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Space Forum / Astronomy / March 2007Schwarzschild Black Hole - any density?
I have mentioned elsewhere that the Schwarzschild Black Hole can be of any density, as that was my long time assumption (I assumed that this was common knowledge and an accepted concept). But there seems to be some opposition to this. Consider a low density spatial extension with average density of 10-29g/cm^2. According to the Schwarzschild radius equation r=(3c^2/8Gdpi)^.5 the radius should be 13.4022 Billion light years, total mass of 8.5395^52kg. We would expect to see objects close to the event horizon cascade toward the centre under the influence of the gravitational force (curvature of space) of all that accumulated mass, unless the entire Black Hole is rotating at a sufficiently high speed or some force counter to the gravitational force, such as dark energy, prevents the collapse. We can also consider a magically 'just formed' Schwarzschild Black Hole that has not begun to collapse just yet. The bottom line is that, given enough mass, a Schwarzschild Black Hole can form. I don't see where the singularity fits in to this type of Black Hole. A relative singularity may be observed at some point near the centre by an observer near the event horizon, but as one approaches the centre one would find no such singularity. We can track part of this illusion by noting the time dilation at various points from the event horizon to the centre. An observer near the event horizon notes that clocks near the centre seem to have stopped completely. But observers near the centre notice nothing unusual about their own clocks but note that clocks near the event horizon are running almost infinitely faster than their own. But back to my original point - the simple math for a Schwarzschild Black Hole, r=(3c^2/8Gdpi)^.5, indicates that a Black Hole can form from matter of any density. The escape velocity, by the same math, is c, which is what we expect of any Black Hole. I think where some General Relativists have difficulty with this concept is that they want all spacetime curvature to be actual and not relative. Thus in a spatial extension many times greater than the Schwarzschild Radius they see no Black Hole form, even though the math says that one does form but only relative to the observers position in space ie two observers spatially separate will observe Black Holes in different places, possibly encompassing the other observer. One important component of spacetime is time, and we know from SR just how relative time can be. But can a Black Hole form which is not a Black Hole independent of the observer ie could there be a Black Hole at some point in space according to the measurements and observations made by one observer but not another? I say yes.  SignatureKind Regards Robert Karl Stonjek > I have mentioned elsewhere that the Schwarzschild Black Hole can be of any > density, as that was my long time assumption (I assumed that this was common [quoted text clipped - 17 lines] > > I don't see where the singularity fits in to this type of Black Hole. A motionless distribution of matter that is not infinite in extent cannot remain static -- it must collapse. An observer sitting outside the mass distribution would see a black hole with an event horizon, and would be able to observe nothing of the internals. Since a spherical distribution of matter behaves as a point mass to the external observer, it behaves to him as though all the mass were concentrated at a singularity. He would calculate (but could not observe) that all of the matter inside, regardless of its starting distribution, must collapse to a singularity in a finite time. However, an observer within the body would note that all of the matter was heading inevitably towards the center, that is, the mass distribution is collapsing. As it does so the density rises and the boundary of his observable universe collapses too. He is headed irresistably toward the center along with everything else, and at some point the density interior to his location will be sufficient for him to observe an event horizon below him, one that is growing as matter is falling in. Eventually this growing event horizon will grow to meet the event horizon that the external observer sees; the space within this horizon will be devoid of matter save for the singularity at the center (ignoring spontaneous particle pair generation). > A motionless distribution of matter that is not infinite > in extent cannot remain static -- it must collapse. This is only true for densities above a critical density (which depends on the size of the matter distribution). For instance, there's no expectation that earth or sun must collapse. But a star well over the Chandrasekhar limit (~1.5 solar masses) must collapse, unless it sheds enough mass to get below the limit. That is a limit on mass, not density. Tom Roberts > > A motionless distribution of matter that is not infinite > > in extent cannot remain static -- it must collapse. [quoted text clipped - 4 lines] > Chandrasekhar limit (~1.5 solar masses) must collapse, unless it sheds > enough mass to get below the limit. That is a limit on mass, not density. I was referring, of course, to matter unsupported by other means such as electromagnetic forces as you find in condensed matter. But you are correct in that I should have been more specific. > This is only true for densities above a critical density (which depends > on the size of the matter distribution). For instance, there's no > expectation that earth or sun must collapse. But a star well over the > Chandrasekhar limit (~1.5 solar masses) must collapse, unless it sheds > enough mass to get below the limit. That is a limit on mass, not density. As I understand it, in order for the type Ia supernova to occur, it must be a neutron star siphoning mass from some external source most likely from a companion start. A neutron star has a very small volume. So, the electron degeneracy will definitely occur if the added mass to this neutron star reaches a critical limit --- the Chandrasekhar mass. However, other stars, starting out with larger volume and not necessarily neutron stars, are not bound by this 1.5 solar mass threshold. > > A motionless distribution of matter that is not infinite > > in extent cannot remain static -- it must collapse. [quoted text clipped - 6 lines] > > Tom Roberts A Schwarzschild radius need not be a collapsed star. If space can be curved around a galaxy then there must be some escape velocity to overcome that curvature. If the curvature surrounding a cluster of galaxies is such that the escape velocity from the cluster is greater than the escape velocity of a single galaxy then it stands to reason that if a sufficient number of objects is mapped the escape velocity will eventually reach that of light at which point the Schwarzschild