Classial Trajetories in Rindler Spae and Restrited Struture of Phase Spae with PT-Symmetri Hamiltonian Soma Mitra 1 and Somenath Chakrabarty 2 Department of Physis, Visva-Bharati, Santiniketan 731 235, West Bengal, India 1 somaphysis@gmail.om 2 E-mail:somenath.hakrabarty@visva-bharati.a.in Abstrat The nature of single partile lassial phase spae trajetories in Rindler spae with the PTsymmetri hamiltonian have been studied. It has been shown that only a small portion of the phase spae is aessible to the partiles, whereas the major part of the phase spae region is ompletely forbidden. It has also been notied that the area / volume of the forbidden region of phase spae inreases with the inrease in the strength of gravitational field. The physial signifiane of suh squeezing of phase spae in strong gravitational field has been disussed. PACS numbers: 03.65.Ge,03.65.Pm,03.30.+p,04.20.-q 1
It is well known that in the ase of a system of inertial frame of referenes, the spaetime transformations in the relativisti senario are given by the Lorentz transformations [1, 2]. However, for a frame undergoing a uniform aelerated motion in an otherwise flat Minkowski spae-time geometry, the Lorentz type transformation an also be derived to relate the oordinate sets in aelerated frame with that of inertial set of oordinates [3 7]. These are the so alled Rindler oordinate transformations and are given by: ( ) ( ) 2 αt t = α + x sinh and ( ) ( ) 2 αt x = α + x osh (1) Hene one an show that the inverse transformations an be written as: ( ) t = 2 x + t 2α ln x t and x = (x 2 (t) 2 ) 1/2 2 α (2) Where α is the uniform aeleration of the frame. The Rindler spae-time oordinates as shown above (eqns.(1) and (2)) are then just an aelerated frame transformation of the Minkowski metri of speial relativity. It an very easily be shown that the Rindler oordinate transformation hanges the Minkowski line element ds 2 = d(t) 2 dx 2 dy 2 dz 2 to ( ) 2 ds 2 = 1 + αx d(t ) 2 dx 2 dy 2 dz 2 The general form of metri tensor may then be written as ( ( g µν = diag 1 + αx ) ) 2, 1, 1, 1 (3) (4) whereas in the ase of onventional flat Minkowski spae-time we have g µν = diag(+1, 1, 1, 1) (5) A survey of literature shows that the onept of priniple of equivalene is essential to obtain the Rindler oordinate transformation. Our study is therefore based on the priniple of equivalene, aording to whih a frame of referene undergoing an aelerated motion in absene of gravitational field is equivalent to a frame at rest in presene of a gravitational field. Therefore a frame undergoing a uniform aeleration motion is equivalent to a rest frame in presene of a onstant gravitational field of strength equal to the magnitude of the 2
aeleration. Then α in the present senario may be treated as the onstant gravitational field for a frame at rest. Now following the onept of relativisti dynamis of speial theory of relativity [1], the ation integral may be written as b b S = α 0 ds Ldt (6) a a Then putting α 0 = m 0, as has been done in speial theory of relativity, where m 0 is the rest mass of the partile and is the veloity of light, the Lagrangian of the partile is given by L = m 0 [ ( 1 + αx ) 2 v 2 ] where v is the three veloity of the partile. The three momentum of the partile is then given by Hene the Hamiltonian of the partile may be written as p = L, or v (8) m 0 v p = [ (1 ) ] 2 1/2 + αx v 2 2 (9) (7) H = p. v L or (10) ( H = m 0 1 + αx ) ( ) 1/2 1 + p2 (11) m 2 0 In the lassial level, the quantities H, x and p are treated as dynamial variables. Further, it an very easily be verified that the Hamiltonian H eqn.(11) is not hermitian. However the energy spetum has been observed to be real [8]. This is found to be solely beause of the fat that H is PT-invarient. Now it is well know that PxP 1 = x, PpP 1 = p, whereas, TpT 1 = p and PαP 1 = α but TαT 1 = α, therefore it is a matter of simple algebra to show that PT H (PT) 1 = H PT = H. As has been shown by several authors [9] that if H PT = H, then the energy spetrum will be real. Here P and T are respetively the parity and the time reversal operators. Whih has also been reported in one of our publiations [8]. Of ourse with the replaement of hermitiity of the Hamiltonian with the P T-symmetry, we have not disarded the important quantum mehanial key features of the system desribed this Hamiltonian (eqn.(11)). This point was also disussed in an elaborate manner in referene [9] and in some of the referenes ited there. 3
