Do black holes exist?

Vesselin Petkov, 27.02.2016

Minkowski Institute, Montreal, Canada

This is another example of the kind of questions that are being analyzed at the Minkowski Institute – on the one hand, it is believed that black holes exist, especially after the recent detection of gravitational waves (initially almost unanimously regarded as a signature of black holes, but now it was shown that that might not be the case [1]); on the other hand, however, a careful (Minkowski-type) examination of the Schwarzschild solution of the Einstein equations in general relativity reveals that the two major problems with it cannot be easily brushed aside:

1. The Schwarzschild solution contains a singularity and perhaps most relativists feel very uneasy about this fact, because the existence of a singularity in a physical theory [2] is a clear sign that the theory breaks down in the circumstances where the singularity appears. The authors of a recent attempt [3] to free general relativity of singularities stated their motivation clearly: “We believe that no acceptable physical theory should have a singularity (!), not even a coordinate singularity of the type discussed above! The appearance of a singularity shows the limitations of the theory.”

2. The Schwarzschild solution implies that there exist two realities inside and outside the Schwarzschild sphere of radius r = 2m (where m is the gravitating mass) [4] – one for observers falling together with the constituents of the collapsing body (of mass m), and another for distant observers far away from the Schwarzschild sphere. The problem is most clearly formulated by Dirac [6]:

“We see that the Schwarzschild solution for empty space can be extended to the region r < 2m. But this region cannot communicate with the space for which r > 2m. Any signal, even a light signal, would take an infinite time to cross the boundary r = 2m, as we can easily check. Thus we cannot have direct observational knowledge of the region r < 2m. Such a region is called a black hole, because things may fall into it (taking an infinite time, by our clocks, to do so) but nothing can come out.

The question arises whether such a region can actually exist. All we can say definitely is that the Einstein equations allow it. A massive stellar object may collapse to a very small radius and the gravitational forces then become so strong that no known physical forces can hold them in check and prevent further collapse. It would seem that it would have to collapse into a black hole. It would take an infinite time to do so by our clocks, but only a finite time relatively to the collapsing matter itself.”

So, here is the second major problem with black holes explicitly formulated – as we are (and always have been) distant observers it will take an infinite time for black holes to form, which, in ordinary language, means that black holes will never exist for us.

In the second reality of the falling observers, according to general relativity, black holes will form.

I wonder whether there will be physicists and (perhaps more probably) philosophers who would claim that reality is observer-dependent. But even if this were the case, black holes would never exist in our reality, because this follows from general relativity – the same theory that predicts the very existence of black holes. More rigorously stated, the Schwarzschild solution of the Einstein equations in general relativity predicts both (i) black holes, and (ii) that they will never form for distant observers.

What strikes me is that there are colleagues who freely talk about black holes perhaps without realizing that they are employing double standards with respect to the meaning of “take an infinite time”:

  • From Dirac’s quote above: “Any signal, even a light signal, would take an infinite time to cross the boundary r = 2m” – in this case “take an infinite time” is interpreted to mean “never“; so only according to this interpretation even light will never escape from the Schwarzschild sphere, which allows one to talk about black holes.
  • Again from Dirac’s quote above: “It would seem that it would have to collapse into a black hole. It would take an infinite time to do so by our clocks” – in this case “take an infinite time” inexplicably is not taken to mean “never” as above; instead, it is assumed without any justification that black holes exist.


For more on whether or not black holes exist see [7-9].



[1] Vitor Cardoso, Edgardo Franzin, and Paolo Pani, “Is the Gravitational-Wave Ringdown a Probe of the Event Horizon?” Phys. Rev. Lett. 116, 171101 (2016) and also Synopsis: Did Black Hole “Mimickers” Produce LIGO Signal?