radius will have been reached. Where the singularity. especially if the collection of objects is rotating and no infalling occurs (as there is no centre where significantly more mass is found than at any other point in the spatial extension.) SignatureKind Regards Robert Karl Stonjek > > > A motionless distribution of matter that is not infinite > > > in extent cannot remain static -- it must collapse. [quoted text clipped - 14 lines] > objects is mapped the escape velocity will eventually reach that of light at > which point the Schwarzschild radius will have been reached. Only if the mass-energy density exceeds a critical value. If the objects are spacially spread out and the density does not exceed the critical value for any given volume, then no event horizon will form. > Where the > singularity. especially if the collection of objects is rotating and no > infalling occurs (as there is no centre where significantly more mass is > found than at any other point in the spatial extension.) The metric is given by the mass-energy density. Objects in free-fall follow spacetime geodesics. If the density exceeds the critical value, there are no trajectories that can prevent collapse. > > > > A motionless distribution of matter that is not infinite > > > > in extent cannot remain static -- it must collapse. [quoted text clipped - 29 lines] > exceeds the critical value, there are no trajectories > that can prevent collapse. Are you saying that there is no escape velocity for the Milky Way? For the local group? For a universe sized collection of objects? I say there is, and the simple math, at least, backs me up on that. If the escape velocity reaches the speed of light then we have what you might call a 'Black Galaxy', 'Black Cluster' and so on. The event horizon is a direct consequence of the escape velocity, and the escape velocity is directly effected by the mass of the object or collection of objects and the radius of that mass. Consider this: is the escape velocity of the Milky Way greater or less than the escape velocity of the Earth, or even of our sun? For a Milky Way approximation I consider a spherical gas of matter with a radius of 50,000 light years and a mass of 5.8*10^11 solar masses Escape Velocity=Ve=(2Gm/r)^.5 18.04*10^6m/s For Earth I consider a radius of 6,372,767m and a mass of 5.9736*10^24kg 11.18*10^3m/s Thus the escape velocity of the galaxy is over a thousand times higher than the escape velocity of the Earth. At 617.54*10^3m/s, the sun's escape velocity is also far lower. Considering that the speed of light is around 300*10^6m/s, the escape velocity of our Milky way only needs to be twenty times higher and it would be totally dark as light would not be able to escape. Of course the Milky way is not spherical, being around 1,000 light years thick and around 100,000 light years in diameter. SignatureKind Regards Robert Karl Stonjek [snip] > > The metric is given by the mass-energy density. Objects > > in free-fall follow spacetime geodesics. If the density [quoted text clipped - 3 lines] > Are you saying that there is no escape velocity for the Milky Way? For the > local group? For a universe sized collection of objects? That is obviously not what I am saying. Read the paragraph again. Whether or not there is an escape velocity from any given body or collection of mass-energy depends upon the mass-energy density. The Milky Way does not have sufficient overall mass-energy density to form an event horizon. As for a universe-sized collection of objects, it again depends upon the density. > I say there is, and the simple math, at least, backs me up on that. If the > escape velocity reaches the speed of light then we have what you might call > a 'Black Galaxy', 'Black Cluster' and so on. If a galaxy were to achieve the mass-energy density sufficient to form an event horizon (escape velocity equal to or exceeding the speed of light), then the contents would collapse. That's what General Relativity's math says. > The event horizon is a direct consequence of the escape velocity, and the > escape velocity is directly effected by the mass of the object or collection > of objects and the radius of that mass. Consider this: > is the escape velocity of the Milky Way greater or less than the escape > velocity of the Earth, or even of our sun? The event horizon and escape velocity are a consequence of the geometry of spacetime as it is shaped by the mass-energy density of the region. Escape velocity is not the causative agent, it is a characteristic that is the result of the underlying physics of space. > For a Milky Way approximation I consider a spherical gas of matter with a > radius of 50,000 light years and a mass of 5.8*10^11 solar masses [quoted text clipped - 16 lines] > way is not spherical, being around 1,000 light years thick and around > 100,000 light years in diameter. I'm not sure what your point is here. Sure, if the Milky Way were dense enough then it could form a black hole. But it's not, so it isn't, and it is nice place to live. Of course, we also should take into account that the radius of the extended halo is much larger than 50,000 LY. If we take the orbits of the Large and Small Magellanic clouds to be indicative, then the radius is more like 180,000 LY. With these numbers the galactic escape velocity becomes about 5.7 x 10^5 m/s . This is lower than your figure. It might be more instructive to look at escape velocity in terms of the density of a spherical region. The relevant formula is: Vesc = r * sqrt((8*pi/3) * rho * G) We can see that the escape velocity is proportional to r for a given density rho. > [snip] > [quoted text clipped - 23 lines] > contents would collapse. That's what General Relativity's > math says. It won't collapse if it is rotating sufficiently fast. And if a collapse is inevitable, that process could take billions of years for a large object. > > The event horizon is a direct consequence of the escape velocity, and the > > escape velocity is directly effected by the mass of the object or collection [quoted text clipped - 7 lines] > not the causative agent, it is a characteristic that > is the result of the underlying physics of space. Causation has not been discussed - escape velocity is a simple but indicative number that can be used to identify a Schwarzschild radius from the inside or outside. > > For a Milky Way approximation I consider a spherical gas of matter with a > > radius of 50,000 light years and a mass of 5.8*10^11 solar masses [quoted text clipped - 25 lines] > we take the orbits of the Large and Small Magellanic clouds > to be indicative, then the radius is more like 180,000 LY. And the mass would be larger as well. There is no reason why one can not calculate the escape velocity for the entire local group as well. > With these numbers the galactic escape velocity becomes > about 5.7 x 10^5 m/s . This is lower than your figure. You