The Blak Hole, the Big Bang A Cosmology in Crisis Stephen J. Crothers Queensland, Australia thenarmis@gmail.om Introdution It is often laimed that osmology beame a true sientifi inquiry with the advent of the General Theory of elativity. A few subsequent putative observations have been misonstrued in suh a way as to support the prevailing Big Bang model by whih the Universe is alleged to have burst into existene from an infinitely dense point-mass singularity. Yet it an be shown that the General Theory of elativity and the Big Bang model are in onflit with well-established experimental fats. Blak holes are not without osmologial signifiane in view of the many laims routinely made for them, and so they are treated here in some detail. But the theory of blak holes is riddled with ontraditions and has no valid basis in observation. Nobody has ever found a blak hole, even though laims for their disovery are now made on an almost daily basis. Nobody has ever found an infinitely dense point-mass singularity and nobody has ever found an event horizon, the tell-tale signatures of the blak hole, and so nobody has ever found a blak hole. In atuality, astrophysial sientists merely laim that there are phenomena observed about a region that they annot see and so they illogially onlude that the unseen region must be a blak hole, simply beause they believe in blak holes. In this way they an and do laim the presene of a blak hole as they please. But that is not how siene is properly done. Moreover, all blak hole solutions pertain to one alleged mass in the Universe, whereas there are no known solutions to Einstein s field equations for two or more masses, suh as two blak holes. In other words, the astrophysis ommunity has no solution to Einstein s field equations that an aount for the presene of two or more bodies, yet they laim the existene of blak holes in multitudes, interating with one another and other matter. Owing to the very serious problems with the Big Bang hypothesis and the theory of blak holes, it is fair to say that neither meets the requirements of a valid physial theory. They are produts of a peer review system that has gone awry, having all the harateristis of a losed aademi lub of mutual admiration and benefit into whih new members are stritly by invitation only. The upshot of this is that the majority of the urrent astrophysis ommunity is imbued with the dogmas of the aademi lub and the voie of dissent onveniently ignored or ridiuled, ontrary to the true spirit of sientifi inquiry. This method has proteted funding interests but has done muh harm to siene. Infinite Density Forbidden Like the Big Bang progenitor, the blak hole is alleged to possess an infinitely dense point-mass singularity. The blak hole singularity is said to be produed by irresistible gravitational ollapse (see for example [1-6]). Aording to Hawking [5]: 1
The work that oger Penrose and I did between 1965 and 1970 showed that, aording to general relativity, there must be a singularity of infinite density, within the blak hole. Dodson and Poston [1] assert: One a body of matter, of any mass m, lies inside its Shwarzshild radius m it undergoes gravitational ollapse... and the singularity beomes physial, not a limiting fition. Aording to Carroll and Ostlie [3], A nonrotating blak hole has a partiularly simple struture. At the enter is the singularity, a point of zero volume and infinite density where all of the blak hole s mass is loated. Spaetime is infinitely urved at the singularity.... The blak hole s singularity is a real physial entity. It is not a mathematial artifat... eall that an inertial frame is just somewhere Newton s First Law holds: A body will remain at rest or move in a straight line with a onstant veloity unless ated upon by an outside fore. Einstein s postulates for Speial elativity are: 1. The speed of light in vauum is the same for all inertial frames;. The laws of physis are the same for all inertial frames. It follows from these two postulates that infinite density is forbidden beause infinite energy is forbidden, or equivalently, beause no material body an aquire the speed of light in vauo. General elativity annot violate Speial elativity sine the former is a said to be a generalisation of the latter; so it too forbids infinite density. That infinite density is forbidden by the Theory of elativity is easily proven with nothing more than simple high shool algebra. Aording to Einstein absolute motion does not exist; only the relative motion between bodies is meaningful. Consider two masses M o and m o at rest, i.e. their relative veloity is zero. These masses are therefore alled rest masses. Let both masses be uboid in shape, sides L o and X o respetively. The rest volumes are just L 3 o and X 3 o respetively. Now if the relative veloity has magnitude v > 0, from the perspetive of mass M o the other mass inreases by mo m (1) v 1 where is the speed of light in vauo. In addition, from the perspetive of mass M o the length of the side of the other mass, in the diretion of motion, is dereased by
v X X o 1 () The other sides of the mass m o do not hange. So mass M o sees a volume v 1. 3 V X o Now reall that density D is the mass divided by the volume. Hene, the density mass M o sees is, m D V mo. (3) v X 3 o 1 These three relations are reiproal, i.e. the perspetive of m o is desribed by the same equations exept that M o and L o replae m o and X o in them, so it doesn t matter who wathes whom; the results are the same. Now note that aording to eq. (3), as v, D. Sine, aording to Speial elativity, no material objet an aquire the speed (this would require an infinite energy), infinite densities are forbidden by Speial elativity, and so point-mass singularities are forbidden. Sine Speial elativity must manifest in suffiiently small regions of Einstein s gravitational field, and these regions an be loated anywhere in the gravitational field, General elativity too must thereby forbid infinite densities and hene forbid point-mass singularities. It does not matter how it is alleged that a point-mass singularity is generated by General elativity beause the infinitely dense point-mass annot be reoniled with Speial elativity. Point-harges too are therefore forbidden by the Theory of elativity