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______ Holes, Singularities & Wormholes
Contents
______ Holes
Singularities
Wormholes
______ Hole
A massive astrophysical object that is theorized to be created from the collapse of a neutron star. The gravitational forces are so strong in a ______ hole that they overcome neutron degeneracy pressure and, roughly speaking, collapse to a point (known as a singularity). Even light cannot escape the gravitational pull of a ______ hole within the ______ hole's so-called Schwarzschild radius.
Uncharged, zero angular momentum ______ holes are called Schwarzschild ______ holes. Uncharged nonzero angular momentum ______ holes are called Kerr ______ holes. Nonspinning charged ______ holes are called Reissner-Nordström ______ holes. Charged, spinning ______ holes are called Kerr-Newman ______ holes. The ______ hole no hair theorem shows that mass, charge, and angular momentum are the only properties which a ______ hole can possess.
Features of a ______ Hole
A ______ hole forms when an object collapses to a small size (perhaps to a singularity) and the escape velocity in its neighborhood is so great that light cannot escape. The boundary of this region is called the event horizon because any event that occurs inside is invisible to outside observers. The radius of the region is RS, the Schwarzschild radius.
Strange Stuff Explained: ______ Holes
A massive star starts to collapse when it exhausts its nuclear fuel and can no longer counteract the inward pull of gravity.
The crushing weight of the star’s overlying layers implodes the core, and the star digs deeper into the fabric of space-time.
Although the star remains barely visible, its light now has a difficult time climbing out of the enormous gravity of the still-collapsing core.
The star passes through its event horizon and disappears from our universe, forming a singularity of infinite density.
Pile enough matter into a small enough volume and its gravitational pull will grow so strong that nothing can escape from it. That includes light, which travels at the absolute cosmic speed limit of 186,000 miles per second. In a stroke of descriptive genius, physicist John Wheeler named these objects “______ holes.” The radius of a ______ hole is called the event horizon because it marks the edge beyond which light cannot escape, so any event taking place inside the event horizon can never be glimpsed from outside—in effect, the inside of the ______ hole is cut off from our universe. It has even been speculated that ______ holes could be pathways into other universes. Gravity is so strong at the center of a ______ hole, that even Einstein’s gravitational laws must break down. The theory that governs the incredibly dense matter and strong gravitational fields at the center of a ______ hole is not yet known.
How ______ Holes Might Appear in Nature
Micro-______ Holes
Formation: Possibly formed in small regions of high density during the first few seconds of the Big Bang.
Unusual Features: Micro-______ holes aren't so ______! Quantum mechanics and thermodynamics tell us that they should radiate all forms of energy due to the strongly varying gravitational fields just outside their event horizon. These ______ holes slowly radiate themselves away, finally disappearing in a flash of high-energy radiation.
Size: If a micro-______ hole is created with a mass of greater than 10^17 grams, it would still be around today. This mass corresponds to about a cubic kilometer of water. In terms of size, it would be smaller than an electron!
Observations: Once believed to be a possible source of gamma-ray bursts, there is no current observational evidence for their existence.
Stellar-Mass ______ Holes
Formation: When a star with a mass of greater than 10 times the mass of the Sun explodes in a supernova, the central core left behind will collapse into a ______ hole.
Unusual Features: When this type of ______ hole is part of a binary system, matter can sometimes stream from the other star into a disk surrounding the ______ hole. This is called an accretion disk.
Size: These ______ holes would range from about 5 to 100 times the mass of the Sun, giving them diameters of 20 to 400 miles.
Observations: At present, there are over 25 objects astronomers suspect may be ______ holes near or in our own galaxy.
A partial list: Cygnus X-1, A0620-00, V404 Cygni, LMC X-1, LMC X-3.
What does an X-Ray binary star look like?
Where is Cygnus X-1?
Super-Massive ______ Holes
Formation: Formed via accretion from smaller bodies in the centers of many galaxies.
Unusual Features: On this scale, some of the infalling matter may become redirected into jets and ejected over the poles of the ______ holes. This phenomenon may explain the radio jets observed from the centers of some galaxies.
Size: HUGE! One million to one billion times the mass of the Sun. They would be about the size of our solar system.
Observations: Active Galactic Nuclei (AGN), Seyfert Galaxies, Quasars.
HST's View fo the Jet in Galaxy M87
HST's view of the accretion Disk in Galaxy NGC 4261
The ______ Hole Next Door
Some theorists believe ______ holes are part of every day life and are even formed by nature right here on Earth. In Science News Online you can find the article The ______ Hole Next Door by Peter Weiss about micro ______ holes. It also explains why the CERN Large Hadron Collider (which will be online in the spring or summer 2007) is important for current theories about multiple dimensions.
Final ‘Proof’ Provided for Milky Way’s Central ______ Hole
"We could not believe our eyes."
-- Astronomer Thomas Ott on the surprising discovery of a star swinging around the very heart of the Milky Way.
Surprising observations of a star swiftly orbiting the cloudy heart of the Milky Way Galaxy have verified with near certainty the existence of a central ______ hole, a theoretical object that still eludes direct detection.
Astronomers watched the star for a decade, tracking two-thirds of its path around the galactic center. No object has ever before been seen so close to the center of any galaxy, nor has any other object previously been observed making more than a small fraction of its orbital trek around a galaxy.
"Our work proves that there is indeed a supermassive ______ hole in our own galaxy," said Rainer Schoedel, a PhD student at the Max-Planck Institute for Extraterrestrial Physics (MPE) in Germany.
An international team of astronomers photographed the star as it zoomed around the galactic center at speeds ultimately exceeding 11 million mph (5,000 kilometers per second). Early this year, the star flitted precariously close to the ______ hole, coming within 17 light-hours, or just three times the distance from the Sun to Pluto.
The observations rule out nearly all other possible explanations for the tremendous amount of matter -- equal to some 2.6 million suns -- packed into a tight spot at the center of our galaxy.
