Appendix C: Can Gravity beQuantized?

Vesselin PetkovInstitute for Foundational Studies ‘Hermann Minkowski’

Montreal, Quebec, Canadahttp://minkowskiinstitute.org/

vpetkov@minkowskiinstitute.org

AbstractIt may turn out that we have been stubbornly ignoring a crucial

message coming from the unsuccessful attempts to create a theory

of quantum gravity – that gravity is not an interaction. This op-

tion does not look so shocking when gravity is consistently and rig-

orously regarded as a manifestation of the non-Euclidean geometry

of spacetime. Then it becomes evident that general relativity does

imply that gravitational phenomena are not caused by gravitational

interaction. The geodesic hypothesis in general relativity and par-

ticularly the experimental evidence that confirmed it indicate that

gravity is not a physical interaction since particles which appear to

interact gravitationally are actually free particles whose motion is in-

ertial (i.e. interaction-free). This situation has implications for two

research programs – quantum gravity and detection of gravitational

waves. First, the real open question in gravitational physics appears

to be how matter curves spacetime, not how to quantize the apparent

gravitational interaction. Second, the search for gravitational waves

should explicitly take into account the geodesic hypothesis according

to which orbiting astrophysical bodies (modelled by point masses) do

not radiate gravitational energy since their worldlines are geodesics

representing inertial (energy-loss-free) motion.

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6.1 Introduction

Since the advent of general relativity and quantum mechanics theirunification has been the ultimate goal of theoretical physics. So far,however, the di↵erent approaches aimed at creating a theory of quan-tum gravity [1] have been unsuccessful. It seems a possible reasonfor this – that gravity might not be an interaction – has never beenconsistently examined. What also warrants such an examination isthat an experimental fact – falling bodies do not resist their appar-ent acceleration – turns out to be crucial for determining the truenature of gravitational phenomena, but has been e↵ectively neglectedso far. Taking it into account, however, makes it possible to refinenot only the quantum gravity research (by recognizing that the gen-uine open question in gravitational physics is how matter determinesthe geometry of spacetime, not how to quantize what has the appear-ance of gravitational interaction) but also to fine-tune the search forgravitational waves by showing that astrophysical bodies, modelled bypoint masses whose worldlines are geodesics (representing inertial orenergy-loss-free motion), do not give rise to radiation of gravitationalenergy.

As too much is at stake in terms of both the number of physi-cists working on quantum gravity and on detection of gravitationalwaves, and the funds being invested in these worldwide e↵orts, eventhe heretical option of not taking gravity for granted should be thor-oughly analyzed. It should be specifically stressed, however, that suchan analysis will certainly require extra e↵ort from relativists who aremore accustomed to solving technical problems than to examining thephysical foundation of general relativity which may involve no calcu-lations. Such an analysis is well worth the e↵ort since it ensures thatwhat is calculated is indeed in the proper framework of general relativ-ity and is not smuggled into it to twist it until it yields some featuresthat resemble gravitational interaction.

The standard interpretation of general relativity takes it as virtu-ally unquestionable that gravitational phenomena result from gravi-tational interaction. However, the status of gravitational interactionin general relativity is far from self-evident and its clarification needsa careful analysis of both the mathematical formalism and the logical

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structure of the theory and the existing experimental evidence.Taken according to its logical structure general relativity demon-

strates that what is traditionally called gravitational interaction isdramatically di↵erent from the other three fundamental interactions,successfully described by the Standard Model, and is nothing morethan a mere manifestation of the curvature of spacetime. Unlike theelectromagnetic interaction, for example, which is mediated by theelectromagnetic field and force, the observed apparent gravitationalinteraction is not caused by a physical gravitational field and a grav-itational force. By the geodesic hypothesis in general relativity, theassumption that the worldline of a free particle is a timelike geodesicin spacetime is “a natural generalization of Newton’s first law” [2, p.110], that is, “a mere extension of Galileo’s law of inertia to curvedspacetime” [3, p. 178]. This means that in general relativity a particle,whose worldline is geodesic, is a free particle which moves by inertia.

