*
What is a wormhole? What does it mean to say that the wormhole
solution is unstable?
*

If one takes the Schwarzschild metric and performs a somewhat complicated coordinate transform, one will discover that the Schwarzschild solution to Einstein's equations is more extensive than first thought. The new solution consists of two asymptotically flat spaces connected together at the event horizon. At first blush, it appears as if one can travel from one of these spaces into the other, thus this wormhole seems to provide a bridge between two universes. However: no timelike worldline can bridge the gap. The other universe is strictly elsewhere from our universe. The best you could hope for is to meet someone from the other universe as you were falling into the singularity. Once inside the horizon, both universes lie in your past light cone.

Now for the wormhole's stability: If we examine the wormhole solution in our modified coordinates, we find that the solution evolves as measured in the modified time coordinate. Basically, the Schwarzschild "throats" form in both universes, connect, and then pinch off into singularities. Viewed from outside, using "ordinary" time coordinates, the whole thing just sits there, but in the modified time coordinates, those which would be more appropriate for someone falling in, time is very finite and ends at the singularity.

Are wormholes real? They are a real solution, but that doesn't mean
that they exist in the universe. They do not result from the collapse
of a star. They just have to be built in * ab initio*. There are
several known weird solutions to Einstein's equations that probably
don't have much to do with the real world. They are worth studying
though because they provide fascinating insights into properties of
the theory general relativity.

If you want to know more about wormholes, may I recommend Kip Thorne's
well-written book *Einstein's Outrageous Legacy.*

Technically, no, because the black hole solution is a * vacuum *
solution that has an external region that goes to flat space outside of
the hole. The universe is not a vacuum and there is no known (or
required) external spacetime. However, the closed model (see Chapter
11) shares some features with the internal solution of a black hole, in
that gravity is going to bring everything to a collapse and a
singularity, and no worldline can escape that fate.

We never see the ship cross the horizon, that is true. Time does seem to come to a halt. But we don't watch the ship hovering there forever. Very quickly the last photon that we will be able to see from the ship would reach us and we would see the ship no more. The redshift effectively goes to infinity in a finite time.

Actually a black hole could exist inside a black hole. Imagine turning a whole galaxy into a black hole by bringing all its stars extremely close together. All these stars might themselves be black holes. The individual black holes won't even be touching when they are all surrounded by a larger event horizon with a radius corresponding to the mass of the galaxy. However, once they are all within the event horizon they will end up merging together in a final collapse to a singularity.

Black holes would be hard to detect, and a great deal of unseen mass could be stored inside black holes. However, based on how we think black holes must form (from collapsing cores of relatively rare supermassive stars) there can't be that many black holes in comparison to more normal stars. Hence we expect that the total mass in black holes is only a small fraction of the mass we can see, so that they don't constitute a major component of the unseen, or "missing" mass in the universe. (See Chapter 14)

They aren't actually moving faster than light. The simply * appear
* to be moving faster than light. The jet is beamed toward us,
nearly along our line of sight. It moves only slightly slower than
light, so the light carrying the history of its motion tends to be
bunched up when it gets to us. Imagine somebody flying from Alpha
Centari at close to c while you watch through a telescope. By the
time the light gets here telling you that they have left, they are
about to arrive, so to you the journey speed appears to be superluminal.
But by subtracting out the known light travel time, you can compute
the true speed, and verify that it is less than c.

It is possible that tiny black holes could have formed and that they would be radiating away via the Hawking process by now. But none has ever been detected. Most cosmologist don't regard this as very likely.

Somebody once seriously suggested that a mini black hole could account for the Tunguska event in Siberia. Depending on the size of the black hole (let's assume it truly is a mini-hole with a Schwarzschild radius that is submicroscopic) it would pass straight through the Earth, possibly wreaking a little havoc at the entry and exit points due to its strong tidal forces. It would actually accrete very little mass in its passage.

Over the lifetime of the solar system (10 billion years or so) there is virtually no chance that we will encounter a black hole. (Or any other star, for that matter.) If the universe expands forever, the matter that presently makes up the Earth may one day end up in a black hole. Most things may very well end up in black holes in the incredibly huge distant future. Then in an even longer period of time those black holes will evaporate.

We might say that the singularity is where all matter was crushed to infinite density (finite matter in zero volume). A better definition might be a point at which world lines come to an abrupt end, marking the end of time and space as it were.

It is generally regarded as a finite mass packed into zero volume. Most physicists regard zero volume as a not well-posed idea, hence the expectation that there must be a way within a larger theory (quantum gravity) to avoid zero volume and instead end in some finite, albeit extreme, state. However, we don't know for sure.

Pretty close to zilch. The only way we know how to form a black hole is through the collapse of a supermassive star, and there aren't any such stars close enough that their black holes would affect us. (The supernova itself could affect us, however, by raining down high energy radiation on the Earth.)