"If one accepts the universal validity of the laws of physics," Schoedel told SPACE.com, "it is extremely hard to avoid the conclusion that the supermassive ______ hole in the Milky Way does indeed exist."
Chandra Catches Milky Way Monster Snacking
Einstein's Relativity Theory predicted ______ Holes. ______ holes are formed by the gravitational collapse of solar masses. Astronomical observations suggest a super-massive ______ hole containing millions to billions of solar masses in the center of most galaxies. As it turns out our Milky Way is no exception.
Also see Animations & Video and The Chandra Spacecraft.
New Twists on the Milky Way's Big ______ Hole
The supermassive ______ hole at the center of our Milky Way Galaxy is heftier than thought and rotates at an amazing clip, new research shows.
For years scientists said the ______ hole contained about 2.6 million times the mass of the Sun. They now believe the figure is somewhere between 3.2 million and 4 million solar masses.
And a new study suggests all that mass, confined to an area about 10 times smaller than Earth's orbit around the Sun, spins around about once every 11 minutes. The Sun, for comparison, takes about a month to make a revolution on its axis. Earth spins once every 24 hours.
______ holes can't be seen or measured directly, because light passing near them gets trapped. So astronomers measure a ______ hole's mass by observing the orbital speed of nearby stars.
The new mass estimate was made by two separate groups, one at the University of California, Berkeley, and another at the University of California, Los Angeles, UC Berkeley physicist Reinhard Genzel told SPACE.com.
Astronomers Watch a ______ Hole Eat a Meal
Astronomers from the Institute of Astronomy (IoA) in Cambridge, England have watched a bundle of matter at the heart of a galaxy 100 million light-years away as it orbited a supermassive ______ hole four times on its way to being destroyed. The material was approximately the same distance as our Earth is from the Sun, but instead of taking a year, it only took a quarter of a day, because of the massive gravity of the ______ hole. By tracking the matter's doomed orbit, astronomers were then able to calculate the mass of the ______ hole: between 10 and 50 million solar masses.
Static, Uncharged - Schwarzschild ______ Holes.
The first kind of ______ holes hypothesized, and hence, the "simplest". In 1916, Karl Schwarzschild hypothesized the existence of ______ holes from Einstein's theory of relativity. However, Schwarzschild only considered mass as a factor.
The Schwarzschild ______ hole non-moving, and does not have any charge. Put simply, it is brought about by the collapse of a stationary star. Light bends around the edge of the ______ hole, an observation that involves the theory of a photon sphere. A photon sphere occurs at a distance of 1.5 times the Schwarzschild radius. At this distance, light rays orbit unstably around the ______ hole. Photons orbit in a sphere around the gravity pull (assuming that the pull is equal in all directions).
Moving closer into the cosmic phenomena, we eventually reach the mathematical construct: an event horizon. It marks the distance where the gravity pull of the ______ hole is so strong that even light cannot escape it, and occurs at the Schwarzschild radius.
When an object is compressed below its Schwarzschild radius (RS), it becomes a ______ hole.
Static, Charged - Reissner-Nordström ______ Holes
Unlike the simpler Schwarzschild ______ hole, the Reissner-Nordström ______ hole contains charge. It has more than one event horizon. A second, inner horizon forms just above the singularity. It is termed the Cauchy horizon. Electrons appear to "hover" around this second horizon. In other words, a charged ______ hole has two radii where time seems to stop.
The more charge the ______ hole has, the smaller the outer event horizon is, and the larger the inner event horizon. If enough charge is amassed to equal the mass of a ______ hole, then both the outer and the inner horizon would merge. However, this would be near impossible as the minimum mass that a ______ hole is calculated to have is 10^30 times the speed of light. If charge becomes greater than the mass of the ______ hole, then both horizons would vanish, revealing the singularity ("naked singularity"). When this happens, we may actually be free to move around the singularity and resist being sucked into it without having to go faster than the speed of light. However, in order for this to occur, we have to generate charge equal to or more than the value of at least 3 solar masses, or 6 x 10^30. This would be near impossible to do.
Kerr ______ Holes
The Schwarzschild reference frame is static outside the ______ Hole, so even though space is curved, and time is slowed down close to the ______ Hole, it is much like the absolute space of Newton. But we will need a generalized reference frame in the case of rotating ______ Holes. Roy Kerr generalized the Schwarzschild geometry to include rotating stars, and especially rotating ______ Holes. Most stars are rotating, so it is natural to expect newly formed ______ Holes to process significant rotation too.
Features of a Kerr ______ Hole
The Kerr ______ hole consists of a rotating mass at the center, surrounded by two event horizons. The outer event horizon marks the boundary within which an observer cannot resist being dragged around the ______ hole with space-time. The inner event horizon marks the boundary from within which an observer cannot escape. The volume between the event horizons is known as the ergosphere.
What is a Kerr ______ hole?
The usual idealised "static" ______ hole is stationary, unaccelerated, at an arbitarily-large distance from the observer, is perfectly spherical, and has a point-singularity at its centre.
When one of these idealised ______ holes rotates, it gets an extra property. It's no longer spherically symmetrical , the receding and approaching edges have different pulling strengths and spectral shifts, and the central singularity is no longer supposed to be a dimensionless point.
The equatorial bulge in the event horizon can be deduced in several ways
... as a sort of centrifugal forces effect. Since it's possible to model the (distantly-observed) hole as having all its mass existing as an infinitely-thin film at the event horizon itself (i.e. where the mass is "seen" to be), you'd expect this virtual film to have a conventional-looking equatorial bulge, through centrifugal forces.
... as a sort of mass-dilation effect. Viewed from the background frame, the "moving film" of matter ought to appear mass-dilated, and therefore ought to have a greater gravitational effect, producing an increase in the extent of the event horizon. Since the background universe sees the bh equator to be moving faster than the region near the bh poles, the equator should appear more mass-dilated, and should have a horizon that extends further.
