Saturday, May 09, 2020

A brief history of black holes

Today I want to talk about the history of black holes. But before I get to this, let me mention that all my videos have captions. You turn them on by clicking on “CC” in the YouTube toolbar.

Now about the black holes. The possibility that gravity can become so strong that it traps light appears already in Newtonian gravity, but black holes were not really discussed by scientists until it turned out that they are a consequence of Einstein’s theory of general relativity.

General Relativity is a set of equations for the curvature of space and time, called Einstein’s field equations. And black holes are one of the possible solution to Einstein’s equations. This was first realized by Karl Schwarzschild in 1916. For this reason, black holes are also sometimes called the “Schwarzschild solution”.

Schwarzschild of course was not actually looking for black holes. He was just trying to understand what Einstein’s theory would say about the curvature of space-time outside an object that is to good precision spherically symmetric, like, say, our sun or planet earth. Now, outside these objects, there is approximately no matter, which is good, because in this case the equations become particularly simple and Schwarzschild was able to solve them.

What happens in Schwarzschild’s solution is the following. As I said, this solution only describes the outside of some distribution of matter. But you can ask then, what happens on the surface of that distribution of matter if you compress the matter more and more, that is, you keep the mass fixed but shrink the radius. Well, it turns out that there is a certain radius, at which light can no longer escape from the surface of the object, and also not from any location inside this surface. This dividing surface is what we now call the black hole horizon. It’s a sphere whose radius is now called the Schwarzschild radius.

Where the black hole horizon is, depends on the mass of the object, so every mass has its own Schwarzschild radius, and if you could compress the mass to below that radius, it would keep collapsing to a point and you’d make a black hole. But for most stellar objects, their actual radius is much larger than the Schwarzschild radius, so they do not have a horizon, because inside of the matter one has to use a different solution to Einstein’s equations. The Schwarzschild radius of the sun, for example, is a few miles*, whereas the actual radius of the sun is some hundred-thousand miles. The Schwarzschild radius of planet Earth is merely a few millimeters.

Now, it turns out that in Schwarzschild’s original solution, there is a quantity that goes to infinity as you approach the horizon. For this reason, physicists originally thought that the Schwarzschild solution makes no physical sense. However, it turns out that there is nothing physically wrong with that. If you look at any quantity that you can actually measure as you approach a black hole, none of them becomes infinitely large. In particular, the curvature just goes with the inverse of the square of the mass. I explained this in an earlier video. And so, physicists concluded, this infinity at the black hole horizon is a mathematical artifact and, indeed, it can be easily removed.

With that clarified, physicists accepted that there is nothing mathematically wrong with black holes, but then they argued that black holes would not occur in nature because there is no way to make them. The idea was that, since the Schwarzschild solution is perfectly spherically symmetric, the conditions that are necessary to make a black hole would just never happen.

But this too turned out to be wrong. Indeed, it was proved by Stephen Hawking and Roger Penrose in the 1960s that the very opposite is the case. Black holes are what you generally get in Einstein’s theory if you have a sufficient amount of matter that just collapses because it cannot build up sufficient pressure. And so, if a star runs out of nuclear fuel and has no new way to create pressure, a black hole will be the outcome. In contrast to what physicists thought previously, black holes are hard to avoid, not hard to make.

So this was the situation in the 1970s. Black holes had turned from mathematically wrong, to mathematically correct* but non-physical, to a real possibility. But there was at the time no way to actually observe a black hole. That’s because back then the main mode of astrophysical observation was using light. And black holes are defined by the very property that they do not emit light.

However, there are other ways of observing black holes. Most importantly, black holes influence the motion of stars in their vicinity, and the other stars are observable. From this one can infer the mass of the object that the stars orbit around and one can put a limit on the radius. Black holes also swallow material in their vicinity, and from the way that they swallow it, one can tell that the object has no hard surface. The first convincing observations that our own galaxy contains a black hole came in the late 1990s. About ten years later, there were so many observations that could only be explained by the existence of black holes that today basically no one who understands the science doubts black holes exist.

What makes this story interesting to me is how essential it was that Penrose and Hawking understood the mathematics of Einstein’s theory and could formally prove that black holes should exist. It was only because of this that black holes were taken seriously at all. Without that, maybe we’d never have looked for them to begin with. A friend of mine thinks that Penrose deserves a Nobel Prize for his contribution to the discovery of black holes. And I think that’s right.

* Unfortunately, a mistake in the spoken text.


  1. The singularities of ordinary General Relativity can be avoided by considering the (mathematically well defined) Einstein-Yang-Mills-Dirac-Higgs System which is (heuristically) the super-classical limit of the (not mathematically well-defined) Standard Model. This system has complete solutions without singularities, solitons, and a Cyclic Universe solution. (The system has negative energy density; hence doesn't satisfy the positivity conditions in the Penrose-Hawking Singularity Theorems.) The E-Y-M-D-H equations provide an alternative approach to a Cyclic Universe which Penrose has recently been advocating. They also imply that the massive compact objects now classified as Black Holes are actually Quark Stars, possibly with event horizons, but without singularities. A Super version of the above-including super-neutrinos-might be needed to explain Dark Matter. The E-Y-M-D-H is also a totally geometricized theory as a non-commutative geometry; the charge e and the mass m of the electron are geometric invariants of the non-commutative geometry analogous to pi. Unfortunately, there are quantum phenomena, such as EPR, for which this beautiful theory doesn’t make adequate predictions.

  2. For anyone interested in reading what I'm fairly sure is the key Penrose paper referred to by Sabine, try Google Scholaring:

    Gravitational collapse and space-time singularities 1965

    To paraphrase Sabine, this was the paper the got the collective GR community of the 1960s off of their first-let's-assume-a-spherical-cow butts and into a broader, more meaningful exploration of the fascinating and tricky extrema cases of GR.

    Anyone who has read my poor comments here knows I'm adamant about the importance of the intuitive side of mathematics. You may think that's just my personal bias, but there's a bit more to it than that. From my non-trivial background in AI, I'm keenly aware of the incredible waste of computational effort that occurs, both in computers and in biology, when poorly selected initial parameterizations cause even the most solidly reasoned formal systems to proliferate unchecked and without any reality-indexed branching constraints.

    For fellow ancients who recall the once-terrifying technological market expansion of Japan in the 1960s and 1970s, their seeming invincibility saw its first huge chink when Japan at national levels decided they could be the first into artificial intelligence by exploiting the unfettered expansion and exploration of formal descriptions of problems, specifically by building machines that would directly execute a variant of predicate logic (Prolog).

    It was an unmitigated disaster. The machines proved to be incredibly efficient at exploring branches that, at least on paper, looked very pretty and promising. But there was no overriding "understanding" or "intuition" of which branches were likely to have real meaning. Consequently, their very costly machines proved superb only at exploring the nearly infinite domain of precisely stated non-solutions.

    (I will resist the temptation to mention string theory... oops!)

    What I like most about Sabine's invocation of this particular Penrose paper... which is just one of many examples of a truly amazing lifetime full of remarkable accomplishments in physics... is that Penrose clearly re-inserted some well-based intuitive guidance back into a process that had drifted badly, via a poor choice of initial parameters, into wasting time on exploratory branches that were not at all representative of the actual GR problem.

    Think of it this way: Even Newton's physics predicted that a sufficiently massive object would at some point turn black and unable to emit light, since even in Newton's theory light was bent to some degree by gravity. Thus a much better parameterization of the exploration launch point for GR mass-concentration extrema would not have been the assumption of perfectly symmetric spheres, which from Sabine's description seems to have caught on more-or-less by accident, but rather the question of "what does happen to the GR equations if you put too darned much mass in a single spot?" Assuming that this cannot happen solely because there exists an argument against it that begins with the impossible "perfectly spherical" assumption makes as much sense as asserting that all trains leaving at 12 noon will be empty because no one can get on the train at exactly at 12 noon... QED.

    From Penrose's short 1965 paper it looks to me that this sort of let's-get-back-to-real-cases thinking is exactly what he did... and history shows it made a huge difference.

    So kudos for Sir Roger Penrose for giving the GR community of that era a good swift kick of pick-a-better-starting-point intuition. I can think of very few people who have done as much in physics as Sir Penrose, or who has been as innovative coming up with novel approaches to hard problems, without getting a Nobel Prize for some aspect of their total body of work.

  3. I am still searching for an intuitive explanation of how the Schwarzschild solution can be produced physically by throwing matter at a heavy mass. Given that the gravitational potential increases near that mass, an external observer sees increasing time dilation as the heavy mass contracts. For the external observer then, making a black hole is like a Zeno process where it takes more time the closer you are to the Schwarzschild solution. As a result, you never reach your goal.

    One 'intuitive' way to see how the process works could be to reverse time and see how the Schwarzschild solution can be disassembled into a lower density mass assembly, but that can't work because nothing leaves a black hole...

    Serious discussions about this always seem to avoid the question by saying that for a local observer the time to cross the horizon is finite. So what? We are all external observers to the black holes we're looking at!

    Louis Marmet

    1. Louis,

      As you say, the time it takes an infalling observer to cross the horizon is finite. What you learn from this is that the inside of the horizon is physically real, it's not a boundary at infinity.

      Another way to see this is to do what I sketched in my video. Take a distribution of matter and compress it. If you compress it below its own Schwarzschild radius, the solution will have a horizon. The time it takes to reach that state is finite in local coordinates. It becomes very large for an outside observer. (It is generally believed that it's not strictly speaking infinitely large, but fapp it doesn't make a difference.)

    2. Btw, I should have added, the formal proof are Hawking and Penrose's singularity theorems. Google will tell you details.

    3. I have the same problem as Louis. In our frame as observers on the Earth we have to use our time scale. When we say that a black hole was created one billion years ago then we ask what happened to this black hole during this one billion years. Maybe that for an in-falling observer only one minute has passed during this time and so for him anything looked ‘normal’. But we here on Earth ask for the material that can have moved into it during ‘our’ one billion years.

      I have heart this argument about the in-falling observer repeatedly, but it does not make logical sense to be. And for this understanding I had a respected co-thinker. Some time ago the Nobel prize laureate Gerard 't Hooft gave a lecture at our university. And he addressed just this problem and he said: For us as observers here there is no motion in the event horizon and nothing can have moved into the black hole.

    4. In order to obtain an infinite infalling time, one assumption is that the variable "t" in the Schwarzchild solution represents time as seen by a distant observer.
      This assumption is usually justified by the fact that when r (the radial coordinate) goes to infinity, the Schwarzchild metric coincides with the "flat" metric from special relativity.
      But there is at least one other time variable with this property, namely Eddington's time t_E (already mentioned in another comment).
      If one assumes now that t_E is the time for a distant observer, then, lo and behold, the infalling time becomes finite!
      I cannot post a reference here because it would violate the comment rules on this blog. So, if anyone's interested please send me an email (

    5. This is exactly my concern: the time it takes, as measured by an external observer, to compress a mass below its own Schwarzschild radius could be infinite. I can't decode the word 'fapp', but for the observer outside the black hole in a finite T=4x10^17s-old universe, it does make a big difference if the compression time is infinite.
      The Schwarzchild solution is one of many possible solutions of Einstein's equations, but if no physical process can produce it the solution is not physical. (I saw this in another of your videos!)

