The total energy of two bodies in gravitational interaction must be
(m1 + m2) c^2 - G m1 m2 / r ,
where r is the distance between them. When r is G/c^2 times the reduced mass, the total energy and hence the total mass vanish! It is the Schwarzschild radius, so a black hole may form. Does it necessarily have zero mass? Is this not contradictory?
Don't forget that as seen by distant observers, matter in a collapsing star is never seen crossing the horizon. So the proper (nonrelativistic) calculation involves computing the gravitational potential energy of a spherical shell with the given mass and the Schwarzschild radius as its radius. The gravitational potential energy U of a shell of mass M and radius R is given by is given by U = −3GM2/5R. If you substitute the Schwarzschild radius, R = 2GM/c2, you get U = −3c2M/10. So only about 30% of the collapsing object's mass is released as gravitational energy.
To V. T. Toth: Thanks for the answer. At any rate, I can argue that the total energy vanishes, in general. The argument is connected with Einstein's 1939 argument against black holes, which actually states that the particles forming a black hole must reach the speed of light. As Thorne says, this is not really an argument against black holes, but it is an argument for black holes to be massless.
Not only they must not-they shouldn't: if a black hole did have zero mass, it would be a naked singularity. In general relativity this runs afoul of the cosmic censorship principle-which, while it's not been proved, in general, it has been proved in certain cases. Under certain, different, circumstances quantum effects can be shown to give rise to a horizon, about certain, clssically naked, singularities, thereby ``cloaking'' them-cf. http://arxiv.org/abs/hep-th/0410076.
Incidentally, the expression displayed is that of the potential energy. The fact that it vanishes, in a non-relativistic context, doesn't mean anything particular.That the potential energy vanishes does not mean that the mass vanishes, in this setting. The reason is that the constant, in Newtonian mechanics, doesn't have any physical meaning. The fact that it's unbounded from below is more problematic and indicates that additional terms-e.g. the angular momentum contribution-are necessary to stabilize the classical ground state.
Toth cited the 'Schwarzschild radius'. However, in Schwarzschild spacetime this quantity is neither the radius nor even a distance, and so it is not the radius of anything therein. thus the 'Schwarzschild radius' is meaningless.
In General Relativity gravity is not a force, because it is spacetime curvature. The cosmologists maintain that their black holes have a (non-zero) finite mass. They claim that this mass is concentrated at their black hole 'singularity' where density is infinite and spacetime curvature is infinite. Thus, they claim that a finite mass produces infinite gravity. Marvellous!
They also claim that their black holes have and do not have an escape velocity simultaneously at the same place. However, this is impossible.
Einstein and his followers assert that material sources for his 'gravitational field' are both present and absent by the very same mathematical constraint! That too is impossible.
Nobody claims that the mass of a black hole is concentrated at the singularity, in particular since it's false. Nor does anyone claim anything about infinite gravity. The singularity, indeed, is hidden by the event horizon.While classical gravity can make statements about the existence of the singularity, since the physical processes that could probe it imply quantum effects, what meaning could be given to ``infinite gravity'' isn't, for the moment, known-just that this expression is a measure of ignorance. In certain special cases (cf. previous message above), how quantum effects do give rise to an event horizon, hiding the singularity, can be calculated in a consistent way.
Black holes don't have an escape velocity-that's a Newtonian notion, that doesn't make sense here.
General relativity is science, not religion, so the notion of ``followers'' is meaningless. A black hole is formed by gravitational collapse of matter, as calculated already by Oppenheimer and collaborators in the 1930s and a black hole has a definite mass, as can be found by standard calculations in any course on general relativity. The spacetime of a black hole isn't the vacuum and motion of matter in it can be calculated-that's how the existence of astrophysical black holes is deduced. In particular, that a black hole spacetime is characterized by a length scale is a fact.This length scale can be identified with the black hole radius.What this means is that there exists a region of spacetime that's causally disconnected, within the framework of general relativity, from infinity.
I don't really understand what this equation is supposed to represent.
When you say,
"The total energy of two bodies in gravitational interaction must be
(m1 + m2) c^2 - G m1 m2 / r ,"
, do you have a supporting reference for that, or an explanation of why?
Okay, so "(m1 + m2) c^2" is obviously the total energy equivalent of two masses that might be used to make up a two-body system, and "- G m1 m2 / r" seems to relate to the Newtonian calculation of the energy that you have to put in in order to wrench those two masses apart and take them out of effective gravitational range if their initial distance at the start of the experiment is r.
But black holes (and Newtonian "dark stars") don't form by our ripping systems apart and flinging the pieces to null infinity, they form by matter coalescing under the force of gravity, so perhaps if you have two masses m1 and m2 that collide and merge, and the distance between their centres when they hit is r, then if the two bodies somehow manage to absorb and retain the full energy of the impact, perhaps the Newtonian calculation for the "ballpark" energy of the combined body might be something like
(m1 + m2) c^2 ... + ... G m1 m2 / r
?
Robert: Actually, E=mc^2 is an exact result under Newtonian theory as well. It's not a particularly SR-specific result.
Einstein may have given a slightly misleading impression in his 1905 followup paper, but since it was a cool result, and it was "his", and he was trying to promote special relativity at the time, I suppose that it was understandable that he might have chosen to present it as a wonderful new SR result and "forgotten" to mention that if you deleted the new Lorentz/Einstein relationships in his argument and replaced them with the older Newtonian equations, you still got exactly the same result.
So he basically announced:
"If SR, then ... E=mc^2 ! "
and forgot to mention:
"On the other hand, if not SR, and if NM instead, then ... Still E=mc^2 ! "
:D
Stephen: "Schwarzchild radius" is in common use as the default name for the radius of a black hole event horizon, even if one is not using Schwarzchild spacetime. It's also the radius calculated by John Michell in 1783 under Newtonian theory as the radius of a surface whose escape velocity equalled lightspeed.
However, the big difference between the Newtonian and GR1916-based models is that in the Newtonian model, light can leave the critical surface for a limited distance, for a limited time before being recaptured, whereas with GR1916, the signals from the critical surface or lower are more emphatically -- they never venture outside the surface at all. So dark stars and black holes both have horizons at r=2M ... but the dark star horizon fluctuates and leaks, whereas the black hole horizon is a perfect one-way barrier.
Your question is elementary school fourth grader fare: e.g. "A particle dropped into a singularity generates infinite energy (1/r) by time it reaches the center." [a fellow fourth grader "pupil" ca 1960-61]
But conservation of mass-energy was also studied in fourth grade [I studied Glasstones' Sourcebook on Atomic Energy, etc.]: It was a result shown by M. Curie, refined by Einstein's SR...
So, I'd find a way to do the calculations completely rather than ask a one-bit-question: Maybe Einstein's GR already gives an answer that Newton could accept; Potential energy is power-limited...even back-inductiing...
REF: There's more to this story: (Note the dark-energy-consequence!)
http://www.youtube.com/watch?v=fhVXG-jl8Uk
http://lanth.us/gravity.html (b.g. and f.g. discussions)
Jose: having Googled, the problem definitely lies with the (slightly counterintuitive) convention of using a minus sign.
Relative gpe's can be positive or negative, and the absolute "physical" gpe of a system is assumed to be positive. However, when we want to do global calculations and need absolute values to cross-reference, we find that we can't establish an agreed lower gravitational level to call "zero" (in the way that we can have an agreed "zero Kelvin" as a reference for temperature), and in fact, there are some arguments (see: Raymond above) that there might be no such finite lower level.
So instead, we take a situation that seems to us to be a reasonable candidate for being an agreed //upper// level – the situation where two masses are so far away from each other than their gravitational attraction is negligible – and we use that apparent maximum effective gravitational height as our reference.
Since we're using that reference situation as a sort of "datum line", and the number that we assign to it is purely arbitrary, we use normal "datum line" conventions and call it "zero".