have not given the mass and radius figures you are working with. Did you add the mass of the Magellanic clouds, plus dust and dark matter (dead suns etc) known to inhabit the halo? Note that the escape velocity applies to the edge of the radius ie if you calculate the radius to be 180ly, then the escape velocity you calculate is from a distance of 180ly from the centre of the object (the escape velocity is often given as 'r+h' where the 'h' is the height above the radius of the object eg the escape velocity from 1 km above the earth's surface is much lower than at the surface). Thus the escape velocity I calculated is still valid at the radius I quoted, which was just a loose average erring on the side of conservatism. > It might be more instructive to look at escape velocity > in terms of the density of a spherical region. The [quoted text clipped - 4 lines] > We can see that the escape velocity is proportional to r > for a given density rho. You offer no numbers for your equation - why? Please give the numbers you are working with ie density and radius. SignatureKind Regards Robert Karl Stonjek On Mar 29, 7:04 am, "Robert Karl Stonjek" <ston...@ozemail.com.au> wrote: [snip] > > If a galaxy were to achieve the mass-energy density > > sufficient to form an event horizon (escape velocity [quoted text clipped - 4 lines] > It won't collapse if it is rotating sufficiently fast. And if a collapse is > inevitable, that process could take billions of years for a large object. If the contents are rotating sufficiently fast so as to prevent collapse, then the mass-energy density will be less than that required for an event horizon to form. Energy adds in different ways -- some, like potential energy, is negative. In non-static systems you have to consider more than just the masses, since energy of all types contributes to the stress-energy tensor of GR. > > > The event horizon is a direct consequence of the escape velocity, and > the [quoted text clipped - 13 lines] > indicative number that can be used to identify a Schwarzschild radius from > the inside or outside. I'm sorry, I must have misinterpreted your intended meaning when you said "The event horizon is a direct consequence of the escape velocity." > > > For a Milky Way approximation I consider a spherical gas of matter with > a [quoted text clipped - 12 lines] > > > the escape velocity of the Earth. At 617.54*10^3m/s, the sun's escape > > > velocity is also far lower. I just re-ran your numbers. I find, using your mass and radius figures, an escape velocity of just 5.7 x 10^5 m/s for the galaxy. This is only about 50X the Earth's escape velocity, not the 1000X you obtained. > > > Considering that the speed of light is around 300*10^6m/s, the escape > > > velocity of our Milky way only needs to be twenty times higher and it [quoted text clipped - 22 lines] > you add the mass of the Magellanic clouds, plus dust and dark matter (dead > suns etc) known to inhabit the halo? Your numbers: M = 5.8 x 10^11 Msun Radius: r = 50,000 ly Vesc = sqrt(2*G*M/r) > Note that the escape velocity applies to the edge of the radius ie if you > calculate the radius to be 180ly, then the escape velocity you calculate is [quoted text clipped - 5 lines] > Thus the escape velocity I calculated is still valid at the radius I quoted, > which was just a loose average erring on the side of conservatism. Except that it was off by several orders of magnitude. > > It might be more instructive to look at escape velocity > > in terms of the density of a spherical region. The [quoted text clipped - 7 lines] > You offer no numbers for your equation - why? Please give the numbers you > are working with ie density and radius. The equation above is quite general. I offered it as a suggested way to look at the escape velocity in terms of the density of a region of uniform density. For example, setting Vesc = c, the radius at which a black hole must form is given by r = c/sqrt((8*pi/3)*rho*G) Similarly, you could solve for the density, rho, beyond which collapse must occur for a region of a given radius. > On Mar 29, 7:04 am, "Robert Karl Stonjek" <ston...@ozemail.com.au> > wrote: [quoted text clipped - 18 lines] > the masses, since energy of all types contributes to the > stress-energy tensor of GR. True, but it makes you wonder if something else may be afoot when Neutron Stars rotate at over one thousand RPM. > > > > For a Milky Way approximation I consider a spherical gas of matter with > > a [quoted text clipped - 17 lines] > This is only about 50X the Earth's escape velocity, not the 1000X > you obtained. Yes, you are right - I had the wrong number for the length of a light year. SignatureKind Regards Robert Karl Stonjek >> I have mentioned elsewhere that the Schwarzschild Black Hole can be of >> any [quoted text clipped - 45 lines] > universe collapses too. He is headed irresistably toward > the center along with everything else, Maybe; maybe not. I can't see why a body couldn't be in an orbit around the central mass, but lie within the event horizon. I can't even see why it couldn't just be held up way above the central singularity by some kind of scaffolding arrangement. (Assuming a really, really strong scaffold). Nor am I convinced that the observable Universe for somebody within the event horizon is constrained by the event horizon - matter has no problem falling into the hole, and from the perspective of an observer inside the event horizon this actually occurs - they can see matter and photons falling through the event horizon. >and at some point > the density interior to his location will be sufficient for [quoted text clipped - 4 lines] > be devoid of matter save for the singularity at the center > (ignoring spontaneous particle pair generation). Are you saying that within the black hole there may be one or more regions that form a black hole within the black hole? On the face of it, this appears possible (I have never really considered it before), but I can't see why the singularities around the little black holes should necessarily increase until they merge together to form a single black hole with the same event horizon as the system as a whole. Nor can I see why the matter inside a black hole should collapse to a "single point". Apart from anything else, the Pauli exclusion principle would seem to make this impossible if the black hole contains 3 or more electrons. If all matter was made up of objects with no physical dimensions (eg point masses), and they were allowed to occupy the same piece of space, and they had no interactions other than gravity, and they had no angular momentum, then, yes, they would collapse to a point. At least one of these is impossible even in in principle (zero dimensions, due to Heisenberg). Maybe I am misunderstanding your argument? > >> I have mentioned elsewhere that the Schwarzschild Black Hole can be of > >> any [quoted text