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Inequivalence between Active Gravitational Mass and Energy for a Composite Quantum Body 21st International Conference on General Relativity (New-York, July 11, 2016) Andrei G. Lebed Department of Physics, University of Arizona and L.D. Landau Institute for Theoretical Physics NSF grant DMR-1104512 Active gravitational mass of a composite body in a classical physics r G electron (-e) R proton (+e) t G m p me R m p me E1 / c 2 R K. Nordtvedt, Class. Quantum Grav., v. 11, p. A119 (1994) S. Carlip, Am. J. Phys., v. 66, p. 409 (1998) Weak field approximation g h - Minkowski metric , 1 h h h 2 | h | 1 h h T (t | x x ' | / c, x ' ) 3 h (t , x ) 4G d x' | x x'| h (t , R) 3 4G T ( t , x ' )d x ' R Active gravitational mass 2 g 00 1 2 c GM R 1 g 00 00 h00 00 h00 00 h 2 1 g 00 1 h00 h11 h22 h33 2 M (h00 h11 h22 h33 ) / 4c 2 1 m me 2 c a g T kin 3 pot (t , r ) T (t , r ) d r Stress-energy tensor mv ( t ) v (t ) 3 T (r , t ) [r rp (t )] 2 2 1 v / c Tem 1 4 1 F F F F 4 T mv2 3 2 Tpot e2 2 r kin mv2 e 2 2 mv2 e 2 2 m me / c 2 / c r r 2 2 a g Classical virial theorem and Einstein’s equation 2 2 2 2 mv e mv e mga me / c 2 2 / c 2 r r 2 2 2 K t P mv2 e 2 m t me 2 r a g Einstein’s equation t mv2 e 2 /c 2 2 r t 2 / c2 t E m t me 2 c a g K. Nordtvedt, Class. Quantum Grav., v. 11, p. A119 (1994) S. Carlip, Am. J. Phys., v. 66, p. 409 (1998) Einstein’s equation in semi-classical gravity Einstein’s equation 1 8G ˆ R Rg 2 T 2 c 2 2 2 2 2 p e p e a 2 mg me / c 2 / c 2me r 2me r pˆ 2 e 2 2 pˆ 2 e 2 2 a mg me / c 2 / c 2me r 2me r a mg me pˆ 2 e 2 pˆ 2 e 2 2 /c 2 / c2 2me r 2me r Quantum virial theorem and Einstein’s equation Macroscopic ensemble of hydrogen atoms with energy E1 E1 Quantum virial theorem mga me pˆ 2 e2 2 2me r 2 2 ˆ pˆ 2 e 2 p e / c2 2 / c2 2me r 2me r Einstein’s equation mga me E1 c2 A.G. Lebed, Int. J. Mod. Phys. D, vol. 24, p. 1530027 (2015). A.G. Lebed, Adv. in High Ener. Phys., vol. 2014, p. 678087 (2014). Breakdown of Einstein’s equation for superposition of stationary states E2 1, 2 (r , t ) E1 1 1 r exp( iE1t / ) 2 r exp( iE2t / ) 2 E me c 2 E1 E2 2 E1 E2 V1, 2 ( E1 E2 )t mˆ me 2 cos 2 2c c a g Einstein’s equation mˆ t E c2 A.G. Lebed, Int. J. Mod. Phys. D, vol. 24, 1530027 (2015). A.G. Lebed, Adv. High Ener. Phys., vol. 2014, 678087 (2014). Possible Positive Einstein’s reaction Drawing of Natalia Lebed Summary • 1) The Einstein’s equation, E mg c , is established for averaged over time active gravitational mass in classical physics; a 2 • • • • New results: a 2 2) The Einstein’s equation, E mg c , is established for the expectation value of active gravitational mass and energy for stationary quantum states; 3) For superpositions of stationary quantum states, it is shown that the expectation value of active gravitational mass exhibits time-dependent oscillations even in the case, where the expectation value of energy is constant; 4) Is it possible to experimentally observe these time-dependent oscillations for a macroscopic ensemble of the superpositions? This would be the first direct observation of quantum effects in General Relativity 5) After time average procedure, we recover the Einstein’s equation. What about passive gravitation mass? A.G. Lebed (Poster): Breakdown of the Equivalence between Passive Gravitational Mass and Energy for a Quantum Body: Theory and Suggested Experiment Realistic experiment on the Earth’s orbit by using of spacecraft v R’ Earth (M) A.G. Lebed, Int. J. Mod. Phys. D, v. 24, 1530027 (2015); J. Phys.: Conf. Ser. v. 490, 012154 (2014); Adv. High Ener. Phys. v. 2014, 678087 (2014); Cent. Eur. J. Phys. v. 11, p. 969 (2013). Inside the Spacecraft E2 E1 10ev 120,000 K T V N mg V 1 rB 22 10 2 c 2 R0 exp 400 10175 exp kbTR exp 6000 102500 exp kbTB