This means that all the lower positive amounts of gpe that appear in lower-energy situations end up being assigned negative numbers that relate to how much lower they are than this assumed maximum. But this doesn't mean that these values relate to absolute negative energies – you're free to assume that there's also a larger, positive, unstated gpe offset in play that keeps everything positive in real terms.
Thanks for the answers, most of them very sensible. A few remarks: There is a definition of mass (or energy) in general relativity, the problem being that the gravitational energy cannot be localized (no energy-momentum tensor for it). Of course, the two-particle example with Newtonian potential is simplistic and might be misleading. However, the exact relativistic gravitational energy fulfills the essential properties: (1) It is negative. (2) It is not extensive but grows faster with volume. Both these properties suggest that my reasoning holds for realistic cases.
Eric Baird said: “‘Schwarzchild radius’ is in common use as the default name for the radius of a black hole event horizon, even if one is not using Schwarzchild spacetime. It's also the radius calculated by John Michell in 1783 under Newtonian theory as the radius of a surface whose escape velocity equalled lightspeed.”
First, the quantity r used by Michell is the radial quantity because he used Newton’s theory. However, it is neither the radius nor even a distance in any alleged black hole metric. If you think otherwise, please provide your proof. Second, the Michell-Laplace dark body does not share any of the features of the ‘black hole’, other than possessing mass, on the assumption that any black hole universe (and related metric) even contains a mass (they don’t!). Third, Michell determined the maximum radius for any given mass at which the hypothetical dark body would form, and this maximum radius fixes the escape speed at that of light. So the dark body has an escape speed. Newton’s theory can’t be used to calculate anything for a black hole because black holes do not exist in Newton’s universe (see my second point above).
Eric Baird said: “So dark stars and black holes both have horizons at r=2M ... but the dark star horizon fluctuates and leaks, whereas the black hole horizon is a perfect one-way barrier.”
This is precisely the means by which the cosmologists give black holes an escape velocity and no escape velocity simultaneously at the same place. Since such a split personality is impossible the black hole is nonsense.
Stam Nicolis said: “Nobody claims that the mass of a black hole is concentrated at the singularity, in particular since it's false. Nor does anyone claim anything about infinite gravity.”
Well Stam, you are wrong. Here are just a few examples. According to Hawking [1],
“The work that Roger Penrose and I did between 1965 and 1970 showed that, according to general relativity, there must be a singularity of infinite density, within the black hole.”
According to Carroll and Ostlie [2],
“A nonrotating black hole has a particularly simple structure. At the center is the singularity, a point of zero volume and infinite density where all of the black hole’s mass is located. Spacetime is infinitely curved at the singularity. . . . The black hole’s singularity is a real physical entity. It is not a mathematical artifact . . .”
According to Dodson and Poston [3],
“Once a body of matter, of any mass m, lies inside its Schwarzschild radius 2m it undergoes gravitational collapse . . . and the singularity becomes physical, not a limiting fiction.”
According to Penrose [4],
“As r decreases, the space-time curvature mounts (in proportion to r^−3), becoming theoretically infinite at r = 0."
I could cite many more sources but the above is sufficient.
Thus, as I previously remarked, it is claimed by the cosmologists that a black hole has a finite mass, that this mass is concentrated at its ‘singularity’ at the ‘centre’ where r = 0, where density is infinite and spacetime curvature infinite. Since Einstein’s gravity is not a force but instead spacetime curvature, the cosmologists do indeed assert that a finite mass produces infinite gravity. Since that is just bullshit, their black hole is therefore a fallacy, and my citation of their ‘infinite gravity’ is not, as you say, “a measure of ignorance”.
Furthermore, the r = 0 they talk about is not the centre of anything, and the quantity r they talk about is neither a radius nor a distance in the relevant metric. As I said before, the ‘Schwarzschild radius’ is not the radius of anything because it is not even a distance in the relevant metric. The black hole is again a fallacy.
Stam Nicolis said: “Black holes don't have an escape velocity-that's a Newtonian notion, that doesn't make sense here.”
No Stam, you are wrong again. The cosmologists do in fact assert that all black holes have and do not have an escape velocity simultaneously at the same place. Since this is impossible their black hole is again a fallacy. Here are some examples. According to the Collins Encyclopædia of the Universe [5],
“black hole A massive object so dense that no light or any other radiation can escape from it; its escape velocity exceeds the speed of light.”
According to Gerardus ‘t Hooft, Nobel Laureate [6],
“A black hole is characterized by the presence of a region in space-time from which no trajectories can be found that escape to infinity while keeping a velocity smaller than that of light.”
According to Bland-Hawthorn [7],
“A black hole is, ah, a massive object, and it’s something which is so massive that light can’t even escape. … some objects are so massive that the escape speed is basically the speed of light and therefore not even light escapes. … so black holes themselves are, are basically inert, massive and nothing escapes.”
According to Hawking [1],
“I had already discussed with Roger Penrose the idea of defining a black hole as a set of events from which it is not possible to escape to a large distance. It means that the boundary of the black hole, the event horizon, is formed by rays of light that just fail to get away from the black hole. Instead, they stay forever hovering on the edge of the black hole.”
According to O’Neill [8],
“No particle, whether material or lightlike, can escape from the black hole.”
According to Chandrasekhar [9],
“The problem we now consider is that of the gravitational collapse of a body to a volume so small that a trapped surface forms around it; as we have stated, from such a surface no light can emerge.”
I could cite many more sources but the above is sufficient.
Thus, as I previously remarked, it is claimed by the cosmologists that a black hole has and does not have an escape velocity simultaneously at the same place. Since this too is impossible their black hole is a fallacy.
Stam Nicolis said: “General relativity is science, not religion, so the notion of ``followers'' is meaningless.”
Wrong again Stam. General Relativity is blind faith. Blind faith is not science. Faiths have followers. So my use of the word ‘followers’ is pertinent.
Stam Nicolis said: “The spacetime of a black hole isn't the vacuum and motion of matter in it can be calculated-that's how the existence of astrophysical black holes is deduced.”
That’s not true either Stam. First you have to specify which black hole you think you are talking about. For instance, in the case of the supposed charge-neutral non-rotating type of black hole, Ric = 0, and so there is no matter. Second, no alleged black hole metric space pertains to configurations of two or more masses. Third, there are in fact no known solutions to Einstein’s field equations for two or more masses and there is no existence theorem by which two or more masses can even be contemplated by means of General Relativity. Fourth, the Principle of Superposition does not hold in General Relativity, because it is a nonlinear theory. So superposition is not allowed. Let X be some supposed black hole universe and let Y be some supposed big bang universe. Then X + Y is not a universe. So from where do the cosmologists get their multiple black holes inside some big bang universe? They superpose. But that’s a really big no-no! Thus, once again, the black hole universes are fallacious, along with the big bang universes.
Stam Nicolis said: “This length scale can be identified with the black hole radius.What this means is that there exists a region of spacetime that's causally disconnected, within the framework of general relativity, from infinity.”
Wrong again Stam. First you must specify what is the radial quantity in any alleged black hole metric. You haven’t done so. It can’t be the quantity r used by the cosmologists, since that is not even a distance in their associated metric. Second, the notion of extending from r = 2m to r = 0 to generate a black hole is false, because the supposed extension is demonstrable nonsense [10].
