Particle properties outside of the static limit in cosmology
aa r X i v : . [ g r- q c ] A p r Particle properties outside of the static limit in cosmology
A. A. Grib
A. Friedmann Laboratory for Theoretical Physics, Saint Petersburg, Russia;Theoretical Physics and Astronomy Department of the Herzen University,Moika 48, Saint Petersburg 191186, Russiaandrei [email protected]
Yu. V. Pavlov
Institute of Problems in Mechanical Engineering, Russian Academy of Sciences,Bol’shoy pr. 61, St. Petersburg 199178, Russia;N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University,18 Kremlyovskaya St., Kazan 420008, [email protected]
Received Day Month YearRevised Day Month YearIt is shown that in the rest frame of the observer in expanding Universe states of particleswith negative energy exist.The properties of such states are studied. The comparison withthe case of negative energies of particles in black holes and rotating coordinates out ofthe static limit is made.
Keywords : Negative energy; Penrose process; expanding Universe.PACS numbers: 04.20.-q, 98.80.Jk
1. Introduction
The existence of particles with negative energies and the possibility of observationsof consequences of their existence is well known in the black hole physics due tothe Penrose process.1 , mc ( c is the velocity of light) then there is an illusionthat the negative energy is possible only in case of the very strong external (forexample gravitational) field. Really, for velocity v ≪ c the full negative energy E of the particle moving on the distance r from the attracting massive body with themass M E = mc + mv − G mMr < ⇒ r < GMc = r g , (1)where G is gravitational constant, r g is the gravitational radius. However the defini-tion of the energy depends on the choice of the Killing vector which corresponds totranslation in time of the reference frame. This leads to the possibility of existenceof negative energy in noninertial reference frame. A. A. Grib & Yu. V. Pavlov
In papers3 ,
2. Negative Energies in Expanding Universe
Take the interval of the Friedmann homogeneous and isotropic expanding Universein the Robertson-Walker form ds = c dt − a ( t ) (cid:18) dr − kr + r d Ω (cid:19) , (2)where k = ± , d Ω = dθ + sin θ dϕ . Incoordinates t, r, θ, ϕ the background matter defining the metric of space-time is atrest. For simplicity take the case k = 0. Then the distance from the observer at restrelative to the background matter at the point r = 0, for example of the astronomeron the nonrotating Earth, to the object with the coordinate r will be defined by theproduct ar (see Refs. 6, 7). Let us use the new coordinates t, D, θ, ϕ : D = a ( t ) r, dD = ˙ ar dt + a dr. (3)Then dr = dDa − D ˙ aa dt (4)and the interval (2) becomes ds = (cid:18) − D ˙ a c a (cid:19) c dt + 2 D ˙ aa dDdt − dD − D d Ω . (5)Consider particle movement in metric of the general form ds = g ( dx ) + 2 g dx dx + g ( dx ) + g ΩΩ d Ω , (6)where g < g ΩΩ < g g − g <
0. From the condition ds ≥ g − q g − g g − g g ΩΩ (cid:0) d Ω dx (cid:1) − g ≤ dx dx ≤ g + q g − g g − g g ΩΩ (cid:0) d Ω dx (cid:1) − g . (7) article properties outside of the static limit in cosmology One can see from (7) that the surface g = 0 plays the role of the static limit.In the chosen coordinate system in the region g < g > a > g < a < p i E = p c = mcg k dx k dτ = mc dx dτ (cid:18) g + g dx dx (cid:19) . (8)Here τ is the particle proper time. If the metric does not depend on time then theenergy is conserved. The necessary condition for the energy to be negative is dx dx < − g g . (9)Using the limitation (7) one obtains the limitations on possible values of the energyof particle mc − g dx dτ g − g g − | g | s g − g g − g g ΩΩ (cid:18) d Ω dx (cid:19) ≤ E ≤ mc − g dx dτ g − g g + | g | s g − g g − g g ΩΩ (cid:18) d Ω dx (cid:19) . (10)From inequalities (10) one has that at the region where g > g < D s = ca | ˙ a | = c | h ( t ) | , (11)where h ( t ) = ˙ a/a is the Hubble parameter. The energy of the freely moving particleis E = E ′ (cid:18) − ˙ a a D c + ˙ aa Dc dDdt (cid:19) , (12)where E ′ = mc dt/dτ . So in these coordinates particles with negative energies aremoving in coordinate D > D s so that the velocity is slower than some definite value dDdt < c aD ˙ a (cid:18) D D s − (cid:19) . (13)Let us rewrite the inequality (13) in terms of coordinates t, r, θ, ϕ . Then we obtainin the case ˙ a > v = a drdt < − c D s D , D > D s . (14) A. A. Grib & Yu. V. Pavlov
This has a meaning similar to that obtained by us for the case of rotating coordinateframe t, r, θ, ϕ : particles with negative energies close to the static limit ( D → D s )must move with velocities close to the light velocity in direction of the observer inexpanding Universe.Now let us discuss a special case of the de Sitter Universe with the scale factor a = a exp Ht , where a and H are constants. The Hubble constant is constantand the static limit D s = c/H is constant and it is equal to the cosmological eventhorizon for the observer at the origin L H = a ( t ) c ∞ Z t dt ′ a ( t ′ ) = cH . (15)So processes (Penrose processes) with particles with negative energy are not seenby the observer. This is analogous to situation of nonrotating black holes whereparticles with negative energy exist only inside the event horizon.8 However forscale factor a = a t α (0 < α <
1) the cosmological horizon does not exist and somevisible consequences of existence of particles with negative energies can be observed.In this paper we consider particles as classical particles. Surely if they are quan-tum new features appear.9 , Acknowledgments
This research is supported by the Russian Foundation for Basic research (Grant No.18-02-00461 a). The work of Yu.V.P. was supported by the Russian GovernmentProgram of Competitive Growth of Kazan Federal University.
References
1. R. Penrose,
Gravitational Collapse: The Role of General Relativity , Rivista Nuovo Ci-mento I , Num. Spec., 252–276 (1969).2. R. Penrose and R. M. Floyd, Extraction of rotational energy from a black hole , NaturePhys. Sci. , 177–179 (1971).3. A. A. Grib and Yu. V. Pavlov,
Comparison of particle properties in Kerr metric andin rotating coordinates , Gen. Relativ. Gravit. , 78 (2017).4. A. A. Grib and Yu. V. Pavlov, Static limit and Penrose effect in rotating referenceframes , Theor. Math. Phys. , 1117–1125 (2019).5. R. H. Boyer and R. W. Lindquist,
Maximal analytic extension of the Kerr metric ,J. Math. Phys. , 265–281 (1967).6. G. F. R. Ellis and T. Rothman, Lost horizons , Am. J. Phys. , 883–893 (1993).7. A. A. Grib, Early Expanding Universe and Elementary Particles (Friedmann Lab.Publ., St. Petersburg, 1995).8. A. A. Grib and Yu. V. Pavlov,
Particles with negative energies in black holes , Int. J.Mod. Phys. D , 675–684 (2011).9. A. Vilenkin, Quantum field theory at finite temperature in a rotating system , Phys. Rev.D , 2260–2269 (1980).10. V. P. Neznamov and V. E. Shemarulin, Analysis of half-spin particle motion in Kerr-Newman field by means of effective potentials in second-order equations , Grav. Cosmol.24