clipped - 50 lines] > couldn't just be held up way above the central singularity by some kind of > scaffolding arrangement. (Assuming a really, really strong scaffold). The reason is, according to our best theory of space and gravity, within an event horizon all trajectories must lead to a central singularity. The density of the mass-energy curves the space in such a way that this is so. Further, the force of gravity must eventually overcome all other forces. This is due to the fact that gravity is, without exception, a strictly attractive force; there aren't positive and negative gravitational charges that can cancel. So there is ultimately nothing that can withstand the attraction and everything must collapse. > Nor am I convinced that the observable Universe for somebody within the > event horizon is constrained by the event horizon - matter has no problem > falling into the hole, and from the perspective of an observer inside the > event horizon this actually occurs - they can see matter and photons falling > through the event horizon. It is constrained in the sense that there is no way to reach the event horizon from within. Things can fall in, but nothing can move in any direction that leads anywhere but towards the singularity. > >and at some point > > the density interior to his location will be sufficient for [quoted text clipped - 11 lines] > increase until they merge together to form a single black hole with the same > event horizon as the system as a whole. The black hole will accrete matter (including other black holes) and grow in size until all of the matter within the original black hole is within a single singularity. This is because there can be no escape from the original, and everything inside must collapse. > Nor can I see why the matter inside a black hole should collapse to a > "single point". Apart from anything else, the Pauli exclusion principle [quoted text clipped - 6 lines] > > Maybe I am misunderstanding your argument? The Pauli Exclusion Principle is also eventually overcome by gravity. The pressure that resists collapse due to the PEP is called electron degeneracy pressure, and for a mass greater than the Chandrasekar Limit (about 1.44 solar masses), it is not able to prevent gravitational collapse. >> >> I have mentioned elsewhere that the Schwarzschild Black Hole can be of >> >> any [quoted text clipped - 69 lines] > there is ultimately nothing that can withstand the > attraction and everything must collapse. Why must gravity overcome all other forces? Imagine a black hole the size of the Universe (as indeed it could be, if its density exceeds that of the Schwarshild density). Why are you so sure that everything in it must eventually collapse to a single point? >> Nor am I convinced that the observable Universe for somebody within the >> event horizon is constrained by the event horizon - matter has no problem [quoted text clipped - 32 lines] > is because there can be no escape from the original, and > everything inside must collapse. Collapse to a point? Why? >> Nor can I see why the matter inside a black hole should collapse to a >> "single point". Apart from anything else, the Pauli exclusion principle [quoted text clipped - 15 lines] > than the Chandrasekar Limit (about 1.44 solar masses), it is > not able to prevent gravitational collapse. Sure, its unable to prevent a black hole forming, but it is certainly sufficient to prevent the collapse into a single point. Schwarzshild's solutions to GR were derived in 1914-1918 (while he was in the trenches) and published in about 1920 (I think) - long before the Pauli exclusion principle was known. It played no part in Schhwarshild's mathematics. The Pauli exclusion principle plays no direct part in the formation of a black hole - indeed, there is no force which can prevent the collapse into a black hole if the density is high enough. It does have a huge effect on what happens within the black hole, and specifically if all of the matter inside a black hole will collapse to a single point. It can't, due to the Pauli exclusion principle. Or don't you believe that the basic rules of QM apply inside a black hole? If not, why not? [snip] > > The reason is, according to our best theory of space > > and gravity, within an event horizon all trajectories [quoted text clipped - 9 lines] > > Why must gravity overcome all other forces? See above. Gravity is strictly attractive, and the gravitational force can grow without bounds once electron degeneracy pressure is overcome -- there is no other force (that we know of) that matter produces that can hold against it. > Imagine a black hole the size of the Universe (as indeed it could be, if its > density exceeds that of the Schwarshild density). Why are you so sure that > everything in it must eventually collapse to a single point? Because that's what the physics of General Relativity says will happen. Now GR could be wrong, but so far there has never been an empirical observation that has contradicted it in the least. [snip] > > The black hole will accrete matter (including other black > > holes) and grow in size until all of the matter within the [quoted text clipped - 3 lines] > > Collapse to a point? Why? Again, see above. > >> Nor can I see why the matter inside a black hole should collapse to a > >> "single point". Apart from anything else, the Pauli exclusion principle [quoted text clipped - 18 lines] > Sure, its unable to prevent a black hole forming, but it is certainly > sufficient to prevent the collapse into a single point. No! That's the whole point (pardon the pun). The electron degeneracy pressure is overcome for a mass concentration that exceeds the Chandrasekar limit. The result will be a body composed of degenerate matter, neutronium if the total mass is not too large. Add a bit more mass and you overcome the degeneracy pressure of the neutrons too (Tolmann-Oppenheimer-Volkoff limit, analogous to the Chandrasekar limit). And, you can keep on adding mass to exceed any conceivable limit of opposing forces that might arrise beyond that point (say quark degeneracy pressure, if it exists). > Schwarzshild's > solutions to GR were derived in 1914-1918 (while he was in the trenches) and [quoted text clipped - 10 lines] > believe that the basic rules of QM apply inside a black hole? If not, why > not? The Pauli Exclusion Principle has limits regarding the amount of force it can support. Beyond a certain point it ceases to keep electrons with the same quantum numbers apart. See, for example, http://en.wikipedia.org/wiki/Electron_degeneracy_pressure > [snip] > [quoted text clipped - 17 lines] > is no other force (that we know of) that matter > produces that can hold against it. That may be true of gravity, but it ignores current cosmological thinking where a repulsive force, essentially an antigravity (though never actually called that) is an essential part of the