Dynamical evolution of a scalar field coupling to Einstein's tensor in the Reissner-Nordström black hole spacetime
aa r X i v : . [ g r- q c ] S e p Dynamical evolution of a scalar field coupling to Einstein’s tensor in theReissner-Nordstr¨om black hole spacetime
Songbai Chen ∗ , Jiliang Jing † Institute of Physics and Department of Physics, Hunan Normal University,Changsha, Hunan 410081, People’s Republic of ChinaKey Laboratory of Low Dimensional Quantum Structuresand Quantum Control of Ministry of Education, Hunan Normal University,Changsha, Hunan 410081, People’s Republic of China
Abstract
We study the dynamical evolution of a scalar field coupling to Einstein’s tensor in the backgroundof Reissner-Nordstr¨om black hole. Our results show that the the coupling constant η imprints inthe wave dynamics of a scalar perturbation. In the weak coupling, we find that with the increase ofthe coupling constant η the real parts of the fundamental quasinormal frequencies decrease and theabsolute values of imaginary parts increase for fixed charge q and multipole number l . In the strongcoupling, we find that for l = 0 the instability occurs when η is larger than a certain threshold value η c which deceases with the multipole number l and charge q . However, for the lowest l = 0, wefind that there does not exist such a threshold value and the scalar field always decays for arbitrarycoupling constant. PACS numbers: 04.70.Dy, 95.30.Sf, 97.60.Lf ∗ [email protected] † [email protected] I. INTRODUCTION
The dynamical evolution of an external field perturbation around a black hole has been an object ofgreat interest for physicists for the last few decades. One of the main reasons is that the frequencies ofthe quasinormal oscillations appeared in the dynamical evolution carry the characteristic information aboutthe black hole, which could provide a way for astrophysicists to identify whether there exists black hole inour Universe or not [1–3]. The further studies indicate that the quasinormal spectrum could help us tounderstand more deeply about the quantum gravity [4–6] and the AdS/CFT correspondence [7–9]. Moreover,the stability of a black hole can be examined by the study of the dynamical behaviors of the perturbations inthe background spacetime [10–13]. Therefore, a lot of attention have been focused on the dynamical evolutionof various perturbations in the various black holes spacetime.The simplest field in the quantum field theory is scalar field, which associated with spin-0 particles. Thedynamical evolution of the scalar field in the different black hole spacetimes has been investigated extensively.It is found that for usual scalar field the late-time evolution after the quasinormal oscillations is dominated bythe form t − (2 l +3) for the massless field [14–16] and by the oscillatory inverse power-law form t − ( l +3 / sin µt for the massive one [17, 18]. Moreover, the dynamical evolution of the scalar field has also been considered inthe cosmology, which shows that the scalar field can be presented as the inflaton to drive the inflation of theearly Universe [19] and as the dark energy to drive the accelerated expansion of the current Universe [20–22].However, the above investigations are limited to the case where the action has a form S = Z d x √− g (cid:20) R πG + 12 ∂ µ ψ∂ µ ψ + V ( ψ ) + ξRψ (cid:21) + S m , (1)where ψ , R and V ( ψ ) are corresponding to scalar field, Ricci scalar and scalar potential, respectively. In thisaction, the coupling between the scalar field and the spacetime curvature contains only the term ξRψ , whichrepresents the coupling between the scalar field and the Ricci scalar curvature.Theoretically, the general form of the action with more couplings between the scalar field and the spacetimecurvature can be expressed as S = Z d x √− g (cid:20) f ( ψ, R, R µν R µν , R µνρσ R µνρσ ) + K ( ψ, ∂ µ ψ∂ µ ψ, ∇ ψ, R µν ∂ µ ψ∂ ν ψ, · · · ) + V ( ψ ) (cid:21) + S m , (2)where f and K are arbitrary functions of the corresponding variables. Obviously, the nonlinear