sine there an be no harge without mass. The Signatures of a Blak Hole The signatures of the blak hole are an infinitely dense point-mass singularity and an event horizon. But we have already seen that infinite density is forbidden by the Theory of elativity. So the laim for an infinitely dense point-mass singularity is false. This result is suffiient to prove that blak holes are not predited by General elativity at all. In an attempt to esape this dilemma, astrophysial sientists are quik to resort to the argument that at the singularity General elativity breaks down, and so it annot desribe what happens there, so that some kind of quantum theory of gravity is needed. Nonetheless, the blak hole singularity is still said to be infinitely dense. If General elativity breaks down at the alleged singularity, as they laim, then General elativity annot say anything about the singularity, let alone that it is infinitely dense. And there is no quantum theory of gravity to desribe it or anything else gravitational. So the singularity is either infinitely dense, as they laim, or it annot be desribed by General elativity, whih breaks down there, as they also laim. It an t be both, either at the same time or at different times, aording to 3
fany. But in either ase it is inonsistent with the Theory of elativity sine infinite density is stritly forbidden by the Theory. It is noteworthy at this point that Newton s theory of gravitation does not predit blak holes either, although it is often laimed that it does, in some form or another: we will ome bak to this point later. What about the event horizon of the blak hole? Aording to the theory of blak holes it takes an infinite amount of time for an observer to wath an objet (via the light from that objet, of ourse) to fall down to the event horizon. It therefore takes an infinite amount of time for the observer to verify the existene of an event horizon and thereby onfirm the presene of a blak hole. However, nobody has been and nobody will be around for an infinite amount of time in order to verify the presene of an event horizon and hene the presene of a blak hole. Nevertheless, sientists laim that blak holes have been found all over the plae. The fat is, nobody has assuredly found a blak hole anywhere - no infinitely dense point-mass singularity and no event horizon. Some blak hole proponents are more irumspet in how they laim the disovery of their blak holes. They instead say that their evidene for the presene of a blak hole is indiret. But suh indiret evidene annot be used to justify the laim of a blak hole, in view of the fatal ontraditions and physially meaningless properties assoiated with infinitely dense point-mass singularities and event horizons. One ould just as well assert the existene and presene of deep spae uniorns on the basis of suh indiret evidene. Some laim that the energy of a blak hole of mass m is E m. But then they have an infinite density assoiated with a finite energy, whih violates Speial elativity one again. The onept of point-mass is rather meaningless. A point is a mathematial objet, not a physial objet. Thus, a point has no mass and a mass is not a point. One annot go to a shop and buy a bag full of points, but one an buy a bag full of marbles. Nonetheless, the astrophysis ommunity would have us believe that points an be material and of infinite density. The point-mass is a onfounding of a mathematial objet with a physial objet and is therefore invalid. It is also of great importane to be mindful of the fat that no observations gave rise to the notion of a blak hole in the first plae, for whih a theory had to be developed. The blak hole was wholly spawned in the reverse, i.e. it was reated by theory and observations subsequently misonstrued to legitimize the theory. eports of blak holes all over the plae is just wishful thinking in support of a belief; not fatual in any way. Blak Hole Esape Veloity It is widely held by astrophysiists and astronomers that a blak hole has an esape veloity (or, the speed of light in vauo) [4, 5, 7-18]. Chandrasekhar [4] remarked, 4
Let me be more preise as to what one means by a blak hole. One says that a blak hole is formed when the gravitational fores on the surfae beome so strong that light annot esape from it.... A trapped surfae is one from whih light annot esape to infinity. Aording to Hawking [5], Eventually when a star has shrunk to a ertain ritial radius, the gravitational field at the surfae beomes so strong that the light ones are bent inward so muh that the light an no longer esape. Aording to the theory of relativity, nothing an travel faster than light. Thus, if light annot esape, neither an anything else. Everything is dragged bak by the gravitational field. So one has a set of events, a region of spaetime from whih it is not possible to esape to reah a distant observer. Its boundary is alled the event horizon. It oinides with the paths of the light rays that just fail to esape from the blak hole. A neutron star has a radius of about ten miles, only a few times the ritial radius at whih a star beomes a blak hole. I had already disussed with oger Penrose the idea of defining a blak hole as a set of events from whih it is not possible to esape to a large distane. It means that the boundary of the blak hole, the event horizon, is formed by rays of light that just fail to get away from the blak hole. Instead, they stay forever hovering on the edge of the blak hole. However, aording to the alleged properties of a blak hole, nothing at all an even leave the blak hole. In the very same paper Chandrasekhar made the following quite typial ontraditory assertion: The problem we now onsider is that of the gravitational ollapse of a body to a volume so small that a trapped surfae forms around it; as we have stated, from suh a surfae no light an emerge. Hughes [15] reiterates, Things an go into the horizon (from r > M to r < M), but they annot get out; one inside, all ausal trajetories (timelike or null) take us inexorably into the lassial singularity at r 0. The defining property of blak holes is their event horizon. ather than a true surfae, blak holes have a one-way membrane through whih stuff an go in but annot ome out. Taylor and Wheeler [19] assert,... Einstein predits that nothing, not even light, an be suessfully launhed outward from the horizon... and that light launhed outward EXACTLY at the horizon will never inrease its radial position by so muh as a millimeter. 5