current big bang model. Dark matter and/or the driver of inflation (not everyone I talk to seems to agree that the two are the same and I am agnostic on the issue) is represented in equations by the cosmological constant. The bottom line is that one can not simply claim that gravity is all there is unless you explicitly discount dark energy and the possibility of a rapidly rotating object or an extremely large low density Black Hole where the period required for collapse amounts to billions of years due to the great distances involved. Let's not get too hung up on the collapsed star here - at the root of this thread I explicitly float the idea of the low density Schwarzschild Radius which rules out collapsed stars which are very dense by nature (compared to, say, the approximately 10-29g/cm^-3 of the universe). SignatureKind Regards Robert Karl Stonjek > > [snip] > > [quoted text clipped - 35 lines] > which rules out collapsed stars which are very dense by nature (compared to, > say, the approximately 10-29g/cm^-3 of the universe). A collapsed star is only a specific example. Dark energy is a large-scale effect which does not apply to compact objects or small regions of space. When one speaks of the mass-energy density of region of space, however large, it must include *all* of the mass and energy. The total must include any dark energy component, which is negative and tends to 'dilute' the mass-energy of the usual suspects. > >> I have mentioned elsewhere that the Schwarzschild Black Hole can be of > >> any [quoted text clipped - 83 lines] > > Maybe I am misunderstanding your argument? There is no reason why a Schwarzschild Radius can not be a relative beast, so that a Schwarzschild Radius could exist inside a Schwarzschild Radius etc, and in any of those there could be solid little Black Hole of the collapsed star variety. Note that this Russian doll concept of the Schwarzschild Radius is well known in SR with regard to velocity - accelerate up to 99.9% of c relative to your unaccelerated frame and meet a comover who is about to accelerate to 99.9% of c. Some general relativists seem to be stuck in the concrete thinking stage and are unable to conceive of relative relativity, as was the hallmark of Special relativity and the nature of time in general. The general Schwarzschild Radius opens up the relativity of general relativity such that spatial curvature could well be relative rather than absolute ie two observers could measure such a great difference that one measures a (Schwarzschild) Black Hole and one doesn't. This may well occur where the scale of the Schwarzschild Radius is huge and the density is low. SignatureKind Regards Robert Karl Stonjek > > I have mentioned elsewhere that the Schwarzschild Black Hole can be of any > > density, as that was my long time assumption (I assumed that this was common [quoted text clipped - 44 lines] > be devoid of matter save for the singularity at the center > (ignoring spontaneous particle pair generation). Yes, but there are several exceptions that I can think of right off:- -) the matter could be rotating with a rotational speed sufficient to prevent infalling; -) the entire spatial extension encapsulating the Schwarzschild Black Hole could be expanding for whatever reason eg caused by the effect of dark energy; -) the Schwarzschild Black Hole is forming and infalling is yet to be significant. This is quite possible if the density is very low and the distance between significant loci of mass is great eg in a universe full of galaxies. And there are other exceptions. The bottom line is that one need not consider what eventually happens when answering the question at the root of this thread - it is not inconceivable that there is a period between where a Schwarzschild Black Hole has formed but collapse to a singularity has not yet occurred is not and if so, the larger the scale of the Black Hole the greater this period. For a universe sized Schwarzschild radius the period could be billions of years (but one imagines a universe somewhat larger than the one we occupy... SignatureKind Regards Robert Karl Stonjek > > However, an observer within the body would note that all > > of the matter was heading inevitably towards the center, [quoted text clipped - 13 lines] > -) the matter could be rotating with a rotational speed sufficient to > prevent infalling; The original problem statement assumed an initially static distribution. But no matter, if the mass-energy density is sufficient to create an event horizon then the equations of General Relativity indicate that there can be no trajectories that do not end in a singularity. This rules out circular orbits. The only place that stable circular orbits can obtain is *outside* the event horizon. > -) the entire spatial extension encapsulating the Schwarzschild Black Hole > could be expanding for whatever reason eg caused by the effect of dark > energy; Then the mass-energy density would less than that required to form the event horizon in the first place. Dark energy represents a *negative* energy potential, reducing the mass-energy density to less than critical. > -) the Schwarzschild Black Hole is forming and infalling is yet to be > significant. This is quite possible if the density is very low and the > distance between significant loci of mass is great eg in a universe full of > galaxies. There is no such thing as "yet to be significant" in this case; if the mass-energy density is sufficient to form an event horizon, then the space metric has already taken on the properties which must lead inevitably to the formation of the singularity. > And there are other exceptions. The bottom line is that one need not > consider what eventually happens when answering the question at the root of [quoted text clipped - 4 lines] > could be billions of years (but one imagines a universe somewhat larger than > the one we occupy... Nevertheless, the singularity is inevitable and all trajectories within the event horizon must lead to it. > > Yes, but there are several exceptions that I can think of right off:- > > -) the matter could be rotating with a rotational speed sufficient to [quoted text clipped - 8 lines] > circular orbits can obtain is *outside* the event > horizon. I don't see how that can be - Black Holes have angular momentum and that momentum must be within the event horizon. > > -) the entire spatial extension encapsulating the Schwarzschild Black Hole > > could be expanding for whatever reason eg caused by the effect of dark [quoted text clipped - 15 lines] > on the properties which must lead inevitably to the > formation of the singularity. Yes, but in the case of a universe sized Schwarzschild radius, the formation of a singularity could take billion of years - you are still thinking star, where everything is conveniently close - I am thinking big, where distances are so that great that a collapse