[1] Hawking, S. W., The Theory of Everything, The Origin and Fate of the Universe, New Millennium Press, Beverly Hills, CA, 2002
[2] Carroll, B. W. & Ostlie, D. A., An Introduction to Modern Astrophysics, Addison-Wesley Publishing Company Inc., 1996
[3] Dodson, C. T. J. & Poston, T., Tensor Geometry - The Geometric Viewpoint and its Uses, 2nd Ed., Springer-Verlag, 1991
[4] Penrose, R., Gravitational Collapse: The role of General Relativity, General Relativity and Gravitation, Vol. 34, No. 7, July 2002
[5] Collins Encyclopædia of the Universe, Harper Collins Publishers, London, 2001
[6] ‘t Hooft, G., Introduction to The Theory of Black Holes, online lecture notes, 5 February 2009, http://www.phys.uu.nl/~thooft/
[7] Bland-Hawthorn, J., ABC television interview with news reporter Jeremy Hernandez, 24 Sept. 2013, http://www.physics.usyd.edu.au/~jbh/share/Movies/Joss_ABC24_13.mp4
[8] O’Neill, B., Semi-Riemannian Geometry With Applications to Relativity, Academic Press, 1983
[9] Chandrasekhar, S., The increasing role of general relativity in astronomy, The Observatory, 92, 168, 1972
[10] Crothers, S. J., General Relativity: In Acknowledgement Of Professor Gerardus ‘t Hooft, Nobel Laureate, http://viXra.org/abs/1409.0072
Jose, there seems to be a small problem with your formula because the Schwarzschild radius 2GM/c2 implies that the BH would have a mass of m1m2/2(m1+m2) which is half the reduced mass. However, let's set this aside for now because you make an interesting point that black holes may have zero energy. The minuscule Hawking temperatures of macroscopic black holes offer some further support for this view. Could it be that general relativity contains mechanisms prohibiting the formation of negative energy objects? Also, in Oppenheimer-Snyder collapse a black hole with an event horizon only forms asymptotically. Contrast this frozen star viewpoint with the singularity of the Schwarzschild metric. It seems to me that however one might define it, the potential energy of a singularity should be far lower (indeed, infinitely so) than the potential energy of a frozen star of the same mass. However, as the concept of energy conservation is now so unfashionable amongst relativists, it has become quite acceptable to gloss over such piffling concerns.
Concerning comments here about conservation of energy and momentum; General Relativity violates the usual conservation of energy and momentum for a closed system and is thereby in conflict with a vast array of experiments. Relativists do indeed attempt to disregard conservation issues. Einstein attempted to fix his theory to make it accord with the usual conservation laws as determined by experiment, by constructing his pseudotensor. However, (1) his pseudotensor is not a tensor, (2) his pseudotensor is in fact a totally meaningless concoction of mathematical symbols.
Using simple relativistic arguments, to me a mass M obviously has a total relativistic energy Mc^2. Now, the gravitational potential energy of a mass M clasicaly is -GMM/(R/2), where R/2 is the radius of the Mass M assumed to be close to a sphere of diameter R.
Let us see the case for the universe. If initially there was nothing, zero energy, then after this zero initial condition one can thing that Mass M may be created out of nothing as long as a gravitational potential energy (obviously negative -GMM/(R/2) is created joinly with the mass M so that we must have zero total energy:
Mc^2 = GMM/(R/2), i.e. 2GM/c^2 = R
This is the condition to have a black hole of mass M and size R. Then, one can think of the creation of the universe out of nothing, and having a kind of polarization of energies: the positive relativistic energy and the negative gravitational potential energy: total being always zero, like the initial condition. Then the universe is an expanding black hole. taking 2G/c^2 = constant = 1 in certain units, one has M=R the mass-boom I have analysed in the papers here, in the RG.
As far as the black hole in the center of the galaxy, one has to see how this conditions may apply.
Antonio Alfonso-Faus - There is no such thing as a black hole. See my reply above to Stam Nicolis for details.
Stephen Crothers: What about the big black hole at the center of galaxies, mass may be thousands or millions of solar masses? What about quasars with their big black hole in the ceter too? ·
Antonio Alfonso-Faus - I refer you again to my reply to Stam Nicolis. It seems you did not read it. There are no such things as black holes, so there is no black hole at the centre of any galaxy, and quasars are not black holes. Calling them black holes is faith and wishful thinking, not science. You have not adduced a scientific argument.
Crothers, S. J., General Relativity: In Acknowledgement Of Professor Gerardus ‘t Hooft, Nobel Laureate,
http://vixra.org/pdf/1409.0072v2.pdf
Stephen Crothers: Yes. I could agree with you that the many, many, experimetal observations that exists in the scientific literature, are not a DIRECT prove of the existence of the so called Black Holes ( as you probably know, the name coined by the late John Archibald WHEELER). Nevertheless, the INDIRECT evidences are very well known, very numerous, very well treated scientific papers, observations atributed to the existence of black holes quite convincing argumented. I could not repeat here such an amount of informatio, that is very well known.
BTW scientific arguments, which you seem to appreciate very much, are acceptable only when they are based on the aplication of the scientific method, which needs to have a reasonable theory plus predictions, plus observations confirmed by observation. In the case of black holes the scientific literature is full of scientific papers, very profesional most of them, with a sound and well known theory predicting them As I said, the observations are INDIRECT, but with plenty of predictions based on the possible existence of the theoretically predicted black holes, predictions that have ben observed.
You claim black holes do not exist. May be you are right. But I do not know of any scientific work, based on the scientific method, proving your claim. In conclusion, the theoretical basis predicting black hoes to me sounds well treated and acceptable. The confirmation of the existence of black holes by the obervations so far is based on INDIRECT evidence, but very well trated and acceptable. Your claim is just a possibility, with no direct nor indirect observational evidence, so far.
Relativistic gravitational energy is well defined for a spherically symmetric compact body (space-time is asymptotically flat). Unfortunately, the standard references do not care to compute it (some hardly mention it). Of course, I have not checked every reference and, therefore, I welcome information about it. I realize now that my question is related to the "positive energy theorem" (which is not mentioned in standard GR textbooks). In the paper by Gibbons et al about it, I read: "It is also generally assumed that any black hole has positive mass." I have no proof that the mass has to be zero (but only some arguments). Is there a proof that it has to be strictly positive?
Antonio Alfonso-Faus - There are two things that any physical theory must satisfy, (1) logical consistency, (2) correspondence with experiment. The theory of black holes is riddled with logical inconsistencies and so the theory is false. That closes the case completely. A demonstrably false theory cannot be used to describe or model anything, because it is false!
There is no such thing as infinite density. Nothing has an infinite density. Infinite density is inconsistent with experiment and observation.
I again refer you to my reply to Stam Nicolis for the details. They are not complicated. Without a single equation the black holes, and the big bangs, are easily proven to be nonsense.
Jose Gaite - I refer you to my reply to Stam Nicolis. You will see there simple proofs that the black hole is a figment of irrational imagination, and so the question of its mass is of no consequence to science. Black holes simply do not exist.
As for Relativistic gravitational energy, General Relativity violates the usual conservation of energy and momentum for a closed system, and so it is in conflict with a vast array of experiments. Consequently, it is false. I refer you to reference [10] in my reply to Stam Nicolis for the proof. The 'positive mass theorem' is therefore another nonsense. And since General Relativity violates the usual conservation of energy and momentum for a closed system Einstein's gravitational energy can't be localised, which means that his elusive gravitational waves are elusive because they too do not exist.
Stephen Crothers: You say "I again refer you to my reply to Stam Nicolis for the details. They are not complicated. Without a single equation the black holes, and the big bangs, are easily proven to be nonsense"
O.K. I read your details in your reply to Stam Nicolis: you are right saying that without a single equation you prove certain things to be nonsense. That is why I did not read any of your philosophical statemens: no equations, no maths, then it is very difficult to talk physics, very difficult indeed. And no physics means no cosmology. You are right also when you thougt that I did not read your replies.
You see, I am a cosmologyst and you have referred to cosmologysts in many many aspects. It appears that you include all cosmologysts in your sentences. I failed to see my "beliefs" in most of the statements that you have constructed.
Antonio Alfonso-Faus - You have still failed to acknowledge that the black holes and the big bangs are demonstrably false dogmas. The arguments I adduced in my reply to Stam Nicolis were intentionally simple, without any mathematics, because anybody can understand these arguments without any knowledge of the mathematical issues. Nobody has need of the mathematical issues to understand these facts that eliminate black holes and big bangs.
If you had followed up with the references I provided you will have found that reference [10] contains all the mathematical issues as well, in its appendices. Those interested in such mathematical complexities can easily consult that reference to their heart's delight.