... as a shift effect. This tidy ellipsoidal shape isn't necessarily what people actually see - it's an idealised shape that's designed to illustrate an aspect of the hole's deduced geometry independent of the observer's viewing angle. In fact, the receding and approaching sides of the hole (viewed from the equator) might appear to have different radii, because it's easier for light to reach the observer from the approaching (blueshifted) side than the receding (redshifted) side (these shifts are superimposed on top of the normal Schwarzchild redshift).
If we calculate these motion shifts using either the SR shift assumptions f'/f = (flat spacetime propagation shift) × root[1 - v²/c²] or the plain fixed-emitter shift law f'/f = (c-v)/c, and then treat them as "gravitational", then by multiplying the two opposing shifts together and rooting the result, we can get the same averaged dilation factor of f'/f=root(1 - v²/c²) in each case, and by applying the averaged value, we recreate the same sort of equatorially-dilated shape that we got in the other two arguments.
Of course, none of these "film" arguments work for a rotating point, which immediately tells us that the distribution of matter within a rotating ______ hole is important, and that the usual method of treating the actual extent of a body within the horizon as irrelevant (allowing the use of a point-singularity) no longer works when the hole is rotating (a rotating hole can't be said to contain a point-singularity).
In the case of a rotating hole, the simplest state that we can claim is equivalent to the rotating film of matter for a distant observer is a ring-singularity.
Notes
The idea of being able to treat a non-rotating ______ hole as either a point-singularity or a hollow infinitely-thin film is a consequence of the result that the actual mass-distribution is a "null" property for a ______ hole, as long as it is spherically symmetrical. If the mass fits into a Schwarzchild sphere, the usual static model of a ______ hole allows the hole's mass to be point-sized, golfball-sized, or of any size up to the size of the event horizon.
It's usual to treat all the matter as being compacted to a dimensionless point, but sometimes it's useful to go to the other extreme and treat the matter as being at its "observed" position - as an infinitely-thin film at the event horizon (see Thorne's membrane paradigm).
The idea of being able to treat all shifts as being propagation effects is something that probably ought to be part of GR - in the context of ______ holes, the time-dilation effect comes out as a curved-space propagation effect due the enhanced gravitation due to kinetic energy. However, there's a slight "political" problem here, in that GR is supposed to reduce to SR, and SR is usually interpreted as having Lorentz shifts which are supposed to be non-gravitational (because allowing the possibility of gravitational effects upsets the usual SR derivations). A GR-centred physicist might not have a problem with this approach of treating all shift effects as being equivalent, a SR-centred one probably would.
The "bulginess" of a Kerr ______ hole is illustrated on p.293 of the Thorne book (fig 7.9). Thorne says that the effect of the spin on the horizon shape was discovered Larry Smarr in 1973.
Overview of Kerr Spacetime
Kerr spacetime is the unique explicitly defined model of the gravitational field of a rotating star. The spacetime is fully revealed only when the star collapses, leaving a ______ hole -- otherwise the bulk of the star blocks exploration. The qualitative character of Kerr spacetime depends on its mass and its rate of rotation, the most interesting case being when the rotation is slow. (If the rotation stops completely, Kerr spacetime reduces to Schwarzschild spacetime.)
The existence of ______ holes in our universe is generally accepted -- by now it would be hard for astronomers to run the universe without them. Everyone knows that no light can escape from a ______ hole, but convincing evidence for their existence is provided their effect on their visible neighbors, as when an observable star behaves like one of a binary pair but no companion is visible.
Suppose that, travelling our spacecraft, we approach an isolated, slowly rotating ______ hole. It can then be observed as a ______ disk against the stars of the background sky. Explorers familiar with the Schwarzschild ______ holes will refuse to cross its boundary horizon. First of all, return trips through a horizon are never possible, and in the Schwarzschild case, there is a more immediate objection: after the passage, any material object will, in a fraction of a second, be devoured by a singularity in spacetime.
If we dare to penetrate the horizon of this Kerr ______ hole we will find ... another horizon. Behind this, the singularity in spacetime now appears, not as a central focus, but as a ring -- a circle of infinite gravitational forces. Fortunately, this ring singularity is not quite as dangerous as the Schwarzschild one -- it is possible to avoid it and enter a new region of spacetime, by passing through either of two "throats" bounded by the ring (see The Big Picture).
In the new region, escape from the ring singularity is easy because the gravitational effect of the ______ hole is reversed -- it now repels rather than attracts. As distance increases, this negative gravity weakens, just as on the positive side, until its effect becomes negligible.
A quick departure may be prudent, but will prevent discovery of something strange: the ring singularity is the outer equator of a spatial solid torus that is, quite simply, a time machine. Travelling within it, one can reach arbitrarily far back into the past of any entity inside the double horizons. In principle you can arrange a bridge game, with all four players being you yourself, at different ages. But there is no way to meet Julius Caesar or your (predeparture) childhood self since these lie on the other side of two impassable horizons.
This rough description is reasonably accurate within its limits, but its apparent completeness is deceptive. Kerr spacetime is vaster -- and more symmetrical. Outside the horizons, it turns out that the model described above lacks a distant past, and, on the negative gravity side, a distant future. Harder to imagine are the deficiencies of the spacetime region between the two horizons. This region definitely does not resemble the Newtonian 3-spacebetween two bounding spheres, furnished with a clock to tell time. In it, space and time are turbulently mixed. Pebbles dropped experimentally there can simply vanish in finite time -- and new objects can magically appear.
The complete model of Kerr spacetime built in Chapter 3 adds two more horizons to each such interhorizon region (there will be many regions) -- and shows that Kerr spacetime is organized symmetrically around the spatial 2-spheres at which these horizons intersect.