Indeed, two particles that seem to be subject to gravitational forcesin reality move by inertia according to general relativity since theirworldlines are timelike geodesics in spacetime curved by the particles’masses. The acceleration of the particles towards each other is relativeand is caused not by gravitational forces, but by geodesic deviation,which reflects the fact that there are no straight worldlines in curvedspacetime. In general relativity the planets, for example, are freebodies which move by inertia and as such do not interact in any waywith the Sun because inertial motion does not imply any interaction.The planets’ worldlines are geodesics20, which due to the curvature ofspacetime caused by the Sun’s mass are helixes around the worldline ofthe Sun (which means that the planets move by inertia while orbitingthe Sun).

Therefore, what general relativity itself tells us about the worldis that the apparent gravitational interaction is not a physical inter-action in a sense that two particles, which appear to interact gravita-tionally, are free particles since they move by inertia. This readily, butcounter-intuitively explains the unsuccessful attempts to create a the-ory of quantum gravity – it is impossible to quantize what we regardas gravitational interaction since it simply does not exist accordingto what the logical structure of general relativity itself implies (with-

20Only the center of mass of a spatially extended body is a geodesic worldline.

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out importing features to general relativity whose sole justification isthe belief that gravitational phenomena are caused by gravitationalinteraction).

Two main reasons have been hampering the proper understandingof gravitational phenomena. The first reason, discussed in Sect. 6.2,is that the profound consequences of the geodesic hypothesis for thenature of gravitational interaction have not been fully realized mostlydue to the adopted definition of a free particle in general relativity,which literarily posits that otherwise free particles are still subject togravitational interaction – an assumption that does not follow fromthe theory itself. Sect. 6.3 examines the second reason – that sincethe advent of general relativity there have been persistent attempts tosqueeze general relativity and ultimately Nature into the present un-derstanding that gravitational energy and momentum (as energy andmomentum of gravitational interaction and field) are part of gravita-tional phenomena.

6.2 General relativity implies that there isno gravitational interaction

The often given definition of a free particle in general relativity – a par-ticle is “free from any influences other than the curvature of spacetime”[5] – e↵ectively postulates the existence of gravitational interaction byalmost explicitly asserting that the influence of the spacetime curva-ture on the shape of a free particle’s worldline constitutes gravitationalinteraction.

However, if carefully analyzed, the fact that particles’ masses curvespacetime, which in turn changes the shape of the worldlines of thoseparticles, does not imply that the particles interact gravitationally.There are two reasons for that. First, the shape of the geodesic world-lines of free particles is determined by the curvature of spacetime alonewhich itself may not be necessarily induced by the particles’ masses.This is best seen from the fact that general relativity shows both thatspacetime is curved by the presence of matter, and that a matter-freespacetime can be intrinsically curved. The latter option follows fromthe de Sitter solution [4] of Einstein’s equations. Two test particles

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in the de Sitter universe only appear to interact gravitationally but infact their interaction-like behaviour is caused by the curvature of theirgeodesic worldlines, which is determined by the constant positive in-trinsic curvature of the de Sitter spacetime. The fact that there are nostraight geodesic worldlines in non-Euclidean spacetime (which givesrise to geodesic deviation) manifests itself in the relative accelerationof the test particles towards each other which creates the impressionthat the particles interact gravitationally (test particles’ masses areassumed to be negligible in order not to a↵ect the geometry of space-time).

