In 1783 John Michel,
a Cambridge professor, pointed out the possibility that a sufficiently massive
and compact star could have an escape
velocity greater than that of light. If so, light would be unable to
escape from such a star.
In 1907 Einstein had a pivotal insight: A falling observer
does not experience gravity…
He described this as the “happiest thought of my life.” This insight led Einstein to propose the principle of equivalence:
A local frame of reference (e.g., the inside of a small elevator) in free fall in a gravity field is indistinguishable from a local frame of reference far from a gravity field.
Einstein explained his reasoning with his famous elevator gedanken (i.e., thought) experiment. Suppose that you are unfortunate enough to be in an elevator in free fall. Because the floor of the elevator accelerates towards the center of the Earth as fast as you do your body cannot exert a force on the elevator's floor. That is, you have become weightless. This is true of all objects within the elevator. If you take a pen from your pocket and let go of it the pen will remain suspended in the air because the pen is accelerating at the same rate as everything else.
An amazing consequence of the principle of equivalence is that light must bend in a gravitational field! This is illustrated in this diagram. It shows three scenes: the view from within a rocket in deep space (far from gravitational influences); the view from within the same rocket but in free fall near a gravitating object (say the Earth) and the view from the vantage of an observer who is stationary with respect to the gravitating object and who, therefore, experiences gravity.
According to the principle of equivalence, the rocket occupant cannot tell whether he is in deep space or in free fall. In particular, a laser fired across the cabin will appear to him to travel in a straight line from the laser gun to the light detector whether the rocket is in deep space or in free fall. There are two events to consider: a) the laser is fired and b) the light is detected.
(Abstractly, an event is a place at a given time. An event can be labeled by specifying four coordinates: (x,y,z,t), the actual values of which are a matter of arbitrary convention.)
We now consider the same scene as viewed from an observer on the ground. Since events a) laser fired and b) light detected are elements of reality, the ground observer must see the same events. But, by the time the laser light reaches the detector the rocket has fallen some distance. The only way to reconcile the observations of the two observers is to suppose that somehow gravity bends light!
Finally, in 1915 after much arduous thinking Albert Einstein introduced a new theory of gravity called General Relativity.
Einstein's great insight was to recognize that objects move the way they do in a gravitational field not because of a mysterious force, called the force of gravity acting at a distance (as Newton at first suggested), but because the space and time in which they move is warped by the presence of matter and energy.
Elephants and students, if they are sufficiently close together, fall towards the Earth with the same acceleration simply because they experience the same space and time geometry. On the other hand, if the elephants and students are far apart they will, in general, experience different space and time geometries, in which case they need not fall at the same rate.
Spacetime
Space and time are tightly bound together, so much so that we prefer to think of them as a single smooth 4-dimensional entity called spacetime.
Planets orbit the Sun because they are forced to move in a spacetime that has been warped by the Sun's mass.
It is important to note that not only is space warped but also time. Indeed, clocks run slowest where the warping of spacetime is greatest. Since the gravity at the Sun's surface is stronger than that at the Earth's surface a clock on the Sun would run slower than one on Earth. As we shall see, the temporal distortions near a black hole can be extreme.
Critical Radius
The general theory of relativity predicts that if any object, for example a star, is squeezed below a certain critical radius gravity will overwhelm all known forces. If this happens the inescapable conclusion is that the object will be crushed to infinite density in a singularity.A singularity is a place or time when the spacetime curvature is infinite. It can be thought of as an edge to spacetime in the sense that this is where space and time cease to exist. The shape of this edge can be a point or a circle.
Many scientists believe that true singularities probably cannot exist because the quantum nature of matter and energy must play a role. It may be that on extremely small distance scales the very notion of spacetime breaks down and the putative singularity dissolves into a chaotic quantum foam.
Karl Schwarzschild
In 1916 Karl Schwarzschild found the first exact solution to Einstein's general relativity equations. This solution was interpreted as describing the spacetime geometry around a stationary mass.
Later it was understood to also describe the spacetime geometry of a spherically symmetric object called a black hole, a term coined by John Wheeler in 1969.
Remarkably, Schwarzschild found his solution within months of Einstein's publication of his equations of general relativity. Even more astonishing, he did this work while a frontline soldier in World War I. Sadly, shortly after his impressive calculations, Karl Schwarzschild died at the front.
The critical radius is called the Schwarzchild radius, in his honor.
Schwarzchild Radius
The Schwarzchild radius is
the radius at which the escape velocity from an object is equal to the speed of light.
Recall that the escape velocity is the initial speed needed for an object to escape to infinity from another object, for example a star. On Earth you would need an initial speed of at least 11 kilometers per second (7 miles/second) to escape to infinity.