In the Ditionary of Geophysis, Astrophysis and Astronomy [9], one finds the following assertions: blak hole A region of spaetime from whih the esape veloity exeeds the veloity of light. In Newtonian gravity the esape veloity from the gravitational pull of a spherial star of mass M and radius is GM v es, where G is Newton s onstant. Adding mass to the star (inreasing M), or ompressing the star (reduing ) inreases v es. When the esape veloity exeeds the speed of light, even light annot esape, and the star beomes a blak hole. The required radius BH follows from setting v es equal to : GM BH... In General elativity for spherial blak holes (Shwarzshild blak holes), exatly the same expression BH holds for the surfae of a blak hole. The surfae of a blak hole at BH is a null surfae, onsisting of those photon trajetories (null rays) whih just do not esape to infinity. This surfae is also alled the blak hole horizon. A. Guth [0] tells us,... lassially the gravitational field of a blak hole is so strong that not even light an esape from its interior... And aording to the Collins Enylopaedia of the Universe [17], blak hole A massive objet so dense that no light or any other radiation an esape from it; its esape veloity exeeds the speed of light. Now, if its esape veloity is really that of light in vauo, then, by definition of esape veloity, light would esape from a blak hole, and therefore be seen by all observers. If the esape veloity of the blak hole is greater than that of light in vauo, then light ould emerge but not esape; there would therefore always be a lass of observers that ould see it. Not only that, if the blak hole had an esape veloity, then material objets with an initial veloity less than the alleged esape veloity ould leave the blak hole, and therefore be seen by a lass of observers, but not esape (just go out, ome to a stop and then fall bak), even if the esape veloity is. Esape veloity does not mean that objets annot leave; it only means they annot esape if they have an initial veloity less than the esape veloity. Hene, on the one hand it is alleged that blak holes have an esape veloity, but on the other hand that nothing, inluding light, an even leave the blak hole. The laims are ontraditory - nothing but a meaningless play on the words esape veloity [4, 5]. Furthermore, esape veloity is a two-body onept (one body esapes from another), whereas the blak hole is derived not from a two-body gravitational interation, but from an alleged one- 6
body onept (but whih is in fat a no-body situation). The blak hole has no esape veloity. The Mihell-Laplae Dark Body It is also routinely asserted that the theoretial Mihell-Laplae (M-L) dark body of Newton s theory, whih has an esape veloity, is a kind of blak hole [3-5, 7, 9, 10, ] or that Newton s theory somehow predits the radius of a blak hole [19]. Hawking remarks [5], On this assumption a Cambridge don, John Mihell, wrote a paper in 1783 in the Philosophial Transations of the oyal Soiety of London. In it, he pointed out that a star that was suffiiently massive and ompat would have suh a strong gravitational field that light ould not esape. Any light emitted from the surfae of the star would be dragged bak by the star s gravitational attration before it ould get very far. Mihell suggested that there might be a large number of stars like this. Although we would not be able to see them beause light from them would not reah us, we ould still feel their gravitational attration. Suh objets are what we now all blak holes, beause that is what they are blak voids in spae. In the Cambridge Illustrated History of Astronomy [3] it is asserted that, Eighteenth-entury speulators had disussed the harateristis of stars so dense that light would be prevented from leaving them by the strength of their gravitational attration; and aording to Einstein s General elativity, suh bizarre objets (today s blak holes ) were theoretially possible as end-produts of stellar evolution, provided the stars were massive enough for their inward gravitational attration to overwhelm the repulsive fores at work. But the M-L dark body is not a blak hole. The M-L dark body possesses an esape veloity, whereas the blak hole has no esape veloity. Objets an leave the M-L dark body, but nothing an leave the blak hole. There is no upper limit of the speed of a body in Newton s theory, so masses an always esape from the M-L dark body, provided they leave at or greater than the esape veloity. The M-L dark body does not require irresistible gravitational ollapse, whereas the blak hole does. It has no infinitely dense point-mass singularity, whereas the blak hole does. It has no event horizon, whereas the blak hole does. There is always a lass of observers that an see the M-L dark body [4, 5], but there is no lass of observers that an see the blak hole. The M-L dark body an persist in a spae whih ontains other matter and interat with that matter, but the spaetime of the blak hole is devoid of other masses by onstrution and onsequently annot interat with anything. Thus, the M-L dark body does not possess the harateristis of the hypothesized blak hole and so it is not a blak hole. Gravitational Collapse Muh of the justifiation for the notion of irresistible gravitational ollapse into an infinitely dense point-mass singularity, and hene the formation of a blak hole, is given to the analysis due to Oppenheimer and Snyder [30]. Hughes [15] relates it as follows; 7
