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General Relativity without Black Holes

by John G. Cramer

Alternate View Column AV-100
alternative, general, relativity, Yilmaz, theory, black, holes
Published in the April-2000 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 10/19/99 and is copyrighted ©1999 by John G. Cramer
All rights reserved. No part may be reproduced in any form without
the explicit permission of the author.


This column is a milestone. It's the 100th Alternate View column that I've written for Analog over a period of 16 years beginning in 1983. I was on a sabbatical in Berlin when Stan recruited me to write the column after Jerry Pournelle, my predecessor as AV columnist, decided to step down. The AV columns are a soapbox that was too attractive to pass up, and I've used them to promote an interst in science and to feed cutting-edge science ideas, primarily in the areas of physics and astrophysics, to the readers and writers of science fiction.

In this column I want to examine a recent variant of General Relativity that predicts, among other things, that black holes do not exist. General Relativity (GR) was first formulated by Albert Einstein in 1913. Today it remains the standard model for gravitation. Over the past 86 years it has survived many experimental and observational tests and challenges without requiring modification. Even the recent Type 1A supernova observations that are taken as indications that the vacuum itself contains energy (see my column in the May-99 Analog.) have not required a modification of GR. Einstein anticipated the possibility that space contained energy and introduced the "cosmological constant" in the theory to account for it. Most physicists today consider GR to be the epitome of a nearly perfect theory. It was carefully constructed on a foundation of physical reasoning and mathematical elegance, and it has a certain intrinsic beauty. GR sets a very high standard that makes theories in other areas of physics appear contrived and inelegant by comparison.

Nevertheless, a small group of dissident theoretical physicists has recently been pointing out certain problems with orthodox GR and advocating a modification that has interesting consequences. It's this GR variant that I want to focus on here.

In standard GR, gravity is considered to be "geometrical", to be a consequence of the curvature of space produced by nearby mass-energy.. If a mass or an energy-containing field is present in space, GR predicts that the space will become distorted. This distortion or curvature of space produces gravitational effects like the attraction between masses and the gravitational bending of light rays.

The exception to this rule is the gravitational field itself. While there is energy stored in the gravitational field, unlike all of the other known energy fields (the strong, weak, and electromagnetic interactions) the energy present in gravitation does not, in conventional GR theory, produce space curvature. Starting with Einstein, the justification for this is that to have gravitationally-produced curvature would be "double counting", that since gravitation was produced by the curvature, it should not make more curvature.

However, Einstein's choice of excluding gravitational energy as a source of curvature leads to problems with local energy and momentum conservation. With the exception of gravitational energy, the law of conservation of energy applies to all fundamental interactions "locally" at all points in space. Because gravitational energy does not produce curvature, it does not respect local energy conservation. While energy is conserved in a large volume of space in GR, it is not conserved point-by-point.

Another well-known problem with GR is that many of its solutions have space-time "singularities", places where the mathematics "blows up" to give infinities in certain physical quantities. An example of this problem is the event horizon of a black hole, where time "freezes" at a certain distance from a super-massive object. Inside this boundary is a singular region, a place where mathematics cannot take us. Such mathematical anomalies in the solutions of Einstein's equations are very disturbing. They have been taken by some, including Einstein himself, as a signal that something may be fundamentally wrong with the GR formalism in the regime where very strong gravitational fields are present.

A third problem with GR is that we are sure there must be some comprehensive theory (quantum gravity) that describes gravity at the quantum level, yet orthodox GR theory seems to be incompatible with standard quantum mechanics,. Almost all of the attempts to unify quantum mechanics and general relativity have failed, in part because the singularities of general relativity seem to be incompatible with the quantum formalism. The one exception to this incompatibility is superstring theory (see my AV column in the December-1999 Analog), a theory that cleverly avoids the point-like particles that make singularities. However, superstring theory is still in the development phase, and has not yet reached the point where it can be confronted by measurements or make testable experimental predictions.

The revision of general relativity theory that I want to tell you about is the work of a group of dissident physicists led by Hüseyn Yilmaz of Tufts University. They claim that a slight modification of the orthodox GR formalism cures the problems described above and offers other mathematical advantages. The Yilmaz version of general relativity modifies Einstein's equations by introducing the assumption that gravitation, like all other energy fields, produces space curvature. Yilmaz implements this by adds a gravitationally produced "stress-energy tensor" to Einstein's equations. The resulting variant of general relativity conserves energy locally and has no singularities. Yilmaz also claims that it can be quantized and that, unlike GR, it reduces to Newtonian gravitation and mechanics in the weak field limit. It can be shown to be a "gauge theory" (very similar to electromagnetism), a characteristic that makes it more mathematically tractable and easier to obtain multi-body solutions.