Second, the experimental fact that particles of di↵erent masses falltowards the Earth with the same acceleration in full agreement withgeneral relativity’s “a geodesic is particle-independent” [3, p. 178],ultimately means that the shape of the geodesic worldline of a freeparticle in spacetime curved by the presence of matter is determinedby the spacetime geometry alone and not by the matter. This is clearlyseen when the central point of general relativity – the mass-energy ofa body changes the geometry of spacetime around itself – is explicitlytaken into account. The very meaning of changing the geometry ofempty spacetime by a body is that the geodesics of the new spacetimegeometry are set. This is so because what essentially determines thetype of spacetime geometry is the corresponding version of Euclid’sfifth postulate, which is expressed in terms of the geodesic worldlinesof the spacetime geometry. Hence a geodesic is particle-independentbecause a geodesic is a feature of the spacetime geometry itself. Thefact that the worldline of a free particle is influenced by the curvatureof spacetime produced by a body does not constitute gravitational in-teraction with the body since the shape of the free particle’s worldlineis not changed by the body’s mass-energy – the body curves solelyspacetime, regardless of whether or not spacetime is empty, becauseno additional energy is spent for curving the geodesic worldline of thefree particle (or in three-dimensional language – no additional energyis spent for making the particle orbit the body or fall onto it). In short,the mass-energy of a body changes the geometry of spacetime no mat-ter whether or not there are any particles in the body’s vicinity, andthe shape of free particles’ worldlines reflects the spacetime curvatureno matter whether it is intrinsic or induced by a body’s mass-energy.

Another indication that the shape of a geodesic worldline is set

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by the geometry of spacetime alone, irrespective of the origin of thespacetime geometry itself, is the fact that the spacetime curvaturecreated by the mass-energy of a free body determines not only theshape of the worldline of a second nearby body, but also the shape ofthe first body’s own geodesic worldline. This is best seen in the bi-nary star systems where two stars orbit the common center of gravitywhich lies between them. As each star curves spacetime regardless ofwhether or not the other star is there, the resulting spacetime curva-ture is a superposition of the curvatures produced by the two stars.In the so curved spacetime the geodesic worldlines of the starts’ cen-ters are helixes around the worldline of the common center of gravity(curvature). So the worldline of each star’s center is determined bythe spacetime curvature induced by both stars; by contrast, if one ofthe stars were a particle with much smaller mass compared to themass of the other star, the particle’s geodesic worldline would be ahelix around the geodesic worldline of the center of the other star.If it is assumed that the shape of the stars’ geodesic worldlines werenot determined solely by the spacetime geometry, but were a resultof gravitational interaction between the two stars – that is, if it werethe mass-energy of each star that determined through the spacetimecurvature the shape of the other star’s worldline – it would be then amystery how each star would interact gravitationally with itself, i.e.how the mass-energy of each star would also determine the shape of itsown worldline. This problem does not arise when it is recognized thatthe shapes of the stars’ worldlines merely reflect the spacetime curva-ture regardless of whether or not it is intrinsic or created by the stars’masses. Simply, the mass-energy of each star individually and inde-pendently from the other star changes the spacetime geometry, and theshapes of the stars’ geodesics reflect the resultant spacetime curvature.(Of course, this problem does not arise in the Newtonian gravitationaltheory since it regards gravitational phenomena as caused by a force,not as a manifestation of spacetime curvature.)

The essential role of inertial motion in general relativity followsfrom the basic fact that the existence of geodesics is a feature of curvedspacetime itself just like the existence of straight worldlines is a featureof flat spacetime. Straight worldlines represent the inertial motion offree particles of any mass in flat spacetime and the straightness of theirworldlines is regarded as naturally reflecting the spacetime geometry.

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Analogously, geodesics in curved spacetime represent free particles ofany mass that move by inertia. The shape of the geodesics also reflectsthe spacetime geometry and is not an indication of some interactionexactly like the shape of the straight worldlines in flat spacetime is notan indication of any interaction. The equal status of geodesics in flatand curved spacetimes is encoded in the fall of di↵erent masses withthe same acceleration. By the geodesic hypothesis, their fall is inertialand indeed the motion of falling particles is unsurprisingly similar tomotion by inertia in the absence of gravity – particles that move byinertia do so irrespective of their masses.

That a geodesic worldline in curved spacetime represents an uncon-ditionally free particle becomes clearer from a closer examination ofthe geodesic hypothesis itself and particularly from the experimentalevidence which proved it.