Here are the facts in summary:
(1) Black holes are demonstrable nonsense;
(2) Big bangs are demonstrable nonsense;
(3) General Relativity is demonstrable nonsense.
If you contest any of the arguments I have posted to this site, then provide your counter-arguments and we shall see if they hold any water. Anybody on this site who wants to defend holes and bangs is invited to address the arguments I adduced in my reply to Stam Nicolis. Anybody who wants to defend the mathematical niceties I have adduced in reference [10] in my reply to Nicolis is welcome to try. We shall see then if any such rebuttal holds water.
Here is reference [10] again:
Crothers, S. J., General Relativity: In Acknowledgement Of Professor Gerardus ‘t Hooft, Nobel Laureate,
http://vixra.org/pdf/1409.0072v2.pdf
Stephen Crother: Your reference seems to be a letter of congratulations to Prof. 't Hooft. That is OK with me. I agree to congratulate to Nobel Laureates, they have done a good job. BTW I have never seen an abstract completely dedicated to congratulate someone else. But this leaves the reader without any knowledge of what is all about. You see, it is usual when you read a paper first to read the abstract. If it is scientifically interesting then you read the conclusions. If you keep interested then you go to the text in more or less extension. I could only read a few sentences of this original abstract.
Antonio Alfonso-Faus - I see, you prefer not to read anything and to understand an argument without knowing what it is. Since the reference is too much work for you, then just return to my reply to Stam Nicolis. Either acknowledge that the black hole is nonsense or put up your arguments to what I reported in my reply to Nicolis. I'll bet you'll take the last option, do nothing!
Stephen Crothers: I do apologize Stephen. Right now I have no free time to see your work. Very sorry.
Antonio Alfonso-Faus - I regard your last post as disingenuous. This is a forum. I posted a response to Stam Nicolis on this forum. All you need to do is read the posts on this very forum which you are posting to. Are you making comments here without even reading what has been posted? That would be a real waste of time.
I also asked in my original question if a massless black hole is not a contradiction in itself. For example, it would have no horizon, so it should be "naked" (as has been remarked); but a naked singularity is supposed to have positive mass! On the other hand, this does not imply that a zero mass object resulting from the collapse of ordinary particles is non-existent (Minkowski space), because of baryon number conservation. In fact, I have learnt that Zeldovich considered states of vanishing mass and arbitrary baryon number already in 1962! (JETP 42, 641).
At any rate, I do not deny that the Oppenheimer-Snyder 1939 theory of collapse leads to a singularity with positive mass and a horizon. However, the pressureless radial motion is not realistic (and actually this type of collapse is reversible). Einstein's 1939 paper (same year!) attempted to prove that 'the "Schwarzschild singularities" do not exist in physical reality.' Unfortunately, this paper was published in a mathematical journal and is not very realistic either. However, to my knowledge, there is no proof that a realistic collapse leads to a stable, non-zero mass black hole.
Jose, I don't see how a black hole can have zero mass. Without any mass how would it gravitate or cause spacetime to curve? I think your original calculation hints that the total energy (not mass) might be zero. In this context, consider black hole thermodynamics. The surface temperature of a black hole depends on its surface gravity. For an extremal black hole the mass is not zero but the surface gravity is and so the temperature vanishes. However, it is impossible for an extremal black hole to form by any physical process. This makes me wonder whether nature abhors objects of zero or, worse still, negative energy.
Einstein's paper pointed out that the time dilation approaches infinity as a particle approaches the event horizon. Although it takes a finite proper time for a particle to reach the horizon (let's say 1 second) an infinite time must then elapse for any external clock because global relationships always exist between the rates at which differently located and differently moving clocks advance. It is no good saying focus on the infalling particle because the spacetime continuum can contain no worldlines along which the proper time is infinite (or larger). Therefore, proper times beyond 1 second do not exist for the infalling particle - they are unphysical. This is also why no event horizon actually forms during Oppenheimer Snyder collapse, except in an asymptotic sense. The authors pointed out that particles could reach the singularity in finite time but in doing so they ignored the constraint that no clock located anywhere within the spacetime manifold should record more than an infinite pasage of time. It is this kind of muddled thinking that has been responsible for the present confusion concerning the non-existent information paradox. It may not be fashionable to say this nowadays but Einstein was right to seriously doubt that black holes with event horizons can exist in nature.
Jose Gaite - Einstein's theory requires matter to be present to produce a gravitational field. His gravitational field is spacetime curvature induced by the presence of matter. According to Einstein everything except his gravitational field is matter. Since matter must produce his gravitational field Einstein claims that Ric = 0 contains matter, despite there being no matter terms in the equation. This leads to a post hoc insertion of Newton's expression for escape velocity in the solution for Ric = 0. Moreover, according to Einstein and his followers, matter is both present and absent by the very same mathematical constraint, T_{uv} = 0. This is in fact impossible. Thus, Ric = 0 actually contains no matter, and so, once again, the black hole is proven a fallacy.
Most important is the issue of the usual conservation of energy and momentum for a closed system. General Relativity violates the usual conservation of energy and momentum for a closed system and is therefore in conflict with a vast array of experiments. How many experiments does it take to invalidate a theory? Only one. Consequently, General Relativity is untenable. All the details are here:
Crothers, S. J., General Relativity: In Acknowledgement Of Professor Gerardus ‘t Hooft, Nobel Laureate,
http://vixra.org/pdf/1409.0072v2.pdf
I also refer you to my reply to Stam Nicolis (see above). It is proven there in very simple terms that the black hole is a fallacy - nothing but the product of irrational imagination.
Robert, thanks for this. If there are no event horizons then perhaps we can dispense with Hawking radiation altogether. I have no difficulty with the possibility that black holes might not fully evaporate. If all particles eventually find themselves participating in gravitatonal collapse somewhere in the universe there will still be empty regions of spacetime to consider and the proper times along worldlines there should remain finite. Therefore, event horizons still do not (quite) form, preventing anything from plunging through them. The ultimate outcome is that all foms of gravitational collapse stabilise and the spacetime manifold becomes stagnant. No incomplete geodesics anywhere, no drama, no paradoxes.
To Robert: I like Wald's book but I don't have it available now.
I should ask now a very general question: Is there any calculation of a realistic collapse, analytical or numerical, showing the formation of a black hole? I must emphasize the word "realistic".
To Robert: I mean realistic by astrophysical standards. Oppenheimer-Snyder collapse does not comply: otherwise, there would be no stars in the Universe, only black holes! When you say "great extent", you may be thinking of super-massive BHs, but no astrophysicist would take seriously OS collapse model for them.
I have had a quick look at Joshi's papers (he even has a book). I see no proofs of formation of black holes but only of naked singularities (this agrees with my previous knowledge: Choptuik's and related numerical work shows the formation of naked singularities but NOT BHs).
I remember worrying about this when trying to model back holes as collapsing shells (no radiative losses by Birkhoff) and trying to keep track of the mass. I think I have done a reasonable job recently from the continued collapse perspective. It is very hard because of the ill-conditioned limit of the equations. I'll warn you that I'm in the tiny camp of people who believe that anything leading to a true event horizon or singularity has transfinite exterior time problems that render such solutions unphysical.
I would encourage you to look at the meaning of the coordinate r (while spacelike) and the induced measure and how this impacts the mass energy density. This is my recent approach.
Article Globally Causal Solutions for Gravitational Collapse
Robert J. Low - No, an event horizon doesn't really form:
The Parallax Effect on Short Hair
https://www.youtube.com/watch?v=nXF098w48fo
Wormholes: Science Fiction or Pure Fantasy?,
https://www.youtube.com/watch?v=16_GuYobDZ4
General Relativity: In Acknowledgement Of Professor Gerardus ‘t Hooft, Nobel Laureate,
http://vixra.org/pdf/1409.0072v2.pdf
Clifford, many thanks for posting a link to you interesting article. I feel that the set of people lacking Einstein's physical intuitions and insights is much larger than generally appreciated.
