Negative Gravitational Mass in a Superfluid Bose-Einstein Condensate Fran De Aquino

Negative Gravitational Mass in a Superfluid Bose-Einstein Condensate Fran de Aquino

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Fran de Aquino. Negative Gravitational Mass in a Superfluid Bose-Einstein Condensate. 2017. hal- 01516293v2

HAL Id: hal-01516293 https://hal.archives-ouvertes.fr/hal-01516293v2 Preprint submitted on 5 May 2017

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Negative Gravitational Mass in a Superfluid Bose-Einstein Condensate

Fran De Aquino Professor Emeritus of Physics, Maranhao State University, UEMA. Titular Researcher (R) of National Institute for Space Research, INPE Copyright © 2017 by Fran De Aquino. All Rights Reserved.

Newton's 2nd law of motion tells us that objects accelerate in the same direction as the applied force. However, recently it was shown experimentally that a Superfluid Bose-Einstein Condensate (BEC) accelerates in the opposite direction of the applied force, due to the inertial mass of the BEC becoming negative at the specifics conditions of the mentioned experiment. Here we show that is not the inertial mass but the gravitational mass of the BEC that becomes negative, due to the electromagnetic energy absorbed from the trap and the Raman beams used in the experimental set-up. This finding can be highly relevant to the gravitation theory.

Key words: Negative Gravitational Mass, Bose-Einstein Condensates, Superfluids.

1. Introduction

A recent paper described an experiment gravitational masses where the force would be that shows a Superfluid Bose-Einstein of attraction. Condensate (BEC) with negative mass, and In this article, we show that is not the accelerating in the opposite direction of an inertial mass but the gravitational mass of the applied force [1]. The experiment starts with a BEC that becomes negative, due to the BEC of approximately 105 87Rb atoms electromagnetic energy absorbed from the confined in a cigar-shaped trap oriented along trap and the Raman beams used in the the x-axis of a far-detuned crossed dipole trap. experimental set-up. The consequences of this Using an adiabatic loading procedure, the finding can be highly relevant to the BEC is initially prepared such that it occupies gravitation theory. the lowest minimum of the lower spin-orbit coupled (SOC) BEC. By suddenly switching 2. Theory off one of the two dipole trap beams, the condensate is allowed to spread out along the Some years ago I wrote a paper [2] x-axis. Then, the BEC is imaged in-situ for where a correlation between gravitational expansion times of 0, 10 and 14 ms. In the mass and inertial mass was obtained. In the negative x-direction, the BEC encounters an paper I pointed out that the relationship essentially parabolic dispersion, while in the between gravitational mass, mg , and rest positive x-direction, it enters a negative mass inertial mass, m , is given by region. This leads to a marked asymmetry in i0 the expansion. Obviously, negative mass does not ⎧ ⎡ 2 ⎤⎫ mg ⎪ ⎛ Un ⎞ ⎪ mean anti-matter. Anti-matter is simply χ ⎨ ⎢ 121 +−== ⎜ r ⎟ − ⎥⎬ ()11 ⎢ ⎜ 2 ⎟ ⎥ mi0 ⎪ ⎝ i0cm ⎠ ⎪ matter which has the opposite electric charge ⎩ ⎣ ⎦⎭ from normal matter, whereas negative mass means more exactly negative gravitational where U is the electromagnetic energy mass. If one particle had ordinary positive absorbed or emitted by the particle; = vcn gravitational mass, and one had negative r gravitational mass, then the gravitational is the index of refraction of the particle; c is force between the masses would be repulsive the speed of light. differently of in the case of two positive Equation (1) can be rewritten as follows 2