Kerr-Newman ______ Hole
A rotating charged ______ hole. An exact, unique, and complete solution to the Einstein field equations in the exterior of such a ______ hole was found by Newman et al. (1965), although its connection to ______ holes was not realized until later (Shapiro and Teukolsky 1983, p. 338).
Rotating (Kerr) ______ Holes, Charged and Uncharged
Most stars spin on an axis. In 1963, Roy Kerr reasoned that when rotating stars shrink, they would continue to rotate. Kip Thorne calculated that most ______ holes would rotate at a speed 99.8% of their mass. Unlike the static ______ holes, rotating ______ holes are oblate and spheroidal. The lines of constant distance here are ellipses, and lines of constant angle are hyperbolas.
Unlike static ______ holes, rotating ______ holes have two photon spheres. In a sense, this results in a more stable orbit of photons. The collapsing star "drags" the space around it into rotating with it, kind of like a whirlpool drags the water around it into rotating. As in the diagram above, there would be two different distances for photons. The outer sphere would be composed of photons orbiting in the opposite direction as the ______ hole. Photons in this sphere travel slower than the photons in the inner sphere. In a sense, since they are orbiting in the opposite direction, they have to deal with more resistance, hence they are "slowed down". Similarly, photons in the inner ring travel faster since they are not going against the flow. It is because the photon sphere in agreement with the rotation can travel "faster" that it is on the inside. The closer one gets to the event horizon, the faster one has to travel to avoid falling into the singularity - hence the "slower" moving photons travel on the outer sphere to lessen the gravitational hold the ______ hole has.
The rotating ______ hole has an axis of rotation. This, however, is not spherically symmetric. The structure depends on the angle at which one approaches the ______ hole. If one approaches from the equator, then one would see the cross-section as in the diagram above, with two photon spheres. However, if one approached at angles to the equator, then one would only see a single photon sphere.
The position of the photon spheres also depend on the speed at which the ______ hole rotates. The faster the ______ hole rotates, the further apart the two photon spheres would be. For that matter, a ______ hole with a speed equal to its mass would have the greatest possible distance between the two photon spheres. This is because of greater difference in the speed between the photon spheres. As the speed of rotation increases, the outer sphere of photons would slow down as it meets greater resistance, even as the inner sphere would travel "faster" as it is pushed along by the centripetal forces.
Next, we move on to look at the ergosphere. The ergosphere is unique to the rotating ______ hole. Unlike the event horizon, the ergosphere is a region, and not a mathematical distance. It is a solid ellipsoid (or a 3-dimensional ellipse). The ergosphere billows out from the ______ hole above the outer event horizon of a charged ______ hole (a.k.a. Kerr-Newman), and above the event horizon of an uncharged ______ hole (a.k.a. Kerr). This distance is known as the static limit of a rotating ______ hole. At this distance, it is no longer possible to stay still even if one travels at the speed of light. One would inevitably be drawn towards the singularity. The faster the rotation, the further out it billows. When the ergosphere's radius is half the Schwarzschild radius along the axis of rotation, it experiences the greatest distance it can billow out. At this point, even light rays are dragged along in the direction of rotation. Strangely enough, it is postulated that one can enter and leave as one likes since technically, you have not hit the event horizon yet.
For a rotating ______ hole, the outer event horizon switches time and space as we know it. The inner event horizon, in turn, returns it to the way we know it. Singularity then becomes a place rather than a time, and can technically be avoided. When angular velocity increases, both the outer and the inner event horizon move closer together.
In the diagram, you would have noticed that the singularity here is drawn as a ring, and not a point, as it was for the static ______ hole. In the case of a rotating ______ hole, the gravity around the ringed singularity is repulsive. In other words, it actually pushes one away, allowing you to actually leave the ______ hole. The only way to approach the ring singularity would be to come in from the equatorial plane. Other trajectories would be repelled with greater strength, proportional to the closer the angle is to the axis of rotation.
In addition, there would be a third photon sphere about the ring singularity. If light is parallel to the axis of rotation, the gravity and the anti-gravity of the singularity are balanced out. Light then traces out the path of constant distance (which, in the case is an ellipsoid). Technically, this might lead the light into another universe through the singularity, and then back out again. At this point within the ______ hole, we may see three types of light: the light reflected from our universe behind us; the light from other universes; and the light from the singularity.
Beyond the Event Horizon
Of all a ______ hole's bizarre characteristics, none seems stranger than the fact that the solutions to Einstein's equations tell us that these holes in space-time can serve as bridges into other universes. As fans of science fiction are well aware, a parallel universe is a universe entirely separate from our own. Among the many speculations as to the nature of these universes, is the idea that there could be parallel versions of ourselves inhabiting these universes; each living out a slightly different version of our lives. This idea doesn't seem so irrational when viewed in relation to the equally strange world of quantum mechanics [10]. It is important to note, however, that the existence of these other universes is, at present, a purely theoretical construction.
Scientists often use a space-time diagram to demonstrate graphically the
strange world of General Relativity (figure 5).
Figure 5: Space-time diagram [11].
The future is located at the top of the diagram. All motion in space must travel within 45º of the vertical time line. Paths that are inclined greater than 45º are space-like, faster than light trips. Accordingly, the zone below the 45º line is shaded gray and marked "forbidden". Professor Roger Penrose of Oxford University has developed a special type of space-time diagram that is very useful for representing the solutions of ______ hole equations [12]. These diagrams quickly show the ______ hole's connection with parallel universes.
Figure 6 is a Penrose diagram of a simple Schwartzchild ______ hole. Upon first glance, it appears far more complex than the diagram in figure 5, yet it really isn't. Just as before, all paths through space must be inclined at an angle less than 45º from the vertical axis. The singularity of the ______ hole, denoted by the row of shark's teeth at the top, is at a 90º angle, hence it is space-like. The event horizon's one way nature is shown by the sharp bend in it's line. The path of a traveler into the hole is shown by the curved line that passes through the event horizon. Two things stand out as unusual about this diagram: First there is an extra singularity in the past (at the bottom). Secondly, there is the extra universe on the left.