functions f and K provide the more non-minimal couplings between the scalar field and the curvature of the backgroundspacetime. These new couplings modify the usual Klein-Gordon equation so that the motion equation forthe scalar field is no longer generally a second-order differential equation in this case, which yields the morecomplicated behavior of the scalar field in the background spacetime. By introducing the derivative couplingterm R µν ∂ µ ψ∂ ν ψ , Amendola [23] studied recently the dynamical evolution of the coupled scalar field in thecosmology and obtained some new analytical inflationary solutions. Capozziello et al. [24, 25] investigateda more general model with two coupling terms R∂ µ ψ∂ ν ψ and R µν ∂ µ ψ∂ ν ψ , and found that the de Sitterspacetime is an attractor solution in this case. Recently, Sushkov found [26] that the equation of motionfor the scalar field can be reduced to second-order differential equation when it is kinetically coupled to theEinstein tensor. This means that the theory is a “good” dynamical theory from the point of view of physics.Moreover, Sushkov [26] also found that in cosmology the problem of graceful exit from inflation with thederivative coupling term G µν ∂ µ ψ∂ ν has a natural solution without any fine-tuned potential. Recently, Gao[27] investigated the cosmic evolution of a scalar field with the kinetic term coupling to more than one Einsteintensors, and found that the scalar field behaves exactly as the pressureless matter if the kinetic term is coupledto one Einstein tensor and acts nearly as a dynamic cosmological constant if it couples with more than oneEinstein tensors. The similar investigations have been considered in Refs.[28, 29]. We studied the greybodyfactor and Hawking radiation for a scalar field coupling to Einstein’s tensor in the background of Reissner-Nordstr¨om black hole spacetime and found that the presence of the coupling enhances both the absorptionprobability and Hawking radiation of the black hole [30]. These results may excite more efforts to be focusedon the study of the scalar field coupled with tensors in the more general cases. The main purpose of this paperis to investigate the dynamical evolution of the scalar perturbation coupling to the Einstein tensor G µν in theReissner-Nordstr¨om black hole spacetime and see the effect of the coupling on the stability of the black hole.The plan of our paper is organized as follows: in the following section we will introduce the action of ascalar field coupling to Einstein’s tensor and derive its master equation in the Reissner-Nordstr¨om black holespacetime. In Sec.III, we will study the effect of the coupling on the quasinormal modes in the weaker coupling,and then examine the stability of the black hole in the stronger coupling. Finally, in the last section we willinclude our conclusions. II. THE WAVE EQUATION OF A SCALAR FIELD COUPLING TO EINSTEIN’S TENSOR INTHE REISSNER-NORDSTR ¨OM BLACK HOLE SPACETIME
In order to study the dynamical evolution of a scalar field coupling to Einstein’s tensor in a black holespacetime, we must first obtain its wave equation in the background. The action of the scalar field couplingto the Einstein’s tensor G µν in the curved spacetime has a form [26], S = Z d x √− g (cid:20) R πG + 12 ∂ µ ψ∂ µ ψ + η G µν ∂ µ ψ∂ ν ψ (cid:21) . (3)The coupling between Einstein’s tensor G µν and the scalar field ψ is represented by the term η G µν ∂ µ ψ∂ ν ψ ,where η is coupling constant with dimensions of length-squared.Varying the action (3) with respect to ψ , one can find the wave equation of a scalar field coupling toEinstein’s tensor can be expressed as [26, 30]1 √− g ∂ µ (cid:20) √− g (cid:18) g µν + ηG µν (cid:19) ∂ ν ψ (cid:21) = 0 . (4)Obviously, the dynamical evolution of a scalar field depends on the the Einstein’s tensor G µν and the couplingconstant η . Since all the components of the tensor G µν vanish in the Schwarzschild black hole spacetime, wecannot probe the effect of the coupling term on the dynamical behavior of the