Newton’s first law of motion (i.e. Galileo’s law of inertia) describesthe motion of a free particle that is not subject to any interactions.Such a particle moves by inertia, which means that it o↵ers no re-sistance to its motion with constant velocity. If a particle is subjectto some interaction, which prevents it from maintaining its inertialmotion, the particle resists the forced change of its velocity, i.e. theparticle resists its acceleration. The particle’s reaction and its resis-tance to the interaction is captured in Newton’s third and second lawsof motion. The third law reflects the fact that when a free particleis subject to some action it o↵ers an equal and opposite reaction byresisting the action. The profound meaning of Newton’s second lawis that a force is only needed to overcome the resistance the particleo↵ers to its acceleration.

It is the intrinsic feature of a particle to move non-resistantly byinertia when its motion is not disturbed by any influences that consti-tutes an objective criterion for a free particle. That is, non-resistantmotion is a necessary and su�cient condition for a particle to be free.A particle is subject to some interaction only if it resists its motion.

Galileo’s and Newton’s law of inertia was first generalized in specialrelativity by Minkowski who realized that a free particle, which movesby inertia, is a straight timelike worldline in Minkowski spacetime [6].By contrast, the worldline of an accelerating particle is curved, i.e.deformed. Had this generalization of the law of inertia been care-fully analyzed, two immediate consequences would have been realized.

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First, the experimental fact that acceleration is absolute, because it isdetectable due to the resistance an accelerating particle o↵ers to itsacceleration, finds an unexpected but deep explanation in Minkowski’sspacetime formulation of special relativity. The acceleration of a par-ticle is absolute not because the particle accelerates with respect tosome absolute space, but because its worldline is curved and thereforedeformed, which is an absolute geometric property that correspondsto the absolute physical property of the particle’s resistance to its ac-celeration. Second, the resistance an accelerating particle o↵ers toits acceleration can be also given an unforeseen explanation – as theworldline or rather the worldtube of an accelerating particle is curved,the particle’s resistance to its acceleration (i.e. the particle’s iner-tia) can be viewed as originating from a four-dimensional stress whicharises in the deformed worldtube of the particle21 [7, Chap. 9].

Based on Minkowski’s rigorous definition of a free particle in spe-cial relativity, the above criterion for a free particle can be made evenmore precise – in three-dimensional language, a free particle does notresist its motion, whereas in four-dimensional (spacetime) language afree particle is a timelike worldtube, which is not deformed. And in-deed, in Minkowski spacetime straight worldtubes are not distorted,which explains why a free particle, represented by a straight world-tube, o↵ers no resistance to its free or inertial motion. This criterionprovides further justification for the geodesic hypothesis in generalrelativity by clarifying that a timelike geodesic worldtube in curvedspacetime, which represents a free particle, is naturally curved due tothe spacetime curvature, but is not deformed22.

The generalized Minkowski definition of a free particle in spacetime(no matter flat or curved) – a free particle is a non-deformed worldtube(straight in flat spacetime and geodesic in curved spacetime) – indi-cates that a geodesic worldline does represent an unconditionally free

21This explanation of inertia implies that the worldtubes of particles are realfour-dimensional objects completely in line with Minkowski’s view of special rela-tivity as a theory of an absolute four-dimensional world and particularly with hisexplanation of length contraction, which would be impossible if the worldtube ofa relativistically contracted body were not real, i.e. if it were a mere geometricalabstraction [6] (see also [7, 8]).

22Rigorously speaking, this is true only for a small (test) particle. Tidal stresses,caused by geodesic deviation, give rise to some deformation but that is not causedby a gravitational force.

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particle in general relativity. Indeed, no interaction is behind the factthat the worldtube of a free particle in flat spacetime is straight andthe same is true for a free particle in curved spacetime – no interactionis responsible for the curved but not deformed geodesic worldtube of afree particle there (in agreement with the fact that a geodesic worldlineis the analog of a straight worldline in curved spacetime).

What is crucial for testing both the geodesic hypothesis and thegeneralized definition of a free particle in spacetime and for determin-ing the true nature of gravitational phenomena is the experimentalfact that particles falling towards the Earth’s surface o↵er no resis-tance to their fall. This essential experimental evidence has been vir-tually neglected so far, which is rather inexplicable especially giventhat Einstein regarded the realization of this fact – that “if a personfalls freely he will not feel his own weight” – as the “happiest thought”of his life which put him on the path towards general relativity [9].