Figure 6: Penrose Diagram of Schwartzchild ______ Hole [13].
The additional singularity, marked as past space-like singularity on the diagram, is what is known as a white hole. Very simply put, a white hole is the opposite of a ______ hole. Instead of engulfing everything that comes near it, the white hole repels matter. Notice the direction of its event horizon. Some physicists maintain that the singularity of a ______ hole opens into another universe (figure 7). The idea behind a white hole is that matter that falls into a ______ hole in our universe is then belched out in another. It is worth noting, however, that astronomers have never observed a white hole, so their existence is doubted.
Figure 7: ______ Hole Connected to Another Universe [15].
Figure 8 is a Penrose diagram of a Kerr ______ hole. It will be remembered that the spinning ______ hole exhibits several curious features, such as the ring singularity, the region of negative space, and the region through which travels into the past are possible. The ring singularity is denoted by the rounded off sharks teeth to show that it is a bit more forgiving than its non-rotating cousin. It should be obvious that the singularity is vertical (time-like). This means that one could escape from it with slower than light velocities. The area marked negative closed time-like loop is a region just inside the singularity. Very simply put, this is a region in which the normal barriers between past and future lose meaning. A traveler into this region could visit any place in his past or future, were it not for the one way nature of the event horizon. Just beyond the time-like loop region, lies the area of negative space. Unfortunately, this hole has not one, but two one way event horizons that would prevent a traveler from ever re-entering our own universe. However, as shown in the diagram, he would have his choice of many other universes to visit. Figure 8 shows four (three parallel universes, in addition to our own), however, this diagram could be extended an infinite number of times in both the past and future directions. Two examples of a travelers path into the Kerr hole are shown in the diagram. Path "A" takes the traveler into the ring singularity, while path "B" shows his path into another universe.
Figure 8: Penrose Diagram of a Kerr ______ Hole.
More general in-depth information: ______ Holes.
Developments in General Relativity: ______ Hole Singularity and Beyond
Igor Novikov wrote:
At the 20-th Texas Symposium on Relativistic Astrophysics there was a plenary talk devoted to the recent developments in classical Relativity. In that talk the problems of gravitational collapse, collisions of ______ holes, and of ______ holes as celestial bodies were discussed. But probably the problems of the internal structure of ______ holes are a real great challenge. In my talk I want to outline the recent achievements in our understanding of the nature of the singularity (and beyond!) inside a realistic rotating ______ hole. This presentation also addresses the following questions:
Can we see what happens inside a ______ hole?
Can a falling observer cross the singularity without being crushed?
An answer to these questions is probably “yes”.
______ Holes & Thermodynamics
Rong-Gen Cai, Li-Ming Cao and Da-Wei Pang wrote:
Recently Gibbons et al. in hep-th/0408217 defined a set of conserved quantities for Kerr-AdS ______ holes with the maximal number of rotation parameters in arbitrary dimension. This set of conserved quantities is defined with respect to a frame which is non-rotating at infinity. On the other hand, there is another set of conserved quantities for Kerr-AdS ______ holes, defined by Hawking et al. in hep-th/9811056, which is measured relative to a frame rotating at infinity. Gibbons et al. explicitly showed that the quantities defined by them satisfy the first law of ______ hole thermodynamics, while those quantities defined by Hawking et al. do not obey the first law. In this paper we discuss thermodynamics of dual CFTs to the Kerr-AdS ______ holes by mapping the bulk thermodynamic quantities to the boundary of the AdS space. We find that thermodynamic quantities of dual CFTs satisfy the first law of thermodynamics and Cardy-Verlinde formula only when these thermodynamic quantities result from the set of bulk quantities given by Hawking et al. We discuss the implication of our results.
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Singularity
A place where the known laws of physics no longer apply, where, as Caltech physicist Kip Thorne (1940-) puts it, gravity "unglues" space and time. Singularities may be points, one-dimensional lines, or even two-dimensional sheets. The general theory of relativity predicts that singularities form inside ______ holes but are conceled from the rest of the universe behind event horizons. A proper formulation of quantum gravity may well avoid the infinities associated with classical singularities.
Strange Stuff Explained: Singularity
The destiny of all matter that falls into a ______ hole is to get crushed to a point of zero volume and infinite density—a singularity. General relativity also implies that our expanding universe began from a singularity.
A singularity is a region of space-time in which gravitational forces are so strong that even general relativity, the well-proven gravitational theory of Einstein, and the best theory we have for describing the structure of the universe, breaks down there. A singularity marks a point where the curvature of space-time is infinite, or, in other words, it possesses zero volume and infinite density. General relativity demands that singularities arise under two circumstances. First, a singularity must form during the creation of a ______ hole. When a very massive star reaches the end of its life, its core, which was previously held up by the pressure of the nuclear fusion that was taking place, collapses and all the matter in the core gets crushed out of existence at the singularity. Second, general relativity shows that under certain reasonable assumptions, an expanding universe like ours must have begun as a singularity.
Wikipedia: Singularity
A (gravitational) singularity occurs when an astrophysical model, typically based on general relativity, predicts a point of infinite curvature. The term is closely related to the mathematical meaning of "singularity": a gravitational singularity occurs when the equations produce a mathematical singularity.
The Big Bang cosmological model of the universe contains a gravitational singularity at the start of time (t=0). At the "Big Bang Singularity," the model predicts that the density of the universe and the curvature of space-time are infinite. However, the basic Big Bang model does not include quantum effects, so its predictions are valid only shortly after the projected singularity.
A singularity also exists within a ______ hole, where general relativity predicts a region of infinite curvature. In a non-rotating ______ hole, the singularity occurs at a single point in the model coordinates, and is called a "point singularity". In a rotating ______ hole, the singularity occurs on a ring, and is called a "ring singularity". Rotating ______ holes are sometimes referred to as Kerr ______ holes.
