Approaching the Black Hole

Approaching the black hole (89K GIF movie). This is the same movie as on the index page.

Clicking on the image gives you a double-size version of the same movie (same 89K GIF, same resolution, just twice as big on the screen).

So hey, this ain’t Star Wars. But at least it’s scientifically accurate, kind of. All images on this site were calculated mathematically, using standard General Relativity and the Schwarzschild metric.

Here are a few things I did so it would look better:

• The timing in the movie is wrong. In reality, all the action would happen in a tremendous rush at the end.
• The Schwarzschild surface has a red grid on it. In reality, black holes are black.
• The colours of the stars are exaggerated. In reality all the stars would be dazzlingly white, with just a hint of colour.

And a few things I did because it was simpler:

• The black hole has a Schwarzschild (spherical) geometry. In reality most black holes probably rotate, and they are described by a Kerr geometry (see here from Michael Cramer Andersen’s pages or here from Peter Diener’s Relativistic particle trajectories page).
• There are no redshifts or colour changes to the images. In reality there would be a whole range of redshifts and blueshifts and colour changes. Lower down this page is a sample redshift map of the sky.
• The companion blue star is spherical. In reality its shape (its true shape, not just its image) would be tidally distorted by the black hole.
• There is no gas around to impede us. In reality the blue star would probably have a stellar wind, probably feeding an accretion disk of gas around the black hole.
• Only the gravity of the black hole itself, not the gravity of the blue, green, or yellow stars, is included in computing the infall trajectory.
• The black hole and the other three stars remain stationary during the time we take to fall into the black hole. Actually this is not a bad approximation.

This is a plan view of the fictional stellar system and our trajectory into it. The trajectory ends at the location of the black hole.

The system is a quadruple system, a binary binary, consisting of a \(60\) solar mass blue star with a \(30\) solar mass black hole companion, together with a green star plus yellow star binary.

The \(60\) solar mass blue star is a massive main sequence star, about \(20\) times the radius of our Sun. The yellow star is about \(15\) solar radii, and the green star about \(10\) solar radii. The yellow and green stars are larger in radius than main sequence stars with the same colours. They appear to be pre-main sequence stars. This system evidently formed relatively recently, only about \(100{,}000\) years ago.

Sounds arbitrary? Not really. First off, stellar-sized black holes are thought to form from the core-collapse of a massive star, incidentally causing a supernova. So if you want to find a black hole in a stellar system, a good place to look is a system of massive stars. Next, I wanted the stars to provide a nice background, so I pushed the limits on how big and close they could be.

If the blue star were losing mass into an accretion disk spiralling on to the black hole, as it probably would in reality, then the system would be an example of a High Mass X-Ray Binary System (movies) such as Cygnus X-1 (x-ray image; optical image; description).

Looking toward the star system at distance of 1 million Schwarzschild radii, 100 million kilometers, from the black hole.

The Schwarzschild radius of a black hole, the radius from within which not even light can escape, is proportional to its mass, and is equal to \(100\) kilometers for a \(30\) solar mass black hole.

This is a plan view of our trajectory into the black hole. The free-fall trajectory is rather special, because it puts us (temporarily) into a circular orbit with no rocket power required (aside from the manoeuvering thrusters needed to take careful aim at the outset).

The circular orbit at \(2\) Schwarzschild radii is unstable, a type of circular orbit which exists in General Relativity, but not in Newtonian gravity. A tiny forward blast on the thrusters will send us back out to safety; a tiny reverse blast will send us into the black hole. The unstable orbit at \(2\) Schwarzschild radii is that orbit which corresponds to zero kinetic energy (zero velocity) at infinity.

\(100\) Schwarzschild radii from the black hole.

The tidal force, the difference in the gravitational force between between your head and your toes, is now at 1 gee.

The tidal force increases rapidly as we free-fall inward. The tidal force goes as \(M / r^3\) at distance \(r\) from a black hole of mass \(M\).

While stretching us radially, the tidal force also compresses us laterally. What a way to slim.

\(26\) Schwarzschild radii from the black hole.

The blue star image forms an ‘Einstein ring’.

Any mass, not just a black hole, will bend the light from an object precisely behind it into an Einstein ring. However, large rings require deep gravitational potential wells. A number of examples of Einstein rings in astronomy are known.

Illustrating the manner in which images are gravitationally distorted. Mass bends light around it (upper picture).

Thus a foreground lensing mass (red dot) appears to ‘repel’ the image of a background object radially outward (picture to left). The repulsion stretches the image in the transverse direction. Parts nearer the lensing mass are pushed out more, so images appear compressed radially. Multiple images may appear if the gravitational potential of the lens is deep enough.

In the illustrated case, the lensing mass is a black hole. Any light rays which come within \(1.5\) Schwarzschild radii of the black hole fall into the black hole. Here there is a dark region, bounded by the red lines, within which images of background objects cannot appear.

All masses bend light, not just black holes, but large distortions require deep gravitational potential wells. A beautiful example is the arcs seen around rich galaxy clusters such as Abell 2218.

See the Spacetime Wrinkles exhibit on the gravitational bending of light or Pete Newbury’s gravitational lensing demos for more.

\(10\) Schwarzschild radii from the black hole.

The Schwarzschild surface is the horizon of the black hole, from within which nothing, not even light, can escape. The red grid on the Schwarzschild surface is fake, there for your edification. The gravity of the black hole allows us to see all the way around it. Thus we see the entire Schwarzschild surface of the black hole; indeed we see it replicated infinitely many times (of course we see the Schwarzschild surface here only because I put a grid on it; in reality we would see only blackness).

If the Schwarzschild surface did emit any light, from our point of view it would be infinitely redshifted, and infinitely slowed in time. A person falling through the Schwarzschild surface ahead of us would appear to freeze at the surface, taking an infinite time to fall in.

Actually, the Schwarzschild surface is not completely black. It emits Hawking radiation, which is thermal blackbody radiation, produced by quantum mechanical effects: the strong gravitational acceleration near the black hole modifies vacuum fluctuations. For a \(30\) solar mass black hole, the temperature of the Hawking radiation is tiny, only \(2 \times 10^{-9} \, \mbox{Kelvin}\). The wavelength at the peak of the blackbody spectrum is about equal to the Schwarzschild radius. See the page on Hawking radiation for more.

Although classically the light from an infalling person would appear to an outside observer to redshift without limit at the horizon, quantum mechanically the light redshifts only until the wavelength becomes comparable to the size of the black hole. Thereafter the person becomes indiscernible from the thermal Hawking radiation. Thus although classically a person would appear to an outside observer to take an infinite time to fall in, quantum mechanically the person disappears in a finite time.

\(3\) Schwarzschild radii from the central singularity of the black hole.

This radius marks the location of the innermost stable circular orbit around a black hole. Outside \(3\) Schwarzschild radii, all circular orbits are stable, meaning that a small blast on the manoeuvering thrusters by a rocket in circular orbit would not perturb the orbit greatly.

The inner edge of any accretion disk would normally be here.

However, we are not in this orbit: we would have to blast our rockets to slow down into this orbit. We free fall onward and inward.