can take astronomically long time. > > And there are other exceptions. The bottom line is that one need not > > consider what eventually happens when answering the question at the root of [quoted text clipped - 7 lines] > Nevertheless, the singularity is inevitable and all trajectories > within the event horizon must lead to it. That may be in the future, even the far distant future of a very large Schwarzschild Black Hole. It is reasonable, then, to consider the Schwarzschild radius before the formation of a singularity or even before significant compaction has occurred. SignatureKind Regards Robert Karl Stonjek > > > Yes, but there are several exceptions that I can think of right off:- > > > -) the matter could be rotating with a rotational speed sufficient to [quoted text clipped - 11 lines] > I don't see how that can be - Black Holes have angular momentum and that > momentum must be within the event horizon. Angular momentum is a conserved quantity, so yes, even a singularity can have angular momentum. Take, for example, the electron. It has spin yet is as far as we can tell it is a point particle. > > > -) the entire spatial extension encapsulating the Schwarzschild Black > Hole [quoted text clipped - 22 lines] > where everything is conveniently close - I am thinking big, where distances > are so that great that a collapse can take astronomically long time. No, I'm not thinking only star. The equations of General Relativity are, well, general. They apply at all scales from star size to universe size. There's nothing magical about size when it comes to the spacetime metric that results from a given mass-energy density. [snip] > > Nevertheless, the singularity is inevitable and all trajectories > > within the event horizon must lead to it. [quoted text clipped - 3 lines] > Schwarzschild radius before the formation of a singularity or even before > significant compaction has occurred. Sure. From an outside observer there would be no difference in what is observed. Inside, an observer would note that his universe is collapsing (Big Crunch). > > > > Yes, but there are several exceptions that I can think of right off:- > > > > -) the matter could be rotating with a rotational speed sufficient to [quoted text clipped - 16 lines] > example, the electron. It has spin yet is as far as > we can tell it is a point particle. A point particle would have an infinite escape velocity etc etc. Point particle is one way of modelling the electron, but it does not seem to be entirely consistent eg recombination is required in some equations where these infinities are simply crossed out. An electron can also be modelled as a wave. 'Spin' is the angular momentum intrinsic for a particle. I read that one should not confuse the spin of particles with the spin of the massive objects we are familiar with. Just what particles are really doing when they 'spin' is somewhat mysterious. 'Spin' is a property of particles and should be treated as such - not considered as the 'spin' of a spinning top. For one thing, particles can have a spin of two or a half, which means, if the analogue to a spinning top were to be extended, that a top need only spin through 180 degrees or as much as 720 degrees before revealing the same point to an observer (depending on whether the top is a Fermion or Boson). Thus 'spin' as in a spinning top can be ruled out right there. > > > > -) the entire spatial extension encapsulating the Schwarzschild Black > > Hole [quoted text clipped - 30 lines] > > [snip] No, but for very large bodies the period of collapse may be billions of years. Thus a Schwarzschild radius can exist without a singularity and without a substantial increase in density (over say, a few million years). > > > Nevertheless, the singularity is inevitable and all trajectories > > > within the event horizon must lead to it. [quoted text clipped - 7 lines] > in what is observed. Inside, an observer would note that > his universe is collapsing (Big Crunch). Thus a Schwarzschild radius can have any density, which was the questioned posed by this thread - QED - you have agreed with my position. Note that the question was not whether or not any such Schwarzschild radius actually exists - that is a quite separate issue. SignatureKind Regards Robert Karl Stonjek On Mar 29, 7:21 am, "Robert Karl Stonjek" <ston...@ozemail.com.au> wrote: > > > > > Yes, but there are several exceptions that I can think of right > off:- [quoted text clipped - 23 lines] > entirely consistent eg recombination is required in some equations where > these infinities are simply crossed out. I think you are refering to "renormalization", which is a technique employed under certain circumstances when performing calculations in quantum theory. As far as I'm aware (but I am willing to be corrected), renormalization is not required for the determination of the intrinsic angular momentum of the electron. > An electron can also be modelled > as a wave. 'Spin' is the angular momentum intrinsic for a particle. I read [quoted text clipped - 8 lines] > Fermion or Boson). Thus 'spin' as in a spinning top can be ruled out right > there. The spin of particles, although different from what we tend to think of as spin for macroscopic objects, does still represent units of angular momentum. It just behaves a bit differently at the quantum level. Still, it is angular momentum nonetheless, and particles can carry angular momentum from place to place. A system of charges, for example, that lose rotational energy via electromagnetic radiation, must shed angular momentum. Put another way, in order for the conservation of angular momentum to hold, the angular momentum must be carried away by the photons that are being emitted. Photons are spin 1. [snip] > > No, I'm not thinking only star. The equations of General > > Relativity are, well, general. They apply at all scales [quoted text clipped - 7 lines] > years. Thus a Schwarzschild radius can exist without a singularity and > without a substantial increase in density (over say, a few million years). Yes. > > > > Nevertheless, the singularity is inevitable and all trajectories > > > > within the event horizon must lead to it. [quoted text clipped - 13 lines] > Note that the question was not whether or not any such Schwarzschild radius > actually exists - that is a quite separate issue. I suppose you mean that deifferent regions inside an event horizon can have different densities? Regions inside an event horizon may have different densities while the collapse is ongoing, yes. But collapse will happen, and eventually all the matter will end up in the singularity. In order for an event horizon to form, the average density of the whole region inside event horizon must exceed a threshold determined by the radius of the region. Given a density figure, rho, one can calculate the Schwarzschild radius: r = c/sqrt(8*pi*G*rho) > On Mar 29, 7:21 am, "Robert Karl Stonjek" <ston...