    6. Pascal,

      You refer to Eddington’s time. Yes, there is of course another definition of time possible which does not have the mentioned problem. Fact is that we observe that material moves into a black hole. So, there should be a formalism available which describes this fact.
      But the origin of this discussion referred to Einstein’s GR, where the applicable metric is represented by Schwarzschild. And Schwarzschild’s metric says that at the event horizon the time progress reaches zero. So, physically speaking, any motion ceases. And that is in my understanding a conflict of Einstein with the reality, as we can observe a motion of matter through the horizon.

    7. Antooneo: my comment definitely referred to Einstein's GR, and to the Schwarzchild solution. The issue is what is the meaning of the variable "t" in the Schwarzchild solution.

    8. Louis: I think it’s straightforward myself, because I think Oppenheimer and Snyder's original "frozen star" black hole is correct. See On Continued Gravitational Contraction dating from 1939. It was no flash in the pan - see the 1971 Physics Today article Introducing the black hole by Remo Ruffini and John Wheeler. They said “in this sense the system is a frozen star”. There is no central point singularity. But there is a black hole. It grows from the inside out, like a hailstone. Imagine you’re a water molecule. You alight upon the surface of the hailstone. You can’t pass through this surface, but soon you’re surrounded by other water molecules, and eventually you’re buried by more. So whilst you can’t pass through the surface, the surface can pass through you. I can’t explain why Einstein missed this in his 1939 paper On a stationary system with spherical symmetry consisting of many gravitating masses. Or the prediction of gamma ray bursters. Perhaps it was because war was looming and his mind was elsewhere.

      Of course, Einstein's "spatially variable" speed of light must mean Penrose/Hawking singularity theorems are wrong. That's because in a gravitational field the ascending light beam speeds up. In a strong gravitational field, the ascending light beam speeds up even more. It doesn't get dragged back, like Hawking said on page 76 of his 1966 paper Singularities and the geometry of spacetime. See Is The Speed of Light Everywhere the Same? by PhysicsFAQ editor for something contemporary on this.

    9. Pascal,

      I have looked into Eddington’s ‘t’. It is a complicated calculation and I did not come to a final conclusion. But I have a logical point:

      If the direct use of the metric of Einstein / Schwarzschild says that any radial motion at the event horizon is zero, how can it logically be that a modified coordinate system says that there is a radial motion; and that both are based on the same theory of Einstein? Whether there is a radial motion or not, is an elementary physical fact, which cannot depend on a coordinate system.

  4. Hello,
    there are two (related) questions I think about since
    a long time:
    - Let's assume ther exists some force which stops the
    compression of the matter in a black hole at maybe half of
    the Schwarzschild radius or 1/100 th or a millionth of it.
    Could some observer outside measure that?
    - If the mass is compressed to a "point", what about Heisenbergs uncertanty relation? Mass located to some "point" would mean infinite impulse for that mass, wouldn't it?

  5. > too good precision

    "to a good precision". Looks like a speech-to-text issue or something.

    1. Hi Sergei,

      Thanks for spotting! I have fixed this :)

  6. Thank you for the nice article as usual!

    Small typo (I guess): the Sun's Schwartzschild radius is a few miles, not a few thousand miles.

    1. Andrea,

      You are right of course! Sorry for that blunder :/

  7. The curvature is dependent on the inverse cube of the radial distance from the center. This gives the tidal acceleration. The inverse square comes from the geodesic equation.

    One has to admire Karl Schwarzschild. Einstein presented his general theory in November 1915 and Schwarzschild derived his solution the following spring. It is remarkable for anyone to internalize this new theory that quickly, but to compound things Schwarzschild was a colonel in the German army in the east against Russia. WWI was the stupidest and most ugly war fought, and to be an officer and soldier in a war and then on the side to work the first solution to Einstein's field equations is remarkable.

    Schwarzschild did not die of battle wounds, but of an autoimmune disorder called pemphigus. In fact while he worked out this solution he was battling this disorder.

    1. With my first comment I thought I would expand on this. The The Riemann curvature R^r_{trt} = m/r^3, m = GM/c^2, is at the event horizon R^r_{trt} = 1/8m^2. So while the curvature varies as the cube of the radius, at the horizon with r = 2m, it is a quadratic dependency.

      I spotted the comment about the first black hole found in this galaxy. It is the case the first was found by the Uhuru spacecraft in 1972-3, and designated Cygnus X-1. I actually remember as a school kid being shown a newspaper article on this by my father, who's undergraduate degree was engineering. I was somewhat baffled by this as I recall.

      There were a number of stellar black holes found, and one was recently found designated HR 6819 or QV Telescopii about 1100 light years away. It is about 5 solar masses and visible to the naked eye. The existence of this BH was inferred by methods similar to Doppler shift used to find extrasolar planets.

  8. The first convincing observations that our own galaxy contains a black hole came in the late 1990s.

    This is... not really correct, for two reasons.

    First -- I get the impression you're thinking of the supermassive black hole (SMBH) at the center of the Milky Way. But -- leaving aside the fact that supermassive black holes were circumstantially identified as the most likely power sources for active galactic nuclei (quasars, etc.) back in the 1970s and 1980s -- the Milky Way's wasn't the first to be detected.

    The 1995 review article by Kormendy & Richstone listed eight detections of supermassive black holes in galaxy centers (one of those being the Milky Way's). The strongest case on the basis of stellar kinematics, in their view, was for the SMBH in the center of M31 (the Andromeda galaxy); the first convincing case was actually in M31's satellite galaxy M32, from observations in the late 1980s. The strongest cases on the basis of gas kinematics were M87 (whose SMBH was recently imaged by the Event Horizon Telescope) and Messier 106, where the SMBH was detected by measurement of water-vapor maser emission sources in the accretion disk around the SMBH. This last one was really the most convincing case at that point.

    Second, the first the first really "convincing observations" of a black hole anywhere (and in our own galaxy) were those of the stellar-mass black hole in the X-ray binary Cygnus X-1, back in the early/mid 1970s. (Followed by LMC X-3 in the 1980s, though that one is, technically, in another galaxy.)

    1. Hi Peter,

      Thanks for pointing out. I knew that there were several candidates detected simultaneously, but I was under the impression that Sag A* had the highest significance. Sorry for getting this wrong. Now that I think about it, it makes sense that it wouldn't be given that it's comparably small.

      And, yes, there were other observations before that and one or the other astrophysicist might have found those convincing, but they arguably did not convince many. The tide only really changed in the mid-late 90s. With best wishes,


    2. What is the evidence that the supermassive objects at the center of galaxies are actually black holes? I suppose one can estimate their masses. Can one estimate also their radius? If one could estimate it and if it turns out to be no larger than the Schwarzchild radius that would be strong evidence. Is it possible to do this? Do we have other kind of evidence?

  9. Another history of black holes is here:

    An excerpt:

    " The debate over black holes' existence flared up between Indian American astrophysicist, then student of Cambridge University, Subrahmanyan Chandrasekhar, and English astronomer Arthur Eddington in the 1930s.

    Chandrasekhar had calculated that a white dwarf much heavier than the Sun couldn't exist, and that it would undergo a collapse into a singularity with infinite density. On 11th January 1935, with Eddington's apparent approval, Chandrasekhar was to deliver his results to a meeting of the Royal Astronomical Society in London. Little did he know, Eddington had prepared his own talk, and would give it directly following Chandrasekhar's.

    Eddington belittled Chandrasekhar's argument, declaring that due to its solely mathematical basis, it had no bearing on the physical universe. How could something so large as a star effectively disappear? He upheld the view that white dwarfs could not totally collapse. Though Eddington's case was fairly unsubstantiated, the Briton's reputation prevented anyone from daring to openly disagree with him. Even Chandrasekhar was not given the chance to respond to Eddington's retort.

    Their rivalry on the matter continued for some years. At a meeting in Paris in 1939, Eddington maintained his disapproval of Chandrasekhar's ideas, despite the quiet, growing support for Chandrasekhar from the likes of Bohr, Pauli and Dirac. By this stage, it was widely believed that Eddington misunderstood the problem, thus he became largely ignored on this topic. At the meeting, Eddington claimed that there was no experimental test that could determine which of the two notions was correct, which was a valid point, but with astronomer and white dwarf expert Gerard Kuiper on hand to immediately refer to his work supporting Chandrasekhar, "

  10. A few days ago I had the occasion to revisit the book Spinors and Space-Time (Penrose and Rindler-- they keep the hyphen in space-time, which I find appealing). That monograph led me back further to Penrose's Techniques of Differential Topology in Relativity (July 1970). An AIP History interview contains these words from Penrose: "...I also liked the idea of steady state for itself: a universe that always was there. I think this has an aesthetic appeal of its own. There was always a conflict in me because I also felt very strongly that general relativity was right, obviously a beautiful theory. Again, it was aesthetics to some extent. I had strong feelings for general relativity, and I also recognized the conflict between these two sets of ideas, (the steady state model versus general relativity). This conflict was influential in the way that I thought about cosmology. In trying to resolve the conflict, I was finally able to see that it was irresolvable, in a geometrical way. This ultimately led to singularity theorems — by thinking about light cones and how they focused. At an earlier stage, I had had a rather wild idea about how you might be able to make steady state and general relativity consistent with each other." (24 January 1989).

  11. Please consider using in your narration (which is in English) the system of measurement used in all English-speaking countries (other than one lone exception) and in all countries where English predominates as the language of instruction at university level (such as India). Imperial conversion could then be relegated to a visual off to the side.
    I mention this only because your narration packs a bigger punch psychologically than looking at a visual conversion off to the side. It seems therefore more appropriate to use English in the voiced narration for the reasons cited above. This, in addition to the fact that metric (SI) is the established system of measurement in all of the sciences as well.
    Rech vielen Dank!

    1. If you really want to know, the reason I said "miles" instead of "kilometer" is that "miles" is easier to pronounce. And for a theorist like me its the same anyway.

    2. It might be of some trivia historical interest to know the unit miles is actually metric-like. The word comes from the Latin Mille, or a thousand. It is a distance unit based on a thousand strides Roman legionnaires would take in a march.

      I am all for the conversion to metric units in engineering, and the US has been hobbled by not being tooled with the rest of the world. However, I am very much against the idea of abolishing the imperial or English system entirely. I think it is important to have these conversions between systems, for I think it is important to convey the fact units of measurement are ultimately arbitrary. If people have to mentally convert it prevents a bias in assuming there is some absoluteness to a system of units.

      The imperial system of weights and measures is based in part on factors of 2 or halves. It is in some ways then easy to visualize these, and frankly cooking in metric units strikes me as terrible. The Fahrenheit temperature units are terrible for science, but honestly with weather and home temperatures and so forth they are for some reason more mentally workable. Also a liter is slightly larger than a quart, a mile about 1.6 kilometers, an ounce is 28.5 grams, 30 is you are interested in a rough mental conversion, and so forth. So it is not hard to flip between metric and imperial units mentally.

      The English system, which includes the avoirdupois units, is a remnant of a time when different parts of Europe and further the world has local units. It was a cobbled up mess to communicate measures. The metric system was championed by Napoleon and was framed around then fairly exact measures of the Earth. One has to realize French geographers were going around measuring land and sea at a time of war with England, and they were doing this in India under the British Raj. The English most vociferously resisted any French at the time and so resisted the metric system, though converted later on. In the early US there were initiatives to convert as well, but failed in Congress.

