Journey into a Schwarzschild black hole

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Journey into the Schwarzschild black hole. This one shows a map, a clock, and an artificial grid on the black hole's horizon.

The simplest kind of black hole is a Schwarzschild black hole, which is a black hole with mass, but with no electric charge, and no spin. Karl Schwarzschild discovered this black hole geometry at the close of 1915^1,2,3,4,r54, within weeks of Einstein presenting his final theory of General Relativity.

The background is Axel Mellinger's All-Sky Milky Way Panorama (by permission).

Dive in!

For an explanation of what happens in the movie, read on below.

Journey into the Schwarzschild black hole. This one is plain.

This is the same movie as above, but without the map, clock, or grid.

Notice that you cannot tell when you pass through the horizon.

Journey into the Schwarzschild black hole. This one is in stereo.

This stereo version includes only the final part of the voyage, through the horizon to the singularity. To view the stereo, cross your eyes, and relax your focus.

The distance in the stereo movie is the “affine distance”, which is the natural generalization of distance along the past lightcone to an emitter in general relativity⁶.

The stereo movie has been adapted to human binocular vision. The movie does not represent what you would actually see with your two eyes if you visited a real black hole. In reality, the curved spacetime would distort wavefronts of light away from spherical, confusing your binocular perception. The conflicting visual cues might make you feel queasy. But the failure of binocular vision is merely a limitation of beings who have evolved in flat spacetime. Trinocular vision would work fine. The three-eyed ape at right, drawn by my daughter Wildrose, would have no problem leaping from tree to tree in a highly curved spacetime.

General relativity

Almost a century after Albert Einstein proposed it in 1915⁷, the General Theory of Relativity continues to beat its competitors in the Darwinian struggle to be top theory of gravity⁸. Black holes are general relativity's most extreme prediction. Astronomers find ubiquitous evidence for objects in which a large mass is concentrated in a small space, and which are exceptionally bright, energetic, and variable. The extreme properties of these objects seem to leave little room for anything other than a black hole. Of course, astronomers do not see the black hole itself, which is black. Rather, they see the violent effects of the black hole on its environment.

At left is a Hubble Space Telescope image of the center of M87, the giant elliptical galaxy at the center of the Local Supercluster of galaxies. M87 contains the largest black hole whose mass has yet been measured, 6 billion times the mass of our Sun⁹. Emerging from the galaxy is a powerful jet which Hubble has clocked at 6 times the speed of light. Click on 4D Perspective to find out why this does not contradict relativity's assertion that nothing can move faster than light.

“Black holes have no hair”

The “no-hair” theorem states that the geometry outside (but not inside!) the horizon of an isolated black hole is characterized by just three quantities:

• Mass
• Electric charge
• Spin

Black holes are thus among the simplest of all nature's creations. They are simpler than stars, much simpler than planets, and vastly simpler than human beings.

It is this simplicity that gives me the audacity to make movies of black holes, and to have some confidence that they portray black holes accurately, provided of course that the theory of General Relativity is correct.

The movies on this page show the simplest of all kinds of black hole, a Schwarzschild black hole, which has mass, but no charge, and no spin. The Schwarzschild geometry describes the geometry of empty space surrounding any spherical mass.

Real black holes are likely to be more complicated than the Schwarzschild geometry: real black holes probably spin, and the ones that astronomers see are not isolated, but are feasting on material from their surroundings.

When a black hole first forms from the collapse of the core of a massive star, it is not at all a no-hair black hole. Rather, the newly collapsed black hole wobbles about, radiating gravitational waves. The gravitational waves carry away energy, settling the black hole towards a state where it can no longer radiate. This is the “no-hair” state.

Map

The inset to the bottom left of the movie is a map of your trajectory into the black hole. You follow a real free-fall trajectory.

The green region is a “safe” zone where circular orbits are stable.

The yellow region is a “risky” zone where circular orbits are unstable. If you are on an unstable circular orbit, then a tiny burst on your maneuvering thrusters will send you into the black hole, or off into outer space.