8 ⎪⎧ ⎡ ×108.1 ⎤⎪⎫ m 121 +−= −1 m 6 ⎧ ⎡ 2 ⎤⎫ ()BECg ⎨ ⎢ 4 ⎥⎬ 0()BECi () mg ⎛ W ⎞ v ⎪ ⎢ ⎥⎪ ⎩⎪ ⎣⎢ BEC ⎦⎥⎭⎪ χ ⎨ 121 +−== ⎜ ⎟ − ⎬ ()21 mi0 ⎢ ⎝ ρ vc ⎠ ⎥ ⎩⎪ ⎣ ⎦⎭⎪ Note that, for v < .5.109 sm −1 the where ρ is the matter density, v is the velocity BEC of radiation through the particle, and W is the gravitational mass of the BEC ( m ()BECg ) density of absorbed electromagnetic energy. becomes negative. Lene Hau et al., [3] Substitution of the well-known relation showed that light speed through a BEC = 4 vDW into Eq. (2) yields reduces to values much smaller than −1 ⎧ ⎡ 2 ⎤⎫ .100 sm . mg ⎪ ⎛ 4D ⎞ ⎪ χ ⎢ 121 +−== ⎜ ⎟ − ⎥ 31 Consequently, we can conclude that it ⎨ ⎜ 2 ⎟ ⎬ () 87 mi0 ⎪ ⎢ ⎝ ρcv ⎠ ⎥⎪ is the gravitational mass of the BEC of Rb ⎩ ⎣ ⎦⎭ atoms that becomes negative and not its where D is the power density of the radiation inertial mass. absorbed by the particle. Also it was deduced in the reference [2] In order to apply the Eq. (3) to the BEC a generalized expression for the Newton's 2nd previously mentioned, we start calculating the rest inertial mass of the BEC, which is given by law of motion, which shows that the 5 −27 expression for inertial forces is given by m 0()BECi ×≅ (1066.1909187.86101 kg)=×× r r = g amF (7) −20 ×= 104.1 kg The presence of mg in this equation shows Assuming that the average radius of the BEC is that the inertial forces have origin in the approximately 40 μm (See reference [1]), then we gravitational interaction between the particle can calculate the density of the BEC, i.e., and the others particles of the Universe, just m −20 0()BECi ×104.1 kg −− 38 as Mach’s principle predicts. In this way, the ρBEC == 3 ×≅ .102.5 mkg V 4 −6 new equation expresses the incorporation of BEC 3 π()×1040 m the Mach’s principle into Gravitation Theory, Substitution of the values of ρ into BEC and reveals that the inertial effects upon a Eq. (3) gives body can be strongly reduced by means of the ⎧ 2 ⎫ m ()BECg ⎪ ⎡ D ⎤⎪ decreasing of its gravitational mass. Note that χBEC ⎨ ⎢ +−== − ⎥⎬ ()41065.0121 m 4 only when mg reduces to mi0 is that we have 0()BECi ⎪ ⎣⎢ vBEC ⎦⎥⎪ ⎩ ⎭ r r The variable D , in Eq. (4), refers now to the total the well-know expression ( = i amF ) of the power density of the radiation absorbed by the Newton’s law. BEC (from the trap and the Raman beams, used Taking Eq. (6) for an arbitrary value of in the experimental set-up of reference [1]). v < .5.109 sm −1, we obtain m −= Km , According to the authors of the experiment the BEC ()BECg 0 ()BECi power of the Raman beams are of approximately where K is a positive number. Substitution of 3mW (2.9 mW in one of the two beams,3.3 mW in this equation into Eq. (7) yields the other), focused to a beam waist of 120 microns ( 60μm radius); the absorption coefficient is r r FBEC −= 0()BECi aKm (8) EE RR = 4.05.21 . Thus, we can write that

Pabs 4.0 Pbeams The sign − in this expression reveals clearly D == = ( ) −6 2 SBEC π()×10604 m why the BEC accelerates in the opposite direction of the applied force, i.e., ()+× 3.39.24.0 mWmW −24 = 2 ×≅ mW ()5.104.5 π ×10604 −6 m s r r () BEC BEC =−= 0()BECi aKmFF (9) Substitution of this value into Eq. (4) gives