    3. "I am all for the conversion to metric units in engineering, and the US has been hobbled by not being tooled with the rest of the world. "

      Seems to vary by engineering discipline. The semiconductor industry is solidly metric, but you'll still occasionally find non-SI units (eg, "angstroms")

      Well into the 1990s, USA automobiles used mixture of metric and english fasteners & components, but now are solidly metric.

      HVAC design still seems to use cumbersome units such as "BTU/hr", "grains" of water vapor, water vapor permeability "1.0 US perm = 1.0 grain/square-foot·hour·inch of mercury" etc.

  12. Sabine wrote: "physicists accepted that there is nothing mathematically wrong with black holes"
    One thing that still seems wrong with the Schwarzschild solution, or at least with its usual interpretation, is that the metric inside the Schwarzschild radius has the wrong signature.
    In GR the signature is supposed to be (-, +, +, +). Strangely enough this is recalled at the beginning of the wikipedia entry on the Schwarzschild metric, but it apparently did not occur to them that this in contradiction with (part of) the rest of the article. What this sign change strongly suggests is that the points inside the radius do not actually belong to the solution manifold,
    i.e., there are no points in spacetime inside the Schwarzschild radius. This is possible because a curved spacetime can have a nontrivial topology: in contrast to flat spacetime, there is no reason to expect that every value of the coordinate r corresponds to points on the solution manifold.

    Here is a very crude analogy with elementary geometry. Suppose we study the properties of the (real) hyperbola x^2-y^2=1. If we reason from the assumption that there are points on this curve inside the “radius” |x|<1, we will surely be able to reach very strange conclusions…
    All of this begs the question: why on earth do physicists seem to believe that there must be a point on the spacetime manifold for every value of the coordinates?

    1. Pascal,

      You fix the metric signature at infinity, there is nothing wrong with it changing sign at the horizon.

      Yes, a curved space-time can have nontrivial topology, so can flat space-time. But the complete analytic continuation of the Schwarzschild solution doesn't have this property. This isn't a matter of debate, it's just mathematical fact.

    2. As an alternative approach one can try the change of variable r=a(1+log ch p), where a is the Schwarzchild radius, p is a new variable and ch p = (exp(p)+exp(-p)) / 2 is the hyperbolic cosine. You can check for yourself that this results in a metric which has no singularity at all, not even a central singularity! And the points inside the Schwarzchild radius are not part of the solution.

    3. Or you could calculate the Kretschmann scalar of Schwartzchild original article (not Hilbert), and see the absence of singularity. Note that Hilbert metric (the badly copied Schwartzchild metric) has that singularity.

      I personnally don't understand why one should want to find a mathematical solution "inside" the black hole, because talking of an interior has no sense. In the same way that coordinate "r" is not radial position.

      There are theorems (by Michel Mizony) that shows that formation of a black hole during star collapse is impossible.

      Finally, Sabine, thank you for your blog, it is always an ineteresting place for debate.

    4. If you don't like the interior solution, you have to cope with geodesics that just end at finite proper time which makes absolutely no sense.

      Well, it seems that the theorem that you quote disagrees with Penrose and Hawking's. One of them is wrong. Which one do you think it is?

    5. Sabine wrote: “If you don't like the interior solution, you have to cope with geodesics that just end at finite proper time which makes absolutely no sense.”
      This is not quite true. See my previous comment on the change of variables which makes the interior (including the singularity) disappear. The situation is sort of like in my other comment about the hyperbola x^2-y2=1. Imagine a point moving on the hyperbola, starting at some coordinate with x>1 and y>0, with the x-coordinate decreasing as the point moves.
      What happens when the point reaches x=1? It doesn’t stop moving, doesn’t continue with x<1, and doesn’t fall off a cliff! Rather, it continues its trajectory with y<0. To come back to the Schwarzchild solution, remember that the change of variables that I pointed out was: r=a(1+log ch p), where a is the Schwarzchild radius, p is a new variable and ch p = (exp(p)+exp(-p)) / 2 is the hyperbolic cosine.
      The rough analogy with the hyperbola is that if you start with p>0 and p decreases, when p reaches 0 (i.e., you reach the horizon), the trajectory can continue with p<0. Isn’t that wonderful?

    6. Pascal,

      Look, it's called "geodesically incomplete" and it's a math thing and you can look it up and there's a proof and that's that. I don't care at all how you want to glue together different sides of a static metric, because the time-independent solution is not realistic anyway.

    7. I am aware that the standard way of approaching this issue is via the Kruskal-Szekeres coordinates, but it's not the only one.

  13. Do you mean "had turned from mathematical wrong, to mathematically correct but non-physical"?

    1. Yes... Dammit. I really need someone to read these scripts *before* recording the video.

  14. Hi Sabine - little typo
    ". Black holes had turned from mathematically wrong, to mathematically wrong but non-physical" - I assume you meant "to mathematically correct but non-physical"

    1. Hi patfada,

      Yes, that's what I meant... Sorry about that :/

  15. You have made a good point about a Nobel for foundational theoretical work in GR. Not even Einstein got a Nobel for that! Peebles, Thorne and Chandrasekhar each came close though. Proving that black holes are generic and ubiquitous ought to qualify. Not that it is likely to happen at this point, with Hawking gone, and Penrose in his late 80s.

  16. I wonder why they didn't get the Nobel Prize for that? Could it be because Black Holes were already predicted, and so they didn't add anything "new" to the conversation? Were Nobels given for discovering evidence Black Holes exist?

  17. The Schwarzschild radius of the sun, for example, is a few thousand miles…

    I think you meant meters, not miles.

    1. Yes, or a few miles. Sorry about that, I have fixed this to the extent possible. (It's correct in the illustration.)

  18. According to the GR postulate, not only mass, but also every form of energy contributes to the curvature of the postulated spacetime. This includes the energy associated with gravity itself. Therefore the "Einsteinian" field equations are non-linear. Note: The systems of equations (Einstein, Friedmann) of general relativity, on which the statements of the standard model of cosmology are based, do not provide any analytical solutions.

    Only idealizations and approximations lead to calculable solutions to a limited extent. The inevitable contradictions come with the obviously inadmissible idealizations and approximations of the system of nonlinear, chained differential equations. Mathematically, the principle of covariance cannot be “violated” because it is based on axiomatics. Only this axiomatic requirement "disappears with the mutilation" (idealization and approximation) of the actual equations. In other words, the mathematically correct equations have no analytical solutions. The reduced equations (approximations, idealization, for example Schwarzschild metric) have solutions, but they are not covariant in this sence. Thus, no solution has “an absolut” meaning. Because depending on the (self) chosen metric, different results are obtained.

    In other words: One formalizes in a general relativistic way, but then - using idealizations and approximations - generates objects such as black holes based on Schwarzschild metrics or just approximated gravitational waves, and then incorporates them as locally generated events in the flat, time-decoupled, absolute space. Result: The theory that is ultimately used is no longer “general relativistic” in the sense of the causal axiomatics and postulates.

    More specifically: In the usual "physical evaluation" of ds² of the Schwarzschild metric, a zero or pole is interpreted as a Schwarzschild radius. In a more or less associated plausibility analysis, the question is asked of the distance from which photons / electromagnetic waves in the field of a "gravitational mass" can no longer escape energetically.

    Conditions under which the Schwarzschild solution was derived:

    The spacetime and the gravitational source in the center r = 0 are radially symmetrical.

    The Schwarzschild solution for r approaches infinity in the flat Minkowski metric and is compatible in the far field with the Newtonian gravitational potential of a point mass.

    The gravitational mass distribution and the resulting space-time are static.

    The last requirement can be “weakened” by adding time-dependent mass distributions, whereby the Schwarzschild solution turns out to be the only possible space-time outside of any radially symmetrical gravitational source (see Birkhoff theorem)

    But, for example, the so-called Eddington-Finkelstein coordinate transformation eliminates the coordinate singularity of the Schwarzschild solution and "ensures" that for the advanced solution inside and for the retarded solution outside particles can enter and exit the black hole! In other words, the postulated black holes of the Schwarzschild metric were the result based on two integration constants of the selected coordinate system, another coordinate system by Eddington and Finkelstein eliminates the coordinates -artifact and "gives" the supposed black hole the property that particles can leave the black hole. For the sake of completeness it should be mentioned that "one" was "dissatisfied" with the Eddington-Finkelstein coordinates. The Kruskal-Szekeres coordinates followed in 1960. One characterizes the Kruskal-Szekeres coordinates as the maximum analytical continuation of the Schwarzschild solution.

    So please remember:It is far from trivial to consider space and time as physical "objects". Space and time are primarily "ordering patterns of the mind". In order to "obtain" physics from these ordering patterns, phenomenological considerations and explanations are required.

    1. Hi Dirk , this is a silly speculative question; If space time is a physical structure; So if a black hole loses energy then space time will come out of it?

  19. The Haeking Penrose part is not very clear. Their theorems are about the singularities, not about the event horizon. They didn't prove that black holes exist. Only if you invoke the cosmic censorship conjecture you get black holes.

  20. Ha ha. I thought "miles*" was going to lead to a footnote explaining why a scientist, and not even a British or American one, was using miles.

    Regards etc.

  21. Dewey B. Larson somewhere said that the argument that collapse to a black hole would result from exhaustion of stellar fuel and the loss of the expansive pressure it generates is actually illogical. Basically, he suggested, these are pressures between atoms caused by their thermal motions, forces that increase the space between atoms, and these are not the same forces responsible for the basic atomic "volume" itself. Therefore, the absence of that thermal motion would not cause the atomic "volume" itself to collapse. (I think this is true even in conventional theory, but I trust you will correct me if I am mistaken.)

    I put the term 'volume' in warning quotes because according to Larson the atom itself is the size of what Rutherford identified as the "nucleus" (a possibility that should have been investigated!) and the spacing in a solid (comparable to the related minimum distance in a gas) is merely the result of a force equilibrium between inward and outward forces. Heat _adds_ to the outward forces, and external pressure _adds_ to the inward forces, in either case changing the point of equilibrium but not responsible for its existence.

    Larson's basic postulate (originally come upon, by the way, in an effort to explain his expression for the inter-atomic distance for elements and compounds) was that there exists a general reciprocal relation between space and time. Considering this initial thought, he observed that _in motion_ space and time _do_ have a reciprocal relation - more space being equivalent to less time and vice versa. (His postulate also necessitates that space and time have the same dimensions and that space should be increasing in the manner that time does.)

    So what are the forces involved in the atomic equilibrium, according to Larson? (No, they're not coulombic.)

    These are (1) the space-time progression which tends to carry objects outward from unit distance at unit speed and (2) gravitation, which _always_ opposes the space-time progression, bringing objects toward unit distance. Inside unit distance, the space-time progression, always acting in the direction _away_ from unit distance, provides the _inward force_, while gravitation, always opposing the space-time progression, is the force responsible for resistance to compression. [!]

    (Unit distance = s_u = 4.559 x 10^-6 cm = c x t_u, where t_u = unit time = 1 over twice the Rydberg frequency = 1/(2)(3.288 x 10^15 sec^-1) = 1.521 x 10^-16 sec. For atomic scale calculations, s_u is reduced by a factor of 156.444. According to Larson, distances less than s_u in "equivalent space" actually represent increases in coordinate time separations.)