In an idealized but illustrative alulation, Oppenheimer and Snyder... showed in 1939 that a blak hole in fat does form in the ollapse of ordinary matter. They onsidered a star onstruted out of a pressureless dustball. By Birkhof s Theorem, the entire exterior of this dustball is given by the Shwarzshild metri... Due to the self-gravity of this star, it immediately begins to ollapse. Eah mass element of the pressureless star follows a geodesi trajetory toward the star s enter; as the ollapse proeeds, the star s density inreases and more of the spaetime is desribed by the Shwarzshild metri. Eventually, the surfae passes through. At this point, the Shwarzshild exterior inludes an event horizon: A blak hole has formed. Meanwhile, the matter whih formerly onstituted the star ontinues ollapsing to ever smaller radii. In short order, all of the original matter reahes and is ompressed (lassially!) into a singularity. Sine all of the matter is squashed into a point of zero size, this lassial singularity must be modified in a omplete, quantum desription. However, sine all the singular nastiness is hidden behind an event horizon where it is ausally disonneted from us, we need not worry about it (at least for astrophysial blak holes). Note that the Priniple of Superposition has been arbitrarily applied by Oppenheimer and Snyder, from the outset. They first assume a relativisti universe in whih there are multiple mass elements present a priori, where the Priniple of Superposition however, does not apply, and despite there being no solution or existene theorem for suh onfigurations of matter in General elativity. Then, all these mass elements ollapse into a entral point (zero volume; infinite density). However, suh a ollapse has not been given any speifi general relativisti mehanism in this argument; it is simply asserted that the ollapse is due to self-gravity. But the ollapse annot be due to Newtonian gravitation, given the resulting blak hole, whih does not our in Newton s theory of gravitation. A Newtonian universe annot ollapse into a non-newtonian universe. Moreover, the blak hole so formed is in an empty universe, sine the Shwarzshild blak hole relates to i 0, a spaetime that by onstrution ontains no matter. Nonetheless, Oppenheimer and Snyder permit, within the ontext of General elativity, the presene of and the gravitational interation of many mass elements, whih oalese and ollapse into a point of zero volume to form an infinitely dense point-mass singularity, when there is no demonstrated general relativisti mehanism by whih any of this an our. Furthermore, nobody has ever observed a elestial body undergo irresistible gravitational ollapse and there is no laboratory evidene whatsoever for suh a phenomenon. Einstein s Field Equations In this setion we will have need of the so-alled Shwarzshild solution : 1 Gm Gm ds dt dr r ( dθ + sin θ dϕ ). r r (4) The quantity in expression (4) is the speed of light in vauum and G is Newton s onstant of universal gravitation. 8
The omponents of the metri tensor are easily read off expression (4), as follows: Gm Gm g00, g11, g r, g33 r sin θ. r r The signature is the ordered set of the signs of the omponents of the metri tensor. In the above ase the signature is ( +,,, ). Some writers use the equivalent signature, +, +, + and write the metri as ( ) 1 Gm Gm ds dt + dr + r ( dθ + sin θ dϕ ). r r In either ase the signature is orrespondingly fixed and annot hange. The astrophysis ommunity usually sets 1 and G 1 in equation (4) to get 1 ( dθ sin θ d ) 1 m m 1 dt 1 dr r ϕ ds + r r (5) with orresponding hanges in the expressions for the omponents of the metri tensor. The quantity m in these expressions is alleged to be mass the mass ausing the assoiated gravitational field. The quantity r appearing in these expressions has never been orretly identified by the theoretiians, as we shall soon see. It is from expressions (4) and (5) that the blak hole was first onjured. Note that only one mass appears in these expressions and that the said mass m is alone in the Universe and so the assoiated blak hole is alone in the Universe. This expression is not a solution for two or more masses. Nonetheless, the astrophysis ommunity talks of the existene of blak holes in multitudes. It should be noted that neither expression (4) nor expression (5) is in fat Shwarzshild s solution. They are a orruption of Shwarzshild s solution, due to the mathematiian David Hilbert. For omparison here is Shwarzshild s atual solution: 1 α α ds 1 dt 1 d ( dθ + sin θ dϕ ), 3 3 ( r + α ) 1 3, 0 < r <, onst. α There is no blak hole in Shwarzshild s solution. Indeed, his solution preludes the blak hole, and for this reason he never spoke of the blak hole. We will also have need of the following expression for Minkowski spaetime, in whih the laws of Speial elativity operate: One again it is usual to set 1 to get, ds dt dr r ( dθ + sin θ dϕ ). 9
ds dt dr r ( dθ + sin θ dϕ ). (6) Note there is no appearane of any quantity representing matter in this expression (alled a metri). It ontains no matter. The quantity is a speed, not a photon, the maximum speed that a moving point is allowed to aquire in this geometry. Note that the signature of Minkowski spaetime is ( +,,, ) and annot hange: for instane, to, +,, (otherwise it would not be Minkowski spaetime). ( ) The foregoing expressions are alled line-elements, or metris, whih are nothing but fany names for a distane equation, like that learnt in high shool. In the foregoing expressions ds denotes an element of distane in spae-time. In eah metri ds is made up of a time-like quantity, t, and three spae-like quantities, r,,. In this way it is laimed in astrophysial irles that time t is the fourth dimension of a 4- dimensional spae-time ontinuum. The terms ontaining r, and olletively onstitute the spatial setion of the spae-time. Thus, in expression (6) the spatial setion is desribed by d ρ dr + r ( dθ + sin θ dϕ ). In the ase of expression (5) the spatial setion is given by ( dθ sin θ d ) 1 m 1 dr + r ϕ d ρ + r In the usual interpretation of Hilbert s [34-37] orrupted version of Shwarzshild s solution, the quantity r has never been properly identified by astrophysis. It has been variously and vaguely alled a distane [7, 38] the radius [1,, 7, 8, 9-14, 1, 8, 39, 40-4], the radius of a -sphere [4, 43] the oordinate radius [44], the radial oordinate [3, 4, 14, 19, 8, 45], the Shwarzshild r-oordinate [45], the radial spae oordinate [46], the areal radius [3, 4, 15,, 44], the redued irumferene [19], and even a gauge hoie: it defines the oordinate r. In the partiular ase of r Gm / it is almost invariably referred to as the Shwarzshild radius or the gravitational radius [45]. However, none of these various and vague onepts of r are orret beause the irrefutable geometrial fat is that r, in the spatial setion of Hilbert s version of the Shwarzshild/Droste line-elements, is the inverse square root of the Gaussian urvature (see Appendix) of the spherially symmetri geodesi surfae in the spatial setion [3-34], and as suh, it does not itself determine the geodesi radial distane from the entre of spherial symmetry loated at an arbitrary point in the related pseudo-iemannian metri manifold. It does not denote any distane at all in the spherially symmetri metri manifold for Shwarzshild spaetime. It must also be emphasized that a geometry is ompletely determined by the form of its line-element [50, 51], of whih signature is a harateristi. The orret geometri identifiation of the quantity r in Hilbert s solution ompletely subverts all laims for blak holes. Aording to Einstein, matter is the ause of the gravitational field and the ausative matter is desribed in his theory by a mathematial objet alled the energymomentum tensor, whih is oupled to geometry (i.e. spaetime) by his field. 10