However, if you want to worry about black holes, consider the chance encounter with a wandering black hole, one moving through the galaxy relative to the rotating plane of the disk in which we live. While the likelihood of this is small, it is at least conceivable. In fairness, it wouldn't have to be a close encounter with a black hole. Any star would do to cause possibly catastrophic problems with the Earth's orbit or the orbit of other planets (catastrophic as far as our personal existence goes at least). Pleasant dreams.

It is almost certain that there are black holes in binary systems in our galaxy. Another question is whether or not there is a really big black hole in the center of our galaxy. In recent years the answer has emerged: yes! Direct observations of the center of the Galaxy have been able to follow the orbits of massive stars over a number of years. Using Kepler's laws we can calculate the mass of the thing that they are orbiting and it comes out to about 3 million solar masses. Thus, although we don't directly observe the black hole, we see the effects of its gravitational field.

The singularity in a Kerr hole forms as a ring, strangely enough. Toroidal, prolate, oblate, etc. event horizons can form, but black hole horizons always radiate away gravitational waves until they settle down into spherical symmetry.

Construction is complete at both the Hanford Site and the Livingston site. Test data runs have been carried out. You can follow the developments through the LIGO home page, Interesting pictures from each site are available in the on-line LIGO newsletters.

A black hole can shrink by Hawking radiation (see chapter 9) but this effect is extremely small for normal sized black holes; at the present time in the universe black holes only grow in size.

These are questions astrophysicists would like to know the answers to in some detail. The exploding star must interact with the black hole, at least by losing some of its ejected matter down the hole. The hole would grow in mass. The transfer of gas with a relative rotation (angular momentum) to the hole could also affect its rotation, since angular momentum is one of the fundamental properties of the hole. Possibly the binary system would be disrupted in the explosion, though it might also hold together. Systems such as the binary pulsar system had to form somehow. How many such systems might there be in our galaxy?

It is crushed into the singularity at the center and its mass-energy goes into the gravitational field of the black hole.

It has been speculated that there could be other universes on the "other side" of singularities. However, there really isn't any support for this based on what we do know about singularities. But since GR must break down into some quantum gravity that we don't yet understand, there remains room for speculation.

As we have seen with the study of General Relativity, time dilation and length contraction (and related effects) occur due to gravitational fields. Time and space are altered by gravity. The gravitational field near a black hole is particularly strong, so these relativistic effects become quite large.

Presumably you mean that gravity itself seems to be escaping from the hole. But a better way to think about it is that the gravity is left behind when the hole forms, and no message could get out to tell gravity that the mass has disappeared behind an event horizon!

The gravitational field of a black hole is left behind in the collapse of the star into a black hole. The field itself generates gravity, i.e. gravitons. Gravity equals energy equals mass. A black hole is a self-generating gravitational field.

Your freefall trajectory in a gravitational field is determined
by the metric (e.g. Schwarzschild, eqn. 9.2). Because your velocity is
generally much less than light the (c dt)^{2} interval is much
larger than the (dx)^{2} interval. Hence the metric coefficient
multiplying the time factor causes this term to have much more of an effect on
determining your worldline's trajectory through spacetime than
does the spatial term.

Black holes pull matter in the same manner as other gravitating bodies, and indeed they eventually clear out their immediate region. Other matter can move into that vacated region only if diverted (by some force) onto trajectories leading into the hole's vicinity. This "refilling" process will be very slow. However, if the universe goes on forever, then black holes will most likely eventually consume everything.

Quantum mechanics must come into play directly near the singularity in the center of the black hole. Quantum mechanics is also involved in the process of Hawking radiation at the horizon of the hole. Note that Hawking radiation is quantum mechanical process taking place in the background gravitational field of the black hole which can still be regarded as determined by Einstein's equations. Near the singularity quantum gravity determines the gravitational field directly.

Yes, but the principle of mass conservation is not strictly valid, and has to be replaced by the principle of mass-energy conservation. Mass and energy are equivalent by Einstein's famous formula. You can create mass if you have sufficient energy; when virtual particles become real it means that they obtained the energy to become real, so mass-energy is conserved. In Hawking radiation the necessary energy comes from the gravitational field itself.

We can't sample the conditions directly, but we can apply physical theory to the problem. If we believe that the theory is sound, and completely supported by observations that we can make, then we can use the process of deduction to learn about things that we cannot directly observe.

Who knows? That's one of the problems with white holes as postulated. There doesn't seem to be any constraint on what they could be producing.

Inside the horizon you could travel backward along the coordinate direction labeled "time" by a distant observer. You couldn't travel into that distant observer's past, however. If you are outside the hole's horizon you could use gravitational time dilation to travel forward in time relative to the distant observer, but not backward.

They would have to be compressed down to the size of their respective Schwarzschild radii. You would need to implode them inward, which, presumably you could do with a focused beam of energy (including gravity waves). However the total energy required would probably exceed the rest mass energy of the Earth or Sun (I haven't calculated it). Or you could just pile more mass onto them until their total mass exceeds that which can be held up by any available force. But that is sort of a cheat since the original mass of the Earth or sun would be negligible in the resulting black hole.

They are equivalent for a nonrotating black hole. Event horizon is a more general term and refers to a lightlike (null) surface in spacetime constituting a boundary between what can be observed and what can't.