[2] Robert Geroch, What is a Singularity in General Relativity? Annals of Physics 48 (1968) 526-540

[3] P.O. Hess, M. Schäfer, W. Greiner, Pseudo-Complex General Relativity (Springer, Heidelberg 2016). See the essence of the approach: W. Greiner, P.O. Hess, M. Schäfer,T. Schönenbach and G. Caspar, “There are No Black Holes – Pseudo-Complex General Relativity”, in: P. Nicolini, M. Kaminski, J. Mureika, M. Bleicher (eds), 1st Karl Schwarzschild Meeting on Gravitational Physics (Springer, Heidelberg 2016) pp. 33-42

[4] In addition to this problem, Papapetrou also and particularly emphasizes the serious anomaly on the Schwarzschild sphere, whose physical meaning, I think, has not been thoroughly examined – “But these geodesics are space-like for r > 2m and time-like for r < 2m. The tangent vector of a geodesic undergoes parallel transport along the geodesic and consequently it cannot change from a time-like to a space-like vector. It follows that the two regions r > 2m and r < 2m do not join smoothly on the surface r = 2m.” [5]

[5] A. Papapetrou, Lectures on General Relativity (Reidel, Dordrecht 1974) pp. 85-86

[6] P.A.M. Dirac, General theory of relativity (Princeton University Press, Princeton 1996) pp. 35-36

[7] Stephen Hawking: ‘There are no black holes’

[8] Backreaction of Hawking Radiation on a Gravitationally Collapsing Star I: Black Holes? (Black holes cannot form)

[9] No Such Thing as a Black Hole? and Researcher shows that black holes do not exist


Do gravitational waves carry gravitational energy and momentum?

Vesselin Petkov, 14.02.2016

Minkowski Institute, Montreal, Canada


A Minkowski-type analysis of the mathematical formalism of general relativity implies that there is no gravitational energy and momentum, which in turn implies that gravitational waves cannot carry gravitational energy and momentum.

The recent detection of gravitational waves reported earlier this week (12 February 2016) in the Physical Review Letters paper “Observation of Gravitational Waves from a Binary Black Hole Merger” should stimulate a closer scrutiny of the century-old open question – why the mathematical formalism of general relativity stubbornly refuses to yield a valid mathematical expression (representing a real feature of the world) for gravitational energy and momentum.

Had he lived longer Minkowski would have certainly loved to examine this question and might have arrived at a reformulation of Einstein’s general relativity in a way similar to his reformulation of Einstein’s special relativity.

This is precisely the type of open questions that are being thoroughly analyzed at the Minkowski Institute – on the one hand, it is taken for granted that gravitational energy and momentum self-evidently exist, but there is some annoying difficulty to represent them in a proper mathematical form; on the other hand, however, an analysis of the mathematical formalism of general relativity (following Minkowski’s example of analyzing the mathematical formalism of Newtonian mechanics that led him to revealing the true physical nature of Einstein’s special relativity as a theory of flat spacetime) demonstrates that there is no room for gravitational energy and momentum in general relativity:

  • There is no proper tensorial expression (which represents a real physical quantity) for gravitational energy and momentum; for 100 years no one has managed to find such an expression.
  • Gravitational phenomena are fully explained in general relativity as mere effects of the non-Euclidean geometry of spacetime and no additional hypothesis of gravitational interaction (and therefore of gravitational energy and momentum) is necessary (as Eddington put in 1921 “gravitation as a separate agency becomes unnecessary” [1]):
    • according to the geodesic hypothesis [2] in general relativity, a particle, whose worldline is geodesic,is a free particle moving by inertia; therefore the motion of bodies falling toward the Earth’s surface and of planets orbiting the Sun (whose worldlines are geodesic) is inertial, i.e., interaction-free, because the very essence of inertial motion is motion which does not involve any interaction whatsoever;
    • if changing the shape of a free body’s geodesic worldtube (from straight geodesic to curved geodesic) by the spacetime curvature induced, say, by the Earth’s mass (which causes the body’s fall toward the Earth’s surface) constituted gravitational interaction, that would imply some exchange of gravitational energy and momentum between the Earth and the body, but such an exchange does not seem to occur because the Earth’s mass curves spacetime irrespective of whether or not there are other bodies in the Earth’s vicinity (which means that, if other bodies are present in the Earth’s vicinity, no additional energy-momentum is required to change the shape of the geodesic worldtubes of these bodies and therefore no gravitational energy-momentum is transferred to / exchanged with those bodies; see [5]). In other words, the Earth’s mass changes the geometry of spacetime around the Earth’s worldtube and it does not matter whether the geodesics (which are no longer straight in the new spacetime geometry) around the Earth are “empty” or “occupied” by particles of different mass, that is, in general relativity “a geodesic is particle independent” [4].
  • The fact that “in relativity there is no such thing as the force of gravity” [3, p. 109] implies that there is no gravitational energy either since such energy is defined as the work done by gravitational forces.

