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.”

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”, http://philsci-archive.pitt.edu/11857 (preprint).

**References**

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, R*elativity: 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, https://en.wikipedia.org/wiki/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).