@ozemail.com.au> > wrote: > > > > I don't see how that can be - Black Holes have angular momentum and that > > > > momentum must be within the event horizon. [quoted text clipped - 14 lines] > corrected), renormalization is not required for the determination > of the intrinsic angular momentum of the electron. Yes, renormalization. Magnetic moment of the electron was the calculation that Feynman mentions in one of his books (can't recall which one - he was among those who developed the technique of renormalisation). > > An electron can also be modelled > > as a wave. 'Spin' is the angular momentum intrinsic for a particle. I read [quoted text clipped - 21 lines] > angular momentum must be carried away by the photons that > are being emitted. Photons are spin 1. The 'spin' specification gives the '*intrinsic* angular momentum'. SignatureKind Regards Robert Karl Stonjek > [...] A Schwarzschild black hole has density=0 EVERYWHERE and EVERYWHEN. That is, it is a vacuum solution of the Einstein field equation. A Schw. black hole has a central singularity which is characterized by a constant M, and outside its horizon it behaves pretty much like an object with mass M. But there is actually no mass anywhere. Like so many around here, you really should learn something about the subject before attempting to write about it. <shrug> Tom Roberts > > [...] > [quoted text clipped - 9 lines] > > Tom Roberts Any locus of mass has an associated escape velocity and so there must be an associated escape velocity for galaxies, clusters of galaxies, and spatial extensions of a size approaching that of our visible universe. If that escape velocity reaches c then we have a Schwarzschild radius. Why you are incapable of considering anything but collapsed stars is quite mysterious - didn't you know that there was spatial curvature around, say, the Milky Way? If the Milky Way were massive enough and rotated at sufficient speed then it could have an escape velocity of c, which would make it a Schwarzschild radius, would not be collapsing and would still be habitable. But as you are such an expert, why not correct those loose thinkers over at Wikipedia who wrote, under the title of 'Schwarzschild Radius', the following: "The Schwarzschild radius of a sphere with a uniform density equal to the critical density is equal to the radius of the visible universe." http://en.wikipedia.org/wiki/Schwarzschild_radius I don't kown exactly who wrote it, but here are the references for the article:- ^ K. Schwarzschild, "Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie", Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fur Mathematik, Physik, und Technik (1916) pp 189. ^ K. Schwarzschild, "Uber das Gravitationsfeld einer Kugel aus inkompressibler Flussigkeit nach der Einsteinschen Theorie", Sitzungsberichte der Deutschen Akademie der Wissenschaften zu Berlin, Klasse fur Mathematik, Physik, und Technik (1916) pp 424. ^ Hamed Moradi, "An Early History of Black Holes", (2004) Monash University ^ A. Einstein, "On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses", Annals of Mathematics, (1939) ^ J.R. Oppenheimer, H. Snyder, "On Continued Gravitational Contraction", Physical Review 56 (1939) p455. Perhaps each of the above deserve a shrug as well? SignatureKind Regards Robert Karl Stonjek > Any locus of mass has an associated escape velocity and so there must be an > associated escape velocity for galaxies, clusters of galaxies, and spatial [quoted text clipped - 5 lines] > could have an escape velocity of c, which would make it a Schwarzschild > radius, would not be collapsing and would still be habitable. No, it seems that you don't fully understand the mechanics of General Relativity as it applies to the spacetime metric. Once an event horizon forms, there are no longer any stable orbits inside -- all trajectories must lead inevitably toward a central collapse. Within an event horizon, space takes on a time-like quality. Just as outside of an event horizon time moves relentlessly forwards towards the future, inside an event horizon all trajectories lead inevitably towards the singularity. [snip] > Within an event horizon, space takes on a time-like > quality. I think it's best not to phrase it this way as it suggests this "time- space switch" is physical whereas it really is just a quirk of a certain coordinate chart (Schwarzschild's). Not even that, it's more like a quirk of coordinate labels (literally, letters). -- Jan Bielawski > > Any locus of mass has an associated escape velocity and so there must be an > > associated escape velocity for galaxies, clusters of galaxies, and spatial [quoted text clipped - 19 lines] > > [snip] That may be the case for stars where the amount of mass increases toward the centre of the object. But there is nothing in the rule book to prevent a hollow shell of matter forming and rotating and for that object to have an escape velocity of c and zero gravitational potential at the centre of the hollow area. That the formation of such an object may be beyond any reasonable probability is irrelevant to its 'in principle' existence. You seem to be unable to think of the Schwarzschild Radius in terms other than that of stars. SignatureKind Regards Robert Karl Stonjek > > No, it seems that you don't fully understand the mechanics > > of General Relativity as it applies to the spacetime [quoted text clipped - 19 lines] > You seem to be unable to think of the Schwarzschild Radius in terms other > than that of stars. You seem to think that size counts in terms of mass-energy density and General Relativity. You also seem to think that Newtonian physics will hold in a domain where the mass-energy density is such that General Relativity must be used --- You need to change rule books when things get extreme! > > > No, it seems that you don't fully understand the mechanics > > > of General Relativity as it applies to the spacetime [quoted text clipped - 26 lines] > be used --- You need to change rule books when things > get extreme! You haven't shown beyond your opinion that I am wrong. Whilst a Black Hole may or may not be hollow (though a somewhat unlikely eventuality) the spin of Neutron stars can be very high indeed. In fact the calculated limit appears to have been exceeded by recent observations (though more data needs to be collected to confirm the find). SignatureKind Regards Robert Karl Stonjek In sci.astro Robert Karl Stonjek <stonjek@ozemail.com.au> wrote: > [...] But there is nothing in the rule book to prevent a > hollow shell of matter forming and rotating and for that object to have an > escape velocity of c and zero gravitational potential at the centre of the > hollow area. That's true, although "escape velocity" is not the right way to think about it. An "escape velocity of c" describes a Newtonian black hole, but a black hole in general