    According to Larson, all of the observed phenomena of extremely high density - from white dwarfs and pulsars to "black holes" - are actually the result of faster-than-light speeds (or we may say, speeds "on the far side of unity") which cause expansions in coordinate time (e.g. of the inner materials of a supernova, which start out, pre-explosion, with thermal motions already close to the speed of light). According to the reciprocal postulate, more time is equivalent to less space, hence higher apparent density for the resulting object. (In line with this, the inner part of 1987A was until recently invisible but has recently re-condensed - in coordinate time, according to Larson's theory - to the point that it is visible as a pulsar.)

    1. I feel compelled to respond to this. The theory that internal heat prevents the collapse of a star dates back to Eddington. The Navier-Stokes equation ∂p/∂z + ρg = 0 for hydrostatic equilibrium can be used for a star. A given layer obeys a rule

      ∂p/∂r - Gρ/r^2 = 0,

      Where we have to employ an equation of state p = ρ(γ – 1)CT. Now temperature enters into the picture. The higher the temperature the higher the pressure. It is not hard to transform this to

      dp = p[(γ – 1)CT]^{-1}dr/r^2.

      Now consider the temperature as dependent on volume pV = NkT and one can crank out some model calculations. I leave that fun to readers.

      Clearly temperature with classical thermodynamics plays a role. For collapsed bodies some strange physics comes into play. The collapsed core of a star such as the sun ordinarily becomes a dense material held up from further collapse by the degenerate pressure of electrons. This is due to the Pauli exclusion principle. For more massive stars the electron can be forced into protons, with the emission of a neutrino, so they become neutron stars. These are complex objects where a neutron fluid fills the center and outer layers are complicated structures. For more dense objects the center can be a quark-gluon plasma. If density and pressure become greater this will become a black hole.

      The astrophysics of ordinary stars and the nuclear processes in the interior is classic stuff. Of course, there is research on the nature of convection and radiative layers. Also, there are complex issues with deviations from spherical symmetry. However, the basic idea still remains; stars are held up from collapse by the generation of thermal energy in the interior.

    2. Steven Athearn wrote:
      > Dewey B. Larson somewhere said...

      In case anyone is wondering, Dewey Larson was a crackpot whose works included The Case Against the Nuclear Atom, an attack on the conclusion normally drawn from the famous Rutherford-Geiger-Marsden alpha particles scattering off gold foil experiment.

      Steven, modern quantum physics has been successful at everything from neutral-current scattering via the weak-force in high-energy experiments to solid-state physics to quantum optics to chemical molecular-orbital theory.

      To be taken seriously, those of you who follow Larson would have to show that his "theories" do at least as well in all those areas.

      You have not done so.

      Doing so would have required using a great deal of math to correctly calculate a while lot of experimental results.

      You have not done so.

      You are therefore going to be dismissed as crackpots.

      I know you think this is unfair.

      But, for better or for worse, you are just wasting your time peddling this among any group of people conversant with modern physics.

      No doubt flat-earthers think it is unfair that most of us are so silly as to think the world is round.

      Anyone interested in this crackpottery can google Larson's "Reciprocal System" and come up with page after page by Larson and his acolytes. Words and words and words and words -- but I at least could find no actual quantitative confrontation with physical reality.

      What I did find most enlightening is that the "true" value of π is actually 4, a fact that certainly does make it easier to convert from radians to degrees!


    3. One can look up Larson and his ideas, that are hopelessly stillborn. The first of these links below is a more neutral overview and the second is a more serious indictment. Larson was a more serious crackpot than someone suggesting a wrong idea on stellar astrophysics.

    4. Dave,

      I take it that

      (1) you do not have in mind any specific flaw in Larson's case against the nuclear atom model

      (2) you are not contending that the hypothesis that atoms themselves are much smaller than had been thought is not one that should have been investigated

      (3) you are not asserting that this possibility was _not_ overlooked historically

      (4) you are not denying that preconceptions (regarding atomic dimensions, the existence of charges in uncharged atoms, and the idea that if electrons can be obtained from atoms they must have been atomic constituents) had a lot to do with the adoption of the nuclear-atom model and the failure to examine the obvious alternative

      (5) you are not disputing that the failure to consider this alternative was the basis of the perceived necessity for introducing several further assumptions of a more questionable character - ad hoc evasions of contradictions with experimental facts or well-established theory, viz. the repulsion between like charges, the instability of the neutron, the instability of classical electron orbits, and annihilation of unlike charges in proximity.

      Rather, you seem to contend that modern quantum mechanics has been so successful in so many areas that it can't be the case that a basic error could be buried so deep in the theoretical structure and you seem to believe that others should be dissuaded from thinking about such questions.

      I agree with you that I am personally not well qualified to bring Larson's ideas to the attention of scientists well versed in modern physics. This is so with regard to both my very limited familiarity with the quantitative details of Larson's system and with those of conventional modern theory. You can hold me responsible for Chapter 1 of McQuarrie and Simon's Physical Chemistry, say, and, with regard to the Reciprocal System, for about three dozen calculations of inter-atomic distances for elements (none for compounds, none for compressibility), a similar small number of calculations using Ronald Satz's equations for the masses of different isotopes (much simpler than the semi-empirical mass formula, btw.), and replications of some of the calculations in Satz's paper on atomic spectra and ionization energies.

      But I can assure you that _extensive_ RS work exists (far beyond what I personally have closely examined) containing "quantitative confrontation with physical reality."

      I see no reason to accept your contention that in order to be taken seriously proponents of a new theory must show that the new theory does at least as well as the modern theories in all the areas in which the modern theories have been successful. That seems like a bar way too high to allow consideration of any new theory.

      Rather, it is reasonable to require a new physical theory to explain _some_ body of experimental facts quantitatively or conceptually.

      And fair consideration of a _critical_ work like The Case Against the Nuclear Atom (1963), is logically independent of whatever case can be made for the new theory - but might well be a psychological prerequisite for willingness to look at Larson's ideas. I doubt whether you have ever ordered that book from your library or read it. (People interested in a precis of the case there can search "Just how much do we really know? Dewey Larson philpapers [dot] org" - especially the concluding six pages). Though few in number, scientists who actually read at least one of Larson's books conceded that he was well-informed about physics research (Mario Girolama Fracastoro, Isaac Asimov, R.D. Redin), found consideration of Larson's work "a useful exercise" or "a service" (Fracastoro; Asimov; Arthur Adamson) or considered proper evaluation of Larson's proposals a difficult but worthwhile undertaking (Felix Schmeidler,
      J. Edward Anderson). They did not display a need to dissuade others with absurd (and frankly, unbecoming) claims that Larson ever suggested that π equals 4.


    5. Lawrence,

      Thank you for your considerate elaboration of the conventional theory, from which I may well learn. From what I can tell, however, "the basic idea" that you well summarize is precisely the one Larson was objecting to, his argument being that collapse of the volume of the star resulting from the loss of thermal energy in the interior does not in itself argue for the _further collapse_ of (what is considered to be) the internal structure of the atom.

      Forgive me if I may have appeared to imply that conventional theory has no further ideas about the nature of that _further collapse_ in terms of the "strange physics" you have also well elaborated. Although Larson would have objected to a number of these ideas, he would not have denied that the electron and proton transformation to neutrons and neutrinos is one that exists - only that this is actually the process responsible for super high observed densities.


    6. Steven Athearn wrote to me:
      >I take it that ...

      Steve, you over-psychoanalyze me!

      There are over 7 billion people on this planet. A significant fraction of them are quite nuts.

      So, any professional, when he sees something about his field, has acquired built-in filters to check to see if the guy pontificating on his subject is in fact a nutjob.

      And Larson and his acolytes trigger pretty much all of those nutjob filters.

      I am, by the way, much more tolerant of outsiders than most professionals are, which is why I actually did look at some of Larson's "work."

      If Larson and his crew had anything of value, they would start off saying, "Look we have found this specific scientific (not deep philosophical) error in this part of physics. And, if you fix that error, then you can get all these quantitative results that the standard theory cannot get. And, we can also get all of the results quantitatively correct that the standard theory does manage to get correct."

      That needs to be upfront in the first few paragraphs or, better still, the abstract. And there needs to be a demonstration right away that they really can do it.

      Larson doesn't.

      Instead, Larson has words and words and words about approaches to the philosophy of science, etc. And he has ex cathedra pronouncements about space and time and all the rest that seem to be utterly meaningless and, in any case, do not seem to lead to any scientific results.

      Typical bona fide crackpot.

      Of course, there is also the issue of Satz' goalpost moving on capacitance and the faster-than-light neutrinos. But I suppose Larson should not be blamed for his idiot acolytes.

      But, no, I did not consider even a single one of the questions you raise. I just spent half an hour or so glancing over Larson's and his followers' "work," and it all cried out "CRACKPOT" as loudly as anything I have ever seen.

      I do not think that Larson or any of his followers would engage in brain surgery or try to fly a commercial airliner without appropriate background or training.

      But, for some reason, they, and quite a few other crackpots, think that they can do real work in physics without any significant background or understanding of existing physics.

      They can't. Physics is not easier than brain surgery or flying an airliner.

      How can I convince you? Well, I have done things in physics and engineering that actually worked. But you probably know so little about STEM subjects that you could not understand what I have done.

      So... about all I can do is tell you what books to start studying to learn real math, science, and engineering if you have any real interest. And if you do that for a decade or so, then we can talk.

      Otherwise, all I can tell you is that Larson and his cub scouts are quite bizarrely hilarious to anyone who knows actual science.

      I probably cannot convince you of that. But then I do not have to.


    7. To argue these points is in line with other science denial trends active today. Popular trends from "scientific creationism," anti-vaxers, geocentrism, flat Earth, moon hoax, big foot (popular in the US), homeopathy, and so forth have roared forwards in the last decade or so. To argue against standard atomic and nuclear physics, which is well tested by experiments and "battle tested," is to put yourself in league with creationists or flat Earthers.

      I am not going to waste my time looking into the details of Larson's so called theory. A quick wiki-search clearly shows this stuff to be nonsense. What does worry me more these days is that this sort of alt-science is growing into some sort of popular crescendo that could overwhelm actual science. This sort of thinking is becoming political. My nation has a cultish following of Don-the-Con t'Rump, a person who has no depth of understanding and has expressed confusion over whether medical tests might prevent disease. He cites anything critical of him as fake news, and is setting up a popular trend of an alt-reality where t'Rump is always right. This is even though he is most often terribly wrong. I see these trends as some sort of social-psychological disorder.

      I am not going to argue the particulars of these errant conjectures of Larson. I wish the really intelligent Larsson, who penned the Far Side would render a comic on this, though indirectly he did. My advice is to look at critiques of this model and to look more deeply into the actual physics. I can't make you or anyone do that, and if you don't then I just hope you have fun in this dismal pseudo-intellectual cul de sac.

    8. There wasn't any attempt to psychoanalyze Dave on my part. Since he had referred to the title of Larson's second book, which I suspected correctly he had never read - all the while conveying the impression that there was no rational basis for its claims - it was clearly in order to present some arguments based on it that I knew could hardly be rationally confuted.