scalar perturbation. In thegeneral relative theory, the simplest black hole with the non-zero components of the tensor G µν is Reissner-Nordstr¨om one, whose metric has a form ds = − f dt + 1 f dr + r dθ + r sin θdφ , (5)with f = 1 − Mr + q r , (6)where M is the mass and q is the charge of the black hole. The Einstein’s tensor G µν for the metric (5) hasa form G µν = q r − f f − r − r sin θ . (7)Defining tortoise coordinate dr ∗ = 1 /f ( r ) dr and separating ψ ( t, r, θ, φ ) = e − iωt R ( r ) Y lm ( θ,φ ) r √ r + ηq , we can obtain theradial equation for the scalar perturbation coupling to Einstein’s tensor in the Reissner-Nordstr¨om black holespacetime d R ( r ) dr ∗ + [ ω − V ( r )] R ( r ) = 0 , (8)with the effective potential V ( r ) = f (cid:18) r − ηq r + ηq (cid:19)(cid:20) l ( l + 1) r + f ′ r (cid:21) + f r ηq (3 r + ηq )( r + ηq ) . (9)Obviously, as the coupling constant η = 0 the radial equation (8) reduces to that of the scalar one withoutcoupling to Einstein’s tensor. In the case η = 0, the coupling constant η emerges in the effective potential,which means that coupling between the scalar perturbation and Einstein’s tensor will change the dynamicalevolution of the scalar perturbation in the background spacetime. In Fig.1, we plot the changes of the V H r L - V H r L - V H r L FIG. 1: Variety of the effective potential V ( r ) with the polar coordinate r for fixed l = 0 (left), l = 1 (middle) and l = 2 (right). The solid, dashed, dash-dotted and dotted lines are corresponding to the cases with η = 0 , , , M = 1 and q = 0 . effective potential V ( r ) with the coupling constant η for fixed l and q . With increase of η , the peak height ofthe potential barrier increases for l = 0 and decreases for other values of l . Moreover, one can find that forthe smaller η the effective potential V ( r ) is positive definite everywhere outside the black hole event horizon.This implies that the solution of the wave equation (4) is bounded and the black hole is stable in this case.However, for the larger η , we find that the effective potential V ( r ) has negative gap, and then the stability isnot guaranteed. In the following section, we will check those values of η for which the negative gap is presentand study the stability of the black hole when the scalar perturbation is coupling to Einstein’s tensor. III. THE INSTABILITY OF SCALAR FIELD COUPLING TO EINSTEIN’S TENSOR IN THEBACKGROUND OF A REISSNER-NORDSTR ¨OM BLACK HOLE
In this section, we first consider the quasinormal modes in the weaker coupling case in which the effectivepotential V ( r ) is positive definite and study the effects of the coupling on the quasinormal frequencies. Then,we shall study the evolution of the scalar field coupling to Einstein’s tensor in time domain using a numericalcharacteristic integration method [31] and check the instability of the black hole in the stronger coupling.Let us now to study the effects of the coupling constant on the massless scalar quasinormal modes inthe Reissner-Nordstr¨om black hole spacetime in the weaker coupling case. In Fig.2 and 3, we present thefundamental quasinormal modes ( n = 0) evaluated by the third-order WKB approximation method [32, 33].It is shown that with the increase of the coupling constant η the real parts of the quasinormal frequenciesdecrease and the absolute values of imaginary parts increases for fixed l and q . This means that the presence Ω R Ω R Ω R FIG. 2: Variety of the real parts of the fundamental quasinormal modes with q for scalar field coupling to Einstein’stensor in the Reissner-Nordstr¨om black hole spacetime. The figures from left to right are corresponding to l = 0 , η = 0 , , ,
6, respectively.We set 2 M = 1. - Ω I - Ω I - Ω I FIG. 3: Variety of the absolute value of imaginary parts of the fundamental quasinormal modes with q for scalarfield coupling to Einstein’s tensor in the Reissner-Nordstr¨om black hole spacetime. The figures from left to right arecorresponding to l = 0 , η = 0 , , ,