It wouldn't be at the present moment. Space is much warmer than the temperature of a solar mass black hole. (Smaller black holes could in principle be warmer than the 3K of space.) If the universe is open, at some remote future date space will be colder than the black holes and then they will begin to decay. This will occur at unfathomably remote times in the future. But all things come to those who wait.

If there were a lot of black holes their total mass would affect the overall gravity of the universe, but this is no different from any other object in the cosmos. If the universe lives long enough it is likely that most of the matter will end up in one black hole or another.

The matter and energy are left behind in the gravitational field of the black hole.

All redshifts are equivalent in their effect, that is, all you can measure from a given line is its wavelength and hence its shift. You can't tell just from that what is cosmological, what is doppler, and what is due to gravity. However, to get a large redshift from gravity you need to be right up next to the event horizon when the light is emitted. Also you would expect light to be emitted from a continuum of radii around the black hole resulting in not one unique redshift but a smeared spectral line.

The limit is the amount of mass that neutron degeneracy pressure can hold up against gravitational collapse. That value is somewhere between maybe 2 to 3 solar masses.

If wormholes connected different parts of the universe it would be difficult to say much about the overall geometry of the universe. (How many wormholes are there? How do they connect things up?) Wormholes might well make time travel possible, which would create a big problem for causality and were certainly violate the cosmological principle.

Conceptualizing such a complex 3 dimensional space geometry is difficult. Conceptualizing higher dimensional spaces is very difficult except abstractly in terms of the mathematics that describes such spaces. One usually thinks about 2 dimensional surfaces curved in the third dimension, although even that is somewhat difficult and may not be adequate for the most complex set of wormholes one might imagine.

Light has energy, and it falls in a gravitational field (Equivalence principle). Gravity operates even on particles like the photon that have zero rest mass.

The Stars do. The power that radiates from near a black hole comes from a different process. Material orbiting near a black hole is accelerated by the strong gravity to velocities near the speed of light. Mass moving near the speed of light has a lot of energy. If that gas becomes turbulent it will dissipate the energy of motion into heat, and the temperatures associated with that heat will be very high, causing the gas to radiate high energy photons.

If the sun collapsed to a black hole it would not be radiating any light (and heat is a form of light). So no, life on Earth would not survive.

From the singularity at the center of the white hole. But this is one of the reasons why we don't think white holes exist.

As far as we know from classical general relativity. If your world line comes to an end you cease to exist.

Wormholes represent a complete Schwarzschild type solution to Einstein's equations. They are permitted by the equations, but there do not seem to be any ways in which they could form (whereas a black hole alone can form from the collapse of a massive stellar core).

The Sun is never going to become a black hole on its own. However, the white dwarf that the Sun ends up as may one day in the incredibly vast distant future fall into a black hole.

I don't know if it would be possible or not to create such a thing, but a mini black hole with the size of an atom could exist. You would just have to have the right amount of mass compressed down to the desired Schwarzschild radius.

The radiation is coming from matter falling into and orbiting around the supermassive black hole, not from the black hole itself.

It is possible in the sense that one can imagine such a thing and nobody has been able to show that the laws of GR absolutely forbid such a thing.

Please read his book, *Einsteins Outrageous Legacy*. It is
highly readable and makes an excellent follow on for people interested
in learning more about GR, black holes and wormholes.

Blackbody radiation is the type of radiation emitted by matter in a state of statistical thermal equilibrium. It means that even though the black hole is naught but a great gravitational field, it acts as if it were a big chunk of matter at some very very low temperature radiating photons out into space. Isn't that strange.

Black holes in the centers of galaxies are the most massive. In some galaxies, hole masses in the billions of solar masses have been inferred. Motions of the stars in the core of the Andromeda galaxy suggest it contains a black hole of 10 million solar masses.

There are some black hole binary systems in the Milky Way. One example is Cygnus X-1 which is located about 8000 lightyears away. We can't "see" the black holes in the sense that you might be thinking, but we can see the stars that are their companions, and we can see the gas falling into the black hole.

Your body could be flattened if the tidal forces were strong enough, and they could be depending on the mass of the hole and your location relative to the horizon.

In principle, given enough time almost everything would end up in black holes. Because the universe is expanding, however, not all black holes would end up merged into one black hole. Unless the universe turns out to be the big crunch type.

In quantum mechanics it is impossible to know the energy of a system perfectly at some moment in time. This includes the vacuum. The energy in the vacuum isn't perfectly zero for all time, but fluctuates around zero in the short amounts of time required by the Heisenberg Uncertainty principle of quantum mechanics.

Not likely. They are arriving from this universe. However, they don't carry any information anyway (except the temperature of the blackbody, hence the mass of the hole that created them), so even if they were "coming from another universe" it wouldn't mean anything.

It is an interesting question: if a theory of physics (such as GR) permits something does that thing have to occur in the universe? If it doesn't occur in the universe does the presence of something so permitted mean that the theory is incorrect in allowing that thing?

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Copyright © 1998 John F. Hawley