      Lawrence thinks that nevertheless, attacking standard atomic and nuclear physics is akin to creationism and flat Earth, given vast experimental evidence the existence of which he assumes Larson must have been ignorant. In contrast, the reviewer of Larson's book for Discovery, The Magazine of Scientific Progress (London, Vol. XXIV No. 7, July 1963), accepted that Larson showed that most of the experimental evidence "is equally consistent with many other hypotheses besides the nuclear atom, and therefore no proof of any hypothesis." Arthur Adamson, later chair of the Department of Chemistry, University of Southern California, responding to an objection Isaac Asimov raised against Larson in Chemical and Engineering News, also accepted this point, while critical reviewers like Asimov, Fracastoro and Redin all explicitly conceded that Larson "shows himself to be well-informed on the current status of physics research" (in Redin's words, Chemical Engineering, July 22, 1963).

      Dave is evidently unfamiliar with even the existence of Larson's early papers, distributed to individual scientists, which have very extensive tabulations of comparisons between calculated and experimental values (e.g. of specific volumes under changing pressure at several specific fixed temperatures for numerous organic liquids), with minimal reference to his ideas about space and time (pretty much only as needed to describe the constant inward force involved in the solid and liquid initial pressures). These were his paper "The Compressibility of Solids" (1959) and his eleven papers on the liquid state (early 1960s).

      As he concludes the introductory paper in the latter series:

      "No satisfactory theoretical system for the calculation of the numerical values of these liquid properties has ever been developed heretofore, although a vast amount of effort has been devoted to the task. Many ingenious and useful mathematical expressions have been developed to facilitate interpolation and extrapolation of the experimental data but in most cases it has been impossible to attach any theoretical significance to these expressions. As one observer puts it, referring specifically to the property of volume, "The quantitative representation of the volumetric behavior of fluids over both gas and liquid regions has proven to be an unusually difficult problem". The nature of the obstacle that has stood in the way of a solution to this problem is revealed by the discussion in the foregoing paragraphs. It has been taken for granted that a liquid is a complex structure requiring complex mathematical expressions for accurate representation of its properties. According to the theory developed herein this concept is erroneous; the liquid aggregate is not a complex structure but a composite in which relatively simple structures coexist in definite proportions. This theory eliminates the need for any complex mathematical treatment and the subsequent papers in this series will show that in each case accurate results can be obtained by very simple mathematics."

    9. Steven Athearn wrote to me:

      >There wasn't any attempt to psychoanalyze Dave on my part.

      Steve, you made a long list of assertions about what I was thinking:
      >(1) you do not have in mind...
      >(2) you are not contending...
      >(3) you are not asserting...
      >(4) you are not denying...
      >(5) you are not disputing...

      all of which are nonsense and all of which pretended to get inside my head and tell me what I am or am not asserting or denying or whatever.

      You are not smart enough to get inside my head.

      Steve also wrote:
      >it was clearly in order to present some arguments based on it that I knew could hardly be rationally confuted.

      Steve, you presented no arguments. Not a one.

      You do not think coherently enough to formulate an argument.

      You are a presumptuously obnoxious person who thinks that you can impose upon those of us who actually know something about science to claim that we should satisfy your standards of proof and discussion.

      But you see we do not have enough respect for people like you or Larson to care.

      We have standards that you and Larson do not meet. You don't have to, of course: you can live your life in as crackpotty a way as you wish.

      But we do not have to play your games. Anyone who takes seriously what you have written here and who thinks Lawrence and I are treating you and Larson unfairly is a person whom I do not respect. I do not care what such people think.

      As I said above, there are over seven billion people on this planet. Millions of them, at least, are certifiably nuts. It is physically impossible for me or Lawrence (or even you!) to take seriously any reasonable fraction of the nutjobs on this planet.

      Whether we are right or wrong to dismiss nutjobs like Larson is really irrelevant: it is physically impossible to do otherwise.

      By the way, the quotes you give from Larson are extremely unimpressive.

      But, since you seem to have essentially zero knowledge of real STEM subjects, I cannot convince you of that.

      Fortunately, I do not have to.

  22. Several people have tried to submit comments to this thread which contain links to websites I don't recognize. I want to remind you all that I don't approve links except those to journals, the arXiv, or well-known newspapers. When in doubt, please consult the comment rules.

  23. Happily enough we live more than 6.000 kilometers away from the nearest Schwarzschildradius and are able to fly to the moon and beyond. But at this moment I need to have a valid reason to go abroad.

  24. Luis,
    phenomenologically I would refer spacetime creation to matter creation. »No mass without space, no space without mass«, so to speak.

    But: Spacetime cannot be experienced sensually and cannot be measured by apparatus. Spacetime is a mathematical construct. The four-dimensional »spacetime« was not developed by Albert Einstein but by the mathematician Hermann Minkowski (1864-1909). Minkowski gave his lecture "Space and Time" on September 21, 1908 in Cologne at the 80th meeting of the German Society of Natural Scientists and Doctors. In this lecture Minkowski introduces the mathematical notations with which Einstein's special theory of relativity can be expanded to general relativity.

    I think an interesting note came from Halton Arp*. According to Arp's statements, mass is generated locally in the interior of the galaxies and moves at almost the speed of light and is then slowed down with increasing mass due to the conservation of momentum. This mass escapes or orbits the galaxy.

    Source: page 231, Seeing Red, Redshifts, Cosmology and Academic Science 1997 by Halton Arp

    *Halton Arp (1927 – 2013) questioned one of the foundations of the Big Bang theory. He became known for the controversial theory that the redshift especially of quasars has a previously unknown non-cosmological cause and is therefore not suitable for determining cosmological distances.

  25. There is an issue here of whether spacetime exists. Does either space or time exist or are they simply illusions. It is an often heard rather trite comment that time does not exist. It has to be stated that physics is not primarily concerned on the existential status of objects; that is more of a philosophical or metaphysical issue. However, we can say that something is ontic if it has a state that remains outside of observation and if a community of observers agree on the same outcome of observations. This leads into the issue of quantum mechanics; where the wave function in ψ-epistemic interpretation has no reality and only measurement outcomes are.real. There are ψ-ontic interpretations that state otherwise, the biggest is the many worlds interpretation.

    I think there is a new sort of equivalence principle that states two inertial frames are such that entangled particles on either will remain in an entangled state. This ignores other quantum noise or thermal processes that induce decoherence. An N-tangle of N particles remains entangled if these are defined on N inertial frames. For N → ∞ this describes in effect a spatial manifold or vacuum with harmonic oscillators defined at each point. If spacetime symmetry or Poincare symmetry is violated there is then equivalently a loss of entanglement.

    This qualitative argument runs into some issues with holography. In particular it runs into questions with the holographic idea there are Planck units of area on horizons or holographic screens so that a black hole with a finite horizon area will have a finite number of quantum states. This is a manifestation of a duality between the locality of fields on say the AdS boundary and the nonlocality of the quantum gravitational states in the AdS bulk. The locality of quantum fields just means it can be localized to an arbitrarily small region or ideally a point. This does have some overlap with matter of quantum nonlocality, but this is largely ignored. In this situation this subtle question over nonlocality vs locality raises its head. With quantum gravitation or nonlocality of fields in spacetime we have difficulties in identifying what is meant by a quantum state at a point in spacetime or what is meant by an event. Even for a black hole a quantum system approaching a black hole is never seen to cross the event horizon, but only asymptotically approach it. With Hawking radiation, we have a curious issue of having quantum events observed near the horizon and at a point of space or spacetime removed from the black hole. Yet at the same time, based on my qualitative argument above, I can build up a spatial manifold with nonlocal entanglements in the N → ∞ limit.

    So is space, time or spacetime real or not? Based on my working idea of an equivalence principle surrounding the equivalence principle I would argue that spacetime either exists or not depending on how an observer chooses to measure things or maybe even what interpretation one puts on measurements. Again, physics does not explicitly tell us whether something exists or not, but what is meant by existence is the mental interpretation an analyst or observer imposes on nature.

    1. Sorry, I have been recently writing these strange tautological statements. I wrote:

      Based on my working idea of an equivalence principle surrounding the equivalence principle

      I meant to write:

      Based on my working idea of an equivalence principle surrounding quantum entanglement.

      I hope this is not some creeping onset of mental dysfunction. Then again if this is the start of my decline into the grave, based on what I see going on of late in the world it might not be that bad.


    2. I sometimes like to think of spacetime as a block with worldlines threading through it, and wonder about the status of events (worldline intersections) vs the worldlines of unobserved or not yet measured "objects". Tempting to consider only the events as real and everything in between those events as emergent, derived somehow from the information content of the events. But then I remember that any massive object, no matter how minute, appears, in the GR equations, to have a sort of continuous event structure-- "mass telling spacetime how to curve and spacetime telling mass how to move" is how it is usually stated. And this implies to me that even an electron has always a well-defined path and a well-defined location. But we know that can't be the case. I Have no idea how to solve that conundrum. Maybe I'm not grasping some essential point? That would not surprise me at all!

    3. To Rick Lubbock,

      Electrons aren't classical objects! there are electron phenomena and electron theories, but in quantum theory there are no "electrons"!

      By the way, one can develop relativistic quantum theories of events, trajectories, etc.!

    4. @ Rick Lubbock, the block world view is a very classical idea. It is not clear whether this works in a quantum setting. I suppose an MWI maven might say a block world splits into different amplitude paths. This gets really strange if spacetime itself is quantized, for there you really have an ambiguity in what is meant by a point or event.

      In Bohm's interpretation there is a definite path, called an active channel, and all the other paths are inactive paths. I would not however advise pursing a minute.

  26. Sehr geehrte Frau Hossenfelder!
    Was mich zu meiner Kommentar-Frage bei Youtube,
    ob man mit Inflation einem Black Hole entkommen
    könnte gebracht hat ist die Frage, die mich seit
    langem beschäftigt, nämlich, warum denn das Universum bei der Anfangsdichte kein Schwarzes Loch geblieben ist.
    Ich nehme an, daß Inflation der Grund ist.
    Mit freundlichen Grüßen
    Manfred Lehr

  27. I enjoyed reading this article and I also enjoyed reading some of the additional information brought up in comments, thank you. Now you got me wondering what Einstein himself thought about black holes so I'll be doing some research on that.

    1. Surprisingly, I read that Einstein didn't think they existed in reality. He thought they were a mathmattical artifact.

    2. He came to the conclusion that so they could form, matter should reach the speed of light, which means infinite pressure. So the star would bounce back before the black hole state.

    3. The word "so” in the first sentence threw me off, but if you're saying he knew GR predicted they could form then yes, he knew that. He just thought in reality other physical dynamics would prevent them from forming he published his thoughts on that in a 1939 paper in the Annals of Mathematics that I think you are referring too.

  28. 11-MAY-2020

    Re: Lawrence Crowell's comment on the reality of space and entanglement. . .

    Imagine you're standing in a place with absolutely no light
    or sound; there's air to breathe but it's still. You
    sense nothing around you. You shout and there's no
    echo; you assume no boundaries except the hard, flat
    surface supporting your mass. You sense a normal 1-G
    environment, but you have no idea how large a mass you're
    planted on.