Here's a little thought experiment. Fire a photon at a black hole. Ask an intrepid friend to slide a mirror between the photon and the event horizon but tell him to wait until your clock reads ~ 1099 years. Wait a bit longer, keeping an eye out for the reflected photon. Alternatively, just calculate what happens using any old coordinates you like. You should find that the photon can indeed return. Therefore, it could never have penetrated (or even reached) the event horizon. It must have been suspended fractionally outside the black hole all along.
Robin, I completely agree. The interesting question is then what is the duration of electric and magnetic multipoles that form as charged particles fall in from the external observers point of view (the actually important point of view). Doesn't this suggest that mass multipoles from accretion disks also persist? Unfortunately, the current approach to numerical relativity has a bit of a fudge on the interior and tackling the problem with the lapse-shift approach seems impossible. I've tried a new approach but it is going to take some work if it will work at all.
Robert, I agree that Schwarzschild coordinates are a problem. Isotropic coordinates are what I used in my paper because of this.
Stephen, thanks for these interesting and thorough links. I am a little distressed that 't Hooft is editing Foundations of Physics. His big accomplishment is in finding the kind of mathy tricks it takes to get calculations out of QFT. This is an accomplishment but gets us no further to the fundamental problems underlying the construction eg Haag's thm. He has a rather kneejerk dismissive attitude and not very willing to see his own mistakes. I am convinced that there is a numerical approach to collapsing bodies that has no transfinite exterior time problem and never allows collapse that will resolve a lot of these problems. When this arrives we will be less in a soup of words and I suspect the no-hair conjecture will be universally rejected along with wormholes, black hole thermodynamics and many other trendy topics.
One reason this problem interests me is that in every other physical situation I know of you can choose freely between Eulerian and Lagrangian approaches. In this case, I have come to the conclusion that using Lagrangian observers leads to contradictions yet the Eulerian approach begs the question of describing the system globally in reference to what? Is anyone here a fan of the Lasenby, Doran and Gull approach which gives meaning to such a background?
Clifford, I'm enjoying reading your article very much but I don't get why you think a choice is necessary between these two approaches you mention. Are you suggesting that from the Lagrangian perspective the photon would not bounce off the mirror in the little experiment I described? If so, what could have changed to affect the calculation? The way I see it there is no inconsistency or contradiction because infalling particles never quite make it to the event horizon. Therefore, we are free to switch viewpoints between infalling observers and external observers without difficulty. Also, the idea that an event horizon might be present at all is only due to the availability of simplistic toy models which cannot realistically form via gravitational collapse. Rather, the stationary black hole metrics are artificial constructs, which is why one should not be at all surprised if they radiate no genuine information.
Sorry, I should have been more specific. In the case of black holes, the Lagrangian observers cross the event horizon. This is because they use their own proper time. Global considerations seem to show that this is inconsistent as you mention. I agree with your perspective and in an asymptotically universe without black holes to start with, they never arise in a completed fashion. The infalling matter simply stalls out and in place of an event horizon is a region of finite volume with limiting degenerate metric.
Your mirror problem is interesting. I think it has the problem that everything is continuously falling in so I don't think it gets out either but the problem is that the mirror doesn't get to stay "still" either. Getting even a simple class of infalling solutions where I could make sense of the mass density in terms of the number of infalling particles in these limiting coordinates was what the paper was about. I think you could make the shells reflecting to check the mirror idea and I think the infalling photon just forms part of the the next infalling "shell."
Cifford, fortunately in these little thought experiments there's no need to fret about engineering challenges as long as we avoid the manifestly impossible. So we're free to imagine a mirror being suddenly thrust in the path of the photon after it has been travelling for 1099 years. Since the mirror would be located outside the event horizon its radial velocity at the moment of reflection could be zero.I don't see any problem. The photon would escape, redshifting back to its original energy as it does so - proving that it never crossed the horizon. So where's the information paradox? Does it perhaps only arise at future timelike infinity? Perhaps problems that pertain to future timelike infinity can only be solved after an infinite time has elapsed!
Robin, I see what you mean. The way you have stated it makes sense. If there is an event horizon and the mirror has not crossed it, the photon will escape. I think it is worth fretting about what you mean by "zero velocity" though. Coordinate velocity relative to what coordinates? Many of us don't like Schwarzchild near the horizon for common reasons. Isotropic coords are my favorite. If you have a static hole then static coordinates are very useful relative to some asymptotic notion of time. One still needs a way to unambiguously assign position for the body in terms of static bodies at infinity but that is not so tough in this situation.
I don't believe there is any information paradox and never have.
Clifford, I meant zero radial velocity in the sense that the mirror would neither be approaching nor receding from the event horizon, the distance to the horizon would, at least momentarily, be constant. Due to its spherical symmetry, I think the Schwarzschild radius would do for this.
You have clearly thought more deeply about the information paradox than I have. Can you imagine a frozen star configuration evaporating away, even partially? What of energy conservation in such situations?
Robin, I think the problem with evaporation of a frozen star is that time has exponentially dilated so that they don't evaporate for the same reason they don't fall further. Relative to the external observers the matter at the horizon (or, more properly, forming horizon) is the most boring place in the universe. Literally nothing is happening. What this means for external perturbations or near grazings or collisions with other black holes is a mystery. Can some matter escape its frozen state in this case? The numerical relativists use a patchwork solution that ensures it won't and Hawing and Ellis gave a treatment (that a bunch of us think is faulty) that shows the hole must form and pull everything in. I'm comfortable enough to not know here.
Energy like mass needs a careful definition to talk about conservation. For a test particle in a static global field you can define a conserved energy but it need not be the energy you expect based on local intuition. Reminds me of the momentum/pseudomomentum problems where conserved quantities can have units and names that are misleading.
Clifford, I tend to agree. Also, if there's no horizon, there can be no capture of negative energy virtual particles. The infalling matter may be hot but its black body radiation would of course be subjected to exponentially increasing time dilation. However, since the matter is not hidden behind a horizon, I suppose a frozen star might conceivably compete with a fully formed black hole in terms of luminosity, even if it isn't actually emitting Hawking radiation. I'm just hazarding a rather idle guess here though.
You mentioned the complex question of what happens during black hole mergers. I wonder whether numerical simulations have tried to dynamically adjust the spacetime grid spacing according to the local time dilation (or its logarithm). I would also impose the requirement that the time dilation remains finite everywhere, but my impression is that Kruskal-like coordinates are routinely used and the locations of horizon(s) are only traced once the simulation has completed.
I am sure no current methods in numerical relativity can do this. The equations become profoundly ill-conditioned here. That is why they have to piece interior solutions or make implicit assumptions about the infinite exterior time behavior and argue vaguely that the way they treat the grid puncture is consistent.
I suspect you are right about the asymptotics but am not knowledgeable about those details.
After reading the posts on black hole horizons, I have to say that the physics of black holes is all right with me, in spite of its paradoxical aspects. In particular, I think that the Oppenheimer-Snyder collapse is a good solution of GR. However, following Einstein 1939, I think that the result (a BH) may not be realistic. The problem is not the absence of large clouds of pressureless dust that are suitable for spherical collapse. Quite the contrary: the formation of large scale structure in standard cold dark matter cosmology is based on them, and the spherical collapse model is indeed studied in cosmology. However, astrophysicists know that pressureless spherical collapse is unstable. Unstable processes do not occur (are unrealistic).
The peculiarity of OS collapse is that it takes place without emission of energy: all the gravitational energy is converted into kinetic energy of the infalling particles. In contrast, in a realistic collapse, most of the energy is emitted: consider core collapse supernova, for example. Arguably, in a realistic collapse to a black hole, all the energy is emitted. I have an argument in the case that the collapse takes place through a succession of quasi-equilibrium states.
