ries in a feasible future. In this dificult sitution, it seems to be
International Journal of Scientific & Engineering Research, Volume 4, Issue 3, April-2013 1
ISSN 2229-5518
Equivalence Between Energy Content And Weight In A Weak Gravitational Field For A Quantum Body
Dr. Luiz Carlos Costa Neto
—————————— ——————————
ormulation of a successful quantum gravitation theory is considered to be one of the most important problems in physics and the major step towards the so-called "Thory of Everything". On the other hand, fundamentals of the General Relativity and quantum mechanics are so diferent that there is a possibility that it will not be possible to unite these two theo-
composite bodies. In particular, for electrstatically bound two bodies, it is shown that gravitational field is coupled to a cobi- nation 3K 2U , where K is kineic energy, U is the Cou- lomb potential energy. Since the classical virial theorem states that the following time average is equal to zero,
ries in a feasible future. In this dificult sitution, it seems to be
2K U
0 , then we conclude that averaged over time
t
important to suggest a cobination of the quantum mechanics and some non-trivial approximaion of the General Relativity. In particular, this is important in case, where such theory can be experimentally tested. To the best of our knowledge, so far
gravitational mass is proportional to the total amount of enegy
[8] [9]:
3K 2U
only quantum variant of trivial Newtonian approximation of mg the General Relativity has been studied experimentally in the t famous Colella et al [1]; Nesvizhevsky et al [2]; and Voronin et
al [3] experments. As to such important and non-trivial qua- tum efects in the General Relativity as the Hawking radition [4] and the Unruh efect [5], they are still very far from their direct and unequivocal experimental confirmations.
t
c2
K U
t
c2
E
(1)
A notion of gravitational mass of a composite body is c2
known to be non-trivial in the General Relativity and related
to the following paradoxes. If we consider a free photon with energy E and apply to it the so-called Tolman formula for
gravitational mass [6], we will obtain mg 2E ce
(i.e., two
The main goal of our papper is to study a quantum prob-
times bigger value than the expected one) [7]. If a photon is confined in a box with mirrors, then we have a composite body at rest.
In this case, as shown in [7], we have to take into account a
lem about weight of a composite body.
As the simplest example, we consider a hydrogen atom in the Earth gravitational field, where we take into account only kinetic and Coulomb potential energies of an electron in a
negative contribution to mg
from stress in the box walls to
curved spacetime.
restore the equation mg E
ce . It is important that the later equation is restored only after averaging over time. A role of the classical virial theorem in establishing of the equivalence between averaged over time gravitational mass and energy is discussed in detail in [8] and [9] for diferent types of classical
We claim three main results in the papper.
The first result is that the weak equivalence between weight in a weak gravitational field and energy in the absence of the field may survive at a macroscopic level in a quantum case [10]. More strictly speaking, we show that the expectation
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ISSN 2229-5518
value of the weight is equal to
E c2
for stationary quantum
The classical Lagrangian and action of an electron in
states due to the quantum virial theorem.
The second result is a breakdown of the weak equivalence
between weight in a weak gravitational field and energy at a
a hydrogen atom have the following standard forms:
2
microscopic level for stationary quantum states due to the fact
L ' m c2 1 m
v '2 e
that the weight operator does not commute with energy oper- ator, taken in the absence of gravitational field.
As a result, there exist a non-zero probability that a meaurement of the weight gives value, which is diferent from
e 2 e r '
S ' L ' dt '
(5)
E c2 .
where me is the bare electron mass, e and v '
are the elecron
We suggest to detect this weak inequivalence of weight in a weak gravitational field and energy by measurements of
charge and velocity, respectively;
eletron and proton.
r ' is a distance between
electromagnetic radiation, emitted by a macroscopic ensemble
of hydrogen atoms, supported and moved in the Earth gravi-
tational field.
It is possible to show that the Lagrangian (5) can be rewrit-
en in coordinates (x; y; z; t) as:
2 2 2
The third result is a breakdown of the weak equivalence
L m c2 1 m ve e
m 3m
v 2 e
(6)
between the expectation values of the weight and energy at a
e 2 e
r e e 2
r c2
macroscopic level for a superposition of stationary quantum states.