When the gravitational fields are relatively weak, the Yilmaz version of general relativity makes predictions that are observationally indistinguishable from Einstein's version. It is only in the limit of strong gravity that the differences between the two theories become apparent in their predictions. This happens when the extra space-time curvature of the gravitational field becomes important. The most dramatic difference is that the Yilmaz version of general relativity is better behaved mathematically and contains no singularities or event horizons. In particular, the Yilmaz theory predicts that there are no black holes. A massive star may collapse to a state more dense than a neutron star, but it never reaches the pathological black hole state of a time-frozen event horizon cloaking a singularity.

At first glance, this prediction would appear to be fatal to Yilmaz relativity. The headlines from recent astronomical observations, particularly those with the new x-ray and gamma ray telescopes, are said to have confirmed the existence of black holes. However, careful examination shows that the new data confirms the existence of collapsed stars that have extremely hot accretion disks and are too massive to be neutron stars. That observation is compatible with Yilmaz relativity. There has never been an indication of actual event horizon. In fact, up to now there have been no astronomical observation that would falsify the Yilmaz version of general relativity.

There is, however, the possibility of observational tests. When a massive star uses up its nuclear fuel and begins to cool, it goes into a catastrophic collapse called a supernova. For stars of about the mass of our Sun, the collapse process is halted by nuclear forces, and after the supernova explosion a neutron star is left behind. For more massive stars the nuclear forces are insufficient to overcome gravitation, and the star continues to collapse to something much smaller and denser than a neutron star (call it a "black hole candidate"). The Yilmaz version of general relativity predicts a larger maximum mass for neutron stars than does orthodox GR. Thus, observation of a very massive neutron star would tend to support the Yilmaz theory. In this context it is interesting that recent fast X-ray observations (see my AV column in the November-1998 Analog) suggest a neutron star with about 2.3 times the mass of the Sun. This is a very large mass for a neutron star. It is at the very outer limits of what standard GR can accommodate and requires considerable tinkering with nuclear forces at high densities to make it possible. This is not definitive evidence, but it does tend to provide some support for the Yilmaz theory. There are similar suggestive data from the spectral shapes of X-rays from neutron stars.

The advocates of the Yilmaz theory list the following additional advantages (not discussed further here) of the Yilmaz theory over conventional GR: (1) it predicts a definite stress-energy tensor while GR does not; (2) it provides exact solutions for gravity waves of arbitrary field strength while GR does not; (3) it has a true Lagrangian while GR does not; (4) it implies Einstein's equivalence principle, while GR must take equivalence as a separate assumption; (5) it is quantizable while GR is not.

The Yilmaz theory is not widely accepted among general relativity theorists. Several critics have published detailed criticisms of the new formalism and its interpretation, and a heated debate has developed in the literature between the Yilmaz group and its critics (see the references).

It is also worth noting that many theorists, the most prominent example being Steven Hawking, have established their reputations based on theoretical calculations that involve black holes. Much of the recent progress in string theory has come by realizing that there is a duality between strings and black holes. What are the implications for theoretical physics in general and string theory in particular, if it were shown that black holes are not real objects, but only artifacts of an unfortunate omission by Einstein in the formulation of general relativity? An unbiased observer can only say that it is a very interesting controversy that must ultimately be resolved by careful calculations combined with observational tests.

The controversy also raises a question that should be of interest to the SF community. Do black holes exist? Or are they only the products of an inadequate theory? The plot lines of many works of hard science fiction, indeed many that have appeared in this magazine, depend on the existence of black holes and on the interesting violence that they do to space-time. Perhaps gravity near collapsed stars is much different than we had imagined. Perhaps there are new effects that become apparent only through application of the Yilmaz version of general relativity. Perhaps there is material for a whole new generation of hard SF here.

John G. Cramer's 2016 nonfiction book (Amazon gives it 5 stars) describing his transactional interpretation of quantum mechanics, The Quantum Handshake - Entanglement, Nonlocality, and Transactions, (Springer, January-2016) is available online as a hardcover or eBook at: or

SF Novels by John Cramer: Printed editions of John's hard SF novels Twistor and Einstein's Bridge are available from Amazon at and His new novel, Fermi's Question may be coming soon.

Alternate View Columns Online: Electronic reprints of 212 or more "The Alternate View" columns by John G. Cramer published in Analog between 1984 and the present are currently available online at: .


The Yilmaz Theory:

Hüseyn Yilmaz, Phys. Rev. 111, 1417-1426 (1958).

Hüseyn Yilmaz, Annals of Physics (NY) 101, 413-432 (1976).

Hüseyn Yilmaz, Il Nuovo Cimento 107B, 941 (1992)

Carroll O. Alley, Per Kennett Aschan, and Hüseyn Yilmaz, preprint gr-qc/9506082 in the LANL Archive, (30 June, 1995).

Laro Schaltzer, "There are no Black Holes!", commentary on the Web only at

Criticisms of the Yilmaz Theory:

Charles W. Misner, Il Nuovo Cimento B114, 1079-1085 (1999).

F. I. Cooperstock and D. N. Vollick, Il Nuovo Cimento 111B, 265 (1996).

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