Despite the above facts, the prevailing view among relativists is that there exists indirect astrophysical evidence for the existence of gravitational energy – coming from the interpretation of the decrease of the orbital period of the binary pulsar system PSR 1913+16 discovered by Hulse and Taylor in 1974 (and other such systems discovered after that), which is believed to be caused by the loss of energy due to gravitational waves emitted by the system (which carry away gravitational energy).

This interpretation that gravitational waves carry gravitational energy should be carefully scrutinized by taking into account the above arguments against the existence of gravitational energy and momentum and especially the fact that there does not exist a rigorous (analytic, proper general-relativistic) solution for the two body problem in general relativity. I think the present interpretation of the decrease of the orbital period of binary systems contradicts general relativity, particularly the geodesic hypothesis and the experimental evidence which confirmed it, because by the geodesic hypothesis the neutron stars, whose worldlines had been regarded as exact geodesics (since the stars had been modeled dynamically as a pair of orbiting point masses), move by inertia without losing energy since the very essence of inertial motion is motion without any loss of energy. For this reason no energy can be carried away by the gravitational waves emitted by the binary pulsar system. Let me stress it as strongly as possible: the geodesic hypothesis and the assertion that bodies, whose worldlines are geodesic, emit gravitational energy (carried away by gravitational waves), cannot be both correct.

In fact, it is the very assumption that the binary system emits gravitational waves which contradicts general relativity in the first place, because motion by inertia does not generate gravitational waves in general relativity. The inspiralling neutron stars in the binary system were modeled (by Hulse and Taylor) as point masses and therefore their worldlines are exact geodesics, which means that the stars move by inertia and no emission of gravitational radiation is involved; if the stars were modeled as extended bodies, then and only then they would be subject to tidal effects and energy would be involved, but that energy would be negligibly small (see next paragraph) and would not be gravitational (see the explanation of the origin and nature of energy in the sticky bead argument below). So, the assertion that the inspiralling neutron stars in the binary system PSR 1913+16 generate gravitational waves is incorrect because it contradicts general relativity. Gravitational waves are emitted only when the stars’ timelike worldlines are not geodesic [6], that is, when the stars are subject to an absolute (curved-spacetime) acceleration (associated with the absolute feature that a worldline is not geodesic), not a relative (apparent) acceleration between the stars caused by the geodesic deviation of their worldlines. For example, in general relativity the stars are subject to an absolute acceleration when they collide (because their worldlines are no longer geodesic); therefore gravitational waves – carrying no gravitational energy-momentum [7] – are emitted only when the stars of a binary system collide and merge into one – “Inspiral gravitational waves are generated during the end-of-life stage of binary systems where the two objects merge into one.”

In fact, gravitational waves are emitted when the two objects are still approaching each other (before they collide), because, due to huge tidal effects the worldlines of the objects’ constituents are deformed, which means that those constituents are subjected to absolute curved-spacetime acceleration. The recent detection of gravitational waves clearly shows that gravitational waves are not emitted by objects moving by inertia – after the two objects coalesce the resulting object moves by inertia (its  worldtube is geodesic) and there is no signal of gravitational waves.

Let me repeat it: when the stars follow their orbits in the binary system, they do not emit gravitational waves since they move by inertia according to general relativity (their worldlines are geodesic and no absolute acceleration is involved); even if the stars were modeled as extended bodies, the worldlines of the stars’ constituents would not be geodesic (but slightly deviated from the geodesic shape) which will cause tidal friction in the stars, but the gravitational waves generated by the very small absolute accelerations of the stars’ constituents will be negligibly weak compared to the gravitational waves believed to be emitted from the inspiraling stars of the binary system (that belief arises from using not the correct general-relativistic notion of acceleration, but the Newtonian one).