equations, so that matter auses spaetime urvature (his gravitational field) and spaetime onstrains motion of matter when gravity alone ats. Aording to the astrophysis ommunity, Einstein s field equations,... ouple the gravitational field (ontained in the urvature of spaetime) with its soures. [31] Sine gravitation is determined by the matter present, the same must then be postulated for geometry, too. The geometry of spae is not given a priori, but is only determined by matter. [5] Again, just as the eletri field, for its part, depends upon the harges and is instrumental in produing mehanial interation between the harges, so we must assume here that the metrial field (or, in mathematial language, the tensor with omponents g ) is related to the material filling the world. [38] ik... we have, in following the ideas set out just above, to disover the invariant law of gravitation, aording to whih matter determines the omponents Γ of the gravitational field, and whih replaes the Newtonian law of attration in Einstein s Theory. [38] Thus the equations of the gravitational field also ontain the equations for the matter (material partiles and eletromagneti fields) whih produes this field. [9] Clearly, the mass density, or equivalently, energy density ρ ( x,t) must play the role as a soure. However, it is the 00 omponent of a tensor T ( x), the mass-energymomentum distribution of matter. So, this tensor must at as the soure of the gravitational field. [41] In general relativity, the stress-energy or energy-momentum tensor T ab ats as the soure of the gravitational field. It is related to the Einstein tensor and hene to the urvature of spaetime via the Einstein equation. [8] Mass ats on spaetime, telling it how to urve. Spaetime in turn ats on mass, telling it how to move. [3] Qualitatively Einstein's field equations are: Spaetime geometry - ausative matter where ausative matter is desribed by the energy-momentum tensor and is a onstant. The spaetime geometry is desribed by a mathematial objet alled Einstein s tensor, G, (, 0, 1,, 3) and the energy-momentum tensor is T. Einstein s field equations are therefore 1 : G 1 g. (7) κt 1 The so-alled osmologial onstant is not inluded. α βι 11
is alled the ii tensor and the ii urvature. If T 0 then one finds that 0 and this expression allegedly redues to and desribes a universe that ontains no matter. 0 (8) In the transition from Minkowski spaetime of Speial elativity to Shwarzshild spaetime for the blak hole, matter is not involved. The speed of light that appears in the Minkowski spaetime line-element is a speed, not a photon. For this speed to be assigned to a photon, the photon must be present a priori. Similarly, for the relations of Speial elativity to hold, multiple arbitrarily large finite masses must also be present a priori. Minkowski spaetime is not Speial elativity beause the latter requires the presene of matter, whereas the former does not. Similarly, the presene of the onstant in the line-element for Shwarzshild spaetime does not mean that a photon is present. The transition from Minkowski spaetime to Shwarzshild spaetime is thus not a generalisation of Speial elativity at all, merely a generalisation of the geometry of Minkowski spaetime. In the usual derivation of Shwarzshild spaetime, mass is inluded by means of a sophisti argument, viz. 0 desribes the gravitational field outside a body. When one inquires of the astrophysis ommunity as to what is the soure of this alleged gravitational field outside a body, one is told that it is the body, in whih ase the body must be desribed by a non-zero energy-momentum tensor sine Einstein s field equations ouple the gravitational field with its soures [31]. Dira [3] tells us that the onstant of integration m that has appeared is just the mass of the entral body that is produing the gravitational field. We are told by Einstein [53] that, M denotes the sun's mass entrally symmetrially plaed about the origin of oordinates. Aording to Weyl [38], the quantity m o introdued by the equation mkm o ours as the field-produing mass in it; we all m the gravitational radius of the matter ausing the disturbanes of the field. Foster and Nightingale [31] assert that the orresponding Newtonian potential is V GM / r, where M is the mass of the body produing the field, and G is the gravitational onstant we onlude that k GM / and Shwarzshild s solution for the empty spae outside a spherial body of mass M is After the Shwarzshild solution is obtained there is no matter present. This is beause the energy-momentum tensor is set to zero and Minkowski spaetime is not 1