6, respectively. We set 2 M = 1. of the coupling parameter η makes the decay of the scalar perturbation more quickly in this case. From Fig.2and 3, one can easily obtain that with increase of q the real parts ω R increases for the smaller η and decreasesfor the larger η . The changes of the absolute values of imaginary parts with q are more complicated. For l = 0, it decreases for the smaller η and increases for the larger η , while for other values of l , it increase with q for all η . These results imply that the presence of the coupling terms modifies the standard results in thequasinormal modes of the scalar perturbations in the background of a Reissner-Nordstr¨om black hole.We are now in a position to study the dynamical evolution of the scalar field coupling to Einstein’s tensorin time domain and examine the stability of the black hole in the stronger coupling cases. Adopting to thelight-cone variables u = t − r ∗ and v = t + r ∗ , one can find that the wave equation − ∂ ψ∂t + ∂ ψ∂r ∗ = V ( r ) ψ, (10)can be rewritten as 4 ∂ ψ∂u∂v + V ( r ) ψ = 0 . (11)This two-dimensional wave equation (11) can be integrated numerically by using the finite difference method l =
0, q = - - - -
202 t L og È Ψ È l =
1, q = - - - - - L og È Ψ È l =
2, q = - - L og È Ψ È FIG. 4: The dynamical evolution of a scalar field coupling to Einstein’s tensor in the background of a Reissner-Nordstr¨om black hole spacetime. The figures from left to right are corresponding to l = 0 , η = 1 , , , M = 1.The constants in the Gauss pulse (13) v c = 10 and σ = 3. suggested in [31]. In terms of Taylor’s theorem, it can be discretized as ψ N = ψ E + ψ W − ψ S − δuδvV ( v N + v W − u N − u E ψ W + ψ E O ( ǫ ) = 0 , (12)where we have used the following definitions for the points: N : ( u + δu, v + δv ), W : ( u + δu, v ), E : ( u, v + δv )and S : ( u, v ). The parameter ǫ is an overall grid scalar factor, so that δu ∼ δv ∼ ǫ . Since the behavior of thewave function is not sensitive to the choice of initial data, we can set ψ ( u, v = v ) = 0 and use a Gaussianpulse as an initial perturbation, centered on v c and with width σ on u = u as ψ ( u = u , v ) = e − ( v − vc )22 σ . (13)In fig.4, we present the dynamical evolution of the scalar field coupling to Einstein’s tensor in the backgroundof a Reissner-Nordstr¨om black hole. For the coupling constant η = 1, the decay of the coupling scalar fieldis similar to that of the scalar one without coupling to Einstein’s tensor, which indicates that the black holeis stable in the weaker coupling. It is expectable because the effective potential V ( r ) is positive definite inthis cases. For l = 0, we also note that the scalar field always decays for any value of the coupling constant η . This means that the lowest l are stable, which can be explained by a fact that for l = 0, the higher η raiseup the peak of the potential barrier so that the potential is always positive definite. Moreover, for the highermultipole numbers l , we find that the scalar field grows with exponential rate as the coupling constant η is l - Η c q = = = FIG. 5: The change of the threshold value η c with the inverse multipole number l − for fixed q . The points l =1 , , , , ,
100 were fitted by the function η c = al − . + b . The values of ( a, b ) for q = 0 . , . . . , . , (73 . , .
10) and (24 . , . larger than the critical value η c , which means that the instability occurs in this case. The main reason is thatfor l = 0 the large η drops down the peak of the potential barrier and increases the negative gap near theblack hole horizon so that the potential could be non-positive definite. In the instability region, the larger η , the instability growth occurs at the earlier times, and the growth rate is the stronger. Furthermore, weplotted the change of the threshold value η c with l in fig.5, and found that the threshold value can be fittedbest by the function η c ≃ al − . + b, (14)where a and b are numerical constants. It is easy to obtain that the lowest l are stable because the threshold a b FIG. 6: The changes of the numerical constants a, b with the charge q , which were fitted by the functions a = q − . and b = r /q , respectively. value η c → ∞ as l →