As we show below, time dependent oscillations of the ex- pectation values of the weight are expected to exist in this
Let us calculate the Hamiltonian, corresponding tothe La- grangian (6), by means of a standard procedure,
case, and, the equivalence is restored after averaging of these oscillations over time.
Below, we derive the Lagrangian and Hamiltonian of a
H ( p, r ) pv L(v, r ) , where
As a result, we obtain:
p L(v, r)
v .
2 2 2 2
hydrogen atom in the Earth gravitational field, taking into
H m c2
p e
p
m
e
(7)
account couplings of kinetic and potential Coulomb energies
e 2m r
e 3 2m
2 r c2
of an electron with a weak gravitational field.
Note that we keep only terms of the order of 1=c2 and dis- regard magnetic force, radiation of both electromagnetic and
gravitational waves as well as all tidal and spin dependent
e e
where canonical momentum in a gravitational field is
e 
p m v 1
3 c2 .
efects.
From the Hamiltonian (7), averaged over time electron
g
Let us write the interval in the Earth gravitational field, us-
ing the so-called weak field approximation [11], [12]:
weight in a weak gravitational field,
expressed as:
me
, can be
t
2 2 2 2
2
2
2 2 2 
mg
m
p c
2 p e
ds 1 2 c2
cdt
1 2 c2 dx
dy
dz
(2) e t e
2m r c2
2m r c2
e
E
t e t
(8)
where:
me c2
GM
(3)
where
E p2 2m
e2 r
is an electron energy. Note that
R
where G is the gravitational constant, c is the velocity of light, M is the Earth mass, R is a distance from a center of the Earth.
Then in the local proper spacetime coordinates:
e
averaged over time third term in Eq. (7) is equal to zero due to
the classical virial theorem. Thus, we conclude that in classical
physics averaged over time weight of a composite body is
equivalent to its energy, taken in the absence of gravitational
field [8], [9].
The Hamiltonian (6) can be quantized by substituting a
momentum operator, momentum, p .
pˆ i r , instead of canonical
x ' 1 x, y ' 1 y
(4)
It is convenient to write the quantized Hamiltonian in the following form:
z ' 1 c2 z, t ' 1 c2 t
pˆ 2 e2
Hˆ mˆ g
(9)
2me r
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ISSN 2229-5518
where we omit term m c2
and introduce weight operator of
iE t
an electron in a weak gravitational field,
1 r, t 1
r exp
1
(12)
pˆ 2
e2
pˆ 2
e2
mˆ g m
2
(10)
In a weak gravitational field Eq. (2), wave function Eq. (12)
e e 2m r c2
2m r c2
is not anymore a ground state of the Hamiltonian (9), (10)
e e
Note that, in Eq. (10), the first term corresponds to the bare electron mass, me , the second term corresponds to the ex- pected electron energy contribution to the weight operator, whereas the third non-trivial term is the virial contribution to
from point of view of an inertial observer, located at infinity. For such observer, in accordance with Eq. (3), a general solu- tion of the Schrodinger equation, corresponding to the Hamil- tonian (9), (10), can be written as:
the weight operator. It is important that the operator (10) does
r t
a
r iE
t
(13)
not commute with electron energy operator, taken in the ab-
,
n n 1
2 exp
n 1
2
sence of gravitational field. It is possible to show that Eqs. (9),
(10) can be obtained directly from the Dirac equation in a
n 1
c
c
curved spacetime, corresponding to a weak gravitational field
[Here factor
1 c2
is due to a curvature of space,
[2].
whereas the term En
1
c2 reflects the famous red shift
in gravitational field and is due to a curvature of time.
n r is a normalized wave function of an electron in a hy-
drogen atom in the absence of gravitational field, correspond-
ing to energy En [13].