Until the early 1990s, it was widely believed that general relativity hides every singularity behind an event horizon, making naked singularities impossible. This is referred to as the cosmic censorship principle. However, in 1991 Shapiro and Teukolsky performed computer simulations of a rotating plane of dust which indicated that general relativity allows for naked singularities. What these objects would actually look like is unknown. Nor is it known if singularities would still arise if the simplifying assumptions used to make the simulation tractable were removed.
Many physicists believe that gravitational singularities are "unphysical", meaning that general relativity ultimately ceases to be an accurate description of gravity somewhere in the vincinity of what would otherwise be a singularity. It is generally assumed that a theory of quantum gravity - a theory that unifies general relativity with quantum mechanics - will provide a better description of what actually occurs where general relativity predicts a singularity. However, no theory of quantum gravity has been experimentally confirmed to date.
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What exactly is a wormhole?
Wormholes are hypothetical entities that show up in theoretical analyses of Einstein's theory of gravity (general relativity). Nobody has yet seen a wormhole, nor are we certain that they exist, but they seem to show up so easily when we do calculations that many physicists suspect that they might actually be out there in the real universe.
There are two main types of wormhole of interest to physicists: Lorentzian wormholes (general relativity) and Euclidean wormholes (particle physics).
Lorentzian wormholes are essentially short-cuts through space and time. They are mainly studied by experts in Einstein gravity, and if they exist in real life would be more-or-less similar to the wormhole on Star Trek: Deep Space 9. (But remember, the show is just entertainment, so don't try to extract detailed physics from DS9; at best it will give you a vague general idea of what is going on.)
The good news about Lorentzian wormholes is that, after about ten years of hard work, we cannot prove that they do not exist. The bad news is that they are very strange objects: If they exist at all they need large amounts of negative mass to hold them open and stop them from collapsing. (Negative mass is not anti-matter, it's a region where the energy of the universe is less than that of ordinary vacuum - definitely weird stuff.) We can get small amounts of negative energy in the laboratory (the Casimir effect), but getting the large amounts needed to hold a decent size Lorentzian wormhole open looks to be hopeless with current technologies. (And there may be deep issues of principle preventing us from collecting a lot of negative energy in one place.)
If Lorentzian wormholes do exist, then it seems classically to be relatively easy to turn them into time machines. This embarrassing feature has led Stephen Hawking to promulgate his Chronology Protection Conjecture. According to this conjecture, quantum effects will conspire to effectively prevent time travel even when it looks like classical physics might allow time travel to occur.
Euclidean wormholes are even stranger: they live in "imaginary time" and are intrinsically virtual quantum mechanical processes. These Euclidean wormholes are of interest mainly to the particle physicists (quantum field theorists). You cannot give them a nice classical interpretation in terms of a well-behaved classical gravitational field, and unfortunately have to know a lot of quantum physics to appreciate even their basic properties.
Strange Stuff Explained: Wormholes
A beam of light traversing a path between two points in curved space-time can take longer to complete the journey than a hypothetical spaceship taking advantage of a wormhole’s shortcut connection between the two distinct regions of space-time.
Although they may seem more the stuff of science fiction than science fact, physicists first dreamed up the idea of wormholes. In 1935, Albert Einstein and Nathan Rosen realized that general relativity allows the existence of “bridges,” originally called Einstein-Rosen bridges but now known as wormholes. These space-time tubes act as shortcuts connecting distant regions of space-time. By journeying through a wormhole, you could travel between the two regions faster than a beam of light would be able to if it moved through normal space-time. As with any mode of faster-than-light travel, wormholes offer the possibility of time travel.
Until recently, theorists believed that wormholes could exist for only an instant of time, and anyone trying to pass through would run into a singularity. But more recent calculations show that a truly advanced civilization might be able to make wormholes work. By using something physicists call “exotic matter,” which has a negative energy, the civilization could prevent a wormhole from collapsing on itself. The stuff of science fiction, to be sure. But perhaps some day in the far future, it could also turn into science fact.
Wikipedia: Wormhole
A wormhole, also known as an Einstein-Rosen Bridge, is a hypothetical topological feature of spacetime that is essentially a "shortcut" from one point in the universe to another point in the universe, allowing travel between them that is faster than it would take light to make the journey through normal space. The name "wormhole" comes from an analogy that was used to explain this; imagine that the universe is the skin of an apple, and a worm is travelling over its surface. The distance from one side of the apple to the other is equal to half the apple's circumference if the worm stays on the apple's surface, but if it instead burrows a wormhole directly through the apple the distance it has to travel is considerably less.
Theoretical basis
It is not known if wormholes are possible. A solution to the equations of general relativity which would make wormholes possible has not been discovered, without exotic matter, a theoretical substance which has negative energy density. However, neither wormhole-accommodating solutions to relativity equations, nor the existence of exotic matter have been disproven. Many physicists, including Stephen Hawking (see Hawking's Chronology Projection Conjecture), believe that due to the problems a wormhole would theoretically create, including allowing time travel, that something fundamental in the laws of physics would prohibit them. However, this remains speculation, and the notion that nature would censor inconvenient objects has already failed in the case of the cosmic censorship principle.
Timeholes
A wormhole could potentially allow time travel. This could be accomplished by accelerating one end of the wormhole relative to the other, and then sometime later bringing it back; relativistic time dilation would result in less time having passed for the accelerated wormhole mouth compared to the stationary one, meaning that anything which entered the stationary wormhole mouth would exit the accelerated one at a point in time prior to its entry. The path through such a wormhole is called a closed timelike curve, and a wormhole with this property is sometimes referred to as a "timehole." It is thought that it may not be possible to convert a wormhole into a time machine in this manner, however; some mathematical models indicate that a feedback loop of virtual particles would circulate through the timehole with ever-increasing intensity, destroying it before any information could be passed through it. This has been called into question by the suggestion that radiation would disperse after traveling through the wormhole, therefore preventing infinite accumulation. The debate on this matter is described by Kip S. Thorne in the book ______ Holes and Time Warps [1], and will likely require a theory of quantum gravity to resolve.
