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Free Fall and Self-Force: an Historical Perspective

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Mass and Motion in General Relativity

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 162))

Abstract

Free fall has signed the greatest markings in the history of physics through the leaning Pisa tower, the Woolsthorpe apple tree and the Einstein lift. The perspectives offered by the capture of stars by supermassive black holes are to be cherished, because the study of the motion of falling stars will constitute a giant step forward in the understanding of gravitation in the regime of strong field. After an account on the perception of free fall in ancient times and on the behaviour of a gravitating mass in Newtonian physics, this chapter deals with last century debate on the repulsion for a Schwarzschild–Droste black hole and mentions the issue of an infalling particle velocity at the horizon. Further, black hole perturbations and numerical methods are presented, paving the way to the introduction of the self-force and other back-action related methods. The impact of the perturbations on the motion of the falling particle is computed via the tail, the back-scattered part of the perturbations, or via a radiative Green function. In the former approach, the self-force acts upon the background geodesic; in the latter, the geodesic is conceived in the total (background plus perturbations) field. Regularisation techniques (mode-sum and Riemann–Hurwitz z function) intervene to cancel divergencies coming from the infinitesimal size of the particle. An account is given on the state of the art, including the last results obtained in this most classical problem, together with a perspective encompassing future space gravitational wave interferometry and head-on particle physics experiments. As free fall is patently non-adiabatic, it requires the most sophisticated techniques for studying the evolution of the motion. In this scenario, the potential of the self-consistent approach, by means of which the background geodesic is continuously corrected by the self-force contribution, is examined.

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Notes

  1. 1.

    Leonardo spent his final years at Amboise, nowadays part of the French Région Centre, under invitation of François I, King of France and Duke of Orléans.

  2. 2.

    His book presents the contributions by several less-known researchers in the flow of time, being a well argued and historical – but rather uncritical – account. An other limitation is the neglect of non-Western contributions to the development of physics.

  3. 3.

    Indeed, it has been stated by Synge [196] ‘...Perhaps they speak of the principle of equivalence. If so, it is my turn to have a blank mind, for I have never been able to understand this principle...’

  4. 4.

    For a review on experimental status of these fundamental laws, see Will’s classical references [211, 212], or else Lämmerzahl’s alternative view [113], while the relation to energy conservation is analysed by Haugan [99].

  5. 5.

    For the first definition, it is worth mentioning the following observation [196] ‘...Does it mean that the effects of a gravitational field are indistinguishable from the effects of an observer’s acceleration? If so, it is false. In Einstein’s theory, either there is a gravitational field or there is none, according as the Riemann tensor does not or does vanish. This is an absolute property; it has nothing to do with any observer’s world-line. Space-time is either flat or curved...’ Patently, the converse is also far reaching: if an inertial acceleration was strictly equivalent to one produced by a gravitational field, curvature would be then associated to inertial accelerations. Rohrlich [173] stresses that the gravitational field must be static and homogeneous and thus in absence of tidal forces. But no such a gravitational field exists or even may be conceived! Furthermore, the particle internal structure has to be neglected.

    The second definition is under scrutiny by numerous experimental tests compelled by modern theories as pointed out by Damour [46] and Fayet [79].

    First and last two definitions are correct in the limit of a point mass. An interesting discussion is offered by Ciufolini and Wheeler [38] on the non-applicability of the concept of a locally inertial frame (indeed a spherical drop of liquid in a gravity field would be deformed by tidal forces after some time, and a state-of -the-art gradiometer may reach sensitivities such as to detect the tidal forces of a weak gravitational field in a freely falling cabin). Mathematically, locality, for which the metric tensor g μν reduces to the Minkowski metric and the first derivatives of the metric tensor are zero, is limited by the non-vanishing of the Riemann curvature tensor, as in general certain combinations of the second derivatives of g μν cannot be removed. Pragmatically, it may be concluded that violating effects on the EP may be negligible in a sufficiently small spacetime region, close to a given event.

  6. 6.

    Again, this opinion is comforted [196] ‘...the principle of equivalence performed the essential office of midwife at the birth of general relativity...I suggest that the midwife be now buried with appropriate honours...’.