According to the special theory of relativity no material object can travel faster than light. Therefore, any material object that approaches closer than the Schwarzschild radius, of a collapsed object, would be trapped forever because to escape the object would have to travel at a speed greater than that of light, which, according to relativity theory, is impossible.
We believe that if no anti-gravity forces exist then any star more massive than about 10 solar masses will collapse to a black hole.
Schwarzchild Radius Formula
The Schwarzchild radius, rS, is calculated usingwhere G is Newton's gravitational constant, M the mass of the collapsed object and c the speed of light.rS = 2GM/c2
Notice that the Schwarzchild radius increases in direct proportion to the mass of the collapsed object. Therefore, an object that is twice as massive as another will have a Schwarzchild radius that is twice as big.
Example: The Sun's Schwarzschild radius is about 3 km. That is, if the Sun (whose radius is 700,000 km) were crushed into a sphere of radius less than 3 km the Sun would become a black hole.
A black hole of mass 50 million solar masses would have a Schwarzchild radius 50 million times bigger than that of the Sun, that is 150 million km, or 1 AU. Such a black hole would just fit within the Earth's orbit!
Event Horizon
The Schwarzschild radius defines a one-way boundary (a membrane) that separates the inside of the black hole from the rest of the universe. This boundary, which is not made of any material, is called the event horizon.All events that occur in the spacetime enclosed by the boundary are forever hidden from outside observers. Therefore, in a real sense a black hole is disconnected from the rest of the universe in both space as well as in time.
Gravitational Time Dilation
Gravity slows time down.
Suppose that you are far away from a black hole, but a friend of yours decides to spend some time near its event horizon and is able, somehow, to return to you. You would find that she had aged less than you. How much less depends on how close to the event horizon she traveled.
We can get an idea of how much time slows down in a gravitational field from the formula
t = t0Ö(1 - rS/r)
where t is the time she has spent at a radius r from the singularity and t0 is your elapsed time and rS is the Schwarzschild radius.
Example: Suppose she was able to venture within 0.01% of the event horizon (that is, r = 1.0001*rS) and stay there for t0 = 100 days, according to your clock. Upon her return you would find that she had aged only single day!
Thus we see that two observers, one who is far away from and one who is close to an event horizon perceive time differently. From the point of view of the distant observer someone approaching the event horizon requires an infinite amount of time to get there! This is another way of seeing why nothing that has dropped below the event horizon can ever return back to the universe from which it came: the latter no longer exists since it has evolved into its infinite future. The only way to return back from whence you came would be to travel backwards in time so that you would appear to return from the infinite future!
Yet from the point of view of the intrepid black hole explorer she gets to the event horizon in a finite amount of time.
Time to Oblivion
Suppose that your curiosity got the better of you and you decided to plunge below the event horizon to explore the strange spacetime that's hidden from the universe outside. Unfortunately, you would have only a finite amount of time before your demise. For a black hole of mass M times that of the Sun the time to go from the event horizon to the singularity is given byt = 1.57 x 10-5 M seconds
For the Sun, which has a Schwarzschild radius of 3km (and for which, by definition, M = 1) you would have no more than 16 microseconds before being crushed out of existence!
Schwarzschild black holes
These black holes are formed from non-rotating matter and contain a point singularity shrouded by an event horizon. This diagram shows the overall structure of the black hole. Let's imagine falling towards the event horizon along a radial direction while facing away from the horizon. Far from the event horizon we can see about half of the stars in the sky, namely, those stars in the hemisphere away from the event horizon.However, the closer we get we find that we can see more and more of the sky. This is due to the bending of light from stars behind us. As we continue our descent the spacetime curvature increases, light is bent more and we see ever more of the universe.
When we reach exactly 1.5 times the Schwarzschild radius, not only do we see a highly distorted view of the entire sky but we see light rays in orbit about the black hole. Indeed, if we turn our head to the right or left we would see the back of our heads (highly magnified)! The set of all circular orbits at this radius form a surface called the photon sphere. It is the blue circle in the diagram.
(Here are two animations (1 and 2) of the paths of light rays as you descend towards the black hole. The second animation zooms in on the falling observer and shows the compression of the scene into an ever smaller angular size.)
As we continue our descent to the event horizon we perceive a menacing scene: the entire sky is now a highly distorted image squeezed into a disk that gets progressively smaller surrounded by the darkness of the black hole which seems to be rapidly enveloping us. The light that rain downs on us from the distant stars gets brighter and brighter because of the blue-shifting caused by the increasing spacetime curvature. Finally, we cross the event horizon and the disk of stars continues to shrink. We have now entered a realm that is truly bizarre.