Jose Gaite:- According to the cosmologists their black holes have and do not have an escape velocity simultaneously at the same place. Since this is in fact impossible, their black holes are fallacies. According to the cosmologists their black hole has a finite mass which is concentrated at the 'singularity' of their black hole, where density is infinite and spacetime curvature also infinite. Gravity is not a force in General Relativity because it is spacetime curvature. Thus, the cosmologists claim that a finite mass produces infinite gravity! That too is however just nonsense, and again their black hole is a fallacy. And no finite mass can have infinite density. No equations are required to prove the black hole a figment of irrational imagination.
The problem is indeed hard, since Einstein himself did not succeed in solving it! However, we know now a lot more. The type of collapse "that can actually happen in practice" must be given by genericity and stability arguments. Joshi studies this question in 4.4.1 of his IJMPD review (BTW, he also considers the instability of OS collapse, but I do not think that he hits the nail on the head). I do not think that energy and regularity conditions are sufficient, and one must consider cosmological boundary conditions. Unfortunately, these are not so well known.
For example, depending on the composition of matter, we may have different collapse processes. Non-baryonic dark matter (WIMPs, say) can only dissipate its energy by gravitational interactions (gravitational radiation, in particular) whereas baryons and electrons are much more likely to just get hot and emit electromagnetic radiation. In core collapse supernova, a good deal of energy is carried away by neutrinos. To know the outcome, it is necessary to appeal to general arguments, based on thermodynamics or, perhaps, ergodic theory (my arguments go along these lines).
Generalizing this definition to general relativity, however, is problematic; in fact, it turns out to be impossible to find a general definition for a system's total mass (or energy). The main reason for this is that "gravitational field energy" is not a part of the energy–momentum tensor; instead, what might be identified as the contribution of the gravitational field to a total energy is part of the Einstein tensor on the other side of Einstein's equation (and, as such, a consequence of these equations' non-linearity). While in certain situation it is possible to rewrite the equations so that part of the "gravitational energy" now stands alongside the other source terms in the form of the stress–energy–momentum pseudotensor, this separation is not true for all observers, and there is no general definition for obtaining it.
How, then, does one define a concept as a system's total mass – which is easily defined in classical mechanics? As it turns out, at least for spacetimes which are asymptotically flat (roughly speaking, which represent some isolated gravitating system in otherwise empty and gravity-free infinite space), the ADM 3+1 split leads to a solution: as in the usualHamiltonian formalism, the time direction used in that split has an associated energy, which can be integrated up to yield a global quantity known as the ADM mass (or, equivalently, ADM energy). Alternatively, there is a possibility to define mass for a spacetime that is stationary, in other words, one that has a time-like Killing vector field (which, as a generating field for time, is canonically conjugate to energy); the result is the so-called Komar mass .Although defined in a totally different way, it can be shown to be equivalent to the ADM mass for stationary spacetimes. The Komar integral definition can also be generalized to non-stationary fields for which there is at least an asymptotic time translation symmetry; imposing a certain gauge condition, one can define the Bondi energy at null infinity. In a way, the ADM energy measures all of the energy contained in spacetime, while the Bondi energy excludes those parts carried off by gravitational waves to infinity. Great effort has been expended on proving positivity theorems for the masses just defined, not least because positivity, or at least the existence of a lower limit, has a bearing on the more fundamental question of positivity: if there were no lower limit, then no isolated system would be absolutely stable; there would always be the possibility of a decay to a state of even lower total energy. Several kinds of proofs that both the ADM mass and the Bondi mass are indeed positive exist; in particular, this means that Minkowski space (for which both are zero) is indeed stable.While the focus here has been on energy, analogue definitions for global momentum exist; given a field of angular Killing vectors and following the Komar technique, one can also define global angular momentum.
Another question on RG asked how does the gravity of a black hole escape the event horizon, if nothing can travel faster than light. Now that the Higgs boson has been found, its mass has been measured and determined to be positive, it is foolish to suppose that the massive objects at the centre of every galaxy possess event horizons. I think the question posed here has some validity, despite its Newtonian inspirations, but the question of how does gravity escape from an event horizon leaves no wriggle room for black hole apologists.
The total energy written in the question is a Newtonian expression. The general relativistic expression would be different from this, so the question is based on an incorrect assumption. There are a number of definitions of mass for a black hole (ADM mass and Komar mass are two popular ones), which are non-zero.
The argument that gravitons could not escape the event horizon, because nothing can that is not faster than light, is well-known to be erroneous. This holds for real particles only. A static gravitational field consists, if you insist on a particle description in quantum mechanical terms (which is a cumbersome description to say the least, the field picture is much more appropriate here) mostly of virtual particles. These are not strictly confined by the speed of light.
Dear Jose Gaite ,
Let's consider only barionic and the motion is radial, no initial angular momentum hence emission of GWs negligible. A straight impact (although quite unlikely to occur in case of merging of big masses) ends up with a certain collision-speed, a function of the gravitational potential.
As you wrote ET = (m1 + m2)c2 - G m1m2 / r = 0
if this condition r = G/c2 *m1m2/(m1 + m2) is satisfied.
In an isolated system, the kinetic energy in COM the center of mass of m1 and m2 , before the impact, at the first order, for the mechanical energy theorem, is equal to the potential energy hence: KE= G m1m2 / R, where R is the distance between the masses. KE , at the first order in this case, is the energy , *not* certainly provided by the masses themselves, since m1 and m2 , right before the impact, are untouched.
ET means only that, as a first approximation, the energy to bring back m1 and m2, after the merge, to infinite distance would exceed the rest energies of the bodies themselves if r < Gm1m2/(m1c2+ m2c2).
ET = E0 - EPgrav
Masses "free fall" towards each-other, the work of the force of gravitation changes the speed (increasing acceleration) of the masses referred to COM from infinite distance to collision.
Excluding reactions of different kinds, if the masses are not very large, there will be an inelastic scattering at the collision resulting in a "hotter" mass of the merger, in thermodynamic sense.
Radiation gets generated ( hence an external observer will detect it), so the energy of the free falling masses is, after a certain time, radiated away. It escapes from the smallest volume containing m1, m2.
By enlarging the masses of m1, m2 , no reason at all that the result is a massless entity, but the merge will result just in something like M= m1 + m2 at the first order. So the BH would have a mass.
Only in one case the result of NO MASS would be reached :
m1 and m2 same mass but one of matter and the other of antimatter, when they come together everything is radiated away...but in such case no BH is formed and that is for sure.
The KE referred to COM is generated by the "work of a force" as a first approximation which pulls the masses together in the Newtonian approximation. The work of such force comes from an unknown "reservoir" where energy is not localized, although somebody would say that it "comes from nothing" or bodies just free fall.
We could open a discussion about the nothingness vs the quantum vacuum or an Active Background, because it is only there that this "apparent PARADOX" can find a solution...GR can provide a more accurate version of the formula used as a first approximation, but it is far from providing a satisfactory solution. That requires a change of paradigm: the system is not properly "isolated" in the classical conception. The background provides the necessary energy which is not a relative quantity since it (RADIATION) cannot be made vanish with a coordinate transformation.
As a conclusion there is no reason at all why the BH outcome of the merge should be mass-less, or should not be any BH.
This first order equation (m1 + m2)c2 - G m1m2 / r =0 does not mean that the total energy, hence the mass of the merge is 0, but just that all the rest energy of the m1 , m2 should be wasted, as a first approximation, to bring them back to infinite distance, if they merged in a way that r = G/c2 *m1m2/(m1 + m2) , considered a very first approximation. The negative potential energy which eventually makes a negative energy density is just a "missing energy". It takes account of the energy which flew away after the collision, missing from a background energy density level which is maximum where masses are not present.
I asked this question in 2014. An update is now due, after the momentous 2015 gravitational wave detection by LIGO, presumably from the inspiral of two orbiting compact bodies (the Weiss-Barish-Thorne Nobel prize).
