Karl Brunstein

<wethinks@netidea.com>

Meadow Creek, British Columbia, Canada V0G 1N0

 

Abstract. Equivalents of the Schwarzschild metric that are static and that do not exhibit black holes are presented for the first time and discussed. Though they do not imply the actual, physical realization of a naked singularity, they nevertheless present one with a serious dilemma: How does one reconcile the simultaneous prediction and lack of prediction of a phenomenon by the same theory?

An Overview

            General Relativity is Einstein’s theory of gravity. In it, the four-dimensional spacetime of Special Relativity is distorted by a massive object, say, a star. It is through this distortion that the star manifests its gravitational field. Einstein’s field equations, as they are called, describe the distortion and gravitational field.

            The most important fundamental situation to be treated by the field equations is that of spherical objects of any mass. An exact solution for this class of problems was found years ago by Schwarzschild. The classical tests of General Relativity are based on his solution.

            The solution has a troublesome feature: an additional singularity, or “infinity”, at a point outside the one it shares with Newton’s gravitational law at . For years, no one thought of giving real, physical meaning to it, probably because singularities generally indicate a theory is inadequate in the region in question. A familiar example is the historically important singularity known as the “ultraviolet catastrophe,” predicted by classical blackbody radiation theory. Simple laboratory measurement made its absurdity clear. As one knows, it led in 1900 to Planck’s quantum hypothesis, an expedient solution that led to quantum theory.

            Interest in the Schwarzschild singularity lay, with one notable exception, nearly dormant for decades. In the 1960s, several individuals seriously began to ascribe physical reality to it. The school grew, their ideas finally blossoming with the “black hole”. In spite of claims to the contrary, no black holes have been discovered. Objects have been found that appear so massive that, according to this school, they must be the black holes expected by the theory.

            The discussion that follows demonstrates that the Schwarzschild singularity is in fact a flaw in General Relativity. The black hole concept leads one directly and inescapably into a gross internal inconsistency.

            To follow the discussion, one needs to understand the concept of metric. A spacetime is completely described by its metric, its four-dimensional analytic measure of “distance”. With the metric, one knows all about gravity and the properties of spacetime in a region. The metric is analogous to an expression for the hypotenuse of a right triangle, generalized to four dimensions, in that it expresses the “distance” between two neighboring event points in terms of four coordinate components. In the absence of gravity, Special Relativity holds. The Pythagorean theorem (extended to four dimensions) defines the metric, with either the time component an imaginary number, or the three spatial components imaginary, arbitrarily. Namely: in two dimensions, ; in four dimensions, either , or .

            Without changing the “distance”, , gravity modifies its distribution among the four components. [Because the time component  dominates the spatial components in most applications, it is conceptually easier, and helps avoid relativistic paradoxes, to think of  as a duration of “proper time” interval (time interval on a local clock affected by speed and/or gravity) rather than as some sort of “distance”. -- Ed.] The spacetime is distorted by the gravitating mass, or star. One is then in the domain of General Relativity. In all cases considered here, the effects enter into the metric as two to four variable coefficients, metric coefficients, in which the gravitating mass appears as a parameter. That is, two, three, or all four terms on the right side of the previous two expressions for  are multiplied by a function, a metric coefficient, in which the star’s mass appears.

            To determine the metric coefficients, one solves Einstein’s field equations. The 10 partial differential equations are cumbersome, so a utilitarian notation and conventions, a shorthand, is used. It employs both subscripts and superscripts, but to avoid ambiguity does not generally use exponents. To square  one writes , for example. The notation is not required in our discussion and is not used after Equation 1.

Introduction

            There is an ever-present danger in theoretical physics that appears to have reached palpable proportions in the field of classical General Relativity. It is the threat that the rising mountain of literature in the field will bury any essential elements that have been earlier overlooked, should they fail to support the dominant schools of thought at the top. Presented here are static solutions to Einstein’s field equations for an uncharged, non-rotating, spherically symmetric mass­­ – the problem treated by Schwarzschild many years ago – that exemplify this present-day counter-productive development. The metrics expressing these solutions do not exhibit black holes. They are fundamental to the understanding of General Relativity. Yet up until now, they are nowhere in print. [1]

            Because of this, many continue to entertain the misconception that General Relativity unavoidably predicts black holes. This is particularly so for the Schwarzschild problem. A few quotations will illustrate the pervasiveness of this belief and the importance attached to it because of its bearing on the issue of naked singularities.

