Open Access Journal of Physics Volume 2, Issue 4, 2018, PP 24-28 ISSN 2637-5826

The Equation of Motion in Rindler Space

Sanchita Das1, Somenath Chakrabarty2 1Department of physics,visva bharati santiniketan 731235,india *Corresponding Author: Somenath Chakrabarty, Department of physics,visva bharati santiniketan 731235,india

ABSTRACT The equation of motion of a massive particle in Rindler space or equivalently in an uniform gravitational field has been studied in both classical and quantum mechanical scenarios. The classical geodesics of motion of the particle and the wave function for the quantum mechanical equation satisfied by the particle are obtained. It has been observed that it is equivalent to the problem of free fall of the particle at the center. Unlike the conventional scenario where the fall occurs at the origin, which is a singularity, here it takes place at the point which is at a finite distance from the origin (singularity). Further, the classical trajectories are not a perihelion type. These are closed type. The massive particle returns to its original position if disturbed from the equilibrium position (having minimum energy). PACS number: 03.65.Ge,03.65.Pm,03.30.+p,04.20.-q

INTRODUCTION local acceleration or local uniform gravitational field. Hence the line element in Rindler space in It is well known that the Lorentz transformations 1+1 –dimension is given by of space time coordinates are between the inertial frame of references [1], whereas the 푑푠2 = −푑휏2 = − 1 + 훼푥 2푑푡2 + 푑푥2 (3) Rindler transformations are between an inertial Where 훕 is the proper time. Then one can define frame and the frame undergoing uniform the classical action integral in the form [1] accelerated motion [2-10]. The space here is 퐵 퐵 called the Rindler space, exactly like the 푑휏 = 퐿 푑흀 (4) Minkowski space in the Lorentz transformation 퐴 퐴 scenario. Further, the Rindler space is locally Where 휆 is an affine parameter which is flat, whereas the flatness of Minkowski space is changing along the trajectory and having the 푑휏 global in nature. Now according to the principle dimension of time. Hence 퐿 = , the 푑휆 of equivalence a frame undergoing accelerated Lagrangian of the particle, explicitly given motion in absence of gravity is equivalent to a by[11-13] frame at rest in presence of gravitational field. 1 The strength of the gravitational field is exactly 푑푡 2 푑푥 2 2 퐿 = 1 + 훼푥 2 + − (5) equal to the magnitude of the acceleration of the 푑휆 푑휆 moving frame. Therefore the Rindler space may be considered to be a space associated with a The aim of the present article is to investigate frame at rest in presence of a uniform the classical motion as well as the quantum gravitational field. The Rindler space-time mechanical motion of the non-zero mass particle transformations in natural unit (푐 = ћ = 1) and in Rindler space. Apparently from the title of in 1+1 dimension are given by [7-10] this article it may sound that it is a very old problem, however so far our knowledge is 1 푡 = + 푥′ sinh 훼푡′ (1) concerned it has not been studied before. 훼 Therefore we expect that this work should be 1 푥 = + 푥′ cosh 훼푡′ (2) reported in some reputed journal. In this article 훼 we have obtained the classical geodesics of Where primed coordinates are in the non-inertial motion and the wave functions for the quantum frame, whereas the unprimed are in the inertial mechanical equation satisfied by the particle. It one. The Rindler space-time transformations are has been observed that the problem is equivalent therefore exactly like the Lorentz transformations to the problem of free fall. We have also but in uniformly accelerated frame. Here α is the investigated this free fall problem of the