relativity is rather different. For example, in Newtonian gravity an object moving at less than the escape velocity can initially move away from a surface, but will ultimately fall back, while in general relativity, an object starting inside the horizon of a black hole can never move outwards at all, but starts to fall inward immediately. In the situation you describe, the interior of the shell can be flat. But the shell is necessarily collapsing -- it cannot be stabilized by rotation, as long as it can't rotate faster than the speed of light -- and observers in the interior will be crushed, typically in a rather short time. This situation is discussed by Lindblom and Brill, Phys. Rev. D10 (1974) 3151. Steve Carlip > In sci.astro Robert Karl Stonjek <stonjek@ozemail.com.au> wrote: > [quoted text clipped - 18 lines] > > Steve Carlip If we are going to consider relativity, then let's not forget the observer. Gravitational time dilation, if extrapolated trivially from non-black objects, is infinite inside the event horizon relative to ANY other observer. Thus a photon does not leave the surface only to fall back nor does it fall back without leaving - the period of transmission is zero events ever ie radiation never occurs as time has effectively stopped relative to all other clocks. This also means that while a Black Hole may have angular momentum, the period of oscillation is infinite or zero oscillations according to any observer outside the event horizon. We may argue that we simply can't know what is going on inside an event horizon, but we can extrapolate from the Neutron star. Measured oscillatory speeds of over 1,000 RPM have been measured. With time dilation, I wonder how fast it is as measured by a surface clock? SignatureKind Regards Robert Karl Stonjek http://amazing-space.stsci.edu/resources/explorations/blackholes/lesson/what isit/index.html http://www.bbc.co.uk/science/space/deepspace/blackholes/index.shtml http://antwrp.gsfc.nasa.gov/htmltest/gifcity/nslens_bh.html http://www.eclipse.net/~cmmiller/BH/blkns.html http://superstringtheory.com/blackh/index.html -- Ahmed Ouahi, Architect Best Regards! > > In sci.astro Robert Karl Stonjek <stonjek@ozemail.com.au> wrote: > > [quoted text clipped - 44 lines] > Kind Regards > Robert Karl Stonjek http://amazing-space.stsci.edu/resources/explorations/blackholes/lesson/what > isit/index.html > [quoted text clipped - 9 lines] > Ahmed Ouahi, Architect > Best Regards! Thanks for that. Those web sites all deal with collapsed star or super dense black holes, not the low density Schwarzschild Black Hole which has been discussed in this thread. SignatureKind Regards Robert Karl Stonjek > Within an event horizon, space takes on a time-like > quality. This is wrong. Space remains space, and 3 dimensional inside any event horizon (indeed at any point within any manifold described by GR). Perhaps you are confused by the way COORDINATE LABELS are conventionally assigned in Schwarzschild spacetime. Yes, in the region r<2M, r is timelike and t is spacelike, but that's merely how we happen to have historically assigned labels to coordinates. Obviously that is an arbitrary human convention of no physical significance. > Just as outside of an event horizon time moves > relentlessly forwards towards the future, inside an event > horizon all trajectories lead inevitably towards the > singularity. All TIMELIKE AND NULL trajectories, yes. This is so because inside the horizon of a Schw. black hole, the singularity is in the future (r is timelike, and -d/dr is future pointing; the singularity is at r=0). The singularity itself is spacelike (i.e. the locus formed by the limit points of ingoing geodesics is a spacelike surface) -- don't let "r=0" fool you into thinking it is "a single space point", because it isn't. Look at a Kruskal diagram, in which the singularities are hyperbolas at the top and bottom of the drawing (future-pointing timelike trajectories move upward within the light cones at +-45 degrees from vertical). Tom Roberts >> Within an event horizon, space takes on a time-like >> quality. > > This is wrong. You don't know what you mean by this, or from what God you obtain such privileged information. > > Within an event horizon, space takes on a time-like > > quality. > > This is wrong. Space remains space, and 3 dimensional inside any event > horizon (indeed at any point within any manifold described by GR). I said time-like quality, not that it becomes time, or that time and space dimenesions are somehow interchanged. > Perhaps you are confused by the way COORDINATE LABELS are conventionally > assigned in Schwarzschild spacetime. Yes, in the region r<2M, r is > timelike and t is spacelike, but that's merely how we happen to have > historically assigned labels to coordinates. Obviously that is an > arbitrary human convention of no physical significance. True. > > Just as outside of an event horizon time moves > > relentlessly forwards towards the future, inside an event [quoted text clipped - 12 lines] > > Tom Roberts > But as you are such an expert, why not correct those loose thinkers over at > Wikipedia who wrote, under the title of 'Schwarzschild Radius', the > following: > > "The Schwarzschild radius of a sphere with a uniform density equal to the > critical density is equal to the radius of the visible universe." That's wrong. I removed it. -- Ben > > But as you are such an expert, why not correct those loose thinkers over at > > Wikipedia who wrote, under the title of 'Schwarzschild Radius', the [quoted text clipped - 6 lines] > > -- Ben I've read that same sentiment (regarding the universe as a Schwarzschild Radius) several times in a variety of places, though I don't have the references at the tip of my fingers right now. The idea was that if the universe was closed then the density of the universe and its radius would comply with the Schwarzschild Radius for an object of that size. Since the expansion of the universe was found to be accelerating, the prospect of the universe being closed has been discounted so we don't hear of that speculation very much now. But the basic idea should still be correct, I would have thought. Tell me why the critical density for a universe sized object is significantly different from a Schwarzschild Radius of the same scale? Is there something in the rule book that disallows it? If so, what? In essence that is what this thread is about - I am unaware of any reason why a universe sized spatial extension can not form a Schwarzschild Radius if the mass is sufficient (expansion should be considered separately ie a none-expanding spatial extension should be considered before ruling it out due to dark energy or expansion or whatever ie an 'in principle' case should be considered before the actual examples, in my opinion). SignatureKind Regards Robert Karl Stonjek |
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