    What meaning does space have in this absence of interaction?
    Whether it's a propagating photon, or an observer's echo while
    trekking in total darkness, something "kinetic" has to happen
    "dynamically" for space to be real. That appears to me to be
    the message of both the special and general theories of relativity.

    If stuff is kinetic (in motion), then space must be dynamically
    "real." General relativity seems to me to take things to the
    reciprocal limit and says, if space is kinetic then stuff is
    dynamically real.

    Potential-driven inflation describes space as the prequel to
    hot big-bang and stuff (mass-energy). The potential must be real
    so the space must be real, since we generally agree that mass-energy
    is real.

    This perhaps is where one begins to see "entanglement" AS spacetime.
    Perhaps an entanglement between motion and manifold is sensible.

    mj horn

    1. If you measure the light coming from a distant burstar different wavelengths of light arrive at the same time. So space appears to be very smooth and continuous with that sort of measurement. If I could perform a Heisenberg microscope experiment near the horizon of a black hole then Hawking radiation becomes a torrent, and Planck units of horizon area become a violent froth of quantum states or micro-black holes. An extremely accelerated frame near the horizon would serve this role. Planck scale physics is where spacetime dissolves away, and in principle at least this should have fingerprints on the CMB.

      One might think of it as what happens if there is a spacetime with only one photon. What is meant by the speed of light? The very distant universe will locally becomes exactly this and in fact those photons will be redshifted beyond the horizon scale.

  29. Sabine,
    I think that Louis, Antooneo, Pascal and I could create a small union for the defense of logic.

    RG makes two very different predictions, one for the distant observer DO for which there is a horizon which prevents that nothing falls into the BH and even that a BH is formed. Another for the CO observer "close to the horizon defined by DO". OC crosses without difficulty this horizon_DO because for him no "near horizon" exists: the horizon recedes as one approaches it. The two propositions cannot be true together.

    In a theory a paradox often consists of a double prediction produced by change of point of view, and only one of which is in accordance with experience. Suffice it to consider the correct prediction and ignore the inadequate is to ignore these paradoxes. If having considered them, we cannot eliminate them correctly, then simple logic forces us to dismiss the theory and / or interpretation that generates them. In this sense, Popper's refutability criterion is necessary but not sufficient. Thus the mathematical consistency of formalisms and the experimental adequacy are not enough. It is also necessary to eliminate theories (formalism provided with an interpretation) which double their unrefuted predictions by manifestly false predictions. Consistency takes precedence over any other criterion, including that of Occam's razor (maximum simplicity, which was not included in Popper's criterion either).

    1. Jean-Paul, I have no idea what you are even talking about, you lost me at "RG". Having said that, if you believe that there is something mathematically inconsistent about black holes I recommend you actually look at the math because then you would find there is nothing inconsistent about it.

    2. Dear Sabine,
      Sorry for RG instead of GR. I try to explain to maths-minded people as you why a lot of more or less ordinary people as me have problem with BH's horizon whatever mathematical consistency of GR. For us its is just a problem of logical consistency. And as physics are not just mathematics, but theory about nature, GR have also to say coherents things about nature.
      We remember from Galileo that "The Book of Nature is written in mathematical language" (although we can doubt that this is the only way to access reality, the second qualities not being reducible to the first qualities). But in his works the logical arguments, their application to experience, thought experiences, take up much more space than calculations. If he did not specify that, even before being formalized in mathematical language, all that this Book says of the Real in a rationally articulated way respects first the cannons of logic, it is that it went without saying for him and his contemporaries.

      I think we can discuss the problem without without knowing and even less mastering GR mathematics. A good popularization is enough to understand.
      If you let beside QM , evaporation effect, information's paradox and all of that, and you look just at GR as a self-coherent theory, horizon would NEVER be crossed for the distant observer DO, because for him "time" STOP on horizon. You cannot merely say that DO has a very stretched vision of the story going on near the horizon. And that is of course a false prediction of GR because of free fall near the "horizon" (the prediction of GR which is true).
      For explaining that, one usually say that "time" is not the same for those different observers. But in GR all observer are equally "right". there is no true or real time opposite to false or apparent time. All clocks have locally the same rythme. The Einstein effect on time is just produce by the differences of positions along curved space-time. More exactly it is the effect of a biased point of view of observer located in x upon what append in y. Biased by the curvature of geosedic between x and y. Just as in SR slowing of clock with relative movement is a biais due to relative simultaneity, a sort of parallax effect. With that difference : in SR, rythme of moving clock never completely stop.
      But following this view, neither Einstein effect nor horizon does'nt actually exist. Because when you approach him, you go from x to x' with x'y < xy ie a less curvature's difference : time in y go faster, and the horizon moves back to y'. He is relative for x.
      However in space-time of GR the curvature between x and y is objective and correspond to the inclination of the light-cone beyond which, for x, the space becomes time and vice versa at y. "Time at y stop really for x". Which in nonsense if you merely thing to the universal time of our universe's history. There is just one univers with its unique Time (and his also unique expanding space ).

    3. @Sabine
      And the fact there is in GR no limit for inclinaison of light-cones implies that mathematical consistance of GR can always be contested. From this fact notably follow the horrific possibility of closed time-lines (see Gödel's universe (*)). And if the "Hadamard catastrophe "in Schwarzschild metric ie the division by zero at RS = 2GM/c2 can be eliminated by Kruskal transformation, it remain that those new coordinate which mixe space and time very skillfully have visibly no physical meaning (see Wikipedia). IMO it swept the problem under the rug.
      (*) Gödel loved these consequence which argued for his idealistic conception of time.

    4. Closed time-lines are not mathematically inconsistent, they are unphysical. Coordinates are not observables, so they never have a "physical" meaning. I think you are very confused and you should know better than to get your wisdom from Wikipedia.

    5. @Sabine
      (1) Ok, i mistake : closed-time lines are only unphysical, they refer nothing real. But why is it not a problem, if Gödel's model is conceivable ? Rather than say that close-time lines forbid Gödel's model, we would have to view this model as a proof by the absurd of an inconsistency in the physical theory which allows it.

      (2) If the aim of singularity-free Kruskal coordinates is to avoid a division by zero which is for GR both a mathematical inconstancy and a limitation physically meaningless, why those coordinates would not have to be physically meaninfull , i mean have a physical justification in GR ? If we are free to create any coordinate without physical justification, the Kruskal solution look like a conjuring trick where the rabbit disappears in the hat.
      Perhaps i ignore those physical (not mathematical) justification, but i read other sources than Wikipedia. So even Jean Eisenstaedt, our french historian of GR, notes that only part of the Kruskal diagram has received a physical interpretation which he believes is convincing. (Einstein and general relativity, p 470). And p 474: "(..) as regards the other zones and in particular the white hole, one cannot deny the speculative character of these topological propositions which for the time being , find in physical reality no justification and which do not predict any phenomenon which it can be envisaged to verify observably ".

    6. Jean-Paul,

      As I already said, coordinates are not observables, they never have a physical meaning, they do not need any physical justification. They are merely handy devices that we use to label observables. The use of the Kruskal Szekeres coordinates is that they provably cover the full analytic continuation of the Schwarzschild solution. Ie, they cover the whole-spacetime and not just parts of it. I frankly don't care what some French historian thinks about General Relativity.

    7. Jean-Paul wrote:
      >f you let beside QM , evaporation effect, information's paradox and all of that, and you look just at GR as a self-coherent theory, horizon would NEVER be crossed for the distant observer DO, because for him "time" STOP on horizon. You cannot merely say that DO has a very stretched vision of the story going on near the horizon.

      Jean-Paul, are you aware that lots of physicists have been debating just this point for a number of years?

      For what it is worth, my own "gut feeling" is that you are probably right. Alas, my gut feeling is often wrong, and people who are at least as smart as me have the opposite gut feeling.

      Jean-Paul also wrote:
      >I think we can discuss the problem without without knowing and even less mastering GR mathematics. A good popularization is enough to understand.

      Unfortunately, the level of debate on this among physicists suggests that you do need to know the mathematical details of GR, and even those of who do know all that still argue the point.

      if you want to really dig into the physics literature on this, let us know, and I and other can point you to papers on the arXiv that debate the point.

      Please note: I am not saying you are stupid for raising the point nor am I trying to discourage you from thinking about it.

      I am just telling you that you will find that the details are far messier than you might expect if you choose to really dig into it.


    8. Sabine, if you want to compute the infalling time in a Schwarzchild black hole, you do need to know what is the meaning of the variable "t" in the Schwarzchild solution. It is usually taken to be the time seen by a distant observer, and this assumption requires a justification.

    9. Pascal,

      If you think about your statement for a moment, it means that you use "t" as a device to connect the measurements of two observers. But "t" itself has no meaning and is not measurable. You can do this calculation in whatever coordinate system you want to. It's just that in some coordinates the calculation will be simpler than in others.

      Look, this isn't exactly a breakthrough insight, so I don't know why we even have to talk about this. Coordinates are labels that you put on space-time. They have no fundamental physical meaning. That's the reason Einstein even came up with relativity!

    10. @PhysicistDave
      I know that in GR the issue of black holes and the horizon has an old and long history, but I wonder if we understood each other. My readings have shown me that the following two statements (1) and (2) are necessary consequences in GR, and if this point is not widely accepted today and is still widely debated, tell me and thank you for arxiv references.
      (1) For a very distant observer DO ie very far from the gravitational potential of BH, time stops on the horizon. In its frame of reference, the time taken to cross the horizon tends to infinity. Any light source which tends towards the horizon is perceived by DO as affected by a gravitational redshift which tends towards infinity.
      (2) For the observer located outside the BH but near its horizon, the horizon is as if it did not exist: everything falls freely in the BH.
      Obviously, one of those two points of view is false, since the two observers belong to the same world (we are not in the MWI of QM).

      Moreover, the concept of horizon comes exclusively from GR and if it poses problems, their solution should not be sought elsewhere than in GR if we consider that it is a self-coherent theory. In particular, the objective reality of the horizon must be well established before looking for whether or not it produces quantum effects. Must we remember that all quantum physics in the vicinity of horizons remains speculative to date ?

    11. Jean-Paul, as the blog-essay is titled "A Brief History of Black Holes," your invocation of historian Jean Eisenstaedt is relevant (as Jean Eisenstaedt wrote the book "The Curious History of Relativity."). One issue with a discussion of these topics is the combination of words used in conjunction: "history" and "black holes." There is no such thing as "a" history. Also, "black holes," is a topic of a larger physical theory (general relativity) which takes years to understand. Jean Eisenstaedt has also published in Physical Review D and American Journal of Physics. Finally, I kindly suggest to all a reading of Einstein's lovely 1961 book, Relativity. There, Einstein explains beautifully "why" coordinates "in themselves...have no significance." (pp. 94).

    12. @Gary
      It is the first popularization book of modern physics that I read a long time ago "not to die an idiot" and that fascinated and intrigued me enough that I did not stop there. But as much the presentation on the RS seemed to me as bright at first glance, as much the presentation on the GR seemed problematic to me, in particular this chapter 23 entitled "the behavior of the clocks and the rules of measurement on a body of reference in rotation".
      It seems to me that when our opinion is based on logical arguments, it is more than a "gut feeling", and that a rational discussion must be possible. For example if in a coordinate system there happens A, and in another it happens non-A, we cannot say that these systems are arbitrary conventions and as such equivalent for the description of the events of nature. For arxiv references I am requesting, thanks.