Open Access Journal of Physics V2● 14 ● 2018 24 The Equation of Motion in Rindler Space particle. Although the gravitational field is (see Appendix for some outline to obtain this considered to be uniform throughout, however expression). Now changing the variable from 푥 the energy of the particle is minimum at the 1 to 푢 = 1 + 훼푥 and redefining 휏 → 퐶2훼흉 , we origin. As a consequence the particle will fall at have from the above equation, the point which is at a finite distance from the 푑2푢 1 origin 푥 = 0 .Now in the Rindler coordinate + = 0 (10) system, the portion 푥 > |푡| of the Minkowski 푑휏 2 푢3 space is called the Rindler wedge. The second This is the classical equation of motion for the wedge 푥 < −|푡|can be obtained by reflection. particle in Rindler space. Integrating both sides We call the first one as right wedge and the with respect to 푢 , this equation may be second one the left wedge of Rindler space. The transformed to the form null rays act as the event horizons for Rindler 1 푑푢 2 1 observers. An observer in the right wedge − = 푐표푛푠푡푎푛푡 (11) 2 푑휏 2푢2 cannot see any eventsin the left wedge. These two regions are casually disjoint two Re-writing 훕 in its original form and defining 푑푢 universes.However, exactly like the Minkowski 푝 = as the momentum of the particle of unit 푑휏 space the past and the present can be defined mass, we have from the above equation and are casually connected. Therefore the actual 2 2 1 푝 퐶훼 singularity 푥 = − is not an accessible to an − 2 = 푐표푛푠푡푎푛푡 = 퐸 (12) 훼 2 2푢 observer in right wedge. The origin (which is Where the first term on the left hand side is like not a singularity) is set at 푥 = 0. To the best of the kinetic energy for a unit mass and the second knowledge this problem has not been done term is an attractive inverse square potential, before.Various sections of the article are whereas the constant on the right hand side may organized in the following manner. In the next be treated as the total energy of the particle. We section we shall develop the formalism of the further assume that the energy of the particle is classical motion of the particle in Rindler space. negative in nature, i.e., we are considering In section-3 we have discussed the quantum bounded motion of the classical particle. Now mechanical motion in Rindler space and finally expressing the above equation in the usual form, in section-4 we have given the conclusion of our given by work. 1 푝2 + 푉 푢 = 퐸 (13) CLASSICAL MOTION IN RINDLER SPACE 2 퐶훼2 Now the well-known Euler-Lagrange equation Where 푉 푢 = − , the inverse square 2푢2 is given by attractive potential. Then depending on the 2 휕 휕퐿 휕퐿 strength 0.5퐶훼 of the attractive potential 푉 푢 , 휕푞 − = 0 (6) the particle may fall at the center( 푥 = 0 or 휕휆 휕 푖 휕푞 푖 휕휆 푢 = 1 ), where the momentum becomes zero Where 푞푖 = 푡 or 푥. Let us first consider 푞푖 = 푡, (푝 = 0, i.e., particle is in the rest condition) and the universal time coordinate. Since t is cyclic in this situation 퐸 = 푉 푢 = 1 = −0.5퐶훼2, the 휕퐿 coordinate, = 0 and therefore maximum depth of the potential, which is also 휕푡 the minimum possible value of the total energy. 휕푡 퐶 = (7) Since it is negative it gives the maximum 휕휏 1+훼푥 2 binding at 푥 = 0 or 푢 = 1 . Since α is the Where 퐶 is a constant and to obtain this uniform acceleration, the parameter 퐶 may be equation we have used the expression for the treated as the strength of the potential 푉 푢 . Lagrangian as given by equation (5). We next One can calculate the two classical turning consider 푞푖 = 푥, then with some little algebra, points in case the motion is oscillatory between we have from the Euler-Lagrange equation two extreme points, where the particle momentum becomes zero. These are at 푢 = 휕푡 2 푑2푥 1 + 훼푥 훼 + 2 = 0 (8) 푢0 = 1 , the central point in the transformed 휕휏 푑휏 1 휕푡 special coordinate and 푢 = 푢 = 0.5퐶|퐸| 2훼 On substituting the expression for from 1 휕휏 for 퐸 negative = −|퐸|. In this case, of course equation (7) we have the motion is periodic. However, if because of 훼퐶 푑2푥 strong attractive gravitational potential at + = 0 (9) 1+훼푥 3 푑휏 2 푢 = 푢0 (for the large value of 퐶) the particle