Several answers to my question criticize the use of a Newtonian expression. Of course, that is not right, but the essence of the question is what happens when the distance between two bodies reduces, e.g., by inspiraling, to the point that their gravitational binding energy is as large as their mutual mass. The consequent loss of energy is supposed to be the origin of a strong burst of gravitational waves.
The total mass emitted as gravitational radiation is assumed to be less than 10% of the mass available. This assumption relies on unreliable numerical simulations and on the observations. However, the total gravitational radiation output cannot be measured.
Therefore, my question can be rephrased as: why is most of the mass not emitted in a compact object merger? If it were, the process would leave an almost massless remnant.
Dear Jose Gaite ,
By re-reading the papers of Taylor about the gravitational radiation of the binary pulsar, there is no mention of "mass loss". Although you can express the radiated GW energy as mass, it was not the mass of the pulsars to be radiated away through GW. What was considered by Hulse and Taylor was the kinetic energy and potential energy of the system through several years (with ARECIBO), by observing the reciprocal distance and speed, estimating the masses and that was enough to give account for the energy radiated through GW quadrupole radiation. The single masses were always there as constant values.
because no rest-mass at all, of the binary pulsars, is emitted through GW waves.
Dear Stefano Quattrini ,
I refer to the GW150914 event, not to the much older an weaker gravitational radiation of the Taylor binary pulsar. The LIGO team actually claims that it is a black hole merger (https://www.ligo.org/science/faq.php#gw150914-bh), but it is the same whether they are neutron stars or black holes.
Dear Jose Gaite ,
I think so too.
Although I don't understand why the "episode" of Hulse and Taylor of the merging binaries should underly a different mechanism than the one of the BH-merge detected by LIGO-VIRGO.
Hulse and Taylor managed to account for the radiated energy according to GR applied in that configuration (pseudotensors) which matched with the decay rate of the orbital energy of the binary.The tested mechanism was not relevant to the variation of the masses of each separate object which was considered constant to a good approximation.
What to my knowledge regarded "masses" is the merger of 2017 of the neutron stars. An amount of mass of gold comparable to some solar masses was expelled in that case and for the first time it was understood how gold or other heavy atoms are generated and why they are so rare. But the gold Suns were outcomes of the collision of neutrons with relevant nuclear reactions to which was provided a huge amount of energy , besides the huge amount of GW radiation.
Dear Stefano Quattrini,
Indeed, Hulse and Taylor proved that the radiated energy is in accord with the loss of energy due to gravitational waves. However, you must consider that it is a very weak decay rate: "the calculated lifetime to final inspiral is 300 million years" (https://en.wikipedia.org/wiki/Hulse%E2%80%93Taylor_binary). To see strong mass losses you have to wait for some million years. Surely, you are not so patient! :-)
I do not know what 2017 neutron star merger you refer to. A reference, please.
Dear Jose Gaite ,
a reference about the merge in 2017
https://www.nationalgeographic.com/news/2017/10/gravitational-waves-discovered-neutron-stars-pictures-science/
and the generation of gold and heavy elements:
"The merger now resolves a long-standing debate about the origin of heavy elements in the periodic table: precious metals, including gold and platinum, and things like the neodymium scientists use when building lasers like LIGO’s. "
no matter how weak is the rate and, as you said in the case of H and T, the observation of the merge is quite far away in earth-time, the mecanism of generation of GW is clear that it does not involve "disappearing of mass/matter" in favor of radiated energy but is fully explained by the energy loss of the orbital decay.
>
could you please give me a reference to it?
Dear Stefano Quattrini,
I'll have a close look at that link.
I do not understand what you mean by the generation of gravitational waves not involving the "disappearing of mass/matter". Whenever energy disappears, so does mass.
I'll give you the reference to the Einstein paper in my next post. I can even send it to you! :-)
Recognizing the simple theory of the electron radius https://wikimili.com/en/Classical_electron_radius and replacing 1/4Pi/epsilon0 e^2 by the G M^2 they only have the mass deficiency. Let the Black Hole be a uniform matter sphere of the Schwarzschild radius. Than its gravitational energy due to self interaction is E_g = - (3/5) G M^2/ r If r is the Schwarzschild radius than G M m /r = m c^2 /2 for the probe mass m so r = 2 G M / c^2 so we have E_g = - 3/10 M c^2. Now let M be the nuclear (rest) energy of all matter (atoms) mutually at infinity which would build the Black Hole sphere than we have:
M c^2 - 3/10 Md c^2 = Md c^2 which is the equation for Md where now Md is the dressed Black Hole Mass. So finally Md = 10/13 M so with 3/13 of the constituent infinity mass deficiency. But it also means from Einstein that part of the mass vaporized as radiation in the formation process and so normally 3/13 of the original rest mass energy is vaporizing in the Black Hole formation process.
Note however that if the Black hole formation was by the free fall of all the particles on each other and than sudden fully non-elastic collisional stop which would generate the radiation by friction which would again convert into mass by for example pair creation than by the total energy conservation potential + kinetic it must be M c^2 - 3/10 Md c^2 + 3/10 Md c^2 = Md c^2 i.e. Md = M which means that the mass of the Black Hole would be finally the same as of the original separated matter particles mutually at infinity.
Dear Jose Gaite ,
yes please send me the reference doc.
Regarding GW waves generation, it is quite clear that their radiated energy does not have anything to do with the rest energy of the merging celestial bodies.
The variation of the metric does involve energy but the masses remain invariant.
Or better, the orbital energy of the binary, from where the GW energy is taken from, producing the orbital decay, is just the gravitational potential energy of the system which trivially depends on the rest mass of the components, but not related at all to its variation. In other words, the kinetic energy of the binary at a certain moment in time, in its center of mass, is not reached the expense of the rest energy of the system.
I 've already proposed the configuration to make the two objects collide radially and merge with an inelastic collision.
They will collide at huge speed, they will not produce GW waves during the approaching, excluding nuclear reactions, the rest of the kinetic energy, in the center of mass of the binary, produces heat in the inelastic collision which is radiated away with EM waves.
I don't see here a relation with the variation of the rest mass of the system, but the mechanism of radiating energy away is eventually similar.
Dear Stefano Quattrini,
I have read the National G. article (at the link) and it is just a popular article about detection of gravitational waves created by mergers of two compact objects (neutron stars or BH). They mention the GW150914 event and some later events (in a figure). The relation with theories of the nucleosynthesis of gold and other heavy elements is interesting but not relevant to our problem.
The reference to the Einstein paper is http://www.jstor.org/stable/1968902?origin=JSTOR-pdf
I think that the paper is free to download. If you cannot do it, tell me. In this paper, Einstein considers the contraction of a system of many small self-gravitating particles and concludes that they reach relativistic speeds before crossing the Schwarzschild radius, as long as they stay self-gravitating. At that point, in the initial (Newtonian) formulation of my question, the total energy and hence the total mass vanish.
Please notice that the condition of continuing self-gravitating equilibrium excludes any radial collapse! This is very important because it distinguishes Einstein's problem from the Oppenheimer-Snyder collapse (also published in 1939!), which is taken as the origin of BH physics.
In an inspiral of two compact objects, the individual masses can remain invariant while the total energy and hence the total mass continuously diminish. Please consider the initial (Newtonian) formulation of my question.
Dear Jose, Jose Gaite
I agree.
thank you, I had a look...
Starting with two masses only and no angular momentum you do have a radial collapse, like in the simplest possible case. You brought forward the following
E= (m1 + m2)c2 - G m1m2 / r
The total energy of the masses right before the radial impact is this one
ETOT= (m1 + m2)c2 + G m1m2 / R
where R is the final distance between the center of masses of m1 and m2.
Since for the mechanical energy theorem the kinetic energy of the two masses, which is positive, is its potential energy which, summed with the rest energy, that gives the total energy available: ETOT = E0 + KE, where E0 = (m1 + m2)c2
If the impact is inelastic and between small masses and the surroundings is close to 0k, all the heat produced during the impact which was its kinetic energy, gets radiated away.
