Particles with negative energies in nonrelativistic and relativistic cases
aa r X i v : . [ g r- q c ] A p r Particles with Negative Energies in Nonrelativisticand Relativistic Cases
Andrey A. Grib , , ∗ and Yuri V. Pavlov , Theoretical Physics and Astronomy Department, The Herzen University,48 Moika, St. Petersburg, 191186, Russia A. Friedmann Laboratory for Theoretical Physics, St. Petersburg, Russia Institute of Problems in Mechanical Engineering of Russian Academy of Sci-ences, 61 Bolshoy, V.O., St. Petersburg, 199178, Russia; [email protected] N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal Uni-versity, Kazan, Russia ∗ Correspondence: andrei [email protected]
Abstract:
States of particles with negative energies are considered for the nonrelativisticand relativistic cases. In nonrelativistic case it is shown that the decays close to theattracting center can lead to the situation similar to the Penrose effect for rotatingblack hole when the energy of one of the fragments is larger than the energy of theinitial body. This is known as the Oberth effect in the theory of the rocket movement.The realizations of the Penrose effect in the non-relativistic case in collisions near theattracting body and in the evaporation of stars from star clusters are indicated. Inrelativistic case similar to the well known Penrose process in the ergosphere of therotating black hole it is shown that the same situation as in ergosphere of the blackhole occurs in rotating coordinate system in Minkowski space-time out of the staticlimit due to existence of negative energies. In relativistic cases differently from thenonrelativistic ones the mass of the fragment can be larger than the mass of thedecaying body. Negative energies for particles are possible in relativistic case in cos-mology of the expanding space when the coordinate system is used with nondiagonalterm in metrical tensor of the space-time. Friedmann metrics for three cases: open,close and quasieuclidian, are analyzed. The De Sitter space-time is shortly discussed.
Key words:
Penrose effect, black hole, metric, expanding Universe
1. Negative Energies and the Penrose Effect in Nonrelativistic Case
It is well known that in the non-relativistic case, energy is determined with accuracyto the additive constant. If the energy of a particle resting on infinity is put equal to zerothen the sign of the energy defines movement of the particle in Kepler problem being eitherlimited (in the case of negative sign) or nonlimited, when the energy is positive or zero [1].If the body arriving in the region with nonzero gravitational field of some other objectdecays there in two parts so that the velocity of one part is smaller than the second cosmicvelocity in the point of decay then its energy will be negative. So the energy of the secondpart becomes larger than the energy of the initial body. This is some realization of thePenrose effect [2, 3] on getting the energy from the rotating black hole due to the decay ofsome body in the ergosphere.Let us give some evaluations for the case when the initial body with mass m and thevelocity v on infinity decays on the distance r from the attracting center of mass M ontwo fragments so that the fragment with mass m is flying relative to the first part of mass m − m in the opposite direction to the initial one with the velocity u . The velocity of theinitial body at the distance r from the attracting mass M is v r = r v + 2 GMr , (1)where G is the gravitational constant. Let us find the velocities of the fragments after thedecay using the conservation of the momentum. So one gets for the velocity of fragmentwith mass m − m v = v r + m m u. (2)The projection of the velocity of the fragment with mass m on the direction of the initialmovement in the point of decay is v = v r − (cid:16) − m m (cid:17) u. (3)It is easy to prove the identity E + E = E + E f , (4)where E = ( m − m ) (cid:18) v − GMr (cid:19) (5)is the energy of the fragment with mass m − m in the gravitational field of the attractingbody. E = m (cid:18) v − GMr (cid:19) (6)is the energy of the fragment with mass m , E = mv E f = m u . (8)This is the energy necessary for the flight of the fragment with the relative velocity u . It isevident interpretation that m is the mass of the fuel flying from the nozzle of the rocketand E f is the energy of the fuel.As it is seen from (4) in case E − E f < m − m (the rocket without fuel)is larger than the initial mechanical energy of the initial object E . From (2)–(8) one canfind the conditions of the realization of such process E − E = E f − E = m (cid:20) v u (cid:16) − m m (cid:17) − v (cid:21) . (9)This expression is positive for small r . Note that formally the profit in energy is unlimited,however it is evident that r must be larger than the Schwarzschild radius r g = 2 GM/c where c is the light velocity. As one can see from (9) to get the energy profit comparablewith the relativistic rest mass of the fuel m c one needs relativistic values of the fuelexpiration velocity and use of gravitational field close to the gravitational radius. Surely inthis case one must do calculations using relativistic theory (change the Newton potentialon the Schwarzschild metric and taking into account that relative velocities are close tothe velocity of light).Note that for m ≪ m and v ≪ v u formula (9) shows that the profit in the energy ofthe rocket in the engine start with the big velocity in the region of movement is explainedby the fact that the engine traction with the fuel expiration relative velocity u is constantand the work A done in this case is proportional to the velocity of the rocket movement: A ≈ um ∆ t v ∆ t = v um . (10)It was H. Oberth [4] in 1929 who was the first to propose this way to increase theefficiency of the rocket engine by start of the engine in the periaster of the trajectory. It isused in astronautics in gravitational maneuvers using the Moon and inter planet flights.Another example of realization of the Penrose effect in nonrelativistic case is the fol-lowing. Let on two meeting circular orbits around the body with attracting mass M twoparticles 1, 2 with masses m ≫ m are rotating on the distance r . Then in case of absoluteelastic collision the particle with mass m will move from the first particle with relativevelocity 2 p GM/r . Its velocity in the system of rest will be 3 p GM/r and so the energywill be 7
GM m / (2 r ) increasing in 4 GM m / ( r ). It is evident that the energy of the firstparticle decreases in the same value. Note that the profit in energy of the order close to m c is possible only near the gravitational radius of the attracting body.If three or more particles are interacting one also can have process similar to the Penroseprocess. For example such decays occur for three interacting stars or evaporation of starsclusters [5].
2. Negative Energies in Rotating Coordinates
In relativity theory negative energies are usually absent.In nonrelativistic limit in the Kepler problem of the massive body with mass m movingaround gravitating mass M on the distance r the full energy of this body taking intoaccount the rest mass energy can be less than zero if this distance is very small E = mc + mv − G mMr < ⇒ r < GMc = r g . (11)Here r g is the gravitational radius.However physics in the ergosphere of the rotating black hole shows that as it was dis-covered by R. Penrose [2] relativistic particles can have negative energies due to dependenceof the energy on the angular velocity of the body rotating around the black hole. In ourpapers [6, 7] it was shown that similar effect occurs in rotating coordinates in Minkowskispace out of the static limit defined by us in analogy with the ergosphere.The interval in Minkowski space in rotating cylindrical coordinates is ds = ( c − Ω r ) dt + 2Ω r dϕ dt − dr − r dϕ − dz , (12)where Ω is the angular rotation velocity.Let us consider this formula for the case of our solar system when the Earth in atrest (nonrotating around its axis) — as ancient Greeks thought. The use of such rotatingcoordinate system is important because after all our observatories and observers are at restin this case. If Ω ⊕ ≈ . · − s − is the angular rotational velocity then from (12) onehas g = 0 for r s = c/ Ω ⊕ = 4 . · km. It is some distance between the orbits of Uranusand Neptune. For distance r > r s the coefficient of metric g < ds < ds > v < c is still possible and no breaking of causality occurs. We call thedistance r s the static limit in analogy of the corresponding surface in the Boyer-Lindquistcoordinates of rotating black hole [8]. For r > r s no body can remain at rest in the rotatingcoordinate system.In cylindrical coordinates of Minkowski space 4-vector of energy-momentum p ′ i = (cid:18) E ′ c , − p ′ r , − L ′ z , − p ′ z (cid:19) , (13)where L ′ z = r p ′ ϕ = mr dϕ ′ dτ = E ′ c r dϕ ′ dt (14)is the projection of the angular momentum on the z axis. Transforming (13) to the rotatingcoordinates one obtains p i = (cid:18) E ′ + Ω L ′ z c , − p ′ r , − L ′ z , − p ′ z (cid:19) . (15)Therefore, the energy in these coordinates is E = E ′ + Ω L ′ z . (16)So the energy E can be negative depending on the sign of L ′ z = L z .As it is known the energy can be written with the help of the Killing vector ζ i as E ( ζ ) = Z Σ T ik ζ i dσ k , (17)where { Σ } is some set of spacelike hypersurfaces orthogonal to ζ i and T ik is the energy-momentum tensor of some matter. Note that negative values of the energy E ( ζ ) can beobtained for the positive energy density T ik if the Killing vector becomes spacelike as it isthe case for the ergosphere of the rotating black hole and in our case in rotating coordinatesin region out of the static limit. This means that the conditions of the Penrose-Hawkingtheorem on singularities in cosmology are not broken in spite of existence of particles withnegative energies considered in this paper. Another remark concerns negative energy ofthe galaxy on the surface of the expanding sphere with homogeneous density of matterinside it resembling the closed Universe in Newton’s approximation. In exact relativisticcase it is not the energy but the energy density of matter is present in Einsteins equations.Energy is not conserved in expanding Universe. However as one can see in next part of ourpaper this non conserved energy of the particle can have in some situations negative sign.For a pointlike particle with the mass m located at point x p one obtains (see [6]) T ik ( x ) = mc p | g | Z ds dx i ds dx k ds δ ( x − x p ) (18)and E ( ζ ) = mc dx i ds g ik ζ k = c ( p, ζ ) . (19)For the rotating coordinates (12) and ζ = (1 , , ,
0) we obtain the energy E ( ζ ) is equalto (16).One can easily obtain the condition for a negative energy beyond the static limit [7] L z < − s p z c + m c Ω − ( c/r ) , vc > cr Ω , r > c/ Ω . (20)Some experiment to observe the consequences of the Penrose process for our solarsystem was proposed in [7]. The idea of the experiment is to show the possibility of suchprocesses which can be interpreted differently in two different coordinate systems — theinertial one and the noninertial rotating coordinates. The particle beyond the static limitdecays on two fragments one with the positive energy, the other with the negative one.The observers on the Earth can not see the particle with negative energy because it exitsonly out of the static limit. But he can see the fragment with positive energy which inrotating coordinates is larger than the energy of the initial decaying particle. From theinertial point of view the energies of all particles are positive but the energy of particlefalling on the Earth is larger than the energy of the initial particle if one takes into accountthe rotation of the Earth around its axis. All quantitative estimates one can find in [7].Further we shall discuss the third possibility of existence of particle states with negativeenergy — that of the expanding Universe.
3. Negative Energies and Static Limit in Expanding Universe
Expanding Universe in the standard model is described in the Friedmann-Robertson-Walker form in synchronous frame [9] as ds = c dt − a ( t ) (cid:18) dr − kr + r d Ω (cid:19) , (21)where d Ω = dθ + sin θ dϕ , k = 1 for closed cosmological model, k = − k = 0 for quasi-Euclidean flat model.The metric (21) can be written also in other coordinates ds = c dt − a ( t ) (cid:0) dχ + f ( χ ) d Ω (cid:1) , (22)where f ( χ ) = sin χ, k = 1 ,χ, k = 0 , sinh χ, k = − r = f ( χ ). In closed model χ is changing from 0 to π , in cases k = 0 , − χ ∈ [0 , + ∞ ). The radial distance between points χ = 0 and χ in metric (22) is D = a ( t ) χ and it’s the same in the metric (21). If t is fixed then the maximal value of D is D max = πa ( t ). In open and flat models D is non limited.Take the new coordinates t, D, θ, ϕ (see also [10, 11]). Then dD = ˙ aa D dt + a dχ, dχ = 1 a (cid:18) dD − ˙ aa Ddt (cid:19) (23)and the interval (22) becomes ds = − (cid:18) ˙ aa Dc (cid:19) ! c dt + 2 ˙ aa D dDdt − dD − a f ( D/a ) d Ω . (24)The interval (24) is the special case of the more general interval ds = g ( dx ) + 2 g dx dx + g ( dx ) + g ΩΩ d Ω , (25)where g < g ΩΩ < g g − g < E = p c = mc g k dx k ds = mc dx ds (cid:18) g + g dx dx (cid:19) . (26)This energy due to limitation from causality 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 (27)is limited by inequality 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) . (28)As one can see the states with zero and negative energy are possible in the