Speial elativity. The astrophysis ommunity merely inserts (Weyl says introdued ) mass and photons by erroneously appealing to Newton's theory through whih they also get any number of masses and any amount of radiation by applying the Priniple of Superposition. This is done despite the fat that the Priniple of Superposition does not apply in General elativity. However, Newton s relations, as explained above, involve two bodies and the Priniple of Superposition. Conversely, 0 ontains no bodies and annot aommodate the Priniple of Superposition. The astrophysis ommunity removes all matter on the one hand by setting 0 and then puts it bak in at the end with the other hand by means of Newton s theory. The whole proedure onstitutes a violation of elementary logi. Einstein asserted that his Priniple of Equivalene and his laws of Speial elativity must hold in suffiiently small regions of his gravitational field, and that these regions an be loated anywhere in his gravitational field. Here is what Einstein [53] said in 1954, the year before his death: Let now K be an inertial system. Masses whih are suffiiently far from eah other and from other bodies are then, with respet to K, free from aeleration. We shall also refer these masses to a system of o-ordinates K, uniformly aelerated with respet to K. elatively to K all the masses have equal and parallel aelerations; with respet to K they behave just as if a gravitational field were present and K were unaelerated. Overlooking for the present the question as to the ause of suh a gravitational field, whih will oupy us later, there is nothing to prevent our oneiving this gravitational field as real, that is, the oneption that K is at rest and a gravitational field is present we may onsider as equivalent to the oneption that only K is an allowable system of o-ordinates and no gravitational field is present. The assumption of the omplete physial equivalene of the systems of oordinates, K and K, we all the priniple of equivalene ; this priniple is evidently intimately onneted with the law of the equality between the inert and the gravitational mass, and signifies an extension of the priniple of relativity to oordinate systems whih are in non-uniform motion relatively to eah other. In fat, through this oneption we arrive at the unity of the nature of inertia and gravitation. For, aording to our way of looking at it, the same masses may appear to be either under the ation of inertia alone (with respet to K) or under the ombined ation of inertia and gravitation (with respet to K ). Stated more exatly, there are finite regions, where, with respet to a suitably hosen spae of referene, material partiles move freely without aeleration, and in whih the laws of speial relativity, whih have been developed above, hold with remarkable auray. In their textbook, Foster and Nightingale [31] suintly state the Priniple of Equivalene thus: We may inorporate these ideas into the priniple of equivalene, whih is this: In a freely falling (nonrotating) laboratory oupying a small region of spaetime, the laws of physis are the laws of speial relativity. Aording to Pauli [5], 13
We an think of the physial realization of the loal oordinate system in terms of a freely floating, suffiiently small, box whih is not subjeted to any external fores apart from gravity, and whih is falling under the influene of the latter.... It is evidently natural to assume that the speial theory of relativity should remain valid in. Taylor and Wheeler state in their book [19], General elativity requires more than one free-float frame. Carroll and Ostlie write [3], The Priniple of Equivalene: All loal, freely falling, nonrotating laboratories are fully equivalent for the performane of all physial experiments. Note that speial relativity is inorporated into the priniple of equivalene. Thus general relativity is in fat an extension of the theory of speial relativity. In the Ditionary of Geophysis, Astrophysis and Astronomy [9], Near every event in spaetime, in a suffiiently small neighborhood, in every freely falling referene frame all phenomena (inluding gravitational ones) are exatly as they are in the absene of external gravitational soures. Note that the Priniple of Equivalene involves the a priori presene of multiple arbitrarily large finite masses. Similarly, the laws of Speial elativity involve the a priori presene of at least two arbitrarily large finite masses (and at least one photon); for otherwise relative motion between two bodies annot manifest. The postulates of Speial elativity are themselves ouhed in terms of inertial systems, whih are in turn defined in terms of mass via Newton s First Law of motion. Shwarzshild s solution (and indeed all blak hole solutions ), pertains to one mass in a universe that ontains no other masses. Aording to the astrophysis ommunity, Shwarzshild spaetime onsists of one mass in an otherwise totally empty universe, and so its alleged blak hole is the only matter present - it has nothing to interat with, inluding observers (on the assumption that any observer is material). In the spae of Newton s theory of gravitation, one an insert into spae as many masses as desired. Although solving for the gravitational interation of these masses rapidly beomes intratable, there is nothing to prevent us inserting masses oneptually. This is essentially the Priniple of Superposition. However, one annot do this in General elativity, beause Einstein s field equations are non-linear. In General elativity, eah and every onfiguration of matter must be desribed by a orresponding energy-momentum tensor and the field equations solved separately for eah and every onfiguration, beause matter and geometry are oupled, as eq. (7) desribes. This is not the ase in Newton s theory, where geometry is independent of matter. The Priniple of Superposition does not apply in General elativity: In a gravitational field, the distribution and motion of the matter produing it annot at all be assigned arbitrarily --- on the ontrary it must be determined (by solving the field equations for given initial onditions) simultaneously with the field produed by the same matter. [9] 14