  7. 7.

    The difference between fall in vacuum and in the air has been the subject of a polemics between the former French Minister of Higher Education and Research Claude Allègre and the Physics Nobel Prize Georges Charpak, solicited by the satirical weekly ‘Le Canard Enchaîné’ [120]. The Minister affirmed on French television in 1999 ‘Pick a student, ask him a simple question in physics: take a petanque and a tennis ball, release them; which one arrives first? The student would tell you ‘the petanque’. Hey no, they arrive together; and it is a fundamental problem, for which 2000 years were necessary to understand it. These are the basis that everyone should know’. The humourists wisecracked that the presence of air would indeed prove the student being right and tested their claim by means of filled and empty plastic water bottles being released from the second floor of their editorial offices…and asked the Nobel winner to compute the difference due to the air, whose influence was denied by the Minister. But in this polemics, no one drew the attention to the Newtonian back-action, also during the polemics revamped in 2003 by Allègre [1] who compared this time a heavy object and a paper ball. Such forgetfulness or misconception is best represented by the Apollo 15 display of the simultaneous fall of a feather and a hammer [4].

  8. 8.

    During the Bloomington 2009 Capra meeting, this state of affairs was presented as ‘the confusion gauge’.

  9. 9.

    Rothman [174] gives a brief historical account on Droste’s independent derivation of the same metric published by Schwarzschild, in the same year 1916. Eisenstaedt [76] mentions previous attempts by Droste [64] on the basis of the preliminary versions of general relativity by Einstein and Grossmann [73], later followed by Einstein’s works (general relativity was completed in 1915 and first systematically presented in 1916 [72]) and Hilbert’s [101]. Antoci [2] and Liebscher [3] emphasise Hilbert’s [102] and Weyl’s [207] later derivations of solutions for spherically symmetric non-rotating bodies. Incidentally, Ferraris, Francaviglia and Reina [80] point to the contributions of Einstein and Grossmann [74], Lorentz [126] and obviously Hilbert [101] to the variational formulation.

  10. 10.

    Earman and Eisenstaedt [69] describe the lack of interest of Einstein for singularities in general relativity. The debate at the Collège de France during Einstein’s visit in Paris in 1922 included a witty exchange on pression (the Hadamard ‘disaster’), see Biezunski [26].

  11. 11.

    The translation of the title and of the introduction to Section 5 of [77] serves best this paragraph ‘The impasse (or have the relativists fear of the free fall?) [..] the problem of the free fall of bodies in the frame of [..] the Schwarzschild solution. More than any other, this question gathers the optimal conditions of interest, on the technical and epistemological levels, without inducing nevertheless a focused concern by the experts. Though, is it necessary to emphasise that it is a first class problem to which classical mechanics has always showed great concern … from Galileo; which more is the reference model expressing technically the paradigm of the lift in free fall dear to Einstein? The matter is such that the case is the most elementary, most natural, an extremely simple problem … apparently, but which raises extremely delicate questions to which only the less conscious relativists believe to reply with answers [..]. Exactly the type of naive question that best experts prefer to leave in the shadow, in absence of an answer that has to be patently clear to be an answer. Without doubts, it is also the reason for which this question induces a very moderate interest among the relativists…’

  12. 12.

    This definition is not faultless (there is no shield to gravity), but it is the most suitable to describe the debate, following Cavalieri and Spinelli [31, 32, 193] and Thirring [199].

  13. 13.

    Generally, the setting of the proper initial conditions may be a delicate issue e.g. when associated with an initial radiation content expressing the previous history of the motion as it will be later discussed; or, in absence of radiation, when an external (sort of third body) mechanism prompting the motion to the two body system is to be taken into account. The latter case is represented by the thought experiment conceived by Copperstock [39] aiming to criticise the quadrupole formula. The experiment consisted in two fluid balls assumed to be in static equilibrium and held apart by a strut, with membranes to contain the fluid, until time t = 0. Between t = 0 and t = t 1, the strut and the membranes are dissolved and afterwards the balls fall freely. Due to the static initial conditions, there is a clear absence of incident radiation, but the behaviour of the fluid balls in the free fall phase depends on how the transition from the equilibrium to the free fall takes place. This initial dependence obscured the debate on the quadrupole formula.