            The general situation with regard to a spherically symmetrical body is well known … the body passes within its Schwarzschild radius . [2] No one who accepts general relativity has found any way to escape the prediction that black holes must exist. [3]

One such question (to be addressed, finally, by numerical methods) is the Cosmic Censorship conjecture and the appearance of naked singularities. This question is probably the most important open question in classical general relativity. [4] Whenever simple configurations of matter collapse according to the rules of general relativity, the collapsed region always seems to be enveloped in a black hole before a singularity forms. [5] [A singularity surrounded by an “event horizon” – defined later in this article – is called a “black hole”, and is responsible for “cosmic censorship” because information cannot get out. Its opposite, a singularity with no event horizon, is called “naked”. – Ed.] How a situation like this has been fostered is a curious and interesting topic, but not properly physical science. The intent here is to present a remedy to the widespread misconception that has resulted.

Essential Background

            To appreciate the discussion, one needs to have certain fundamentals freshly in mind: In General Relativity one assumes that a Riemannian metric may be assigned to spacetime, i.e.,

                                                                   ,                                                             

and that the  are determined by Einstein’s field equations plus the boundary conditions of the problem. For an uncharged, non-rotating, gravitating mass with spherical symmetry in otherwise-empty space, one has as one possible solution the Schwarzschild metric,

                        ,                  

given here in standard form for spherical polar coordinates with the gravitating mass centered at the origin. The mass distribution need not be time independent. For example, the object might pulsate in a spherically symmetric manner with the total mass-energy held constant. To lend simplicity to our subsequent discussion, we rewrite as

                                         ,                                   

where

                                                                                                                           

                                                                                                                         

                                                                                                                                               

            One can use to deduce readily that the velocity of light  in the radial direction, and  perpendicular to it, as inferred by an observer in flat spacetime, are

                                                                                              

                                                                                          

            Plausibly there could exist an object with radius less than , its “Schwarzschild radius”. The surface of a sphere defined by the radius  is labeled an “event horizon” in that . Clearly, no event inside the object’s Schwarzschild radius, or the event horizon, could be communicated to the outside. [6] One should have a Schwarzschild black hole.

            Solutions to the Schwarzschild problem are infinite in number. One might surmise this immediately by noting that yet another solution can be obtained from one previously known by an arbitrary transformation of the four coordinates. Birkhoff has indeed demonstrated that all solutions to the Schwarzschild problem, static or otherwise, are related in this way. [7] Moreover, there is nothing within General Relativity theory to specify which of the infinite solutions is uniquely “physically correct.” This feature of the theory has been universally understood, after Einstein, to reiterate the fact that the choice of coordinates is of no physical consequence. Pauli has stated, for example, “the many possible solutions of the field equations are only formally different. Physically they are completely equivalent.” [8]

            The conclusion proceeds directly from the general principle of relativity: If “all Gaussian coordinate systems are essentially equivalent for the formulation of the general laws of nature” [Einstein’s emphasis], [9] it follows that descriptions of physical events according to these laws must conform to the extent that there be no contradictions between systems with respect to the fundamental nature of the events. It is in this sense that the systems are equivalent. It can be further argued, “the great power possessed by the general principle of relativity lies in the comprehensive limitation which is imposed on the laws of nature in consequence.” [9]

 

 

Solutions Without Black Holes

            A one-parameter family of solutions to Einstein’s field equations for the Schwarzschild problem is given below. One substitutes expressions and into Equation to obtain the solutions in the form of the associated metrics. We have set :

                                                                                                                            

                                                                          

                                                                            

            In the metrics of , and , one has equivalently defined a new radial coordinate  with the relationship

                                               .                                       

Substituting for  in , , and brings the metric form to that characterized by , and , but in the coordinate  with ,  and  unchanged. The prime notation has been dropped. Boundary conditions continue to be satisfied if the exponential terms approach unity as  approaches infinity or  approaches zero, so that the metrics go over into that of flat spacetime for these limits. One can easily verify that this condition is met for . Therefore, one has an infinite number of solutions of this form.