In accordance with the quantum mechanics, probability
that at
t 0
an electron occupies excited state with energy
Below, we discuss some consequences of Eqs. (9), (10). Suppose that we have a macroscopic ensemble of hydrogen atoms with each of them being in a ground state with energy
En 1
c2 is:
Pn 
an 
E1 . Then, from Eq. (10), it follows that the expectation value
of weight operator per atom is:
where:
E pˆ 2 e2
a * r
1
r d 3r
mˆ g
m
1 2
n 1 n c2
e e c2
2m r c2
(11)
(14)
* ' 3
m
E1
c2 1 r r n r d r
e c2
where the third term in Eq. (11) is zero in accordance with the quantum virial theorem [13]. Therefore, we conclude that the weak equivalence between weight in a weak gravitational field and energy in the absence of the field survives at a mac-
for n 1 .
Taking into account that the Hamiltonian is the Hermitian
operator, it is possible to show that:
V
roscopic level for stationary quantum states.
* r ' r d 3r
n,1
Let us discuss how Eqs. (9), (10) break the weak equiva-
lence between weight in a weak gravitational field and energy
1 n
n,1
n,1
(15)
at a microscopic level. First of all, we pay attention that the weight operator (10) does not commute with electron energy operator, taken in the absence of gravitational field. This means that, if we create a quantum state of a hydrogen atom with definite energy, it will not be characterized by definite
where:
En
E1
weight. In other words, a measurement of the weight in such
V ,1
r Vˆ r
r d r
* 3
quantum state may give diferent values, which, as shown, are
n 1 n
2 2
(16)
quantized.
Here, we illustrate the above mentioned inequivalence. Suppose that at t 0 we create a ground state wave function
of a hydrogen atom, corresponding to the absence of gravita- tional field:
Vˆ r 2 pˆ e
2me r
Let us discuss Eqs. (13)-(16). Note that they directly
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International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 4
ISSN 2229-5518
demonstrate that there is a finite probability:
Pn 
an 
For instance, as follows from Eq. (9), for electron wave function:
1
2
V n,1
(17)
1,2 r, t
1 (r )exp iE1t 2 exp iE2t (19)
2
c2 E
E
n 1
which is characterized by the time independent expectation
value of energy,
E E E
2 , the expectation value
for n 1 , that at t 0 an electron occupies n-th energy level.
1 2
of electron weight is the fol win oscillatory function:
In fact, this means that measurement of weight in a weak
gravitational field in a quantum state with a definite energy
lo g
E E V
E E t
(12) gives the following quantized values:
mˆ g
m
1 2
1,2 cos 1 2
(20)
mg n
m En
e e
(18)
2c2 c2
e
e 2
c
Note that the oscillations of the weight directly demon-
strate inequivalence of the weight and energy at a macroscopic
level.
corresponding to the probabilities (17) [14].
[Note that Vˆ r
in Eq. (16) is the virial operator. It is a
It is important that these oscillations are strong (of the or-
der of 2 m
and of a pure quantum origin without classical
part of the weight operator (9), which does not commute with energy operator, taken in the absence of gravitational field.
Due to the fact that Vˆ r presents in Eqs. (9), (16), the proba-
bilities (17) for the quantization law (18) are not equal to zero.
We point out that, although the probabilities (15) are quad-
ratic with respect to gravitational potential and, thus, small,
the changes of the weight (18) are large and of the order of
e
analogs.
If we average the oscillations over time, we obtain the
modified weak equivalence principle between the averaged
over time expectation value of the weight and the expectation
value of energy in the following form:
2 E E
me , where is the fine structure constant.
We also pay attention that small values of probabilities
mˆ g
t
me
1 2
2c2
(21)
(15),
P 0 1018 , do not contradict to the existing Eotvos type
measurements [11], which have confirmed the weak equiva- lence principle with the accuracy of the order of 1012 1013 .
For us, it is very important that the excited levels of a hy- drogen atom spontaneously decay with time, therefore, one can detect quantization law (18) by measuring electromagnetic radiation, emitted by a macroscopic ensemble of hydrogen atoms [15].
The above mentioned optical method is much more sensi- tive than the Eotvos type measurements and we, therefore, hope that it will allow to detect the breakdown of the equiva- lence between energy content and weight in a weak gravita- tional field, suggested in the paper.
