Black holes are formed when massive stars end in core-collapse supernovae. If the matter in the core (and infalling matter) are dense enough for gravity to dominate pressure (including degeneracy pressure) the core will collapse into a black hole. These black holes are called stellar-mass black holes. Some black holes are very much larger, containing millions of stars worth of matter. It is thought that every galaxy contains a supermassive black hole in its center. There are intermediate mass black holes, like the pair that merged, forming the gravity wave signal detected at LIGO last year. It is also theorized that very small micro-black holes could have been created in the early universe just after the big bang.
There are several urban myths about black holes, such as the misunderstanding that black holes are like cosmic vacuum cleaners, sucking in everything. As with other massive objects, it is possible to orbit a black hole and not get drawn in, as long as the orbiting object is far enough away and moving fast enough. Black holes are massive objects so dense that the escape speed goes to the speed of light.
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If you throw a ball upward, it will go up to a height and fall back down. If you throw it harder, it leaves your hand with more speed, and will go higher before stopping and falling back down. The harder you throw it, the higher it goes. If you could throw it hard enough, it would get infinitely high before falling back down. The speed you gave it would be its escape speed. If you threw it a bit harder, it could not fall back down.
We can find the escape speed of an object by setting its gravitational potential energy equal to its kinetic energy and solving for velocity. In the case of a black hole, by definition, the escape speed of an object equals the speed of light. The radius at which the escape speed equals the speed of light is not the "surface" of a black hole, as such. It is called the event horizon.
Black hole embedding diagram
The above diagram is called a spacetime embedding diagram. It is used to describe the spacetime curvature near a black hole.
In flat space, there are two ways to measure the radius of a circle. You can measure the distance from the center of the circle to the edge, or you can measure the circumference of the circle and divide by two pi. The issue with trying to measure the radius of a black hole is that you will get two different measurements for these two methods. This is because if you point a meter stick toward the center of the black hole, spacetime, and everything in it, stretches. Measuring the distance inward doesn't give the same measurement as measuring tangentially.
In order to accommodate the difference in the distances measured tangentially and radially, we could offset the inner circle "downward." The amount of offset gives a measure of the curvature of spacetime.
Animation by Kathy Hadley
As shown in the animation above, the curvature of spacetime gets stronger as you get closer to the black hole, and the slopes of the lines joining the circles get steeper. At the event horizon, the lines would be vertical. This gives the characteristic funnel shape to the embedding diagram of a black hole, and contributes to the incorrect idea that black holes are tunnels leading somewhere.
The embedding diagram above depicts a wormhole. A worm hole is a mathematical structure you can create by changing the sign in the metric and sewing the two solutions of the general relativity equations together, like a piecewise function. It makes for good science fiction, but there is not necessarily a physical counterpart for every equation that can be mathematically written. To go through a wormhole, first you would have to survive falling into a black hole.
If you decided to dive into a black hole, you would get longer as you approached the event horizon, just like the meter stick. We call this elongation "spaghettification."
If you were in a spaceship and dropped a light-emitting probe into a black hole, the light signal would stretch, redshifting as the probe neared the event horizon. As the probe crossed the event horizon, the wavelength of the light would go to infinity. In essence, the next pulse of light would never arrive, and it would look to you like the probe froze on the horizon and never crossed it. If you were on the probe, you would experience something quite different. You would just cross the event horizon.
This sequence of images illustrates what it might look like to fall into a black hole. As you approach the black hole, the gravitational lensing gets stronger, as the light from stars behind the black hole stretches more and more. Falling into the black hole, past the event horizon, all that could be seen would be the light falling in behind you, as no signal could come toward you from the center of the black hole.
Credit for images: Ute Kraus, Max-Planck-Institut für Gravitationsphysik, Golm, and Theoretische Astrophysik, Universität Tübingen
Light emitted from a probe near the event horizon would become more and more redshifted as it traveled through greater distances of curved space. To a person falling into a black hole, the opposite would be true. Light fallin in toward you would become more and more blueshifted.
What would it look like to peer down into a black hole? This artist's animation depicts what you would see if you could journey to the center of our galaxy to view the accretion disk of the supermassive black hole residing there.
This illustration from scientists at the Swift satellite depicts the debris disk of an active supermassive black hole in a galactic center. Supermassive black holes become active during galaxy mergers, when stable orbits of stars can become perturbed by the collision of two galaxies, sending stars too close to the central black holes.
This animation depicts what it might look like to see a star being torn apart as it gets too close to a massive black hole. When the gravitational force of the black hole overwhelms the gravitational force holding the star together, the matter of the star forms into a disk surrounding the black hole. Some of the super-heated matter falls into the black hole, but some escapes along the magnetic poles in hot jets. It is important to understand that this matter did not escape from the black hole itself, just the disk of matter that lies outside the event horizon.
Supermassive black holes may combine to form even larger black holes as smaller galaxies merge into larger galaxies through multiple collisions.
By tracking the orbits of stars near the center of our galaxy, astrophysicist Andrea Ghez was able to locate the supermassive black hole there. Black holes cannot be seen directly, since they emit no light. In essence, the trajectories of the stars allowed us to estimate the mass of the object they orbit, at about four million solar masses. By conjecture, we assume there must be a supermassive black hole there, since that is the only object that could have a mass that large without being seen.