Black holes Topics Black hole basics Black holes
Black holes Topics Black hole basics Black holes, light, and time Black hole structure Falling into a black hole Radiation and other oddities Looking for black holes Motivation Let’s look at black holes! 1
Developing black hole theory Karl Schwarzschild (1873 -1916) We discussed him in our exploration of general relativity. A few weeks after Einstein published general relativity, Schwarzschild applied theory to stars. To simplify Einstein’s extremely complicated field equations, Schwarzschild restricted his solutions to a nonspinning, perfectly symmetrical (spherical) object. In 1915, Schwarzschild wrote two papers in rapid succession, first describing the warpage of space outside the star, then inside the star. These papers were presented by Einstein to the Prussian Academy of Sciences (Germany), as Schwarzschild was serving in the German army. Four months later, Schwarzschild died on the Russian front. 2
Embedding diagram Two dimensional space, illustrated as being warped in a third fictional dimension, is called an embedding diagram. Schwarzschild determined that spacetime is warped around objects with a now-familiar form. The more massive and dense an object, the larger and steeper the dimple in spacetime. Our three spatial dimensions are warped into a fictional hyperspace. While these diagrams are illustrative, it is more accurate to think of warpage as a characteristic of spacetime, instead of our dimensionality intruding into other dimensions. 3
Gravity, time, and photons Time is warped, just as space is. Near the Sun, time flows short by 64 seconds per year (1: 500000 distortion). At the Sun’s core, time flows short by 300 seconds per year (1: 100000 distortion). Since photons move at the same speed all the time, the warpage of space affects them differently than it affects matter… 4
Gravity, time, and photons Imagine a photon leaving the surface of an object that has a circumference somewhat larger than the critical circumference calculated by Schwarzschild. At the object’s surface, time flows more slowly. As a photon climbs out of the gravity well, it enters regions where time is moving faster. So in comparison, the photons would have a lower frequency. This means it would have less energy. This results in a gravitational redshift. 5
Gravity, time, and photons Compact the object even more. The effects of gravitational redshift increase. Recall that for the Sun, Rcrit = 3 km, Ccrit = 18. 6 km. Size of compact object 232, 000×Ccrit (1 R ) 4×Ccrit 2×Ccrit (~ neutron star) 1×Ccrit Gravitational redshift 0. 002% 15% 41% ∞% Wavelength (H, n=3→ 2) 656. 28 nm (red) 754. 7 nm (barely visible) 925. 4 nm (infrared) ∞ nm For a compact object that has been shrunk down to the critical circumference, light cannot escape. Time is frozen at the surface of this object. A photon trying to leave the surface of this critically compact object would be leaving a place in spacetime where time is infinitely slow. Even if it did (impossibly), after an infinitesimal distance its frequency would be infinitely gravitationally redshifted to zero. Robbed of energy, the photon would be dead, and so it never left in the first place. 6
A spacetime rupture Once an object is squeezed to a size smaller than its critical circumference, no outwards pressure can overcome the inwards gravitational force. The object collapses to a single point. The curvature in spacetime becomes infinite. The center point is called the singularity. 7
Frozen in time In 1939, Oppenheimer and Snyder modeled a collapsing star. The model was a simplified version of a real star: – Perfectly spherical; – No spin; – Uniform density; – No radiation or ejected material; – No shocks travelling through the star. They found that as the black hole progenitor shrinks to its critical circumference, time slows at the surface. The black hole will completely shrink to within its critical circumference only in the infinite future! Black holes do not exist! (Yet) Meanwhile, from the perspective of the black hole, it forms within about 90 minutes. 8
Frozen in time Similarly, if you were to watch an object fall into a black hole, you would see it get closer and closer to the critical circumference. As it approaches the black hole, it will traverse spacetime that is increasingly distorted. The time dimension is expanded, and the object will, as a result, appear to slow to a stop just outside the critical circumference. Similarly, it would appear to freeze in time because of the nearly infinite gravitational time dilation. This critical circumference is called the event horizon. If you visited a black hole, and hovered just above the surface of the event horizon, and then managed to return to flat space far from the black hole, you would discover you were (essentially) infinitely far in the future. N. B. The name for black holes used in the U. S. S. R. was “frozen stars. ” 9
A conceptual danger The radius of the black hole In the region near a compact object such as a neutron star, it is no longer true that the length of a curved path is related to the radius of the orbit by C=2πR. Instead, C < 2πR, because the distance R is stretched in warped spacetime. Thinking of a black hole in terms of its “critical radius” has conceptual dangers. You can define an equivalent “radius” (which is, of course, the Schwarzschild radius, Rs), but you should remember the dangers of any definition that includes a measure to the center of the black hole. There lie dragons! We will use the Schwarzschild radius in our discussions, but with caution. It is often best to define black holes in terms of their circumferences, and avoid Rs. 10