Outside the event horizon the geometry was static. However, inside the horizon the geometry is dynamic, that is, it changes with time. Moreover, although the diameter of the black hole, as measured by a distant observer, is finite the volume of the space we now find ourselves in is actually infinite!
As we approach the singularity we find that the space along one dimension expands, while in the other orthogonal directions it contracts. The topology (that is, shape and connectedness) of the space is akin to that of a cylinder. A cylinder can be thought of as a circle multiplied by a line: that is, to every point on the circle we attach a line. The radius of the circle shrinks with time, while the cylinder expands along its length. Of course, instead of a circle we really have a sphere: so the topology of this strange space is a sphere times an infinite line!
Alas, we can't do too much exploring because the tidal forces are gradually tearing our bodies to shreds!
Kerr black holes
In 1963 Roy Kerr (a New Zealander) worked out the structure of a black hole formed from rotating matter. He showed that there is a region outside the event horizons, called the ergo-sphere, that drags space and time around with the rotating black hole, rather like a vortex. Because of the rotation the singularity at the center of a Kerr black hole is a ring, rather than a point. Here is a diagram of such a black hole.Time travel
In 1949 Kurt Godel (famous for his proof of the impossibility of proving all true statements in any logical system that includes the rules of arithmetic) discovered a solution of Einstein's equation that described a rotating spacetime.Einstein was very disturbed by Godel's solution because it predicted that in such rotating spacetimes a spaceship could go off on a journey and return before it set off! Einstein was shocked that his theory of general relativity allowed the possibility of time travel.
We do not know whether or not such spacetimes can exist in our universe. Some, like Hawking, argue that because we have no compelling evidence of visitors from our future, who presumably would have had enough time to develop time travel, we can assume that no such spacetimes exist in the part of the universe to which, in principle, we have access.
Nonetheless, such strange time-warping spacetimes are predicted to exist within Kerr black holes and the possibility of time-travel is being investigated by some physicists, notably Kipp Thorne.
Wormholes
In 1935, Einstein and Nathan Rosen discovered solutions of Einstein's equations that they interpreted as describing bridges between different parts of spacetime. Today we call these Einsten-Rosen bridges wormholes.
The theory predicts that the ring singularity inside a Kerr black hole could be a gateway to another part of spacetime, via a wormhole.
Unfortunately, passage through this ring would be truly a one-way ticket to oblivion: for as you passed through the ring you would witness the entire future history of (a part of ) the universe in a finite amount of your local time. So from your point of view the universe outside the black hole would, in a short time, cease to exist.
More likely, however, you would be vaporized by the infinitely blue-shifted radiation with which you would be bathed, coming from the ever more rapidly evolving universe outside the black hole.
Either way, you have a ticket to oblivion!
Black holes are so strange, that for a long time they were thought to be just a theoretical curiosity with no relevance to our world. But in the 1970s a powerful x-ray source, called Cygnus X-1, was discovered lying about 8000 light years from us.
This x-ray source flickers on time scales of about one hundredth of a second. Moreover, observations suggest that every part of the source changes its brightness at the same time. That can only happen if the source is small enough for some influence to travel from one part to the other in about one hundredth of a second to keep the flickering synchronized across the object.
This implies that Cygnus X-1 must be smaller (probably much smaller) than 1/100 of a light-second across; that is, much smaller than the size of the Earth!
We have some idea of the mass of the x-ray source because if forms a binary with a blue super giant star (HDE 226868) whose mass is expected to be about 30 solar masses. Information about the masses in a binary system can be had by using (a more exact form) of Kepler's 3rd Law:
including information about the radial velocities of the stars. m1 and m2 are the masses of the two stars. The mass of the x-ray source turns out to be about 7 solar masses!(m1+m2)p2 = a3
With such a large mass squeezed into so small a volume (less than the size of the Earth) the best explanation for Cygnus X-1 is that it is a black hole (bound in orbit about the blue super giant) that is stripping material from the giant star. The black hole and its partner orbit each other in about 5.6 days.
As the material approaches the black hole the material is accelerated to enormous speeds and heats up. The material can become so hot that it is able to emit x-rays. It is these x-rays that we believe we are observing.
More recently, strong evidence for the existence of super-massive black holes has been obtained using the Hubble Telescope. For example, the galaxy M87, which is about 50 million light years from Earth, is thought to harbor a black hole at its center with a mass equal to about 2 billion times that of the Sun. The mass of the black hole is inferred from the abnormally high speed with which stars near the galactic center orbit the center. An enormous mass is needed to hold the stars in their tight fast orbits. But the region is so small that one is left with no other explanation other than that the mass must be in a collapsed form; that is, M87 must contain a gigantic black hole at its heart.