Wormholes in General Relativity
Reduced to its most basic elements, a Lorentzian wormhole is a short-cut through space and time. The concept of a Lorentzian wormhole is essentially synonymous with that of a spacewarp -- a warping, beiding, or folding of space and possibly time. ... While we do not have any direct experimental evidence for the existence of such objects, it is commonly believed that such objects might be formed in regions of intense gravitational fields, where the highly curved nature of the spacetime manifold might allow for the existence of nontrivial topology.
Ideas along these lines have been floating around in the physics literature for the past sixty years. The earliest significant contribution I am aware of is the introduction, in 1935, of the object now referred to as an Einstein-Rosen bridge ... The field then lay fallow for twenty years until the period 1955-1957 when Wheeler coined the term "wormhole" and introduced his idea of "spacetime foam" ... A thirty year interregnum followed, punctuated by isolated contributions, until the major revival of interest following the 1988 paper by Morris and Thorn ... The last six years have been a considerable amount of activity, and the field is now sufficiently mature to warrant an overall summary being presented.
Lorentzian wormholes/spacewarp come in at least two varieties:
1. Inter-universe wormholes (wormholes that connect "our" universe with "another" universe).
2. Intra-universe wormholes (wormholes that connect two distant regions of our universe with each other).
Rotating traversable wormholes
Edward Teo wrote:
The general form of a stationary, axially symmetric traversable wormhole is discussed. This provides an explicit class of rotating wormholes that generalize the static, spherically symmetric ones first considered by Morris and Thorne. In agreement with general analyses, it is verified that such a wormhole generically violates the null energy condition at the throat. However, for suitable model wormholes, there can be classes of geodesics falling through it which do not encounter any energy-condition-violating matter. The possible presence of an ergoregion surrounding the throat is also noted.
Ringholes and closed timelike curves
Pedro González-Díaz wrote:
It is shown that in a classical spacetime with multiply connected space slices having the topology of a torus, closed timelike curves are also formed. We call these spacetime ringholes. Two regions on the torus surface can be distinguished which are separated by angular horizons. On one of such regions (that which surrounds the maximum circumference of the torus) everything happens like in spherical wormholes, but the other region (the rest of the torus surface), while still possessing a chronology horizon and non-chronal region, behaves like a coverging, rather than diverging, lens and corresponds to an energy density which is always positive for large speeds at or near the throat. It is speculated that a ringhole could be converted into a time machine to perform time travels by an observer who would never encounter any matter that violates the classical averaged weak energy condition. Based on a calculation of vacuum fluctuations, it is also seen that the angular horizons can prevent the emergence of quantum instabilities near the throat.
On a General Class of Wormhole Geometries
A. DeBenedictis and A. Das wrote:
A general class of solutions is obtained which describe a spherically symmetric wormhole system. The presence of arbitrary functions allows one to describe infinitely many wormhole systems of this type. The source of the stress-energy supporting the structure consists of an anisotropic brown dwarf “star” which smoothly joins the vacuum and may possess an arbitrary cosmological constant. It is demonstrated how this set of solutions allows for a non-zero energy density and therefore allows positive stellar mass as well as how violations of energy conditions may be minimized. Unlike examples considered thus far, emphasis here is placed on construction by manipulating the matter field as opposed to the metric. This scheme is generally more physical than the purely geometric method. Finally, explicit examples are constructed including an example which demonstrates how multiple closed universes may be connected by such wormholes. The number of connected universes may be finite or infinite.
Two Open Universes Connected by a Wormhole: Exact Solutions
Li-Xin Li wrote:
In this paper I present a spacetime of two open universes connected by a Lorentzian wormhole. The spacetime has the following features: (1) It can exactly solve the Einstein equations; (2) The weak energy condition is satisfied everywhere; (3) It has a topology of R² x Tg (g >= 2); (4) It has no event horizons.
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Wormholes and Time Travel? Not Likely
Leonard Susskind wrote:
Wormholes have been advanced as both a method for circumventing the limitations of the speed of light as well as a means for building a time machine (to travel to the past). Thus it is argued that General Relativity may allow both of these possibilities. In this note I argue that traversable wormholes connecting otherwise causally disconnected regions, violate two of the most fundamental principles physics, namely local energy conservation and the energy-time uncertainty principle.
Rebuttal to a Paper on Wormholes
Leonard Susskind wrote:
In a recent paper on wormholes (gr-qc/0503097), the author of that paper demonstrated that he didn't know what he was talking about. In this paper I correct the author's naive erroneous misconceptions.
Leonard Susskind wrote:
None of this means that wormholes make sense. I share the prejudice of the author of [1] that they do not, and I hope to return to the problem.
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______ Holes
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Introduction:
"A luminous star, of the same density as the Earth, and whose diameter should be two hundred and fifty times larger than that of the Sun, would not, in consequence of its attraction, allow any of its rays to arrive at us; it is therefore possible that the largest luminous bodies in the universe may, through this cause, be invisible." -- Pierre Laplace, The System of the World, Book 5, Chapter VI (1798).
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Evolution of Stars
Clouds of Hydrogen begin condensing into more dense clusters due to gravitation.
Eventually the density gets high enough that the Hydrogen begins fusing into Helium. This fusion releases energy, mostly in the form of electromagnetic radiation. Our sun is currently in this phase. Note that the gravitational attraction of the matter of the star is trying to make it smaller; this is balanced by the radiation pressure that is trying to push the matter outward making the star bigger. In class we showed a photograph of a birthplace of stars; the URL is http://www.seds.org/hst/M16WF2.html.