Black holes have no hair Regardless of the object that forms a black hole, very few things distinguish one black hole from another. Matter or antimatter, round or cubical, magnetic field or none, hydrogen or iron— all these features of the black hole progenitor are irrelevant. For example, asymmetries such as big lumps in a collapsing object disappear as the black hole forms. The only three characteristics of black holes that survive are: 1. Mass, 2. Charge, 3. Angular momentum. By the way, the phrase “black holes have no hair, ” coined by John Wheeler took about a decade to be accepted in scientific print in France and the U. S. S. R. 11
Charged black holes are unlikely to persist in nature; the black hole would preferentially draw in oppositely charged particles more rapidly, thus neutralizing the charge. That said, a charged black hole develops a second event horizon interior to the normal one, just around the central singularity, called a Cauchy horizon**. The distance between the two horizons depends upon the charge. The more the black hole is charged, the closer the two horizons get. The mathematics of general relativity suggest that if you charged the black hole enough, the two horizons would approach, merge, and then disappear**. But physicists don’t like the idea of “naked singularities”; it is thought that the cosmic censorship hypothesis (stated by Roger Penrose) would prevent it. The Reissner–Nordström metric is the solution of Einstein’s field equations for the charged black hole problem. 12
Rotating black holes are the most complicated of all, and Einstein’s field equations were first solved for them by Roy Kerr in 1963. Black holes formed from stars should be expected to be spinning because of the conservation of angular momentum. There is a limit to the rotational speed of a black hole—this corresponds to the event horizon (at the equator) traveling at 1 -c. 1 M → 62 microsecond period; 106 M → 62 second period. Models of black holes in accretion disks indicate that once they have absorbed enough material to double their masses, they should be spinning at approximately maximal speed! A rapidly rotating black hole has a very complicated structure… 13
Rotating black holes The rotation produces a complicated shell structure of different zones. Much of this complication comes from the effects of frame-dragging (remember this? ), which means that space is dragged with rapidly rotating, massive objects. Rotation always makes physics a few levels more complicated! 14
Rotating black holes First, a rotating black hole has two photon spheres—a co-rotating and a counterrotating photon sphere. Outer horizon Co -ro ta tin g Second, there are now two event horizons**. Inner horizon Co rot unt ati erng Third, the singularity point is spread into a disk-shaped region**. Singularity Top View 15
Rotating black holes From the side, you can see the interesting fact that the two photon spheres are actually elliptical. Outer horizon Co rot unt ati erng Co -ro ta tin g You can also see the flattened nature of the singularity. Inner horizon Singularity Side View 16
Rotating black holes Consider the fact that the event horizon of a non-rotating black hole marks the boundary where you must move inwardly towards the black hole if you are inside it. But you could hover just above the event horizon of a non-rotating black hole. Because of frame-dragging near a rotating black hole, it is impossible to stand still just above the event horizon. This region, where you cannot stand still, is called the ergosphere. 17
Rotating black holes Interestingly, it is possible to extract energy from a black hole. Fire an object into the ergosphere and program it to break into two portions. If the trajectory is set so one part flies into the black hole, against the direction of rotation, the other portion will be ejected at a much Co rot unt higher velocity. a er Ergosphere Rapidly rotating black holes could be used as a highly efficient (50%) energy source! Compare the efficiency of the Penrose Process with the pathetic 0. 007% efficiency of nuclear fusion. t -ro Co ng i t a tin g Inner horizon Singularity Erg osp Side View Outer horizon her e 18
Descending into a black hole What would happen if you descended into a stellar black hole? As you fell towards the horizon, your regularly-scheduled radio messages to the outside world would be gravitationally-shifted to longer wavelengths, and time-dilated to occur leesssss freeeequeeeeennnnnntlyyyyyyyyy. You would move at ever-quickening pace, although outside observers would see you time-freeze at the event horizon. Tidal forces on an astronaut would be extreme: For a 2 m astronaut falling into a 5 M black hole, Δg = 8× 108 m/s 2, or about 8× 107 g! Human tissue can withstand approximately 1000 g. You’d get dismembered about 650 km from a 5 M black hole. 19
Descending into a supermassive black hole If you fell into a 4× 106 M black hole, as in Sgr A*, the tidal forces would be negligible at the outer event horizon—only about 1. 25× 10 -4 g. Once inside a black hole, there is no turning back. In fact, space is so curved, you can only travel forwards, towards the singularity. (In normal spacetime, this is like time—you can only travel forwards in normal time. ) Meanwhile, by firing retrorockets, you will be overtaken by all the material that enters the black hole after you, allowing you to effectively experience events that happened backwards in time. (In Outer horizon normal spacetime, this is like space—you can travel backwards and forwards in normal space. ) Inside a black hole, space becomes time, and time becomes space. (This switches again at the second event horizon**. ) Inner horizon Singularity But you will run into the black hole soon enough, and die. 20