The famous sticky bead argument has been regarded as demonstrating that gravitational waves carry gravitational energy (because it is converted through friction into heat energy):

“The thought experiment was first described by Feynman (under the pseudonym “Mr. Smith”) in 1957, at a conference at Chapel Hill, North Carolina. His insight was that a passing gravitational wave should, in principle, cause a bead which is free to slide along a stick to move back and forth, when the stick is held transversely to the wave’s direction of propagation. The wave generates tidal forces about the midpoint of the stick. These produce alternating, longitudinal tensile and compressive stresses in the material of the stick; but the bead, being free to slide, moves along the stick in response to the tidal forces. If contact between the bead and stick is ‘sticky’, then heating of both parts will occur due to friction. This heating, said Feynman, showed that the wave did indeed impart energy to the bead and rod system, so it must indeed transport energy.” [8]

However, a careful examination of this argument reveals that kinetic, not gravitational, energy is converted into heat because a gravitational wave changes the shape of the geodesic worldline of the bead and the stick prevents the bead from following its changed geodesic worldline, i.e., prevents the bead from moving by inertia; as a result the bead resists and exerts an inertial force on the stick (exactly like when a particle away from gravitating masses moving by inertia is prevented from its inertial motion, it exerts an inertial force on the obstacle and the kinetic energy of the particle is converted into heat).

It appears more adequate if one talks about inertial, not kinetic, energy, because what is converted into heat (as in the sticky bead argument) is the energy corresponding to the work done by the inertial force (and it turns out that that energy, originating from the inertial force, is equal to the kinetic energy [9]). The need to talk about the adequate inertial, not kinetic, energy is clearly seen in the explanation of the sticky bead argument above – initially (before the arrival of the gravitational wave) the bead is at rest and does not possess kinetic energy; when the gravitational wave arrives, the bead starts to move but by inertia (non-resistantly) since the shape of its geodesic worldline is changed by the wave into another geodesic worldline (which means that the bead goes from one inertial state – rest – into another inertial state, i.e., without any transfer of energy from the gravitational wave; transferring energy to the bead would occur if and only if the gravitational wave changed the state of the bead from inertial to non-inertial), and when the stick tries to prevent the bead from moving by inertia, the bead resists and exerts an inertial force on the stick (that is why, what converts into heat through friction is inertial energy).

I would like to stress it that it is a fact in the rigorous structure of general relativity that gravitational waves do not carry gravitational energy, which, however, had been inexplicably ignored, despite that Eddington explained it clearly in his comprehensive treatise on the mathematical foundations of general relativity The Mathematical Theory of Relativity 2nd ed. (Cambridge University Press, Cambridge 1924), p. 248 [10]: “The gravitational waves constitute a genuine disturbance of space-time, but their energy, represented by the pseudo-tensor t^{\nu}_{\mu}, is regarded as an analytical fiction” (it cannot be regarded as an energy of any kind for the well-known reason that “It is not a tensor-density and it can be made to vanish at any point by suitably choosing the coordinates; we do not associate it with any absolute feature of world-structure” ibid, p. 136).

A systematic critical examination of standard arguments for the received view that gravitational waves carry energy-momentum is found in P. Duerr (Oriel College & Philosophy of Physics, Oxford): “Do Gravitational Waves Carry Energy-Momentum: Critique of a Procrustean Practice”, (preprint).

NOTE (1 June 2017): LIGO Catches its Third Gravitational Wave!

Note the correct wording in the official LIGO GW170104 Press Release: “As was the case with the first two detections, the waves were generated when two black holes collided to form a larger black hole.” No gravitational waves are emitted when the black holes orbit each other before they collide (as the black holes are modelled as point masses, they are geodesic worldlines and no gravitational waves are generated by geodesic worldlines; when the black holes collide their worldlines are no longer geodesic and gravitational waves are emitted). And no gravitational energy is carried by the gravitational waves; as there is no gravitational force in the world, there is no gravitational energy either since such energy is the work done by gravitational forces.