This experimental fact unambiguously confirms the geodesic hy-pothesis because free falling particles, whose worldtubes are geodesics,do not resist their fall (i.e. their apparent acceleration) which meansthat they move by inertia and therefore no gravitational force is caus-ing their fall. It should be particularly stressed that a gravitationalforce would be required to accelerate particles downwards only if theparticles resisted their acceleration, because only then a gravitationalforce would be needed to overcome that resistance.

Thus, the experimental evidence of non-resistant fall of particlesis the definite proof of the central assumption of general relativity –that no gravitational force is causing the gravitational phenomena.This experimental evidence is crucial since it rules out any alternativetheories of gravity and any attempts to quantize gravity (by propos-ing alternative representations of general relativity aimed at makingit amenable to quantization) that regard gravity as a physical fieldwhich gives rise to a gravitational force since they would contradictthe experimental evidence.

The non-resistant fall of particles also confirms the generalized def-inition of a free particle since their geodesic worldtubes are naturallycurved (due to the spacetime curvature) but are not deformed. Afalling accelerometer, for example, reads zero acceleration (in an ap-parent contradiction with the observed acceleration of the accelerom-eter while falling), which is adequately explained when it is taken into

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account that what an accelerometer measures is the resistance it of-fers to its acceleration. The zero reading of the falling accelerometerproves that it o↵ers no resistance to its fall and demonstrates thatit moves by inertia and therefore its acceleration is not absolute (notresulting from a deformation of its worldtube); it is relative due to itsnaturally curved, but not deformed worldtube (that is, the accelerom-eter’s relative acceleration is caused by geodesic deviation which itselfis a manifestation of the fact that the geodesic worldtube of the ac-celerometer and the worldline of the Earth’s center converge towardseach other).

The accelerometer does not resist its fall because its absolute ac-celeration is zero according to general relativity (aµ = d2xµ/d⌧2 +�µ

↵�

(dx↵/d⌧)(dx�/d⌧) = 0), which reflects the fact that its worldtube

is geodesic and is therefore not deformed23. When the accelerom-eter is at rest on the Earth’s surface its worldtube is not geodesic,which by the geodesic hypothesis means that the accelerometer doesnot move by inertia and therefore should resist its being preventedfrom maintaining its inertial motion, i.e. the accelerometer shouldresist its state of rest on the Earth’s surface. Before the advent ofgeneral relativity that resistance force had been called gravitationalforce or the accelerometer’s weight. As implied by the geodesic hy-pothesis the accelerometer’s weight is the inertial force, which ariseswhen the accelerometer is prevented from moving by inertia whilefalling. This is also seen from the fact that the accelerometer’s world-tube is deformed (not geodesic) – the four-dimensional stress in thedeformed worldtube gives rise to a static restoring force that manifest

23Had Minkowski lived longer he might have discovered general relativity (surelyunder another name) before Einstein. Minkowski would have almost certainly no-ticed that inertia could be regarded as arising from the four-dimensional stressin the deformed worldtube of an accelerating particle and therefore inertia wouldturn out to be another manifestation (along with length contraction as correctlyexplained by him) of the four-dimensionality of the absolute world of his spacetimeformulation of special relativity. Then the experimental fact that a falling particleaccelerates (which means that its worldtube is curved), but o↵ers no resistanceto its acceleration (which means that its worldtube is not deformed) can be ex-plained only if the worldtube of a falling particle is both curved and not deformed,which is impossible in the flat Minkowski spacetime where a curved worldtube isalways deformed. Such a worldtube can exist only in a non-Euclidean spacetimewhose geodesics are naturally curved due to the spacetime curvature, but are notdeformed.