    13. Jean-Paul, that is an astute observation (that is, that chapter 23 would seem problematic). While Einstein's book is intended for the layman, his chapter 23 is subtle. John Baez happens to have a nice webpage devoted to it ( Baez writes: "Little wonder that Einstein was 'tormented' by the problem of 'just what coordinates are actually supposed to mean in physics' once they lose their direct physical significance." On his webpage, Baez walks the reader through the subtleties of "The Rotating Disk in Relativity."

  30. Regarding miles, I will always hold a warm place in my heart for expressing c in the most alliterative of English speed units, furlongs per fortnight.

  31. Couple of comments..

    1) I don't think a simple horizon qualifies as a black hole. Indeed many people were aware of the existence of horizons and did not worry about it. So Schwarzschild solution does not qualify as 1st black hole work.

    2) The first genuine work on black holes was by Oppenheimer and H. Snyder, "On Continued Gravitational Contraction", Phys. Rev. 56, p. 455 (9/1/1939)

    3) David Finkelstein deserves more than a mention for a coordinate system shared with Eddington. He was my advisor and we talked a lot about his early work. In fact his paper "Past Future Asymmetry of the Gravitational Field of a Point Particle" Phys. Rev. 110, p. 965 (5/15/1958) was the first paper of the modern age to actually take the idea serious as realizable physics, and should be thought of as the first paper on the black hole as such. It was this recognition of past-future distinction that made it physics and not just a coordinate singularity. The term black hole was coined by J.A. Wheeler. Finkelstein called it a one-way membrane. Finkelstein's actual goal was a topological understanding of spin, something he worked toward his entire life. IMO Finkelstein is the father of the black hole, much as Maxwell is the father of the light wave.


    1. to drl,

      David Finkelstein was also one of my mentors.

      In unified field theories such as the Einstein-Yang-Mills-Dirac-Higgs system there are regions surrounded by event horizons without any singularities in their center!

  32. This comment has been removed by the author.

    1. Sie können Ihre eigenen Kommentare löschen.

    2. Warum bringen Sie sie dann?

    3. Was bring ich? Kapier ich nich, sorry.

    4. This comment has been removed by the author.

    5. Hallo "magnetpendel"

      Der "Löschen" Button ist da, damit Sie Ihre Kommentare löschen können in dem unwahrscheinlichen Fall dass Ihnen auffällt, dass sie kompletten Quatsch geschrieben haben. Natürlich kann ich Ihre Kommentare auch löschen, oder erst gar nicht veröffentlichen. Hab ich aber nicht, weil ich ein höflicher Mensch bin. Zumindest manchmal. Ums nochmal klar und deutlich zu sagen. Jeder hat einen solchen Button für alle seiner seine eigenen Kommentare, bis auf mich, ich hab den Button für alle Kommentare. Ich schlage jetzt mal vor, dass Sie ihren aggressiven Quatsch löschen und mich dann in Ruhe lassen, danke.

    6. Ich hab´ mehrfach auf delete funktioniert nicht.

    7. Mit Verzö ich wechsle jetzt zu Andreas Müller: Universum als Schwarzes Loch.

  33. One of the strangest and most famous features of black holes is that once an observer crosses the horizon, there is no way for him to communicate with the outside. Not even by sending a light signal. But wait. The outside world feels the effect of the gravitational field of the BH. This effect definitely crosses the horizon. So, would it not be possible, at least in principle, for an observer to communicate by modifying the gravitational field? Imagine for instance an observer bringing with him a hypothetical machine that can create gravitational waves. Or even more simply, since the observer has a positive mass it can change the overall gravitational field of the BH by moving inside the horizon. A tiny effect for sure, but in principle it should be detectable outside the horizon. Is there something wrong with this argument?

    1. I've always wondered this myself (although my approach was to take a tiny black hole with you and move it around to signal), but then I assumed the time dilation experienced as you go beyond the event horizon would effectively stop you. But the huge caveat is I am definitely not a physcist! I'd be fascinated to know the informed answer, too.

    2. “but then I assumed the time dilation experienced as you go beyond the event horizon would effectively stop you.” It would be a bad excuse to avoid this discussion. After all, when we are told that communication with the outside is impossible, even with light signals, this issue is usually not brought up. So why bring it up now? Moreover, as a thought experiment we are free to imagine a system having as initial condition a traveller inside the horizon. And don’t ask how the traveller got there : as Sabine likes to point out on this blog, initial conditions do not have to be explained!

    3. @Pascal
      Yes, there are multiple logical problems that arise from the paradoxal both relative and objective status of horizons. For the observer which cross the "objective horizon" of a very big mass without notify any effect, there is no physical, no dynamical reason to prevent him going back because the three dimension of space continuously exist in her neighborough.

    4. > So, would it not be possible, at least in principle, for an observer to communicate by modifying the gravitational field?

      Nope. Not without FTL travel.

      > Or even more simply, since the observer has a positive mass it can change the overall gravitational field of the BH by moving inside the horizon. A tiny effect for sure, but in principle it should be detectable outside the horizon. Is there something wrong with this argument?

      Yes, the argument is completely wrong. You cannot change the static gravitational field, and any gravitational radiation would move at the speed of light and so unable to escape.

    5. Sergei wrote: "any gravitational radiation would move at the speed of light and so unable to escape."
      You’re saying that the effect of the traveller’s mass cannot be felt outside of the horizon. If that’s the case, can you explain why the effect of the mass of the collapsed star, which is also trapped inside the horizon, *can* be felt outside ?
      By the way, the argument that light cannot escape does not apply in this situation (or at least not without a modification). The reason is that in the traveller’s case, you need to allow a small perturbation of the Schwarzchild solution (due to the traveller’s mass) in the interior of the horizon. Then the question is whether this perturbation propagates outside.

    6. This comment has been removed by the author.

    7. > If that’s the case, can you explain why the effect of the mass of the collapsed star, which is also trapped inside the horizon, *can* be felt outside ?

      Yes, yes, I can. It's basic general relativity. But the important part is that no changes inside the horizon can propagate outside. You can conclude as much already after noticing that there is no causal connection from inside to outside (the reverse is not true), as seen by the apparent time dilation to infinity at the horizon, as seen from the outside.

    8. Sergei, you're saying that the effect of the traveller's mass inside the horizon cannot be felt outside. But the sad fate of the traveller is that he will end up in the central singularity, thereby increasing the mass of the black hole. I suppose that the Schwarzchild solution should then be replaced by a new one, taking into account the increase in mass of the black hole. Won't this effect be felt outside?

    9. Pascal, this is actually not a bad question. The answer is that the mass of the central spherical object as seen from the outside includes anything that fall on it, even before it is done falling. With black holes, due to the gravitational time dilation, the infalling matter close to the horizon would be "seen" as smeared all over the horizon, just as it fades into the infrared, effectively becoming a sphere. There is a short ring-down period when some of the infalling mass is carried away as gravitational radiation, but most of the mass is added to that of the original black hole. There is no further "communication" possible without faster than light travel. Again, remember that from the outside the infalling matter never crosses the horizon, only fades into it, so there is no meaningful way in which one can talk about gravitational or any other effects that propagate outward from inside the horizon. This is not an easy concept to grasp, and requires some upper undergrad math skills to fully appreciate, so your confusion is understandable and you are certainly not unique in that.

  34. Sabine,

    Is there any sense in which GR predicts black holes in a stronger sense than NG did?

    1. Hi David,

      Yes... That's because in Newtonian Gravity you only get some kind of black hole if you assume the speed of light is finite, yet Newtonian Gravity itself doesn't tell you that it has to be finite. In General Relativity, on the other hand, you know that the speed of light has to be finite.

    2. Thanks Sabine,

      However I suppose I should have said NG+SR - SR is enough to produce a finite speed of light, I think.

    3. Hi David,

      But NG is incompatible with SR. So I am not sure what you are saying there.

  35. This comment has been removed by the author.

    1. Please avoid submitting comments that are not related to the blogpost, thank you.

  36. An underlying hypothesis of the hypothesis of the existence of "black holes" had to be the hypothesis, that matter is arbitrarily compressible. Isn't it like that?

    So, do we have any hint that this hypothesis is sensible?

    It should be completely clear, that extented matter could not concentrated in a single point. Because a single point consists of nothing more than nothing.

    So, were is the limit of the compressibility of matter? If we have no well founded answer to this question, why should we hold on to the believe of the existence of "black holes"? Only because we like "science fiction"?

    1. If you read Ellis and Hawking Large Scale Structure of space-time you can find an answer. If matter is compressed to a volume bounded by the Schwarzschild radius it is not possible for it to counter further compression. To do so would require some interaction or process that propagates information faster than light.

      Consider a galaxy’s worth of stars, say a trillion of them. Think of the sun as an average star, where really it is larger. The sun has a radius of about .5million km. This is a volume about 10^{17}km^3. A trillion stars has a collective Schwarzschild radius of around 3×10^{12}km. The volume is then about 3×10^{37}km^3. This means it is in principle possible for this many stars to fall below their mutual Schwarzschild radius with no collisions or pressure resisting this infall. Once they fall below that horizon length there is no possible physical process that can result the implosion to the singularity.

      More realistically the implosion of a stellar core below its Schwarzschild radius is possible, It requires a bit of work with equations of state and so forth. The outcome is more or less that a stellar core or more than about 3.4 solar masses can be compressed into a black hole. To learn more requires effort on your part.

    2. Lawrence, you can't be saying that a stellar core can be literally compressed to a point, are you? Such conclusions are drawn within the framework of GR, but we know for a fact that it is not the Theory of Everything. Quantum effects have to kick in at some stage during the collapse, and we don't have an adequate theory to describe what happens next.

    3. The free falling in relation to repulsive matter interactions seems to reveal deeply fundamental boundary conditions for the whole space concept as we can model it as connected four-dimensional continuum.

      In this contex the broken connection by an event horizon and any singularity are obvious apparent artefacts - but of course the length of a world line might be very long, but still finite.

    4. To make things really strange, the singularity in a black hole is not a point. The Schwarzschild singularity is really an entire spatial manifold, it is not exactly a point. The tidal force due to Weyl curvature is such that inside the horizon a Weyl curvature that pulls things apart rather than crushing them down takes over. Things get even more odd with rotating Kerr metric black holes.

    5. Lawrence, what model of black hole are you referring to? In the Schwarzschild solution, the metric is singular only at r=0, and that's a point. I am not counting the horizon, which is an artificial singularity.

    6. It seems the most 'compressible thing' one could hope for is a Bose-Einsten condensate of photons.

    7. Not some kind of condensator I presume.

    8. to Lawrence Crowell7:53 AM, May 13, 2020

      in your example, what is the average distance between the ideal middle points of a pair of adjacent particles which are possible under such conditions? And what kind of forces works between them dominantly? Or is the answer for such a question beyond known physics?

    9. Werisidas: You should be able to make an estimate on your own. It is easy, but the answer would be about 7×10^6km, or about 10 times the diameter of a star. Also this is all Euclidean, for the curvature of spacetime is not huge.