An important harateristi of gravity within the framework of general relativity is that the theory is nonlinear. Mathematially, this means that if g ab and ab are two solutions of the field equations, then ag ab + b ab (where a, b are salars) may not be a solution. This fat manifests itself physially in two ways. First, sine a linear ombination may not be a solution, we annot take the overall gravitational field of the two bodies to be the summation of the individual gravitational fields of eah body. [8] The astrophysis ommunity laims that the gravitational field outside a mass ontains no matter, and thereby asserts that the energy-momentum tensor. Despite this, it is routinely alleged that there is only one mass in the whole Universe with this partiular problem statement. But setting the energy-momentum tensor to zero means that there is no matter present by whih the gravitational field an be aused! As we have seen, when the energy-momentum tensor is set to zero, it is also laimed that the field equations then redue to the muh simpler form, i 0. Blak holes were first disovered as purely mathematial solutions of Einstein s field equations. This solution, the Shwarzshild blak hole, is a nonlinear solution of the Einstein equations of General elativity. It ontains no matter, and exists forever in an asymptotially flat spae-time. [9] However, sine this is a spaetime that by onstrution ontains no matter, Einstein s Priniple of Equivalene and his laws of Speial elativity annot manifest, thus violating the physial requirements of the gravitational field. It has never been proven that Einstein s Priniple of Equivalene and his laws of Speial elativity, both of whih are defined in terms of the a priori presene of multiple arbitrary large finite masses and photons, an manifest in a spaetime that by onstrution ontains no matter. Now eq. (4) relates to eq. (8). However, there is allegedly mass present, denoted by m in eq. (4). This mass is not desribed by an energy-momentum tensor. The reality that the post ho mass m is responsible for the alleged gravitational field due to a blak hole assoiated with eq. (4) is onfirmed by the fat that if, eq. (4) redues to Minkowski spaetime, and hene no gravitational field aording to the astrophysis ommunity. If not for the presene of the alleged mass m in eq. (4) there would be no ause of their gravitational field. But this ontradits Einstein s relation between geometry and matter, sine m is introdued into eq. (4) post ho, not via an energy-momentum tensor desribing the matter ausing the assoiated gravitational field. In Shwarzshild spaetime, the omponents of the metri tensor are only funtions of one another, and redue to funtions of just one omponent of the metri tensor. None of the omponents of the metri tensor ontain matter, beause the energy-momentum tensor is zero. There is no transformation of matter in Minkowski spaetime into Shwarzshild spaetime, and so the laws of Speial elativity are not transformed into a gravitational field by i 0. The transformation is merely from a pseudo- Eulidean geometry ontaining no matter into a pseudo-iemannian (non-eulidean) geometry ontaining no matter. Matter is introdued into the spaetime of i 0 by means of a viious irle, as follows. It is stated at the outset that i 0 desribes 15
spaetime outside a body. The words outside a body introdue matter, ontrary to the energy-momentum tensor, T 0, that desribes the ausative matter as being absent. There is no matter involved in the transformation of Minkowski spaetime into Shwarzshild spaetime via i 0, sine the energy-momentum tensor is zero, making the omponents of the resulting metri tensor funtions solely of one another, and reduible to funtions of just one omponent of the metri tensor. To satisfy the initial laim that i 0 desribes spaetime outside a body, so that the resulting spaetime urvature is aused by the alleged mass present, the alleged ausative mass is inserted into the resulting metri ad ho. This is ahieved by means of a ontrived analogy with Newton s theory and his expression for esape veloity (a two-body relation in what is alleged to be a one-body problem see Appendix), thus losing the viious irle. Here is how Chandrasekhar [4] presents the viious irle: That suh a ontingeny an arise was surmised already by Laplae in 1798. Laplae argued as follows. For a partile to esape from the surfae of a spherial body of mass M and radius, it must be projeted with a veloity v suh that 1 v GM / > ; and it annot esape if v < GM /. On the basis of this last inequality, Laplae onluded that if < GM / s (say) where denotes the veloity of light, then light will not be able to esape from suh a body and we will not be able to see it! By a urious oinidene, the limit s disovered by Laplae is exatly the same that general relativity gives for the ourrene of the trapped surfae around a spherial mass. But it is not surprising that general relativity gives the same s disovered by Laplae beause the Newtonian expression for esape veloity is deliberately inserted post ho by the astrophysiists and astronomers, into the Shwarzshild solution (equation (4) above). Newton s esape veloity does not drop out of any of the alulations to Shwarzshild spaetime. Furthermore, although 0 is said to desribe spaetime outside a body, the resulting metri (4) is nonetheless used to desribe the interior of a blak hole, sine the blak hole begins at the alleged event horizon, not at its infinitely dense point-mass singularity (allegedly at r 0 in equation (4)). In the ase of a totally empty Universe, what would be the relevant energymomentum tensor? It must be T 0. Indeed, it is also maintained by the astrophysis ommunity that spaetimes an be intrinsially urved, i.e. that there are gravitational fields that have no material ause. An example is de Sitter s empty spherial Universe, based upon the following field equations [50, 54]: λg (9) where is the so-alled osmologial onstant. In the ase of metri (4) the field equations are given by expression (8). On the one hand, de Sitter s empty world is devoid of matter ( T 0 ) and therefore has no material ause for the alleged assoiated gravitational field. On the other hand, it is stated that the spaetime 16