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itself as the resistance (inertial) force with which the accelerometer op-poses its deviation from its geodesic path in spacetime. The conceptof inertia in Minkowski’s spacetime formulation of special relativitysheds more light on the physical meaning of the equivalence of inertialand (passive) gravitational masses and forces. They are all inertialand originate from the four-dimensional stress arising in the deformedworldtubes of non-inertial particles (accelerating or being preventedfrom falling) [7, Ch. 9]. So in the framework of relativity the def-inition of mass as the measure of the resistance a body o↵ers to itsacceleration (i.e. to the deformation of its worldtube) becomes evenmore understandable.

6.3 There is no gravitational energy in gen-eral relativity

The second main reason, which has been hampering the proper un-derstanding of gravitational phenomena, is the issue of gravitationalenergy and momentum.

Einstein made the gigantic step towards the profound understand-ing of gravity as spacetime curvature but even he was unable to acceptall implications of the revolutionary view of gravitational phenom-ena. It was he who first tried to insert the concepts of gravitationalenergy and momentum forcefully into general relativity in order toensure that gravity can still be regarded as some interaction despitethat the mathematical formalism of general relativity itself refused toyield a proper (tensorial) expression for gravitational energy and mo-mentum. This refusal is fully consistent with the status of gravityas non-Euclidean spacetime geometry (not a force) in general relativ-ity. The non-existence of gravitational force implies the non-existenceof gravitational energy as well since gravitational energy presupposesgravitational force (gravitational energy = work due to gravity = grav-itational force times distance).

Although the mathematical formalism and the logical structure ofgeneral relativity imply that gravitational phenomena are not causedby gravitational interaction, which entails that there are no gravita-tional energy and momentum in Nature, most relativists regard grav-

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itational energy as a necessary element of the description of gravita-tional phenomena. This position is based not only on the view, whichliterally postulates the existence of gravitational interaction and there-fore of gravitational energy and momentum, but also on two generallyaccepted views.

First, the nonlinearity of Einstein’s equations has been interpretedto support the assumption that like the electromagnetic field, the grav-itational field also carries energy and momentum. However, unlikeMaxwell’s equations, which are linear because the electromagnetic fielditself does not have electric charge and does not contribute to its ownsource, the gravitational field must contribute to its own source if itcarries energy and momentum since in general relativity any energy isa source of gravity. This would be consistent with the fact that Ein-stein’s equations are nonlinear – the nonlinearity would represent thee↵ect of gravitation on itself. However, this interpretation of Einstein’sequations barely hides the major problem of the standard interpreta-tion of generally relativity that there exists gravitational interactionand therefore gravitational field, which has gravitational energy andmomentum. According to the prevailing view in general relativity thecomponents of the metric tensor are the relativistic generalization ofthe gravitational potential. The nonlinear terms in Einstein’s equa-tions are the squares of their partial derivatives, so the energy densityof the gravitational field turns out to be quadratic in the gravitationalfield strength just like the energy density of the electromagnetic fieldis quadratic in the electric and the magnetic fields.

Identifying the gravitational field with the components of the met-ric tensor seems justified only in the limiting case when general rela-tivity is compared with the Newtonian gravitational theory in order todetermine what in general relativity (in that limiting case) correspondsto the gravitational potential and force. However, such an identifica-tion in general relativity itself is more than problematic. There is notensorial measure of the gravitational field in general relativity sinceit can be always transformed away in the local inertial frame [3, p.221]. This is problematic because if the gravitational field existed,then as something real it should be represented by a proper tenso-rial expression. For this reason not all relativists are happy with theidentification of the components of the metric tensor with the gravi-tational field. Synge’s view on this is well known – he insisted that

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the gravitational field should be modelled by “the Riemann tensor,for it is the gravitational field – if it vanishes, and only then, thereis no field” [2, p. viii ]. When gravitational phenomena are properlymodelled by the spacetime curvature, which as something real is rep-resented by the Riemann curvature tensor, it follows that gravitation(the spacetime curvature) makes no contribution to its own source –Einstein’s equations are linear in the Ricci curvature tensor (the con-traction of the Riemann curvature tensor) and the scalar spacetimecurvature (the contraction of the Ricci curvature tensor). So, whengravitational phenomena are adequately modelled by the spacetimecurvature it is evident that the gravitational field is not somethingphysically real, that is, it is not a physical entity. It is a geometricfield at best and as such does not possess any energy and momentum.