The orange region is a “danger” zone where there are no circular orbits, stable or unstable. To remain in orbit in this zone, you must keep firing your rockets. The closer to the horizon you get, the harder you must fire your rockets to keep from falling in.

The red line is the horizon, from within which there is no escape.

Clock

The inset to the bottom right of the movie is a clock, which records the time left until you hit the central singularity, the place where space and time as you know them come to an end.

The clock records your “proper” time, the time that you actually experience in your brain, and that your wristwatch shows. In the movie, the clock slows down not because time is slowing down (à la 1979 movie “Walt Disney's The Black Hole”), but because it is more interesting to run the movie more slowly nearer the singularity, so that you can see more clearly what happens there.

The time is in seconds if the black hole has a mass of 5 millions suns, approximately equal to the mass of the supermassive black hole at the center of our Galaxy. On your trajectory, it takes 16 seconds to fall from the horizon to the singularity.

The background for the movie

Axel Mellinger's spectacular All-Sky Milky Way Panorama provides the background to the movie. Axel has kindly granted permission to use this panorama on this website.

The orange cross at the center of the image at left (click on it for a larger version) marks the position of the supermassive, 4 million solar mass, black hole at the center of the Milky Way, our Galaxy.

Gravitational lensing

The black hole bends light around it, as illustrated at left. A single object produces multiple images.

The resulting gravitational lensing is illustrated at right. The black hole appears to “repel” the image radially, which in turn stretches the image transversely. Parts of the image nearer the black hole are repelled more, so the image appears compressed radially.

Click on the button to see an animation of the Earth in orbit around a black hole. The animation nicely illustrates gravitational lensing. The black hole here is taken to have a radius equal to that of the Earth, which requires that the mass of the black hole be about 2,000 suns. The Earth orbits at 3 Schwarzschild radii (the minimum stable circular orbit), and we observe at rest from a distance of 5 Schwarzschild radii. For these parameters, we would see the Earth orbit the black hole 80 times per second.

This animation is not realistic! The Earth would be tidally torn apart in about one orbit if it were orbiting this close to a black hole of this mass.

Notice that when the Earth recedes from us, it appears reddish (redshifted) and slowed, and conversely when the Earth approaches us it appears blue (blueshifted) and speeded up.

The background to this animation is the 2-Micron All Sky Survey (2MASS).

Any mass, not just a black hole, bends light, but a large distortion requires a deep gravitational potential well. A spectacular example of gravitational lensing is the galaxy cluster Abell 1689 observed by the Hubble Space Telescope.

Einstein ring

If a bright object lies exactly behind the black hole, then light from the bright object will appear as a ring around the black hole, called the “Einstein ring”.

The red grid on the black hole's horizon

If you went up to a real black hole, you would not find a red grid on its horizon. But I figure that any self-respecting spaceship would paint the horizon of the black hole with a heads-up display. After all, black holes are dangerous things. The horizon is painted dark red, as a reminder that anything falling through the horizon would appear to an observer outside the horizon to be dim and redshifted.

You can see the “north” and “south” poles of the black hole simultaneously. That's because the black hole bends light around it, so that in effect you can see around the back of the black hole.

Innermost stable orbit

3 Schwarzschild radii marks the radius of the innermost stable orbit. Outside this radius circular orbits are stable, whereas within it circular orbits are unstable.

Astronomers argue that, if a black hole is accreting, then the inner edge of the accretion disk probably lies at the innermost stable orbit. At that radius, gas peels off from the accretion disk orbiting the black hole, and falls headlong into the black hole.

Unstable orbit

The parameters of your free-fall orbit almost put you on an unstable circular orbit around the black hole. A little more angular momentum, and you would be on an unstable orbit.

But you have chosen a trajectory with not quite enough angular momentum to go into unstable circular orbit, so you continue on into the black hole.