0. Moreover, for the higher l and smaller q , we have the smaller threshold value η c atwhich instability happens. The varieties of the numerical constants a, b with q are presented in fig.6, whichshows that the values of a, b are fitted best by the functions a ≃ q − . and b ≃ r /q , respectively. Thus, as thecharge q vanishes, the threshold value η c tends to infinite for arbitrary l , which means that the Schwarzschildblack hole is stable when it is perturbed by a scalar field coupling to Einstein’s tensor. Actually, since allthe components of the Einstein’s tensor disappear in the Schwarzschild black hole spacetime, the dynamicalevolution of the coupling scalar field is consistent with that of the scalar one without coupling to Einstein’stensor. For the extreme black hole, we find from fig.6 and Eq.(14) that the threshold value η c is the minimumfor arbitrary l = 0, which implies that the instability happens more easily in the extreme black hole. Fromthe previous discussions, we can obtain that in the limit l → ∞ the threshold value η c → b and the effectivepotential (9) has the form V ( r ) | l →∞ = f (cid:18) r − ηq r + ηq (cid:19) l ( l + 1) r . (15)According to the method suggested in [34], the integration Z ∞ r + V ( r ) | l →∞ f dr = Z ∞ r + (cid:18) r − ηq r + ηq (cid:19) l ( l + 1) r dr, (16)is positive definite as η < r /q . It implies that the threshold value has a form η c = r /q as l → ∞ , whichis consistent with the form of the numerical constant b obtained in the previous numerical calculation. IV. SUMMARY
In this paper, we have studied the dynamical evolution of a scalar field coupling to Einstein’s tensor in thebackground of Reissner-Nordstr¨om black hole. Our results show that the the coupling constant η imprintsin the wave dynamics of a scalar perturbation. For the multipole number l = 0, we find that the scalar fieldalways decays for arbitrary coupling constant η . For l = 0, the instability occurs when η is larger than thecritical value η c . Moreover, for the higher l , we have the smaller threshold value η c . In the weak coupling(i.e., η ≪ η c ) case, we find that with the increase of the coupling constant η the real part of the fundamentalquasinormal frequencies decreases and the absolute value of imaginary parts increases for fixed l and q . Withincrease of q the real part ω R increases for the smaller η and decreases for the larger η . For l = 0, the absolutevalue of imaginary parts decreases for the smaller η and increases for the larger η , while for other values of l , it increases with q for all η . Moreover, we find that the threshold value can be fitted best by the function η c ≃ al − . + b and the numerical constants a, b decrease with the charge q . These rich dynamical propertiesof the scalar field coupling to Einstein’s tensor, which could provide a way to detect whether there exist acoupling between the scalar field and Einstein’s tensor or not. It would be of interest to generalize our study0to other black hole spacetimes, such as rotating black holes etc. Work in this direction will be reported in thefuture. V. ACKNOWLEDGMENTS
This work was partially supported by the National Natural Science Foundation of China under GrantNo.10875041, the Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT,No. IRT0964) and the construct program of key disciplines in Hunan Province. J. Jing’s work was partiallysupported by the National Natural Science Foundation of China under Grant Nos. 10875040 and 10935013;973 Program Grant No. 2010CB833004 and the Hunan Provincial Natural Science Foundation of China underGrant No.08JJ3010. [1] S. Chandrasekhar and S. Detweller, Proc. R. Soc. Lond. A , 441 (1975).[2] H. P. Nollert, Class. Quantum Grav. , R159 (1999).[3] K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel. , 2 (1999).[4] S. Hod, Phys. Rev. Lett. , 4293 (1998).[5] O. Dreyer, Phys. Rev. Lett. , 081301 (2003).