According to the second view there is indirect astrophysical evi-dence for the existence of gravitational energy. That evidence is be-lieved to come from the interpretation of the decrease of the orbitalperiod of a binary pulsar system, notably the system PSR 1913+16discovered by Hulse and Taylor in 1974 [10]. According to that inter-pretation the decrease of the orbital period of such binary systems iscaused by the loss of energy due to gravitational waves emitted by thesystems. Almost without being challenged (with only few exceptions[11, 12, 13]) this view holds that the radiation of gravitational energyfrom the binary systems, which is carried away by gravitational waves,has been indirectly experimentally confirmed to such an extent thateven the quadrupole nature of gravitational radiation has been alsoindirectly confirmed.

It may sound heretical, but the assumption that the orbital mo-tion of the neutron stars in the PSR 1913+16 system loses energy byemission of gravitational waves contradicts general relativity, partic-ularly the geodesic hypothesis and the experimental evidence whichconfirmed it. The reason is that by the geodesic hypothesis the neu-tron stars, whose worldlines are geodesics24, move by inertia withoutlosing energy since the very essence of inertial motion is motion with-out any loss of energy. Therefore no energy is carried away by the

24The neutron stars in the PSR 1913+16 system had been “modelled dynami-cally as a pair of orbiting point masses” [14], which means that (i) the tidal e↵ectshad been ignored and (ii) the worldlines of the neutron stars as point masses hadbeen in fact regarded as exact geodesics.

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gravitational waves emitted by the binary pulsar system. So the ex-perimental fact of the decay of the orbital motion of PSR 1913+16(the shrinking of the stars’ orbits) does not constitute evidence forthe existence of gravitational energy. That fact may most probablybe explained in terms of tidal friction as suggested in 1976 [15] as analternative to the explanation given by Hulse and Taylor.

Despite that there is no room for gravitational energy in generalrelativity, it is an experimental fact that energy participates in grav-itational phenomena, but that energy is well accommodated in thetheory. Take for example the energy of oceanic tides which is trans-formed into electrical energy in tidal power stations. The tidal energyis part of gravitational phenomena, but is not gravitational energy.It seems most appropriate to call it inertial energy because it orig-inates from the work done by inertial forces acting on the blades ofthe tidal turbines – the blades further deviate the volumes of waterfrom following their geodesic (inertial) paths (the water volumes arealready deviated since they are prevented from falling) and the watervolumes resist the further change in their inertial motion; that is, thewater volumes exert inertial forces on the blades. With respect to theresistance, this example is equivalent to the situation in hydroelectricpower plants where water falls on the turbine blades from a height(the latter example is even clearer) – the blades prevent the waterfrom falling (i.e. from moving by inertia) and it resists that change.It is that resistance force (i.e. inertial force) that moves the turbine,which converts the inertial energy of the falling water into electricalenergy. According to the standard explanation it is the kinetic energyof the falling water (originating from its potential energy) that is con-verted into electrical energy. However, it is evident that behind thekinetic energy of the moving water is its inertia (its resistance to itsbeing prevented from falling) – it is the inertial force with which thewater acts on the turbine blades when prevented from falling. And itcan be immediately seen that the inertial energy of the falling water(the work done by the inertial force on the turbine blades) is equal toits kinetic energy (see Appendix B: On inertial forces, inertial energyand the origin of inertia).

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Conclusion

The fact that for decades the e↵orts of so many brilliant physicists tocreate a quantum theory of gravity have not been successful seems toindicate that those e↵orts might not have been in the right direction.In such desperate situations in fundamental physics all options shouldbe on the research table, including the option that quantum gravity asquantization of gravitational interaction is impossible because a rig-orous treatment of gravity as a manifestation of the non-Euclideangeometry of spacetime demonstrates that there is no gravitational in-teraction and therefore there is nothing to quantize.

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