I imagine that if, in the distant future, human beings visit the black hole at the center of the Milky Way, they will go into an unstable circular orbit around it. Click on the button to see what things look like on an the unstable circular orbit at 2 Schwarzschild radii. If the black hole's mass is that of the supermassive black hole at the center of the Milky Way, then it would take approximately ten minutes to orbit the black hole in this orbit. Just right for a quick tour! On this unstable orbit, a short forward burst on your thrusters would send you back out into space, while a short retro burst would send you into the black hole.

Photon sphere

At 1.5 Schwarzschild radii is a special location, the photon sphere, where light rays can orbit in circular orbits about the black hole. This is the closest to the black hole that anything can remain in circular orbit without firing its rockets.

Although photons can in principle get “stuck” in circular orbit at the photon sphere, in practice the orbit is unstable, so photons do not concentrate there, and you do not see anything special as you pass through the photon sphere.

Being massive, you cannot travel at the speed of light (it would take you an infinite amount of energy to accelerate to the speed of light). But you could go almost the speed of light. If, going at almost the speed of light, you went into circular orbit just above the photon sphere, it would look like this.

Through the horizon

As you fall through the horizon, at 1 Schwarzschild radius, something quite unexpected happens. You thought you were going to fall through the red grid that supposedly marks the horizon. But no. The red grid still stands off ahead of you.

Instead, the horizon splits into two as you pass through it. Click on Penrose diagrams to understand more about why the horizon splits in two.

Without the grids (animation at right), you would not see any sign that you crossed the horizon. Yes, the animation is boring, isn't it.

The same view in stereo.

Engulfed in blackness? NO!

It is a common misconception that if you fall inside the horizon of a black hole you will be engulfed in blackness. More specifically, the story is that as you fall towards the horizon, the image of the sky above concentrates into a smaller and smaller circular patch, which disappears altogether as you pass through the horizon.

The misconception arises because if you lower yourself very slowly towards the horizon, firing your rockets like crazy just to stay put, then indeed your view of the outside universe will be concentrated into a small, bright circle above you. Click on the button to see what it looks like if you lower yourself slowly to the horizon. Physically, this happens because you are swimming like crazy through the inrushing flow of space (see Waterfall), and relativistic beaming concentrates and brightens the scene ahead of (above) you. See 4D Perspective for a tutorial on relativistic beaming. But this is a thoroughly unrealistic situation. You'd be daft to waste your rockets hovering just above the horizon of a black hole. If you had all that rocket power, why not do something useful with it, like take a trip across the Universe?

If you nevertheless insist on hovering just above the horizon, and if by mistake you drop just slightly inside the horizon, then you can no longer stay at rest, however hard you fire your rockets: the faster-than-light flow of space into the black hole will pull you in. Whatever you choose to do, the view of the outside Universe will not disappear as you pass through the horizon.

Click on the button at left (with the horizon grid) or right (without the horizon grid) for an animation of the appearance of the outside Universe as you lower yourself slowly to the horizon. The Universe appears brighter and brighter as you approach the horizon, tending to infinite brightness at the horizon. But again, no one with any sense would do this.

Inexorably inward

Inside the horizon, space is falling faster than light, carrying you inexorably inward. In this image you are at 0.8 Schwarzschild radii.

As you fall further inside the horizon, the Schwarzschild bubble enlarges.

The same view in stereo.

Tidal forces pull you apart

In a supermassive black hole like this one, the tidal forces are weak enough that you can fall deep inside the horizon before you are torn apart.

The gravity at your feet is stronger than the gravity at your head — as long as you fall in feet first, so that your feet are nearer the black hole than your head. You feel this difference in gravity between your feet and your head as a tidal force, which pulls you apart vertically in a process called “spaghettification”. At the same time as you are pulled apart vertically, you are crushed in the horizontal direction, like a rubber band being pulled. So if you would like to be taller and thinner, then one way to achieve that is to fall into a black hole (and be sure to fall in vertically!). However, like many diets, the improvement to your shape will be only temporary.

Trivium: it is a general fact you will be torn apart by a black hole approximately a tenth of a second before you hit the singularity, independent of the mass of the black hole. This is because the free-fall time goes as the inverse square root of the tidal force, t_ff = g^−1/2 = (G M / r³)^−1/2.