25 Open Access Journal of Physics V2● 14 ● 2018 The Equation of Motion in Rindler Space falls at the center, i.e., at 푥 = 0 or 푢 = 푢0 = 1 Using the changed variable 푢(= 1 + 훼푥) , we then the motion will no longer remain have, oscillatory. In fig.(1) we have shown this ퟏ ퟏ ퟐ ퟐ ퟏ potential well. The coordinate 푥 is plotted along 퐂ퟐ ퟏ+훂퐱 ퟏ α훕 = − ퟏ − − ퟏ − ퟐ (16) 푥 -axis, where the magnitude of depth of 훂 ퟏ 퐂 퐂ퟐ potential |푉 푢 | is plotted along 푦 -axis. Therefor in actual scenario this is not the Since the uniform acceleration α of the frame is potential hill but is potential well. But unlike the completely arbitrary, we shall study the conventional scenario of fall at the center variation of 훼푥 with α훕, keeping in problem, there is no singularity at the origin consideration that α remains constant within this (푥 = 0) , or at the center of the gravitating 2 푥 range and is static in nature. The above object[14-16]. The value of 푉(푢) is −0.5퐶훼 at equation can also be expressed in the following the origin and → −1 for large 푢 values. more convenient form Therefore unlike the conventional case, where 1 2 1 the potential part → −∞ at the origin, here, since 1 2 훼휏 the minimum value of 푢 is positive unity, 훼푥 = 퐶2 1 − 1 − − 1 − 1 (17) 퐶 2 therefore 푉 푢 → −0.5퐶훼2 for 푥 → 0or 푢 → 1. 퐶 In this case the singularity is as if covered by a Which will give the variation of 훼푥 with α훕. point which is at finite distance from the origin Before we obtain numerically the variation of or the actual singularity. Here the origin has α푥 with α훕, we shall impose some constraints on been shifted to 푥 = 0 point. Therefore here the the quantities appearing in the above equation. problem of fall at the origin reduces to the By inspection one can put the following problem of fall at a point which is having finite restrictions: , and if , then . positive special coordinate value. 퐶 > 1 훼푥 > 0 훼흉 > 0 Since we are measuring 푥 along positive direction and the frame is also moving with uniform acceleration α along positive 푥 - direrction, the second constraint is also quit obvious. In fig.(2) we have shown the variation of 훼푥 with α훕 for three different 퐶 values. From the curves one may infer that the geodesics are closed in nature; returned to the initial point, i.e., at 푥 = 0 or 푢 = 1 (Boomerang type geodesic), where the energy of the particle is minimum. The geodesics are not like the perihelion precession type as has been obtained in Schwarzschild geometry [17,18]. Although the gravitational field is constant throughout the region, the turning down of the particle, when it is dynamically disturbed from its equilibrium Figure1. The variation of |푉 푥 | with x position is because of the minimum value of the energy of the particle at the origin 푥 = 0 . For For further investigation of this problem of fall the proper justification of our arguments as at the center to some extent elaborately, let us given above, we next consider the equation of get the geodesics of motion for the particle in motion of the particle under inverse square Rindler space. We start with the equation potential, given by eqn.(10). This equation has 2 2 2 푑푡 푑푥 been solved numerically for the initial 1 + 훼푥 − = 1 (14) 푑푢 푑휏 푑휏 conditions = 0, i.e., the motion is assumed to 푑푡 푑휏 Hence we have after substituting for from be started from rest and for a number of initial 푑휏 eqn.(7) positional coordinates in the scaled form given by 푢 . The boundary value of 푢 is unity, not the 푥 푑푥 0 휏 = 0 1 (15) singular point 푢 = 0 , which is in the left 퐶 −1 2 1+훼푥 2 Rindler wedge. In fig.(3) we have shown the geodesics of motion for three different 푢0 values.

Open Access Journal of Physics V2● 14 ● 2018 26 The Equation of Motion in Rindler Space

well behaved at infinity. Next to study the nature of wave function near the point 푢 = 1, 푅(푢) we substitute 훹 푢 = . Then it can very 푢 easily be been shown that 푅(푢) satisfies the equation

푑2푅 푑푅 푢2 − 2푢 + 훾 − 2 − 퐾2푢2 푅 = 0 푑푢 2 푑푢 (20) Putting 퐾푢 = 휌 as a new variable, we have,

푑2푅 2 푑푅 훾−2 − + 푅 − 푅 = 0 (21) 푑휌 2 휌 푑휌 휌2 To investigate the nature of 푅 near unit point we

write 휌 = 휌0 in the denominator of the second and the third term and then put the limit 휌 → 1. Figure2. The geodesics of motion for various 퐶 0 values: curve (a) is for 퐶 = 2, (b) is for 퐶 = 5, (c) is Then we have for 퐶 = 10, and (d) is for 퐶 = 25. 푑2푅 푑푅 − 2 + 훾 − 3 푅 = 0 (22) 푑휌 2 푑휌 Then for 훾 < 4, we have the solution

1 푢 휌 = exp⁡(휌) 퐴1푒푥푝 4 − 훾 2휌 + 퐴2푒푥푝−4−훾12휌 (23)

Whereas for 훾 > 4, the solution is given by 1 푢 휌 = exp⁡(휌) 퐴1푒푥푝 푖 훾 − 4 2휌 + 퐴2푒푥푝−푖훾−412휌 (24)