    10. Lawrence Crowell: "A trillion stars has a collective Schwarzschild radius of around 3×10^{12}km."

      And the whole cosmos? What might be its Schwarzschild radius? May we live in a black hole? I would suggest that the average distance between particles in the whole cosmos for forming a Schwarzschild radius would be greater than in your example? Right? Or let's take just the visible cosmos. Can we estimate its Schwarzschild radius? Has someone done yet?

  37. Pascal,

    You don't need quantum theory! The classical field theory which is the super-classical limit of the standard model in a curved space-time already has singularity free solutions.

  38. 13-MAY-2020

    @Lawrence Crowell

    For a space (all space not just a subspace) with just a single
    photon, there's no intrinsic mass. It's reduced to the kinetic
    energy. So I see a spacetime manifold conforming to the Gaussian
    envelope of the photon. Space and the photon are the same. A
    shortening wavelength is collapse; otherwise it just expands to

    The speed of light and the shape of space must have something in

    Seems a bit of a cartoon.

    Sorry if this ends up at the end of the
    page rather than attached to the original
    comment. Probably my browser.

    mj horn

  39. Having had my first encounter with a radical rethinking of the black hole paradigm today known as Dark Energy Stars, in which spherical black holes are replaced by donut shaped structures lacking singularities, I’m perplexed to not be able to find any substantial criticisms of such a concept. This idea had its genesis in a 2000 paper by a quartet of authors, posted on the arXiv, with the title – “Quantum Phase Transitions and the Breakdown of Classical General Relativity”. I believe it’s important to have a balanced perspective on new ideas, but strangely, this idea seems to be in a kind of limbo within the scientific community, basically ignored by the great majority, despite the fact that it eliminates singularities and the vexing information loss problem inherent in standard black holes.

    Is it possible that the scientific community, having invested an enormous amount of intellectual capital in the black hole paradigm, regards any alternative to be heresy? But, any controversy may already be mute, with the recent imaging of the 6.5 billion solar mass black hole in the galaxy M87. George Chapline, one of the theory developers, was predicting a “ring of fire” to show up in the image of a black hole. Curiously enough the image of the black hole in M87 does look like a ring of fire. However, as there appears to be no upheaval in the astronomical community about the M87 image, it must conform to the expectation of what a standard black hole should look like.

    1. > spherical black holes are replaced by donut shaped structures lacking singularities

      Has been looked at. Not possible. All black holes horizons are topological spheres. Google "topological censorship" for details.

      > Is it possible that the scientific community, having invested an enormous amount of intellectual capital in the black hole paradigm, regards any alternative to be heresy?

      No. You do not appreciate how competitive the field is. Any half-baked idea gets published. Any wild speculation gets published. All low hanging fruit has been plucked. Dark energy stars are bullshit, the curvature near horizon is not nearly strong enough to result in some quantum phase transition.

      Yes, there is a huge tension between general relativity and quantum mechanics, but any potential breakthrough will not be as pedestrian as that.

      M87 image is of the accretion disk, not of the horizon, you are confused.

    2. David, are you suggesting that social processes in science do not always lead to an optimal outcome? I wish there was a science blog to explore this idea in more detail. :-)

    3. Sergei:

      "Any wild speculation gets published."

      It surprises me that this paper would show up on arXiv, which I thought was something of a gold standard for publication, unlike viXra, where I assume all the junk science, or crackpot ideas, reside.

      "the curvature near horizon is not nearly strong enough to result in some quantum phase transition."

      I thought about that too, but couldn't imagine that in such a sophisticated paper that the authors could have overlooked such a trivial thing. So I was willing to give them the benefit of the doubt until I completely read the paper, and tried to understand it.

      "M87 image is of the accretion disk, not of the horizon, you are confused."

      I worded my comment badly, making it sound like the black hole itself was imaged. From my earliest pop-sci reading, I've long been aware that nothing can escape the event horizon, except Hawking radiation, which is presumably too faint to be detected. I wasn't confused.


      On discovering the Dark Energy Star theory alternative to black holes yesterday, it made me wonder if a community consensus was being challenged, just as MOND is championed by a minority of astrophysicists against the vast majority who support the LCDM, or concordance, model of cosmology. But from what Sergei said it’s apparent this Dark Energy Star theory is a non-starter.

  40. When someone sees a drop of water from rain, did the drop always exist?

  41. I have a question about the Kruskal-Szekeres (KS) coordinates. Any other solution seems to be rejected outright by the experts because the KS coordinates are *the* maximal analytic continuation of the Schwarzchild solution. So there can’t be another sensible solution than KS, and that’s the end of the story. Here there seems to be a uniqueness theorem lurking in the background. Does anyone on this blog know a reference for such a theorem, and in particular what are the precise hypotheses?
    Doing some research on the internet, I stumbled on a post by string theorist Lubos Motl on physics stackexchange, where he writes: “He may also find out that there are several maximal extensions although I am not sure and I cannot mention any well-known examples now.” This is about maximal analytic extensions in general, not specifically about the Schwarzchild solution. The wikipedia article on KS mentions a uniqueness theorem for the Schwarzchild solution in the book “100 years of relativity - spacetime structure: Einstein and beyond”. In chapter 4 they indeed give a uniqueness theorem, without giving a proof or reference. The same chapter mentions that “space–time denotes a smooth, paracompact, connected, orientable and time– orientable Lorentzian manifold”, so these hypotheses presumably apply to the uniqueness theorem (?). I for one have no intuition why a spacetime with a black hole in it should be orientable.

  42. Pascal, the 1960 journal article by Kruskal is informative: Maximal Extension of Schwarzschild Metric (Physical Review, Volume 119, pp. 1743-1745). Another useful reference is "Maximal Analytic Extension of the Kerr Metric," (Journal of Mathematical Physics, Volume 8, Number 2, pp. 265-281). These articles are reprinted in a resource booklet: Black Holes Selected Reprints (1982, American Association of Physics Teachers). Furthermore, Robert Wald has an excellent, pedagogic, discussion of much of this material in his chapter six (1984, General relativity, especially: Kruskal Extension, pp. 148-157)

    1. Thank you for all the references! In the meantime, the author of the chapter cited on wikipedia pointed me to his arxiv paper (see Theorem 4.5). Looks like very serious, non-cranky math so I am sure that Sabine won't mind the link.

  43. If anyone is still following this thread, I have some news about the Krsukal-Szekeres extension and its uniqueness property (or lack thereof). I discussed this issue with colleagues who are actual experts of the maths of general relativity. As it turns out, the uniqueness property mentioned on wikipedia holds only for metrics with signature (-++++) everywhere. In particular, the metric is not allowed to be degenerate at any point of the spacetime. But imho, for the study of black holes it is quite natural to allow a degenerate signature at the horizon. In this case, there is at least one other extension of the exterior Schwarzchild solution than Kruskal-Szekeres.

  44. Sabine, From my A&A view I've long struggled with the increasing gap between theory and observation. Study of AGN & quasar jets since Martin Rees etc in the '70s shws accretion, 'counter wound' toroidal acceleration, and the opposing helical outflows not only ejecting all accreted matter (most re-ionised) but actually resulting in MORE mass than in the accreted galaxy disc! (so likely explaining the 'mass growth' starting from each new open spiral). So does the mathematical "singularity" really exist? And if all matter escapes, is it not the OPPOSITE to a 'black hole'?
    A neighbouring fqXi essay to yours analysis the potential as the 'soliton' or atomic 'Mexican Hat' profile, at each pole. Is this observed process not also viable mathematically?

  45. Canticle wrote: "does the mathematical "singularity" really exist?"
    It certainly does not exist in Einstein-Rosen bridges. This is the alternative to Kruskal-Szekeres mentioned in my previous
    message. When a star collapses, the excess matter could be ejected through an Einstein-Rosen bridge instead of being crushed in a singularity. This is basically the mechanism proposed by Petit and D'Agostini (they use a different parameterization of their solution than Einstein and Rosen, but as far as I understand their solutions are really equivalent).

    1. The singularities of ordinary General Relativity can be avoided by considering the (mathematically well defined) Einstein-Yang-Mills-Dirac-Higgs System which is (heuristically) the super-classical limit of the (not mathematically well-defined) Standard Model. This system has complete solutions without singularities, solitons, and a Cyclic Universe solution. (The system has negative energy density; hence doesn't satisfy the positivity conditions in the Penrose-Hawking Singularity Theorems.) The E-Y-M-D-H equations provide an alternative approach to a Cyclic Universe which Penrose has recently been advocating. They also imply that the massive compact objects now classified as Black Holes are actually Quark Stars, possibly with event horizons, but without singularities. A Super version of the above-including super-neutrinos-might be needed to explain Dark Matter. The E-Y-M-D-H is also a totally geometricized theory as a non-commutative geometry; the charge e and the mass m of the electron are geometric invariants of the non-commutative geometry analogous to pi. Unfortunately, there are quantum phenomena, such as EPR, for which this beautiful theory doesn’t make adequate predictions.

  46. I’ll add that there is a common misconception that an Einstein-Rosen bridge somehow corresponds to the two exterior regions (I and III) in Kruskal-Szeres (KS).
    See for instance the wikipedia page on wormholes. This is horribly wrong, and someone should fix this! One obvious reason why this is wrong is that geodesics cannot go from region I to III in KS, but only from I to II (the black hole interior). Another more subtle reason is that the Einstein-Rosen metric is degenerate, and KS is not. If one of the experts out there can prove me wrong, please raise your hands!

    1. I don't know what you are commenting on, but the Einstein-Rosen bridge is a non-transversable wormhole, so I don't know what your point is with the geodesics.

    2. Sabine, the "non traversable" thing means that the bridge is supposed to collapse if one tries to send matter through it. I don't know if that is right or wrong, but this not relevant here : there is a well-defined metric (you can look up in the 1935 article by Einstein and Rosen), and if there is a metric there are geodesics.

    3. Also, if you want to see what I am commenting on, you can look up the wikipedia page on wormholes. Here is what they say on the way Einstein-Rosen relates to Kruskal-Szekeres:

      In this spacetime [Kruskal-Szekeres], it is possible to come up with coordinate systems such that if a hypersurface of constant time (a set of points that all have the same time coordinate, such that every point on the surface has a space-like separation, giving what is called a 'space-like surface') is picked and an "embedding diagram" drawn depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an "Einstein–Rosen bridge".

    4. I don't have the time to read Wikipedia. I am trying to understand why you are commenting here and it's not clear to me. Yes, if there is a metric, there are geodesics, but that doesn't mean these geodesics will connect any two points. These metrics are all geodesically incomplete, so of course they will not. Could you please be clearer what the connection of your comment is to my video.

    5. Sabine, I am commenting here to try and dispel common misconceptions about black holes. I think this is very relevant to your video. Actually, your answer to my comment contains a related misconception: Einstein-Rosen is geodesically complete! Geometrically, this solution consists of two exterior Schwarwschild solutions glued together at the horizon.
      In that spacetime, a geodesic going through the horizon will consist of two portions of geodesics, one in each of the two exterior Schwarwschild solutions.

    6. You are right about the incompleteness, sorry about that, but this wasn't my point. I merely said just because there is a metric on a space doesn't mean there's a way to travel from any point to an other point. In any case, I don't see how that's relevant to my video which is about the history of black holes, so excuse me but good bye.

  47. Well, maybe you will do a video on wormholes someday and we'll discuss it again then.



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