The same view in stereo.

To the singularity

Near the singularity, the tidal force becomes extreme. The same tidal force that pulls you apart vertically and crushes you horizontally concentrates the view of the outside Universe into a thin band around your waist. The views above and below are dimmed and redshifted, while the view around your waist is brightened and blueshifted.

If you have some transverse motion (some angular momentum) about the black hole, as in the movie, then relativistic beaming concentrates and blueshifts the scene in the direction of your motion.

You never get to see the singularity, because all light is headed towards the singularity, none away from it.

The same view in stereo.

Dive to the singularity

If instead of falling with some transverse motion (some angular momentum) you fall vertically into the black hole (with zero angular momentum), then the approach to the singularity looks like you are landing on a flat plane, not approaching a point. The appearance as a flat plane is caused by the overwhelmingly huge tidal force.

The same view in stereo.

At the singularity

At the center of the black hole is a singularity, a place of infinite of curvature, where space and time as you know them come to an end.

Geometrical intuition, bolstered by pictures like this one would suggest that the center of the Schwarzschild black hole is a point. That intuition is misleading. If you and a friend fall into a black hole at the same time but at different locations (in latitude and longitude), you do not approach each other as you approach the singularity. Rather, the diverging tidal force channels the parts of your body along the inward radial direction. Far from meeting your friend at the singularity, you cannot even put out your arms to touch her.

“The” singularity is not a point. Rather, it is a 3-dimensional spatial boundary where general relativity commits suicide. New physics, presumably quantum gravity in some form, must replace general relativity at singularities. What that new physics is remains a profound unanswered question.

References

1. Tony Rothman (2002) “Editor's Note: The Field of a Single Centre in Einstein's Theory of Gravitation, and the Motion of a Particle in That Field” Gen. Rel. Grav. 34 1541-1543 comments on the fact that the Schwarzschild metric was discovered independently by Johannes Droste in 1916.
2. Karl Schwarzschild (1916) “Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie” Sitzungsberichte der Deutschen Akademie der Wissencshaften zu Berlin, Klasse für Mathematik, Physik, und Technik 1916 189-196;
   an English translation appears as (2003) “On the gravitational field of a mass point according to Einstein's theory” Gen. Rel. Grav. 35 951-959.
3. Karl Schwarzschild (1916) “Über das Gravitationsfeld einer Kugel aus inkompressibler Flussigkeit nach der Einsteinschen Theorie” Sitzungsberichte der Deutschen Akademie der Wissencshaften zu Berlin, Klasse für Mathematik, Physik, und Technik 1916 424-434.
4. Kip S. Thorne (1994) “Black Holes and Time Warps: Einstein's Outrageous Legacy” (W. W. Norton & Co.) recounts the history of Schwarzschild's discovery of his solution, and Einstein's rejection of it.
5. Nathalie Deruelle (2009) “Black Holes Chronology I. up to 1939” (available from Lecture notes from the Institut de Physique Théorique, Saclay, France) summarizes key dates and events in early theoretical understanding of black holes.
6. Andrew J. S. Hamilton & Gavin Polhemus (2010) “Stereoscopic visualization in curved spacetime: seeing deep inside a black hole” New J. Phys. 12 123027 arXiv:1012.4043.
7. Albert Einstein (1915) “Die Feldgleichungen der Gravitation (The Field Equations of Gravitation)” Sitzungsberichte der Deutschen Akademie der Wissencshaften zu Berlin, Klasse für Mathematik, Physik, und Technik 1915 844-847.
8. Clifford M. Will (2006) “The Confrontation between General Relativity and Experiment” Living Rev. Relativity 9 3.
9. Karl Gebhardt, Joshua Adams, Douglas Richstone, Tod R. Lauer, S. M. Faber, Kayhan Gultekin, Jeremy Murphy, Scott Tremaine (2011) “The Black-Hole Mass in M87 from Gemini/NIFS Adaptive Optics Observations” Astrophysical Journal 729 119 arXiv:1101.1954.