Download New-York, July 11, 2016

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
Tpot
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
8G 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  10175
exp  
 kbTR 
  
  exp  6000  102500
exp  
 kbTB 