After about a billion years the Hydrogen fuel is exhausted. It starts fusing the Helium into heavier elements. The temperature of the star goes up, causing it to become much bigger. In this phase the star is called a "red giant."
When this "fuel" is exhausted, there are three possible outcomes, depending on the total mass of the star:
If the mass of the star is less than about 1.2 solar masses, the star becomes a spinning ball of dead slag, gradually cooling down as it radiates away its thermal energy. It is called a "white dwarf." Its radius is on the order of 5000 km, and the density is about 1 ton per cubic centimeter.
If the mass of the star is greater than about 1.2 solar masses but less than about 3 solar masses, it goes through the white dwarf phase, but the gravitational attraction is so strong that the protons and electrons get fused into neutrons. Thus we end up with a spinning ball of neutrons, a "neutron star." The radius is on the order of 10 km, and the density is about 100 million tons per cubic centimeter. Astronomers believe that they have discovered many of these neutron stars; one of the best known is in the Crab nebula and is the remnant of a supernova that was seen on Earth in year 1054 of the current era. They "see" them because as they spin they emit a sweep of radiation, which we observe has having a regular periodicity. Thus before we knew they were neutron stars they were called "pulsars."
If the mass of the star is greater than about 3 solar masses, it goes through the neutron star stage and keeps collapsing. In this case, the gravitational attraction is so intense the neutrons are literally crushed out of existence. We call the object a "______ hole."
The existence of ______ holes is another prediction of General Relatvity, although we saw in the Introduction that Laplace made a similar prediction in 1798. Astronomers believe they have suggestive evidence that ______ holes exist.
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Another Approach to ______ Holes
Above I characterised a ______ hole as an object whose gravitational pressure is so intense that the matter of the object is crushed out of existence. Here is another way of characterising the same phenomenon.
Above is the embedding diagram of a normal star. Of course, the more massive the star, the deeper the "well" in the centre.
Here is the embedding diagram of a ______ hole. The gravitation is so intense that it has punched a hole in the fabric of spacetime. We call the hole in spacetime a singularity.
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The Wormhole Solution
Einstein did not like the singularity in the centre of the ______ hole. In 1935 he and Rosen found another solution to the equations of a ______ hole. It is shown to the above. There is an Einstein-Rosen bridge or wormhole connecting two different regions of spacetime.
The Einstein-Rosen solution does not say anything about the relationship between the two regions of spacetime. One possibility is shown above. The ______ hole in some region of spacetime connects to a ______ hole in another region of spacetime.
The previous figure can be a bit mis-leading in trying to interpret distances. Here is a topologically equivalent version of the first one.
Finally, a sort of wild speculation. The Klein bottle shown above may be a model for how a ______ hole in the universe can connect to the universe as a whole.
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Yet Another Approach to ______ Holes
Above we have a star, and have drawn the path of light rays being emitted from a point on the surface. As we know, the light will be attracted by the mass of the star. For rays that are emitted nearly parallel to the surface of the star, they will end up falling back to the surface of the star. We have drawn an exit cone; light emitted inside the cone escapes from the star, and light emitted outside the cone falls back to the star.
Now we have a more massive star. As we see, the exit cone is now smaller.
For a ______ hole, there is a point at some distance away from the center where the gravitation is so strong that the exit cone totally closes. Thus, nothing closer to the center than this can ever escape. This is why we call these objects ______ holes.
The distance from the center for which the exit cone closes is called the Schwarzschild radius. The sphere surrounding the singularity of this size is called the event horizon since no events that happen inside it can ever be known to us.
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Falling Into a ______ Hole
The figure shows the spacetime diagram of an object falling into a ______ hole. As measured with our rods and clocks we see that it takes an infinite length of time for the object to cross the event horizon at x = 2. Coming out of the mathematics of General Relativity is the worldline of the object inside the event horizon: apparantly it crosses the event horizon at time t = infinity, and then travels backwards in time to finally crash into the singularity at time t = 44.
The figure also includes the light cones of the object. We see that as the object approaches the event horizon, the light cone closes up. This should remind you of the exit cones we discussed above.
You can also see that inside the event horizon the light cones are now about a horizontal axis instead of a vertical one. Here is one way of thinking about this. Out here, we are free to move in any direction we wish in space. However, we can not choose in what direction to go in time: we are constrained to always get older, not younger. Inside the event horizon the object is constrained to fall into the singularity; it has no freedom of movement in space. Whether this means that it has some sort of freedom in time is a good question which I do not know how to answer.
If we are riding with the object as it falls into the ______ hole, we would not experience the infinite time dilation shown in the spacetime diagram: we would just fall through the event horizon. However, if we are falling, say, feet first the gravitational "pull" on our feet is stronger than on our head since it is closer to the singularity. This tidal force is sure to rip us to bits long before we actually get to the event horizon.
Reference for the figure: Misner, Thorne and Wheeler, Gravitation, pg. 848.
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Some Schwarzschild Radii
We can calculate the Schwarzschild radius for the mass of the Earth: it works out to be about one-third of an inch. What this means is that if we could compact the Earth so that its radius gets down to one-third of an inch, the gravitational attraction becomes strong enough that the Earth gets crushed out of existence, creating a ______ hole.
The comparable figure for the mass of the Sun is about 1.1 kilometers.
For a mass of 2.5 x 1053 kg, i.e. a 2 and a 5 followed by 52 zeroes kg, the Schwarzschild radius is about 17 billion light years. This huge mass is an estimate for the total mass of the universe. Also, given that the age of the universe is 15 billion years or so, 17 billion light years is awfully close to the size of the universe. Does this mean that the universe itself is a ______ hole?
It turns out that this question is the same as asking: is the universe closed. If the universe is closed, then it is fairly accurate to say that it is a ______ hole.
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This document is Copyright © 1999, 2000 David M. Harrison. This is version 1.5, date (m/d/y) 02/06/02.
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(Note: the spacing of the above poem is not rendered quite correctly by the Internet Explorer browser.)
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