  14. 14.

    Free fall has also been studied in other contexts. Synge [196] undertakes a detailed investigation of the problem and shows that, actually, the gravitational field (i.e. the Riemann tensor) plays an extremely small role in the phenomenon of free fall and the acceleration of 980 cm/s is, in fact, due to the curvature of the world line of the tree branch. The apple is accelerated until the stem breaks, then the world line of the apple becomes inertial until the ground collides with it.

  15. 15.

    For a critical assessment of black hole stability, see Dafermos and Rodnianski [44].

  16. 16.

    A well-organised introduction, largely based on works by Friedman [84] and Chandrasekhar [33], is presented in the already mentioned book by the latter [34]. Some selected publications geared to the finalities of this chapter are to be listed: earlier works by Mathews [141], Stachel [195], Vishveshvara [204]; the relation between odd and parity perturbations [35]; the search for a gauge invariant formalism by Martel and Poisson [140] complements a recent review on gauge invariant non-spherical metric perturbations of the SD black hole spacetimes by Nagar and Rezzolla [154]; a classic reference on multiple expansion of gravitational radiation by Thorne [200]; the derivation by computer algebra by Cruciani [41, 42] of the wave equation governing black hole perturbations; the numerical hyperboloidal approach by Zenginonğlu [217].

  17. 17.

    Two warnings: the literature on perturbations and numerical methods is rather plagued by editorial errors (likely herein too...) and different terminologies for the same families of perturbations. Even parity waves have been named also polar or electric or magnetic, generating some confusion (see the correlation table, Table II, in [220]). Sago, Nakano and Sasaki [180] have corrected the Zerilli equations (a minus sign missing in all right-hand side terms) but introduced a wrong definition of the scalar product leading to errors in the coefficients of the energy-momentum tensor.

  18. 18.

    There is an editorial error, a numerical coefficient, in the corresponding expressions (2.16) in [133] and (2.8) in [134], which the footnote 1 at page 3 in [139] does not address.

  19. 19.

    For a starting point different from infinity or a non-null starting velocity, but not their combination, see Lousto and Price [133–135], Martel and Poisson [139].

  20. 20.

    The divergence in summing over all l modes is said to be taken away by considering a finite size particle [49].

  21. 21.

    There are editorial errors in the corresponding expressions (3.6) in [134] and (3.4) in [139].

  22. 22.

    There are editorial errors in the corresponding expressions (3.9) in [134] and (3.5) in [139].

  23. 23.

    The Heaviside or step distribution, like the wavefunction of the Zerilli equation, belongs to the C  − 1 continuity class; the Dirac delta distribution and its derivative belong to the C  − 2 and C  − 3 continuity class, respectively.

  24. 24.

    A point-like mass m moves along a geodesic of the background spacetime if m → 0; if not, the motion is no longer geodesic. It is sometimes stated that the interaction of the particle with its own gravitational field gives rise to the self-force. It should be added, though, that such interaction is due to an external factor like a background curved spacetime or a force imposing an acceleration on the mass. In other words, a single and unique mass in an otherwise empty universe cannot experience any self-force. Conceptually, the self-force is thus a manifestation of non-locality in the sense of Mach’s inertia [136].

  25. 25.

    It is currently believed that the core of most galaxies host supermassive black holes on which stars and compact objects in the neighbourhood inspiral-down and plunge-in. Gravitational waves might also be detected when radiated by the Milky Way Sgr*A, the central black hole of more than 3 million solar masses [30, 83]. The EMRIs are further characterised by a huge number of parameters that, when spanned over a large period, produce a yet unmanageable number of templates. Thus, in alternative to matched filtering, other methods based on covariance or on time and frequency analysis are investigated. If the signal from a capture is not individually detectable, it still may contribute to the statistical background [17].

  26. 26.

    In 2002 at the Capra Penn State meeting by Eric Poisson.

  27. 27.

    Given the elegance of this classic approach, the self-force expression should be rebaptised as MiSaTaQuWa-DeWh.