The solutions are standard stationary type and exist for 휌 = 휌0 → 퐾. Further, for bound state solution, i.e., for 퐸 < 0, 퐸퐾 < 푉 , where 퐸퐾 Figure3. The geodesics of motion for various 푢0 and 푉 are the kinetic energy and attractive values: curve (a) is for 푢0 = 15, (b) is for 푢0 = 10 potential respectively. The quantity 푉 is and (c) is for 푢 = 5. 0 maximum for 휌 = 휌0 = 퐾 , i.e., at 푥 = 0 and QUANTUM MECHANICAL EQUATION approaches unity as ρ increases to infinity. Therefore the negative value of 퐸 is maximum Now to complete the study of free fall at the on the unit point. This indirectly indicates the center in our modified formalism , let us next “fall” on the point at 푢 = 1 . The basic investigate the quantum mechanical motion of difference with the usual solution is that the fall the particle in inverse square attractive potential is not at the center (휌 = 0) which is covered by [14,16]. We consider the bound state problem, the unit point. i.e., 퐸 < 0. Re-substituting the exact form of 훕, we have the eigen value equation for a unit mass CONCLUSION particle Therefore we may conclude that in the Rindler 푝2 훽 space the energy of the particle will be 퐻훹 푢 = − 훹 푢 = −퐸훹(푢) (18) 2 푢2 minimum at the point which is the newly 훼2퐶 defined origin or 푥 = 0 or 푢 = 1 . This Where 훽 = . Hence we have 2 indirectly indicate that the particle will occupy 푑2훹 훾 this point because of the minimum valus of the + 훹 푢 − 퐾2훹 푢 = 0 (19) 푑푢2 푢2 energy. This is true for classical as well as 2훽 2퐸 quantum mechanical scenarios. The other Where 훾 = and 퐾2 = . Therefore in the ћ2 ћ2 interesting finding of this work is that unlike the asymptotic region, i.e., 푢 → ∞, 훹~푒−퐾푢 or it perihelion type geodesics these are closed type. goes to zero. The wave function is therefore We call them as boomerang geodesics.

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APPENDIX [3] W. Rindler, Essential Relativity, Springer- Verlag, New York, (1977). In this appendix we shall give a short outline to [4] G.F. Torres del Castillo and C.L. Perez 2 2 1 푑푡 푑 푥 Sanchez, Revista Mexican De Fisika 52,70, show that 1 + 훼푥 훼 + 2 = 퐿 푑휆 푑휆 (2006). 1 푑푡 2 푑휏 2 푑2푥 푑2휏 1 + 훼푥 훼 + = [5] Domingo J. Louis-Martinez, arXiv:10123063. 퐿 푑휏 푑휆 푑휏 2 푑휆 2 훼퐶 푑2푥 [6] M. Socolovsky, Annales de la Foundation + = 0 (25) 1+훼푥 3 푑휏 2 Louis de Broglie 39,1,(2014).

We have [7] Sanchari De, Sutapa Ghosh and Somenath Chakrabarty, Modern Physics Letter A, 30, 푑푡 푑푡 푑휏 푑푡 = = 퐿 1550182, (2015). 푑휆 푑휏 푑휆 푑휏 [8] Sanchari De, Sutapa Ghosh and Somenath (26) Chakrabarty, Astrophys. Space Sci., (2015) Hence 360:8,DOI 10.1007/s10509-015-2520-3. [9] Soma Mitra and Somenath Chakrabarty, EPJ 푑푡 2 푑푡 2 = 퐿2 Plu s,71, 5, (2017). 푑휏 푑휏 (27) [10] C-G Huang, and J-R, Sun, J-R, arXiv:gr- qc/0701078,(2007). 푑2푥 푑 푑푥 푑 푑푥 푑휏 = = [11] J.B. Hartle, Gravity, Benjamin Cumming, 푑휆 2 푑휆 푑휆 푑휏 푑휆 푑휆 (28) (2003). [12] S. Weinberg, Gravitation and Cosmology, John 푑2푥 푑 푑푥 푑휏 = Wiley & Sons, (1972). 푑휆 2 푑휆 푑휏 푑휆 (29) [13] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, Freeman, (1973). 2 2 2 2 푑 푥 푑 푥 푑휏 2 푑 푥 [14] L.D. Landau and E.M. Lifshitz, Quantum 2 = 2 = 퐿 2 푑휆 푑휏 푑휆 푑휆 Mechanics (Non-relativistic Theory), (30) Butterworth Heinemann, (1981). Hence we have the above equality after putting, [15] Vasyl M. Vasyua and Volodymyr M. Tkachuk, 푑푡 퐶 = Eur. Phys. Jour. D70, 267, (2016). 2 푑휆 1+훼푥 [16] Elisa Guillaumin-Espana, H.N. Nunez-Yepez REFERENCES and A.L. Salas-Brito, Jour. Math. Phys. 55, 103509, (2014). [1] Landau L.D. and Lifshitz E.M., The classical [17] S. L. Sapiro and S.A.Teukolsky, Black Holes, Theory of Fields, Butterworth-Heimenann, White Dwarfs and Neutron Stars, John Wiley Oxford, (1975). and Sons, New York, (1983). [2] N.D. Birrel and P.C.W. Davies, Quantum Field [18] R.M. Wald, General Relativity, Cambridge Theory in Curved Space, Cambridge University University press, (1984). Press, Cambridge, (1982).

Citation: Sanchita Das,Somenath Chakrabarty,"The Equation of Motion in Rindler Space", Open Access Journal of Physics, vol. 2, no. 4, pp. 24-28, 2018. Copyright: © 2018 Sanchita Das and Somenath Chakrabarty. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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