^{1,a)}, J. D. Bjorken

^{2}, P. Chen

^{2}and J. S. Liu

^{3}

### Abstract

Most general relativitytextbooks devote considerable attention to the simplest example of a black hole containing a singularity, the Schwarzschild geometry. Only a few discuss the dynamical process of gravitational collapse by which black holes and singularities form. We present two simple analytical models that describe this process. The first involves collapsing spherical shells of light and is analyzed mainly in Eddington-Finkelstein coordinates; the second involves collapsing spheres filled with a perfect fluid and is analyzed mainly in Painleve-Gullstrand coordinates. Our main goal is simplicity and algebraic completeness, but we also present a few more sophisticated results such as the collapse of a light shell in Kruskal-Szekeres coordinates.

This work was supported by NASA Grant No. 8-39225 to Gravity Probe B, and by the U.S. Department of Energy under Contract No. DE-AC02-76SF00515. The authors thank the members of the Gravity Probe B theory group for many critical and stimulating discussions, in particular Francis Everitt, Robert Wagoner, and Alex Silbergleit.

I. INTRODUCTION

II. THIN LIGHT SHELLS

III. THICK LIGHT SHELLS

IV. THIN LIGHT SHELLS IN KRUSKAL–SZEKERES COORDINATES

V. UNIFORM FLUID SPHERES

VI. ZERO PRESSURE FLUID SPHERES

VII. SUMMARY AND FURTHER STUDY

### Key Topics

- Black holes
- 48.0
- Tensor methods
- 26.0
- Red shift
- 16.0
- Surface tension
- 8.0
- Fluid equations
- 7.0

## Figures

(a) A thin light shell falls into a black hole to produce a larger black hole, (b) a thin light shell with flat Minkowski interior collapses to form a black hole.

(a) A thin light shell falls into a black hole to produce a larger black hole, (b) a thin light shell with flat Minkowski interior collapses to form a black hole.

(a) A discrete sequence of light shells forms a black hole, (b) the continuous version of the same process.

(a) A discrete sequence of light shells forms a black hole, (b) the continuous version of the same process.

Some “outgoing” light rays in the thick light shell collapse. The last ray out (B) hovers at the Schwarzschild radius and defines a horizon.

Some “outgoing” light rays in the thick light shell collapse. The last ray out (B) hovers at the Schwarzschild radius and defines a horizon.

Collapse of a thin light shell in Kruskal–Szekeres coordinates, to be compared with Fig. 1(b) in Eddington–Finkelstein coordinates. Only the spacetime region to the right of the line has physical meaning. Compare the light rays A B C to those in Fig. 3.

Collapse of a thin light shell in Kruskal–Szekeres coordinates, to be compared with Fig. 1(b) in Eddington–Finkelstein coordinates. Only the spacetime region to the right of the line has physical meaning. Compare the light rays A B C to those in Fig. 3.

The pure or eternal Schwarzschild geometry in Kruskal–Szekeres coordinates. The entire spacetime region shown is given physical meaning in terms of a white hole region and a second exterior Schwarzschild region. Compare to Fig. 4. See for example Refs. 3 and 4.

The pure or eternal Schwarzschild geometry in Kruskal–Szekeres coordinates. The entire spacetime region shown is given physical meaning in terms of a white hole region and a second exterior Schwarzschild region. Compare to Fig. 4. See for example Refs. 3 and 4.

Collapse of a uniform fluid sphere to form a black hole in Painleve–Gullstrand coordinates.

Collapse of a uniform fluid sphere to form a black hole in Painleve–Gullstrand coordinates.

A thin shell of dust falls into a black hole to form a larger black hole in Painleve–Gullstrand coordinates. This is the analog of Fig. 1 for light.

A thin shell of dust falls into a black hole to form a larger black hole in Painleve–Gullstrand coordinates. This is the analog of Fig. 1 for light.

(a) A discrete sequence of dust shells forms a black hole, (b) a continuous version of the same process forms a black hole. These are analogs of Fig. 2 for light.

(a) A discrete sequence of dust shells forms a black hole, (b) a continuous version of the same process forms a black hole. These are analogs of Fig. 2 for light.

Some “outgoing” light rays in the collapsing dust ball. No ray can escape from the center after the last ray out (B). Compare to Fig. 3 for light shell collapse.

Some “outgoing” light rays in the collapsing dust ball. No ray can escape from the center after the last ray out (B). Compare to Fig. 3 for light shell collapse.

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