To summarize, we have demonstrated that weight of a composite quantum body in a weak external gravitational field is not equivalent to its energy in the weak sense due to quantum fluctuations.
We have also shown that the corresponding expectation values are equivalent to each other for stationary quantum states. In this context, we need to make the following com- ment.
First of all, we stress that, for superpositions of stationary states, the expectation values of the weight can be oscillatory functions of time even in case, where the expectation value of energy is constant.
We pay attention that physical meaning of averaging pro- cedure in Eq. (21) is completely diferent from that in classical time averaging procedure (1) and does not have the corre- sponding classical analog.
In conclusion, we stress that we have considered in the pa- per a point-like [16] composite quantum test body and all our results are due to diferent couplings of kinetic and potential energies with an external gravitational field.
This physical mechanism is completely diferent from those, considered before, where a possibility of a break-down of the weak equivalence principle was discussed due to three mass dependent phenomena: penetration of the de Broglie waves in classically restricted areas, bound states of particles in an ex- ternal gravitational field, and the interference of the de Broglie waves. In addition, we point out that there exists an alterna- tive point of view (see, for example, [17], [18], stating that there cannot be violations due to quantum efects of some gen- eralized weak equivalence principle in any metric theory of gravitation, including the GR.
[1] R. Colella, A.W. Overhauser, and S.A. Werner, Phys. Rev. Lett. 34, 1472 (1975); A.W. Overhauser and R. Colella, Phys. Rev. Lett. 33, 1237 (1974).
[2] V.V. Nesvizhevsky, H.G. Borner, A.K. Petukhov, H. Abele, S. Baebler, F.J.
Rueb, Th. Stoferle, A.Westphal, A.M. Gagarski, G.A. Petrov, and A.V. Strelkov, Nature (London) 415, 297 (2002).
[3] A.Yu. Voronin, H. Abele, S. Baebler, V.V. Nesvizhevsky, A.K. Petukhov, K.V.
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International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 5
ISSN 2229-5518
Protasov, and A. Westphal, Phys. Rev. D 73, 044029 (2006). [4] S.W. Hawking, Nature 248, 30 (1974).
[5] W.G. Unruh, Phys. Rev. D 14, 870 (1976).
[6] L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields (4-th Edition, Butterworth-Heineman, Amsterdam, 2003), p.358.
[7] C.W. Misner and P. Putnam, Phys. Rev. 116, 1045 (1959). [8] K. Nordtvedt, Class. Quantum Grav. 11, A119 (1994).
[9] S. Carlip, Am. J. Phys. 66, 409 (1998).
[10] As usual, in a framework of the weak equivalence principle, we do not take into account gravitational field of a test body (i.e., a hydrogen atom).
[11] See, for example, C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (W.H.
Freeman and Co, New York, 1973), p. 449.
[12] We pay attention that to calculate the Lagrangian (and later - the Hamiltoni-
2
an) in a linear with respect to a small parameter ( R)
c approximation,
2
we do not need to keep the terms of the order of ( R) c
(2), in contrast to the perihelion orbit procession calculations
in metric
[13] See, for example, D. Park, Introduction to the Quantum Theory (3-rd Edition, Dover Publications, New York, 2005), p. 85.
[14] We pay attention that movement with constant velocity, gravitational force,
which acts on each individual hydrogen atom, is compensated by some non- gravitational forces. [This causes very small changes of a hydrogen atom en- ergy levels and is not important for our calculations.] Therefore, the atoms do not feel directly gravitational acceleration, g , but feel, instead, gravitational
potential, R ' vt , changing with time due to motion in the Earth
gravitational field.
[15] Costa Neto, L. C., Equivalence Between Passive Gravitational Mass And Energy For A Quantum Body, International Journal of Scientific & Engineer- ing Research, 2013.
[16] We use the term point-like body in a sense that the Bohr radius, characterizing a typical size of electron wave functions in a hydrogen atom, is much less than both vt and R0. Therefore, we disregard all small tidal efects.
[17] L.Viola and R. Onofrio, Phys. Rev. D 55, 455 (1997).
[18] C. Lammerzahl, Gen. Rel. Grav. 28, 1043 (1996).
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