NOTE (30 June 2017): Strange Noise in Gravitational-Wave Data Sparks Debate



1. A. S. Eddington, “The Relativity of Time,” Nature 106, 802-804 (17 February 1921); reprinted in: A. S. Eddington, The Theory of Relativity and its Influence on Scientific Thought: Selected Works on the Implications of Relativity (Minkowski Institute Press, Montreal 2015). Two years later, in his fundamental work on the mathematical foundations of general relativity The Mathematical Theory of Relativity (Cambridge University Press, Cambridge 1923) [9] Eddington stated it even more explicitly (p. 221): “An electromagnetic field is a “thing;” gravitational field is not, Einstein’s theory having shown that it is nothing more than the manifestation of the metric.”

2. The geodesic hypothesis is regarded as “a natural generalization of Newton’s first law” [3], that is, “a mere extension of Galileo’s law of inertia to curved spacetime” [4]. The geodesic hypothesis has been confirmed by the experimental fact that particles falling toward the Earth’s surface offer no resistance to their fall – a falling accelerometer, for example, reads zero resistance (i.e. zero acceleration; the observed apparent acceleration of the accelerometer is caused by the spacetime curvature caused by the Earth). The experimental fact that particles do not resist their fall (i.e. their apparent acceleration) means that they move by inertia and therefore no gravitational force is causing their fall. It should be emphasized that a gravitational force would be required to accelerate particles downwards only if the particles resisted their acceleration, because only then a gravitational force would be needed to overcome that resistance.

3. J. L. Synge, Relativity: The General Theory (Nord-Holand, Amsterdam 1960) p. 110.

4. W. Rindler, Relativity: Special, General, and Cosmological (Oxford University Press, Oxford 2001) p. 178.

5. V. Petkov, “Physics as Spacetime Geometry,” in: A. Ashtekar, V. Petkov (eds), Springer Handbook of Spacetime (Springer, Heidelberg 2014), Chapter 8. See also “Is Gravitation Interaction or just Curved-Spacetime Geometry?

6. The original prediction of gravitational wave emission, obtained by Einstein (Berlin. Sitzungsberichte, 1916, p. 688; 1918, p. 154), correctly identified the source of such waves – a spinning rod, or any rotating material bound together by cohesive force. None of the particles of such rotating material (except the centre of rotation) are geodesic worldlines in spacetime and, naturally, such particles will emit gravitational waves. This is not the case with double stars; as the stars are modelled as point masses, their worldliness are geodesics (which means that the stars are regarded as moving by inertia) and no gravitational waves are emitted.

7. An immediate and misleading reaction “A wave that carries no energy?!” should be resisted, because it is from the old times of three-dimensional thinking – assuming that a wave really travels in the external world. There is no such thing as a propagating wave in spacetime – what is there is a spacetime region whose “wavelike” geometry is interpreted in three-dimensional language as a wave which propagates in space (exactly like a timelike worldline is interpreted in three-dimensional language as a particle which moves in space); also, keep in mind that there is no such thing as space in the external world, because spacetime is not divided into a space and a time. To address properly (and overcome) another immediate and misleading reaction “Spacetime is nothing more than an abstract mathematical continuum!” read (again) Minkowski’s paper Space and Time or: The world is four-dimensional – Hermann Minkowski’s irrefutable proof to see that the experimental evidence (captured in the relativity postulate at Minkowski’s time and confirming the relativistic effects later) would be impossible if spacetime were nothing more than an abstract mathematical continuum.

8. Sticky bead argument,

9. V. Petkov, “On Inertial Forces, Inertial Energy and the Origin of Inertia,” published as Appendix B in V. Petkov, Inertia and Gravitation: From Aristotle’s Natural Motion to Geodesic Worldlines in Curved Spacetime (Minkowski Institute Press, Montreal 2012).

10. New publication: Arthur S. Eddington, The Mathematical Theory of Relativity (Minkowski Institute Press, Montreal 2016).