Recently it was created a BEC with Then, consider a system in which the ×10506.2 17 23Na atoms [4]. The inertial inertial mass of the hollow sphere is −9 mass of this BEC is m 0()spherei = 000,10 kg ; m 0()BECi ×= 108.3 kg ; 2 −− 19 * ≅ 1 mWD and vBEC = .10 sm , then the m ×= 17 −27 ×=× 108.31066.12310506.2 −9 kg 0()BECi ()( ) gravitational mass of the system will have the following value: The number of atoms/ cm3 , as showed in 13 ()SYSg mm 0 (spherei ) ×−= 105.2 m 0()BECi = −316 −322 reference [4],is n0 cm ×=×= 10742.210742.2 m . 4 Thus, the density of the BEC is given by ×−= 105.8 kg ()12 Thus, if this system is subjected to a gravity −27 22 −− 33 r −1 ρ ()( ×= ) ×=× .1004.110742.21066.123 mkg acceleration = .8.9 smg , as shown in Fig.2, then its weight force will be given by s Substitution of these values into Eq. (3) gives r 4 r 5 ()SYSg ×== 105.8 ggmP , (P ×= 103.8 N ). 2 ⎪⎧ ⎡ D ⎤⎪⎫ This value is greater than the thrust of the fighter m ⎢ −10 −×+−= 1106.1121 ⎥ m ()10 ()BECg ⎨ 4 ⎬ 0()BECi aircraft F-22 Raptor (fifth-generation), which ⎪ ⎢ vBEC ⎥⎪ ⎩ ⎣ ⎦⎭ reaches 160,000N.

s = gmP r ()SYSg

BEC

Fig. 1 – A BEC inside a hollow sphere.

Now consider the system showed in Fig.1. Inside the hollow sphere, whose gravitational gr mass is m ()sphereg ≅ m 0 (spherei ), there is a BEC whose gravitational mass is given by Eq. (10). Fig. 2 – The system sphere-BEC, with negative gravitational mass ( ), in a gravitational Thus, the total gravitational mass of the m ()SYSg < 0 field. system, m (SYSg ), is given by mmm =+= It is important to note that, in spite the light () (spheregSYSg ) (BECg ) speed reach values very close to zero through the BEC in the Lene Hau experiments, the number of ⎧ ⎡ 2 ⎤⎫ ⎪ −10 D ⎪ atoms in the BEC, in this case is very small m0()spherei ⎨ ⎢ −×+−+≅ 1106.1121 ⎥⎬m0()BECi ()11 v4 −1010 65 atoms in comparison with the ⎩⎪ ⎣⎢ BEC ⎦⎥⎭⎪ ( ) 17 23 In 2001 Lene Hau et al., have shown that the ×10506.2 atoms of the BEC of Na, here light speed through a BEC could have values very mentioned. Consequently, the weight force, acting on the BEC in the case of the Lene Hau close to zero. Therefore, vBEC with these −12 magnitudes are not unusual. This shows that the experiments is approximately 10 times smaller than in the case of the BEC of Na atoms. value of m ()SYSg , given Eq. (11), can be strongly reduced even the value of D be small, for * example, of the order of 1 mW 2 . With this velocity the light beam would travel about 0.1mm in a day. 4

References

[1] M.A. Khamehchi et al. (2017). Negative-Mass Hydrodynamics in a Spin-Orbit–Coupled Bose- Einstein Condensate. Phys. Rev. Lett. 118 (15): 155301; doi: 10.1103/PhysRevLett.118.155301 Available at: https://arxiv.org/abs/1612.04055

[2] De Aquino, F. (2010) Mathematical Foundations of the Relativistic Theory of Quantum Gravity, Pacific Journal of Science and Technology, 11 (1), pp. 173-232. Available at: https://hal.archives-ouvertes.fr/hal-01128520

[3] Hau, L.V, et al., (1999) Light Speed Reduction to 17 Meters per Second in an Ultracold Atomic Gas, Nature 397, 594-598.

[4] Pei-Lin You (2016) Bose–Einstein condensation in a vapor of sodium atoms in an electric field, Physica B, 491, 84-92.