            A particularly simple solution is obtained by setting . Relationships , and become:

                                                                                                                            

                                                                                                                                  

                                                                                                                              

The notable feature of the family of solutions characterized by , and is that they exhibit no black holes whatsoever for the infinite subset . [10] One sees this immediately in the particular solution associated with , and . [I.e., there are no infinities other than at . – Ed.]

            All of these solutions exhibit a singularity at the origin. This is not surprising because, in like manner with the standard and isotropic forms, they make use of coordinate systems in which the Newtonian approximation follows in that  is asymptotic to  as  approaches infinity. It should be recognized that the uncertainty principle by itself casts serious doubt on the possibility of a physical singularity developing at the site of this mathematical one. [11] The singularity has merely pointed up the limitations of the theory. It inescapably brings down on one the fact that, if one could indeed momentarily wink at the uncertainty principle and localize the mass at , there would remain the problem of the gross violation of local flatness at that point, a problem these solutions share with both the isotropic and standard metric forms. As with them, one is stuck with the singularity at  with its completely arbitrary properties. To argue that there is a black hole there is untenable. To further argue that the singularity is somehow “within” or “enveloped” by the black hole compounds the blunder. If one wishes nevertheless, in face of this, to suppose a combined black hole and singularity, one is conjuring a physical phenomenon that for all observers is to be realized only after infinite time has elapsed – a concept wholly without meaning.

            As a spherical polar coordinate,  naturally takes on only positive values. It cannot be maintained that the range of physically accessible 3-space should somehow exceed this, that  can take on negative values, and that solutions restricted to positive values are “geodesically incomplete”, as has been suggested to me. To argue this is to give special status to coordinate systems such as those represented by the standard and isotropic metric forms, and thus to throw away the general principle of relativity. One could on this same basis as easily argue that the standard metric form summons up “geodesically fictitious” 3-space. (Nonetheless, we shall take up the opposite supposition, that  can in fact take on negative values, in the Appendix. We shall see, not surprisingly, that it leads to a gross physical inconsistency.) Again in Einstein’s words, “We shall introduce in the general theory of relativity arbitrary coordinates, , , , , which shall number uniquely the space-time points, so that neighboring events are associated with neighboring values of the coordinates; otherwise, the choice of coordinates is arbitrary.” [12] The only primacy that black-hole-exhibiting metrics can have is purely accidental – historical precedence, a logical defense that has not been valid in science since the demise of Scholasticism.

Discussion

            Existence and nonexistence cannot possibly be construed as equivalent formulations of the same event or phenomenon.

            General Relativity’s failure to ensure the existence of black holes does indeed remove much of the significance attached to the latter topic: “If it [cosmic censorship] does not hold, then the formation of a naked singularity during collapse would be a disaster for general relativity theory. In this situation, one cannot say anything precise about the future evolution of any region of space containing the singularity since new information could emerge from it in a completely arbitrary way.” [13] Clearly, in the coordinate systems under consideration, no black hole is formed--nor is any naked singularity likely to be physically realized. There is a disaster here, nevertheless. It is the simultaneous prediction and lack of prediction of a phenomenon by the same theory. At the same time, another door is apparently more widely opened, since there is still much that is germane that is likely to be forthcoming from the study of particle physics under the extreme conditions in the interior of a neutron star. [14]

            Bergmann, in discussing in very general terms issues that would seem to encompass this one, has suggested we take a hard, pragmatic approach to winnowing statements that follow from General Relativity by adopting the following criterion: “There exists a subset of physical variables, the ‘observables’, whose values are independent of the choice of coordinate system employed. Thus, any relationship between observables is ‘meaningful’, and conversely, these are the only relationships that are legitimate.” [15] Perhaps this stern waving away of General Relativity’s encompassing of the tentative phenomenon of black holes is premature; perhaps not. Brushing aside modesty for the moment, I propose instead a more “moderate” approach, a new physical principle, to be placed alongside the cosmic-censor hypothesis. Does General Relativity predict black holes? Yes and no. Voila, the “principle of indecision.”