The resurrection of local realism

Comment on recent articles published in Nature and Physical Review Letters

Louis Vervoort, 10.02.2016

Minkowski Institute, Montreal, Canada

In recent articles [1-3], three research teams, led by Anton Zeilinger at the University of Vienna, Austria [1], Lynden Shalm at NIST in Boulder, Colorado [2], and Ronald Hanson at the Delft University of Technology in the Netherlands [3] have reported on the untimely death of local realism. These teams have succeeded in the experimental tour-de-force to close, in a Bell experiment, the most relevant loopholes for local hidden-variable theories. ‘Realism’ is ‘the assumption that objects have physical properties independent of measurement’, and ‘locality’ is the assumption that there are no influences traveling faster than the velocity of light [1]. The general content of these publications is thus that one of these cornerstones of physics must be given up. If true, such a conclusion would surely represent a landmark in our intellectual history. However, I believe this conclusion is quite premature.

Upon closer inspection the authors appear to be more cautious. In fact, it appears that only a certain class of hidden-variable theories is eliminated by the experiments. Ref. [1] is maybe most explicit, by describing how the experiment would close the ‘freedom-of-choice’ loophole: “This loophole can be closed only under specific assumptions about the origin of [the hidden variables]. Under the assumption that [the hidden variables] are created with the particles to be measured, an experiment in which the settings are generated independently at the measurement stations and spacelike separated from the creation of the particles closes the loophole.”

Now, it is certainly very natural to consider the hidden variables as ‘pertaining’ only to the particle pair, more generally as created at the source with the particles, in the sense of [1]. But it is not the only possibility that is allowed by logic and physical law. My point is that the class of hidden-variable theories still exploiting the freedom-of-choice loophole is vast and natural. For instance, the hidden variables could, besides the particles, also describe a background field or medium in which the particles move and that interacts with particles and analyzers. In this case the full dynamics of the hidden variables is crucial, as I have recently investigated in detail in Ref. [4] (the background field may refer to vacuum fluctuations, an ether, a dark field,…). In [4] it is shown that such background-based theories can reproduce the quantum correlation of the Bell experiment.

Background-based hidden-variable theories are inspired by spectacular recent experiments by Couder, Fort, Bush and co-workers, which demonstrate that background fields (or pilot waves if one prefers) can lead to a surprisingly wide range of quantum-like phenomena in fluid mechanical systems [5-6]. Such background-based theories therefore remind one of Louis de Broglie’s pilot-wave theory. There is a whole series of physicists working on modern variants of this theory, attempting to derive quantum mechanics from a deeper level. One can also refer here to the Cellular Automaton Theory of quantum mechanics of Nobel Laureate Gerard ‘t Hooft [7].

In conclusion, the crucial experiments [1-3] do eliminate a wide and natural class of local-realistic models. But when more general claims are made (‘By closing the freedom-of-choice loophole […] we reduce the possible local-realist explanations to truly exotic hypotheses’ [1]), I disagree. There is a wide and natural class of local-realist theories that is not eliminated by [1-3]. Not only is local realism far from dead; I believe that by prematurely proclaiming its passing one has closed a promising road to finally reconcile quantum mechanics and relativity theory. Some more info can be found in [8].

Acknowledgements. I thank Ronald Hanson for his kind willingness to discuss his group’s experiment and its interpretation.

[1] M. Giustina et al., ‘Significant-loophole-free test of Bell’s theorem with entangled photons’, Phys. Rev. Lett. 115, 250401 (2015)

[2] L. Shalm et al., ‘Strong loophole-free test of local realism’, Phys. Rev. Lett. 115, 250402 (2015)

[3] B. Hensen et al., ‘Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometers’, Nature 526, 682 (2015)

[4] L. Vervoort, ‘No-go theorems face background-based theories for quantum mechanics’, Found. Physics 45, 1 (2015), doi: 10.1007/s10701-015-9973-7

[5] A. Eddi, J. Moukhtar, S. Perrard, E. Fort, and Y. Couder, ‘Level-Splitting at a macroscopic scale’, Phys. Rev. Lett. 108, 264503 (2012)

[6] J. Bush, ‘The new wave of pilot-wave theory’, Phys. Today, Aug. 2015, 47

[7] G. ‘t Hooft, ‘Models on the boundary between classical and quantum mechanics’, Phil. Trans. R. Soc. A 373, 20140236 (2015)

[8] L. Vervoort, Comment on ‘Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons’, arXiv preprint, (2016)