region where g < g > a > g < a < D s = c | h ( t ) | , (29)where h ( t ) = ˙ a/a is the Hubble parameter. This corresponds to the radius of the so-calledlight sphere [10]. In open and quasi-Euclidean models due to non limited D the regionalways exists with D > D s , where particles can’t be at rest in used coordinates and stateswith negative energies are possible. In closed model D ≤ πa ( t ) and the condition for theexistence of such region is | ˙ a ( t ) | > c/π. (30)To understand the meaning of the limit (30) consider closed Universe with dust matter.Then (see [9]) a = a (1 − cos η ) , t = a c ( η − sin η ) , η ∈ (0 , π ) , a ∈ (0 , a ) , (31) D s = a | tan( η/ | and (30) becomes (cid:12)(cid:12)(cid:12) tan η (cid:12)(cid:12)(cid:12) < π. (32)So that in region η ∈ (0 , π ) one has0 < η < − π, π − tan − π ) < η < π. (33)The scale factor in region (33) is in the limits0 < a < π a π (34)and D s = D max π (cid:12)(cid:12)(cid:12) tan η (cid:12)(cid:12)(cid:12) ≤ D max , (35)i.e. the region of existence of negative energies in closed dust Unverse is changing from thealmost entire Friedman universe except of the observer vicinity at η → , π to empty setat η outside the intervals (33).Note that for closed dust Universe for η ≪ a ≈ (cid:18) c a (cid:19) / t / . (36)As it seen from (33) for this case one has the region out of the static limit with states withnegative energies.Static limit in the considered model at η . . π lies in the domain of the particle’shorizon l p = a ( t ) Z t dt ′ a ( t ′ ) = aη. (37)This means that the region with negative particle energies intersects with the region ofcausally connected phenomena for the observer at the origin of coordinate system.The energy of the freely moving particle is E = E ′ (cid:18) − ˙ a a D c + ˙ aa Dc dDdt (cid:19) , (38)where E ′ = mc dt/dτ . So in these coordinates particles with negative energies are movingin 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) . (39)Let us rewrite the inequality (39) in terms of coordinates t, χ, θ, ϕ . Then we obtain in thecase ˙ a > v = a dχdt < − c D s D , D > D s . (40)This has a meaning similar to that obtained by us for the case of rotating coordinate frame t, χ, θ, ϕ : particles with negative energies close to the static limit ( D → D s ) must movewith velocities close to the light velocity in direction of the observer in expanding Universe.The necessary condition for the possibility of observing processes involving particleswith negative energy is D s < L H , where L H = a ( t ) c t max Z t dt ′ a ( t ′ ) (41)is cosmological event horizon, t max is the life time of the universe. In the case of the de SitterUniverse with the scale factor a = a exp Ht , where a and H are constants, t max = ∞ ,the Hubble constant is constant and the static limit D s = c/H is constant and it is equalto the cosmological event horizon. So processes (Penrose processes) with particles withnegative energy are not seen by the observer in this case. This is analogous to situationof nonrotating black holes where particles with negative energy exist only inside the eventhorizon [13]. Note that in the limit t → ∞ , the standard cosmological ΛCDM model tendsto the de Sitter stage.However for models with k = 0 and scale factors a = a t α (0 < α <
1) the cosmologicalhorizon L H = ∞ and some visible consequences of existence of particles with negativeenergies can be observed. In the closed Universe with dust matter (31) t max = 2 πa /c andwe have L H = a ( η )(2 π − η ) . (42)The condition D s < L H is reduced to (cid:12)(cid:12)(cid:12) tan η (cid:12)(cid:12)(cid:12) < π − η. (43)This inequality in particular is true over the entire interval of the existence of the staticlimit in the era of expansion of the closed universe 0 < η < − π and also at the endof the compression era. Thus some processes with the particles with negative energies canbe observed in the models of closed universe also. However, unlike the de Sitter universe,the energy of the particle is not conserved in these cases. This makes the manifestation ofthe Penrose effect less obvious. Author Contributions:
The authors equally contributed to this research work.
Funding:
This research is supported by the Russian Foundation for Basic research (GrantNo. 18-02-00461 a). The work of Yu.V.P. was supported by the Russian GovernmentProgram of Competitive Growth of Kazan Federal University.
Conflicts of Interest:
The authors declare no conflict of interest.
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