desribed by eq. (8) has a material ause, post ho as m in metri (4), even though T 0 there as well: a ontradition. This is amplified by the so-alled Shwarzshild-de Sitter line-element, ( dθ sin θ d ) 1 m λ m λ ds 1 r dt 1 r dr r + ϕ r 3 r 3, (10) whih is the standard solution for eq. (9). One again, m is inserted post ho as mass at the entre of spherial symmetry of the manifold, said to be at. The ompletely empty universe of de Sitter [50, 54] an be obtained by setting in eq. (10) to yield, ( dθ sin θ d ) 1 λ λ 1 r dt 1 r dr r ϕ ds + 3 3. (11) Also, if λ 0, eq. (9) redues to eq. (8) and eq. (10) redues to eq. (5). If both λ 0 and m 0, eqs. (10) and (11) redue to Minkowski spaetime. Now in eq. (10), the term λg is not an energy-momentum tensor, sine aording to the astrophysis ommunity, expression (11) desribes a world devoid of matter [35, 41]. The universe desribed by eq. (11), whih also satisfies eq. (9), is ompletely empty and so its urvature has no material ause; in eq. (11), just as in eq. (9), T 0. Thus, eq. (11) is alleged to desribe a gravitational field that has no material ause. Furthermore, although in eq. (9), T 0, its usual solution, eq. (5), is said to ontain a (post ho) material ause, denoted by m therein. Thus, for eq. (5), it is postulated that T 0 supports a material ause of a gravitational field. At the same time, for eq. (11), T 0 preludes material ause of a gravitational field. T 0 therefore inludes and exludes material ause. This is not possible. The ontradition is due to the post ho introdution of mass, as m, in equations (4) and (5), by means of the Newtonian expression for esape veloity (whih is an impliit two-body relation). Furthermore, there is no experimental evidene to support the laim that a gravitational field an be generated without a material ause. Material ause is odified theoretially in eq. (7). Blak Hole Interations The literature abounds with laims that blak holes an interat in suh situations as binary systems, mergers, ollisions, and with surrounding matter generally. Bearing in mind that all blak holes solutions pertain to a universe that ontains only one mass (the blak hole itself), onepts involving multiple blak holes taitly assume appliation of the Priniple of Superposition, whih however, as we have seen, does not apply in General elativity. Aording to Chandrasekhar [4], From what I have said, ollapse of the kind I have desribed must be of frequent ourrene in the Galaxy; and blak-holes must be present in numbers omparable to, if not exeeding, those of the pulsars. While the blak-holes will not be visible to external observers, they an nevertheless interat with one another and with the outside world through their external fields. 17
In onsidering the energy that ould be released by interations with blak holes, a theorem of Hawking is useful. Hawking s theorem states that in the interations involving blak holes, the total surfae area of the boundaries of the blak holes an never derease; it an at best remain unhanged (if the onditions are stationary). Another example illustrating Hawking s theorem (and onsidered by him) is the following. Imagine two spherial (Shwarzshild) blak holes, eah of mass ½M, oalesing to form a single blak hole; and let the blak hole that is eventually left be, again, spherial and have a mass M. Then Hawking s theorem requires that π π 1 16 M 16 M 8 M π or M M. Hene the maximum amount of energy that an be released in suh a oalesene is M 1 1/ 0.93M. ( ) Hawking [5] says, Also, suppose two blak holes ollided and merged together to form a single blak hole. Then the area of the event horizon of the final blak hole would be greater than the sum of the areas of the event horizons of the original blak holes. Aording to Shutz [6],... Hawking s area theorem: in any physial proess involving a horizon, the area of the horizon annot derease in time.... This fundamental theorem has the result that, while two blak holes an ollide and oalese, a single blak hole an never bifurate spontaneously into two smaller ones. Blak holes produed by supernovae would be muh harder to observe unless they were part of a binary system whih survived the explosion and in whih the other star was not so highly evolved. Townsend [7] also arbitrarily and inorretly applies the Priniple of Superposition to obtain multiple harged blak hole (eissner-nordström) interations as follows: The extreme N in isotropi oordinates is where ds ( dρ + ρ Ω ) V dt + V d M V 1 +. ρ 18
This is a speial ase of the multi blak hole solution where ds partiular V dx dx ds V dt + V dx dx is the Eulidean 3-metri and V is any solution of V 0. In V N 1 + Σi1 M i x x i yields the metri for N extreme blak holes of masses M i at positions x i. Carroll and Ostlie remark [3], The best hope of astronomers has been to find a blak hole in a lose binary system. If a blak hole oaleses with any other objet, the result is an even larger blak hole. If one of the stars in a lose binary system explodes as a supernova, the result may be either a neutron star or a blak hole orbiting the ompanion star. the proedure for deteting a blak hole in a binary x-ray system is similar to that used to measure the masses of neutron stars in these systems. What is the fate of a binary x-ray system? As it reahes the endpoint of its evolution, the seondary star will end up as a white dwarf, neutron star, or blak hole. But Einstein s field equations are non-linear. Thus, despite the laims of the astrophysis ommunity, the Priniple of Superposition does not apply [4, 8, 9]. Therefore, before one an talk of blak hole binary systems and the like it must first be proven that the two-body system is theoretially well-defined by General elativity. This an be aomplished in only two ways: 1. Derivation of an exat solution to Einstein s field equations for the two-body onfiguration of matter; or. Proof of an existene theorem. However, there are no known solutions to Einstein s field equations for the interation of two (or more) masses (harged or not). Furthermore, no existene theorem has ever been proven, by whih Einstein s field equations an even be said to admit of latent solutions for suh onfigurations of matter. The Shwarzshild blak hole is allegedly obtained from a line-element satisfying i 0. For the sake of argument, assume that blak holes are predited by General elativity as alleged in relation to metri (5). Sine i 0 is a statement that there is no matter in the Universe, one annot simply insert a seond blak hole into the spaetime of i 0 of a given blak hole, so that the resulting two blak holes (eah obtained separately from i 0) simultaneously persist in, and mutually interat in, a mutual spaetime that by onstrution ontains no matter! One annot simply assert by an analogy with Newton s theory that two blak holes an be omponents of binary systems, ollide, or merge [4, 5, 9], again beause the Priniple of Superposition does not apply in Einstein s theory. Moreover, General elativity has to date been unable to aount for the simple experimental fat that two fixed bodies will approah one another upon release. Thus, blak hole binaries, ollisions, mergers, blak holes from supernovae, 19