[6] A. Corichi, Phys. Rev. D , 087502 (2003); L. Motl, Adv. Theor. Math. Phys. , 1135 (2003); L. Motl and A.Neitzke, Adv. Theor. Math. Phys. , 307 (2003); A. Maassen van den Brink, J. Math. Phys. , 327 (2004); G.Kunstatter, Phys. Rev. Lett. , 161301 (2003); N. Andersson and C. J. Howls, Class. Quantum Grav. , 1623(2004); V. Cardoso, J. Natario and R. Schiappa, J. Math. Phys. , 4698 (2004); J. Natario and R. Schiappa,Adv. Theor. Math. Phys. , 1001 (2004); V. Cardoso and J. P. S. Lemos, Phys. Rev. D , 084020 (2003).[7] J. Maldacena, Adv. Theor. Math. Phys. , 231 (1998).[8] E. Witten, Adv. Theor. Math. Phys. , 253 (1998).[9] G. T. Horowitz and V. E. Hubeny, Phys. Rev. D , 024027 (2000); B. Wang, C. Y. Lin and E. Abdalla, Phys.Lett. B , 79 (2000) ; J. M. Zhu, B. Wang and E. Abdalla, Phys. Rev. D , 124004 (2001); V. Cardoso and J.P. S. Lemos, Phys. Rev. D , 124015 (2001); V. Cardoso and J. P. S. Lemos, Phys. Rev. D , 084017 (2001);E. Berti and K. D. Kokkotas, Phys. Rev. D , 064020 (2003); E. Winstanley, Phys. Rev. D , 104010 (2001);J. S. F. Chan and R. B. Mann, Phys. Rev. D , 064025 (1999).[10] R. Gregory and R. Laflamme, Phys. Rev. Lett. , 2837 (1993); R. Gregory and R. Laflamme, Nucl. Phys. B ,399 (1994).[11] T. Harmark, V. Niarchos and N. A. Obers, Class. Quant. Grav. , R1 (2007).[12] R. A. Konoplya, K. Murata, Jiro Soda and A. Zhidenko, Phys. Rev. D , 084012 (2008); J. L. Hovdebo and R.C. Myers, Phys. Rev. D , 084013 (2006).[13] S. B. Chen and J. L. Jing, JHEP , 081 (2009).[14] R. H. Price, Phys. Rev. D , 2419 (1972). [15] S. Hod and T. Piran, Phys. Rev. D , 024017 (1998).[16] L. Barack, Phys. Rev. D , 024026 (2000); L. M. Burko and G. Khanna, Phys. Rev. D , 081502 (2003); E. S.C. Ching, P. T. Leung, W. M. Suen and K. Young, Phys. Rev. D , 2118 (1995).[17] H. Koyama and A. Tomimatsu, Phys. Rev. D , 064032 (2001); H. Koyama and A. Tomimatsu, Phys. Rev. D , 044014 (2001); R. Moderski and M. Rogatko, Phys. Rev. D , 044024 (2001); R. Moderski and M. Rogatko,Phys. Rev. D , 084014 (2001); R. Moderski and M. Rogatko, Phys. Rev. D , 044027 (2005); S. Hod and T.Piran, Phys. Rev. D , 044018 (1998). S. B. Chen and J. L. Jing, Mod. Phys. Lett. A , 35 (2008); S. B. Chen,B. Wang and R. K. Su, Int. J. Mod. Phys. A , 2502 (2008).[18] S. Hod, Phys. Rev. D , 104022 (1998) ; L. Barack and A. Ori, Phys. Rev. Lett. , 4388 (1999) ; W. krivan,Phys. Rev. D , 101501(R) (1999); Q. Y. Pan and J. L. Jing, Chin. Phys. Lett. , 1873 (2004).[19] A. H. Guth, Phys. Rev. D , 347 (1981). 1a[20] B. Ratra and J. Peebles, Phys. Rev. D , 3406 (1988); C. Wetterich, Nucl. Phys. B , 668 (1988); R. R.Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. , 1582 (1988) ; M. Doran and J. Jaeckel, Phys. Rev.D , 043519 (2002).[21] C. A. Picon, T. Damour and V. Mukhanov, Phys. Lett. B , 209 (1999); T. Chiba, T. Okabe and M. Yamaguchi,Phys. Rev. D , 023511 (2000).[22] R. R. Caldwell, Phys. Lett. B , 23 (2002) ; B. McInnes, J. High Energy Phys. , 029 (2002); S. Nojiri and S.D. Odintsov, Phys. Lett. B , 147 (2003); L. P. Chimento and R. Lazkoz, Phys. Rev. Lett. , 211301 (2003); B.Boisseau, G. Esposito-Farese, D. Polarski, Alexei A. Starobinsky, Phys. Rev. Lett. , 2236 (2000); R. Gannouji,D. Polarski, A. Ranquet, A. A. Starobinsky, JCAP , 016 (2006).[23] L. Amendola, Phys. Lett. B , 175 (1993).[24] S. Capozziello, G. Lambiase and H. J.Schmidt, Annalen Phys. , 39 (2000).[25] S. Capozziello, G. Lambiase, Gen. Rel. Grav. , 1005 (1999) .[26] S. V. Sushkov, Phys. Rev. D , 103505 (2009).[27] C. J. Gao, JCAP , 023 (2010), arXiv: 1002.4035.[28] L.N. Granda, arXiv: 0911.3702.[29] E. N. Saridakis and S. V. Sushkov, Phys. Rev. D , 083510 (2010), arXiv: 1002.3478.[30] S. B. Chen and J. L. Jing, arXiv: 1005.5601.[31] C. Gundlach, R. H. Price and J. Pullin, Phys. Rev. D , 883 (1994).[32] B. F. Schutz and C. M. Will, Astrophys. J. Lett. , L33 (1985).[33] S. Iyer and C. M. Will, Phys. Rev. D , 3621 (1987); S. Iyer, Phys. Rev. D , 3632 (1987).[34] R. J. Gleiser and G. Dotti, Phys. Rev. D , 124002 (2005); W. F. Buell and B. A. Shadwick, Am. J. Phys.63