Appendix

            Let us, for the sake of argument, momentarily accept the possibility that the solutions characterized by (9), (10) and (11) merely ignore the space “inside the black hole.” That is, let us suppose for the moment that negative values of  are physically meaningful in those solutions. One could argue that the area of the  surface is not zero for these solutions, so there is no reason to restrict r to positive values.

            This reasoning immediately produces a paradox. By allowing negative values of  in the metrics of , and , one has that the mass-independent “Newtonian” singularity is naturally at , while the mass-dependent singularity is at , inside the Newtonian singularity. This is of course just opposite the arrangement that follows from the standard or isotropic form. The mass-dependent or mass-independent qualities are clearly distinguishing physical characteristics, and to have them reverse order like this depending on one’s choice of coordinate system is a blatant absurdity. The hand is inside the glove in the one instance, and the glove is inside the hand in the other, arbitrarily.

            The widely used isotropic metric form reflects much these same considerations. For it, the area of the  surface also is not zero. The inference that  as a consequence takes on negative values breaks rather puzzling new ground. One has for the speed of light

                                         

Setting  (leading to a zero surface area) gives one an infinite value for . So one has, what, a white hole(?) at , which is inside a Newtonian singularity at , which is again inside a black hole at ?

References

[1] These solutions were included as an aside in an unpublished work by D.H. Menzel in 1975. I am indebted to Earle Whipple for making that work available to me.

[2] Penrose, R., “Gravitational Collapse and Space-Time Singularities”, Phys.Rev.Lett. 14, 57-59 (1965).

[3] Misner, C.W., Thorne, K.S. and Wheeler, J.A., Gravitation, Freeman, San Francisco, 620 (1973).

[4] Goldwirth, D.S., Ori, A. and Piran, T., “Cosmic Censorship and Numerical Relativity”, in Frontiers in Numerical Relativity, eds. C.R. Evans, L.S. Finn, and D.W. Hobill, Cambridge Univ. Press, Cambridge, 415 (1989).

[5] Shapiro, S.L. and Teukolsky, S.A., “Black Holes, Naked Singularities and Cosmic Censorship”, Amer.Scientist 79, 330-343 (1991).

[6] Oppenheimer, J.R. and Snyder, H., “On Continued Gravitational Contraction”, Phys.Rev. 56, 455-459 (1939).

[7] Birkhoff, G.D., Relativity and Modern Physics, 2nd ed., Harvard Univ. Press, Cambridge, 253-256 (1927).

[8] Pauli, W., Theory of Relativity, reprinted 1981 by Dover Press, New York, 160 (1921).

[9] Einstein, A., Relativity, reprinted 1961 by Crown Publishers, New York, 97, 99 (1916).

[10] Non-static solutions that do not display singularities other than at the origin have been known for many years. Rightfully or wrongfully, they appear to have been discounted as physically unimportant because of their complicated time dependence. See Moller, C., The theory of Relativity, Oxford Univ. Press, London, 327-328 (1960).

[11] Harrison, B.K., Thorne, K.S., Wakano, M. and Wheeler, J.A., Gravitation Theory and Gravitational Collapse, Univ. of Chicago Press, Chicago, 141-142 (1965).

[12] Einstein, A., The Meaning of Relativity, reprinted 1974 by Princeton Univ. Press, Princeton, 61 (1922).

[13] Shapiro, S.L. and Teukolsky, S.A., “Formation of Naked Singularities: The Violation of Cosmic Censorship”, Phys.Rev.Lett. 66, 994-997 (1991).

[14] Olive, K.A., “The Quark-Hadron Transition in Cosmology and Astrophysics”, Science 251, 1194-1199 (1991).

[15] Bergmann, P.G., “Physics and Geometry”, in Proceedings of the 1964 International Congress for Logic, Methodology and Philosophy of Science, ed. Y. Bar Hillel, North Holland, Amsterdam, 346 (1965).