  28. 28.

    Detweiler and Whiting [59] refer to the contribution inside the light cone via the Hadamard expression [97] of the Green function.

  29. 29.

    The self-force being affected by the gauge choice, the EP allows to find a gauge where the self-force disappears. Again, as in Newtonian physics, such gauge will be dependent of the mass m, impeding the uniqueness of acceleration.

  30. 30.

    Apart from some editorial errors therein, α1, 2, 6 correspond to the A, B, C coefficients in [129, 130], which are not to be confused with the A, B, C coefficients of the mode-sum!

  31. 31.

    Supposing that the relative strengths of the perturbations and the deviations behave as:

    $$\frac{{[{h}^{(1)}]}^{2}} {g} \simeq \frac{{h}^{(2)}} {g} \ll \frac{{h}^{(1)}} {g} < \frac{\Delta \dot{z}} {{\dot{z}}_{p}} \simeq \frac{\Delta z} {{z}_{p}},$$
    (61)

    then, the coordinate acceleration correction would be given by an expansion up to first order in perturbations and second order in deviation [190]:

    $$\begin{array}{rcl} \Delta \ddot{z}& =& {\alpha }_{1}\left (g,{\dot{z}}_{u}\right )\Delta z + {\alpha }_{2}\left (g,{\dot{z}}_{u}\right )\Delta \dot{z} + {\alpha }_{3}\left (g,{\dot{z}}_{u}\right )\Delta {z}^{2} + {\alpha }_{ 4}\left (g\right )\Delta {\dot{z}}^{2} \\ & & +{\alpha }_{5}\left (g,{\dot{z}}_{u}\right )\Delta z\Delta \dot{z} + {\alpha }_{6}\left (h,{\dot{z}}_{u}\right ) + {\alpha }_{7}\left (h,{\dot{z}}_{u}\right )\Delta z + {\alpha }_{8}\left (h,{\dot{z}}_{u}\right )\Delta \dot{z}. \end{array}$$
    (62)

    In Eq. 62: (i) solely second order terms in perturbations are not considered; (ii) the terms \({\alpha }_{2}\left (g,{\dot{z}}_{u}\right )\Delta \dot{z},{\alpha }_{3}\left (g,{\dot{z}}_{u}\right )\Delta {z}^{2},{\alpha }_{4}\left (g\right )\Delta {\dot{z}}^{2},{\alpha }_{5}\left (g,{\dot{z}}_{u}\right )\Delta z\Delta \dot{z}\) represent the background field evaluated on the perturbed trajectory at second order in deviation; (iii) α3 − 5 tend to infinity close to the horizon, conversely to the α1, 2 coefficients; (iv) \({\alpha }_{7}\left (h,{\dot{z}}_{u}\right )\Delta z,{\alpha }_{8}\left (h,{\dot{z}}_{u}\right )\Delta \dot{z}\) represent the perturbed field on the perturbed trajectory, and the α7 − 8 coefficients are larger near the horizon. These last two coefficients may be regularised in l by the Riemann–Hurwitz ζ function as shown in [190].

  32. 32.

    The jump conditions were also dealt with by Sopuerta and Laguna [188].

  33. 33.

    Having suppressed the l index for clarity of notation, after visual inspection of Eq. 26, containing a derivative of the Dirac delta distribution, it is evinced that the wavefunction Ψ is of C  − 1 continuity class and thus can be written as:

    $$\Psi (t,r) = {\Psi }^{+}(t,r)\ {\Theta }_{ 1} + {\Psi }^{-}(t,r)\ {\Theta }_{ 2},$$
    (63)

    where Θ 1 = Θr − z u (t) and Θ 2 = Θz u (t) − r are two Heaviside step distributions. Computing the first and second, space and time and mixed derivatives, Dirac delta distributions and derivatives are obtained of the type δ[r − z u (t)] and δ′[r − z u (t)], respectively. It is wished that the discontinuities of Ψ and its derivatives are such that they are canceled when combined in K, H 2, and H 1. After replacing Ψ and its derivatives in Eqs. 31–33, continuity requires that the coefficients of Θ 1 must be equal to the coefficients of Θ 2, while the coefficients of δ and δ′ must vanish separately. After some tedious computing and making use of one of the Dirac delta distribution properties, f(r)δ′[r − z u (t)] = f[z u (t)]δ′[r − z u (t)] − f′(z u (t))δ[r − z u (t)], at the position of the particle, the jump conditions for Ψ and its derivatives are found. Furthermore, the jump conditions allow a new method of integration, as shown by Aoudia and Spallicci [6].