Efinal = (m1 + m2)c2 with the additional fact that to bring back m1 and m2 to infinite distance one will have to make the work G m1m2 / R.
This equation (same as yours) E0 = G m1m2 / R
means only that the energy needed to bring masses m1 and m2 to infinite distance is their rest energy, which is what occurs in a BH where no mass can come out.
Dear Stefano Quattrini ,
I am not arguing against radial collapse or, in particular, the Oppenheimer-Snyder solution of GR, with no emission of gravitational waves. I already presented my view in my post of Dec 3, 2014. In brief, such a collapse is not realistic.
I should add now that such type of collapse is of little relevance to the physics of inspiralling compact objects.
Dear Jose Gaite,
on one hand, I totally agree with you. On the other hand in the question of the thread about (m1 + m2) c2 - G m1 m2 / r ,
(m1 + m2) c2 - G m1 m2 / r = 0
can be summarized as
E0 = U(r) with r = G/c2 *m1m2/(m1 + m2)
where U is the gravitational potential energy of the system of masses and E0 is their rest energy.
I see no paradox if the meaning of it is the following:
for m1 and m2 at that distance, the energy required to bring them back to an infinite distance is the same as their rest energy. A black hole with mass (m1 + m2) in that case is not unreasonable.
In the attempt to escape to an infinite distance, all the rest energy of the masses will be depleted hence radiated away and nothing will remain rather than radiation.
Dear Stefano Quattrini,
The initial formulation of my question was sketchy (in addition to mixing special relativity with Newtonian gravity). However, its meaning was understandable and it has been clarified in the answers. From two faraway masses (m1, m2) at rest, either a (m1 + m2) mass black hole or a vanishing mass black hole can form, depending on what the process is. In particular, an inelastic radial collision should produce the former whereas an energy-emitting inspiral could produce the latter.
I think that the inelastic radial collision is less realistic and, certainly, of less observational interest, because there is no emission that can be observed.
Recalling history, Einstein apparently argued that matter could never cross the horizon to form a black hole. That is not so, as the Oppenheimer-Snyder solution shows. Nonetheless, Einstein can have been right in a way.
Dear Jose Gaite ,
>
Allow me to disagree.
The only option according to current available knowledge(*) is a merge, ending with a larger mass m1+m2 in both cases, radial or in-spiralling (provided that no mass is expelled during the merge, same as occured for the gold Suns).
If E0 = U(R) with R= G/c2 *m1m2/(m1 + m2) means that the escape energy is the same as the rest energy, I see no Paradox, meaning that a BH is there, whose mass is m1 + m2 .
(*) In fact according to the well known described mechanism of generation of GWs used by Hulse and Taylor, what is radiated away is part of the "orbital energy" (a certain amount of the mechanical energy of the orbit, per unit time).
The tested mechanism of GW has nothing to do with an alleged depletion of the rest energies of the bodies m1c2 and m2c2 , implying "missing of rest mass". It is up to you if you want to suggest other mechanisms for GW generation, then you have to live with the paradox that the final condition is a BH with less mass or no mass at all.
Dear Stefano Quattrini,
The example of generation of GWs by the Hulse and Taylor binary system is misleading, because the emission is too weak. Notice that it can only be detected indirectly.
In the GW150914 event, "the coalescence converted the equivalent of about three times the mass of the Sun (or nearly six million trillion trillion kilograms) into gravitational-wave energy, most of it emitted in a fraction of a second." That was real loss of mass!
Dear Stefano Quattrini,
I guess that you mean that all the released energy can be just kinetic energy of the two compact objects. Not so. As long as the inspiral is a quasi-equilibrium state, it fulfills the virial theorem, which tells us that the kinetic and potential energies are related. A great release of energy has to come from relativistic kinetic energies, which must correspond to larger negative potential energies. Therefore, the released gravitational energy is larger than the released kinetic energy.
Dear Matt Kalinski ,
I have reread your arguments and I am not convinced. You say "generate the radiation by friction". Actually, friction is a generic name for a loss of energy to many degrees of freedom, such that energy is not macroscopically conserved. For two bodies, there must be friction unless they rebound, which is not what we have in mind. However, the energy can stay around the bodies or be emitted away.
Pardon me for jumping into this thread. I read all on the last page, but not old answers.
The "r" is relative, and potential calculated this way is only relative to some other object. If you add mass m1 to black hole m2 and they are both r from some other star system, the energy does not change. This is a non-trivial problem in dealing with black holes, even though supposedly it is only the coordinates that are changing.
There should be nothing solid at the event horizon and nothing should rebound from it. Energy radiated by in-falling mass will be prior to the event horizon. The sum of radiated energy and the final mass energy of the black hole m2' should be the mass energy of m1+m2.
Does that help anyone or just confuse the matter further? I am not trying to address any fundamental mysteries with this answer, like whether mass "actually" crosses the event horizon (a coordinate dependent question), just basic energy accounting. I am not fond of the popular answer listed that says that one does not have a definition of energy in GR. Energy is a real, practical thing which physics is concerned with.
Dear Robert Shuler , you are welcome, by all means! :-)
Of course, conservation of energy is indispensable. Therefore, "the sum of radiated energy and the final mass energy of the black hole m2' should be the mass energy of m1+m2." And, in this regard, I agree with you, we need not worry about GR, because the relevant energies can be measured far away, in the nearly flat space-time, where gravity is weak.
However, I think that a rebound cannot be excluded by "nothing solid at the event horizon". Nonetheless, it cannot happen in an inspiral, because energy is continuously radiated.
Jose Gaite , et al.
I'm glad to see you acknowledged the mixture of GR with Newtonian concepts in your original question.
One formula derived from GR that sheds light on gravitational binding energy and it effect on the gravitational mass is the equation of motion derived by Werner Israel for the case of a thin shell. That formula can be found in Misner Thorne and Wheeler and is (corrected):
M = mu * sqrt (1 + Rdot^2) ^ 1/2 - mu^2 / 2R.
The equation can be interpreted as as expression for the gravitational mass M in terms of rest mass , mu, and the radius R and its radial speed Rdot.
It can be proven that M is strictly greater than zero.
The first term is the kinetic contribution and the second is shell self energy. (binding energy) which reduces the gravitational mass.
If we evaluate the expression for a bound shell at the turning point, Rdot = 0, we get the expression
M = mu - mu^2/ 2 R_max.
which is a constant of motion. The second term is negative of the binding energy.
Regards,
Phil
Dear Phillip Wayne Dennis,
Thanks for your contribution. The expression
M = mu - mu^2/ 2 R_max
is just the Newtonian gravitational binding energy with the addition of the rest mass mu. Thus, GR and Newtonian gravity coincide in this case.
Unfortunately, such coincidence indicates that this example is not useful: there is no loss of energy by gravitational waves. Indeed, no such loss is possible in any collapse (or process) that preserves a spherical symmetry (Misner, Thorne and Wheeler 32.2).
Dear Jose Gaite ,
Thanks for your comments. I agree that there is a formal resemblance to Newtonian physics. But the Israel formula is fully relativistic. I suggest that considerations that mitigate against viewing the GR formula in Newtonian terms is that: (1) in GR there is no gravitational work function as used in Newtonian physics. (2) In Newtonian physics it is the case that M = mu. There is no alteration of gravitational mass due to binding energies (also no alteration due to kinetic energy as in the Israel EoM). Newtonian gravity is linear. The interaction term in the Israel equation is due to the non-linearity of the gravitational field in GR (and was obtained by solving the full Einstein field equations).
I certainly agree that the spherical thin shell has no gravitational radiation as there is no quadrapole. Nonetheless there is a decrease of gravitational mass, but no loss of inertial mass. And it is provable within GR that M > 0, it cannot be reduced to zero.
If the shell is not spherical there can be radiation. Question is does the radiation radiate all of the multipole moments reducing the shell to a sphere at which point the residual gravitational mass would still be greater than zero.
Regards,
Phil
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