Collapse to a Black Hole

This movie shows the collapse of a uniform, pressureless sphere of matter that free falls from zero velocity at infinity. The timing in the movie is correct (unlike the timing in the “Falling into a Black Hole” movie). We observe the collapse from a safe distance of \(10\) Schwarzschild radii.

The outer grid is the surface of the collapsing sphere. The inner grid shows a layer half way in (I made the sphere transparent so you could see in). The changing colours show how the infalling matter becomes redshifted.

The geometry outside the sphere is the Schwarzschild geometry. The geometry inside the uniform sphere is, curiously enough, the same Friedmann-Lemaître-Robertson-Walker geometry that describes the expanding Universe, except that the ‘Universe’ here is collapsing. The fact that the collapse started from zero velocity at infinity means that the interior geometry is flat, and the density of the sphere is the critical density.

The collapse of a real star to a black hole would of course be a great deal more complicated than that of the uniform, pressureless sphere shown here.

Even though the sphere has collapsed to a point from its own point of view, an outside observer (like us) sees the sphere appear to freeze at its horizon, becoming more and more redshifted, and fainter and fainter.

The images of layers at different levels appear to merge together into a single surface, the horizon.

As time goes by, images of layers from close to the centre of the sphere expand and also merge with the images of the outer layers (the movie above doesn’t show this, because it was too confusing to include more than two layers; but the movie below does illustrate how images of inner layers expand and merge with outer layers). The merging of images of inner layers never quite includes the central point itself, which still appears (highly redshifted) as a point at the centre of the sphere.

Answer to the quiz question 9: The star does in fact collapse inside the horizon, even though an outside observer sees the star freeze at the horizon. The freezing can be regarded as a light travel time effect. As described here, space can be regarded as falling into the black hole, reaching the speed of light at the horizon, and exceeding the speed of light inside the horizon. Thus photons that are exactly at the horizon and pointed vertically upwards hang there for ever, their outward motion through space at the speed of light being cancelled by the inward flow of space at the speed of light. It follows that it takes an infinite time for light to travel from the horizon to the outside world. The star does actually collapse: it just takes an infinite time for the information that it has collapsed to get to the outside world.

Answer to the quiz question 10: Gravity does not escape from a black hole. Gravity moves at the speed of light, and cannot get from inside the horizon to the outside world. The gravity felt by a person outside a black hole is the gravity of the stuff that fell long ago into the black hole.

This movie shows a thin spherical shell of matter falling on to a black hole — perhaps the same black hole that collapsed above. The mass of the thin shell here is the same as the initial mass of the black hole. The radius of the horizon of the black hole increases in proportion to its mass, so doubles in size during the collapse.

We, the outside observer, remain stationary during the collapse. The horizon of the black hole expands not because we are getting closer, but rather because the infalling shell of matter gravitationally lenses the image of the black hole beneath it more and more.

The changing colours show how the infalling matter becomes redshifted.

Answer to the quiz question 11: When the mass of a black hole increases, does the horizon appear to engulf stuff that previously fell through? No. Stuff that previously fell through continues to appear frozen and redshifted at the horizon. As the horizon expands, it appears to carry the frozen, redshifted stuff outward with it.

This Eddington-Finkelstein spacetime diagram illustrates the collapse of the uniform, pressureless sphere of mass to a black hole.

As usual, the Finkelstein radial coordinate \(r\) is the circumferential radius, defined so that the proper circumference of a sphere at radius \(r\) is \(2\pi r\), while the Finkelstein time coordinate is defined so that radially infalling light rays (yellow lines) move at 45^o in the spacetime diagram.

The white line shows the worldline of the surface of the collapsing sphere. The pink-red line is the absolute horizon. The absolute horizon is defined to be the surface from within which photons can never escape to the outside. The horizon begins at zero radius, and moves outward until it encompasses the surface of the collapsing sphere.

A singularity forms (cyan line) inside the horizon when the sphere has collapsed to zero radius.

The dark purple lines are lines of constant Schwarzschild time outside the collapsing sphere, and of constant Friedmann-Lemaître-Robertson-Walker time inside the collapsing sphere. The vertical dark blue lines are lines of constant circumferential radius. The line at zero radius before the singularity forms is drawn in bright blue.

Compare this spacetime diagram to the Eddington-Finkelstein spacetime diagram of the plain Schwarzschild geometry.

Watch the Finkelstein spacetime diagram flip between Schwarzschild and collapsing sphere geometries (15K GIF); or same flip, double-size on screen.

Do real black holes have wormholes like the Schwarzschild geometry? No. This can be seen by studying the Kruskal spacetime diagram at left. Compare this Kruskal spacetime diagram to the Kruskal spacetime diagram of the Schwarzschild geometry, which contains a white hole, a black hole, and two Universes connected by a wormhole.

The Kruskal diagram of the Schwarzschild geometry showed it to possess not only a normal horizon but also an antihorizon, beyond which there was, supposedly, another Universe. The Kruskal diagram of the collapsing sphere shows that a normal horizon (pink-red line) forms, but there is no antihorizon. Instead of an antihorizon there is just the surface of the collapsing sphere (white line), within which lies the interior of the sphere, all the way down to zero radius (bright blue line). There is no white hole, no mysterious second Universe, and no wormhole.

Watch Finkelstein morph into Kruskal (40K GIF); or same morph, double-size on screen (same 40K GIF).

The collapse of a real black hole would of course be more complicated than this. However the conclusion remains that the Schwarzschild wormhole is an artefact of the idealized Schwarzschild geometry, and would not occur in reality.

This is a Penrose diagram of the collapse of the sphere to a black hole.

There is no antihorizon. The left edge of the Penrose diagram, in bright blue, marks the centre of the sphere, at zero radius. Typically people draw the zero radius line vertically, but I let the line go where the transformation from Kruskal to Penrose coordinates took it.

Watch Kruskal morph into Penrose (74K GIF); or same morph, double-size on screen (same 74K GIF).

This the same Penrose diagram as above, but with the star collapsed four Schwarzschild time units (\(4 r_s / c\)) further into the past. As time goes by, the Penrose diagram resembles more and more the Penrose diagram of the Schwarzschild geometry. The antihorizon of the Schwarzschild geometry is replaced by the exponentially dimming, redshifting image of the collapsing star, frozen at what appears to be the horizon.

Animated Penrose diagram of the Oppenheimer-Snyder collapse of a star (1.8M GIF); or same animation, double-size on screen (same 1.8M GIF).

Don, this one’s for you.

The animation at left shows a complete sequence of spacetime diagrams from Schwarzschild, to Finkelstein, to Finkelstein with a collapsing star, to Kruskal, to Penrose, to aged Penrose.

Mousing over the animation gives key frames of the animation.