  34. 34.

    Lousto [129] comments only this former part and not the acceleration boost taking place after the Zerilli potential peak.

  35. 35.

    In the Rapid Communication [18] there are seven citations of a yet unpublished material containing mathematical and numerical justifications of the results therein. The author acknowledges private communications by L. Barack.

  36. 36.

    The evolution of an orbit is lately getting the necessary concern. Pound and Poisson [162] apply osculating orbits to EMRI, but unfortunately their method is not applicable to plunge, for two reasons: the semi-latus rectum of the orbit, which decreases for radiation reaction, is smaller than a given quantity, considered as limit in their study case; the velocities and fields in the plunge are highly relativistic and their post-Newtonian expansion of the perturbing force becomes inaccurate. Non-applicability to plunge stands also for the work by Hinderer and Flanagan [104].

  37. 37.

    Indeed, it affirms that it is preferable to apply successively a first order expansion at x 0 and then at x 1, x 2, …,x m , rather then a second or higher order expansion at solely x 0. It is evident, though, that self-consistency and perturbation order are decoupled concepts and that the former may be conceptually applicable to higher orders and more specifically, when, and if, a second order formalism will be available. In the same line of reasoning it would be preferable to apply successively a second order expansion at x 0 and then at x 1, x 2, …, x m , rather then a third or higher order expansion at solely x 0.

  38. 38.

    See Levi-Civita [122, 123], Ciufolini and Wheeler [38], Ciufolini [37].

  39. 39.

    For Mino and Brink [149], the energy and momentum radiated are computed on the assumption that the small body falls in a dynamical time scale, with respect to proper time, well short of the radiation reaction time scale and therefore the gravitational radiation back-action on the orbit is considered negligible. The particle plunges on a geodesic trajectory, incidentally starting from a circular orbit, thus at zero initial radial velocity.

  40. 40.

    For head-on collisions, Price and Pullin have surprisingly shown the applicability of perturbation theory for the computation of the radiated energy and of the waveform for two equal black holes, starting at very small separation distances [164], the so-called close-limit approximation.

  41. 41.

    A non-recent analysis by Simone, Poisson and Will [187], between pN and perturbation methods in head-on collisions, was limited to the computation of the gravitational wave energy flux.

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Acknowledgements

Discussions throughout the years with S. Aoudia, L. Barack, S. Chandrasekhar, S. Detweiler, C. Lousto, J. Martin-Garcia, S. Gralla, E. Poisson, R. Price, R. Wald, B. Whiting are acknowledged. I would like to thank E. Vergès for the support to CNRS School on Mass and the 11th Capra conference held in June 2008 in Orléans. It was my sincere hope that both events could put together different communities working on mass and motion: part of the contributors to this book and their colleagues (Blanchet, Detweiler, Le Tiec and Whiting) have already made concrete steps towards such cooperation [27, 28].

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  1. Observatoire des Sciences de l’Univers en région Centre, Université d’Orléans, LPC2E, Campus CNRS, 3A Avenue de la Recherche Scientifique, 45071, Orléans, France

    Alessandro Spallicci

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  1. Institut d'Astrophysique de Paris, boulevard Arago 98bis, Paris, 75014, France

    Luc Blanchet

  2. Observatoire de la Cote d'Azur, Nice, France

    Alessandro Spallicci

  3. Dept. Physics, University of Florida, Gainesville, 32611-8440, Florida, USA

    Bernard Whiting

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Spallicci, A. (2009). Free Fall and Self-Force: an Historical Perspective. In: Blanchet, L., Spallicci, A., Whiting, B. (eds) Mass and Motion in General Relativity. Fundamental Theories of Physics, vol 162. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3015-3_20

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