Hawking radiation In 1974, Stephen Hawking followed a conceptual framework established by Jacob Bekenstein… Imagine a tiny vacuum fluctuation of energy occurring in space, very close to an event horizon. This fluctuation creates a virtual pair of photons. Tidal forces separate the two particles. One falls into the event horizon, and the other may escape. Far away, we see these escaping photons. The black hole radiates energy! This is now called Hawking radiation. Incidentally, any pair of particle-antiparticles can be formed, not just photons. Therefore, Hawking radiation consists of all kinds of matter and radiation. The vast majority, though, is photonic in nature. 21
Hawking radiation Smaller black holes have stronger tidal forces, and so can separate smaller (shorter wavelength) virtual photon pairs better. Therefore smaller black holes radiate more energetic photons than do large black holes. The photons most efficiently emitted have wavelengths about 1/4 th the black hole’s circumference, and larger. Recall Wien’s law: T = 2, 900, 000/λmax This means that we can look at the characteristics of a black hole’s Hawking radiation, and then give the black hole an equivalent temperature! Furthermore, we can figure out the black hole’s luminosity! What happens to a black hole, as it radiates energy into space? After all, a black hole is simply a hunk of mass-energy, and its corresponding intense gravitational field. The black hole gets smaller—it evaporates with time! 22
Black hole stats The event horizon’s effective temperature: TBH = 6× 10 -8 (M /M) K The black hole’s luminosity: PBH = 9× 10 -29 (M /M)2 W About how long does it take for a black hole to evaporate completely? t. BH = 2. 1× 1067 (M/M )3 years 23
Black hole evaporation To connect these equations to reality, let’s look at some examples: Mass Temperature Power A black hole as we might find in Sgr A*: 4× 106 M 1. 5× 10 -14 K 5. 6× 10 -42 W A black hole formed from a massive star: 5 M 1. 2× 10 -8 K 3. 6× 10 -30 W Evaporation time 1. 3× 1087 years 2. 6× 1069 years A primordial black hole that might live until today; equivalent to a 1. 8× 1011 kg rock 330 m on a side: 8. 8× 10 -20 M 6. 8× 1011 K 1. 2× 1010 W 1. 4× 1010 years A black hole (6 m on a side) evaporating today, with a rapid burst of gammaray energy, but with the luminosity of a small M star: 106 kg 1. 2× 1017 K 3. 6× 1020 W (10 -6 L ) 80 sec 24
Density inside the event horizon… 25
What is our universe? 26
New insights, confusion, and insights Entropy is a quantity measuring the amount of disorder a system has. Entropy is very much an issue of how matter is spread out over the volume of space, and how many ways it can be rearranged in that volume while still looking pretty much the same. The Second Law of Thermodynamics tells us that entropy in the Universe must increase with time. Physicists have developed a number of ways to calculate how entropy increases. Do black holes violate this, by potentially cleaning up the Universe and hiding entropy? NO! Because in time, black holes will radiate the matter and energy back into the Universe as highly entropic (random) Hawking Radiation. --ALSO-Carter, Hawking and Bardeen have found that as black holes draw in matter (and the entropy it contains), the surface areas of their event horizons increase too. Most importantly, these scientists have found a set of laws that say that the surface areas of black holes follow the same laws of thermodynamics that apply to entropy. 27
Entropy, event horizons, holographic universe Again: there is a set of laws that say that the surface areas of black holes follow the same kinds of laws of that apply to entropy. Susskind & ‘t Hoof are particularly interested in that fact that one can draw connections between entropy (which is usually a 3 -D quantity) and the surface area of a black hole (a 2 -D structure). They suggest that information of material falling into a black hole is somehow stored on the event horizon, perhaps as dimples of specific forms, on the event horizon. They next argue that the information (perturbations on the event horizon) are more relevant than the material that passed through the event horizon. Finally, they suggest that perhaps, what we call 3 -D reality is in fact simply a hologram produced by data stored on the 2 -D surface of the boundaries of our Universe. 28
Information paradox and the firewall Black holes absorb not only entropy, but also information. Physics indicates that information cannot be lost either. It is possible that the Holographic Principle ensures the information is not lost. There is also the possibility that as material falls through the event horizon, it is instantly incinerated in a superhot firewall. This would violate general relativity, which says that passing through the event horizon should be unremarkable. Hawking said (in a 2014 paper that included no calculations) that perhaps event horizons (and the firewall) don’t exist. Instead, space distortions blur them into—at most—an apparent horizon. Another resolution is if the Hawking Radiation is somehow entangled with the material inside the black hole, so the information is not really lost. Finally, I wonder—how much of the entire black hole phenomenon might disappear when we manage to integrate gravity and quantum? Will a quantum gravity theory make matters such as singularities disappear? 29
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