hole which gives a more realistic diagram of spacetime of the black hole.

For more technical literature and understanding the embedding part one must refer to the paper 

D. Marolf, arXiv:gr-qc/9806123 for more details.

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