The lassial Hamilton s equation of motion for the partile is then given by [10] ẋ = [H, x] p.x and ṗ = [H, p] p,x (12) where [H, f] p,x for f = x or p is the Poisson braket and is defined by [10] [f, g] p,x = f g p x f g x p (13) Here dot indiates the derivative with respet to time. Now using the expression for Hamiltonian from eqn.(11), the expliit form of the equation of motions are given by ( ẋ = 1 + αx ) p (p 2 + m 2 0 4 ) 1/2 and ṗ = α (p2 + m 2 0 4 ) 1/2 (14) The parametri form of x and p whih give the time evolution of spae oordinate and the orresponding anonial momentum an be obtained after integrating the above oupled equations and are given by x = 2 α [C 0 os(ωt) 1] and p = m 0 tan(ωt nπ) (15) where C 0 is some integration onstant, ω = α/ is the frequeny defined for some kind of quanta in [8] and n is an integer inluding zero. Now for x = 0 at t = 0, we have C 0 = 1. Hene we an write x = [os(ωt) 1] (16) ω Again it is very easy to show that for any integer n, inluding zero, we have p = m 0 tan(ωt) (17) To draw the trajetories for the partile in the phase spae, we start with the relation Integrating over x and p, we have dp dx = ṗ ẋ = ω p 2 + m 2 ( ) 0 4 1 + xω p 2 (p 2 + m 2 0 4 ) 1/2 = C 0 ( 1 + xω (18) ) 1 (19) Now assuming that for x = 0, p = 0, the stable ritial points, the integration onstant beomes C 0 = m 0. Then we an rewrite the above equation in the form [ ( p = m 0 1 + xω ) ] 2 1 4 (20)
This is the mathematial form of lassial trajetory of the partile in the phase spae. It is quite obvious from the above expression that x an not be positive. The positive value of x will make the momentum imaginary, whih is unphysial. Therefore the part of the phase spae with x > 0 are ompletely forbidden for the partile under onsideration in Rindler spae. Further, one an easily verify that as x /ω, the momentum p ±. Beyond whih the momentum will again beome imaginary. Therefore in the Rindler spae only the portion of phase spae with /ω < x 0 will be aessible to the partiles. Sine p is the x-omponent of momentum, it an be negative as well. It is also obvious that if the strength of gravitational field is strong enough the range of x will derease. Therefore near the event horizon of a blak hole sine the gravitational field strength α is extremely high, the aessible phase spae volume / area will aordingly be low enough. In the extreme ase, when the gravitational field strength is infinitely large, the area / volume of the phase spae will be of infinitesimally small in the negative x diretion. In fig.(1) we have shematially shown the appearane of the trajetories. For the sake of illustration we have used /ω = 1 and m 0 = 0.1, 9.5 and 1.0. The upper symmetri pair is for m 0 = 1 and the other two pairs are for 0.5 and 0.1 respetively. It is also obvious from the appearane of the trajetories that all of them are approahing from asymptotially p ± and x 1 to (0, 0) point. Now from the knowledge of statistial mehanis one an infer that with the derease in phase spae volume (or area in low dimension), the abundane of partiles will derease. Therefore the physial signifiane of derease in phase spae volume near the event horizon of a blak hole is that there will be more and more redution in number of partiles at that region. This is of ourse not orret for the eletromagneti waves with semi-lassial quantized form. This is beause of the zero rest mass of the photons. Massless partiles an not exist in pure lassial idea. The derease in abundane of massive partiles with the redution is phase spae area / volume indiretly indiates the absorption of partiles by the blak holes instead of their emission. This is exatly what we expet from the lassial model of a blak hole [11, 12]. Therefore our final onlusion is that at the surfae of event horizon, the phase spae volume or area is vanishingly small and as a onsequene the aomodated partile number is also negligibly small. In other wards, no partile an exist in stable onfiguration very lose to 5
the event horizon of a lassial blak hole. [1] Landau L.D. and Lifshitz E.M., The Classial Theory of Fields, Butterworth-Heimenann, Oxford, (1975). [2] W.G. Rosser, Contemporary Physis, 1, 453, (1960). [3] N.D. Birrell and P.C.W. Davies, Quantum Field Theory in Curved Spae, Cambridge University Press, Cambridge, (1982). [4] C.G. Huang and J.R. Sun, arxiv:gr-q/0701078, (2007). [5] Domingo J Louis-Martinez, Class. Quantum Grav., 28, 036004, (2011). [6] D. Peroo and V.M. Villaba, Class. Quantum Grav., 9, 307, (1992). [7] S. De, S. Ghosh and S. Chakrabarty, Astrophys and Spae Si. (in press, 2015). [8] S. De, S. Ghosh and S. Chakrabarty, Mod. Phys. Lett. A (in press 2015). [9] Carl M. Bender, arxiv:quant-ph/0501052 (and referenes therein) [10] Classial Mehanis, H. Goldstein, Addision Wesley (1972). [11] S.L. Shapiro and S.A. Teukolsky, Blak Holes, White Dwarfs and Neutron Stars, John Wiley and Sons, New York, (1983). [12] S. Weinberg, Gravitation and Cosmology, John Wiley & Sons, New York, (1972). 6
FIG. 1: Shemati diagram for the phase spae trajetories for the typial values of m 0 = 0.1, 0.5 and 1.0 7