A possible quantum probability increase of the cylindrical gravitational field

0.63 MB·English

by  D. Bar

INTRODUCTION 1
A possible quantum probability in
rease of the
ylindri
al gravitational (cid:28)eld
D. Bar
7
0
Abstra
t
0
2

r As known, the
ylindri
al gravitational (cid:28)eld (wave) have been
anoni
ally quantized
a
M
and its wave fun
tion, as the quantum one, interpreted in probability terms. We show

1
in this work, using quantum Zeno methods, that this probability may be substantially
2

2 in
reased and even approa
h unity. For that we (cid:28)rst show, in detailed manner, that the
v
7
ylindri
al gravitational (cid:28)eld may be dis
ussed in the
ommutation number representa-
8
0 tion. We also dis
uss this (cid:28)eld in the transverse-tra
eless (TT) gauge and
al
ulate the
3
0
related trapped surfa
e.
7
0
/
c
q
-
r Key words: Cylindri
al gravitational waves, Quantization, Zeno e(cid:27)e
t
g
:
v Pa
s numbers: 03.70.+k, 04.30.-w, 03.65.Xp
i
X
r
a
1 INTRODUCTION
1
The problem of quantizing the general gravitational (cid:28)eld (GF) (gravitational wave (GW)
[1℄) have, theoreti
ally, been ta
kled by di(cid:27)erent methods and persons beginning from the
earlier works of Rosenfeld [2℄, Bergmann [3℄ and S
hwinger [4℄ to the
anoni
al methods of
Dewitt [5℄, Adm [6℄ and Dira
[7℄. Among the supposed quantum
hara
teristi
s of the GF
(GW) is the probability interpretation [8, 9℄ whi
h is extended from the quantum regime
so that one interprets [10℄ the GF (GW) fun
tion as a probability amplitude [8, 9℄. This
problem of quantizing and interpreting, in probabilityterms, the GF entails in turn a se
ond
interesting problem of how and in whi
h way to in
rease the probabilty of some spe
i(cid:28)
GF.
1
Sin
e the problemof quantizationis found in the literature to refer to gravitational(cid:28)eld [4℄ and alsoto
gravitational wave[10℄ we use here these two
on
epts side by side or inter
hangably on equal basis.
INTRODUCTION 2
Taking the
ylindri
al GF as an example and assuming that one starts from a hypersurfa
e
with a
ylindri
al geometry ane may ask how to in
rease its probability so that this same

ylindri
al geometry may persist upon this hypersurfa
e.
This problem is also found, under di(cid:27)erent terms and terminology, in other dis
iplines of
generalrelativity. Thus, a
entralissueofquantumgravity[11℄istheproblemofsubstantially
in
reasing the probability of some quantum foam [12℄, with a typi
al order of magnitude of
G¯h
the Plan
k-Wheeler length ( c3 ) [13, 14℄, so that it may be realized in full grown human
s
ale with a unity probability. An example of the latter, whi
h is intensely dis
ussed in the
literature [12, 15℄, is that of the theoreti
al in
rease of some foam-like probability tissue in
the form of wormhole to the realized human-s
ale (of unity probability) so that it may be
used for spa
etime and, espe
ially, time travelling [12, 15℄.
Asknown, thementioned(cid:28)rstproblemofquantizingtheGF(GW)(cid:28)ndsitssolutionwhen
one restri
ts the dis
ussion to limited dynami
al regions of geometrodynami
s su
h as the
minisuperspa
e and miniphasespa
e dis
ussed by Dewitt [5℄, Misner [16℄ and, espe
ially, by
Ku
har [10℄ whi
h applied it for quantizing the
ylindri
al GW. Ku
har did it by extending
the extrinsi
time idea, whi
h was (cid:28)rst introdu
ed [6, 17, 18℄ for the linearized theory [13, 14℄
of general relativity, to the nonlinear theory [19℄ and, espe
ially, to the
ylindri
al spa
etime.
This extrinsi
time variable, whi
h is of the Tomonaga-S
hwinger many-(cid:28)ngered time kind
[20, 21℄, is
anoni
ally
onjugate to momentum and not to energy as is the intrinsi
time
whi
h serves more as a label to distinguish between spa
elike hypersurfa
es [13℄ in a one-
parameter family of them.
Thus, using the extrinsi
time variable one may obtain [10℄, as known from [6℄, a formal-
ism whi
h is identi
al to that used for dis
ussing the parametrized [10℄
ylindri
al massless
s
alar (cid:28)eld in a (cid:29)at Minkowskian ba
kground. This suggests, as noted in [10℄, that all the
results obtained from the later
ase may, theoreti
ally, be applied also for the
ylindri
al
GF in a
urved spa
etime. These appli
ations in
lude also the
anoni
al quantization of the

ylindri
al GF (GW) and the derivation [10℄ of a fun
tional time-independent S
hroedinger-
INTRODUCTION 3
type equation for it. Moreover, Ku
har has su

eeded [10℄ to apply for the
ylindri
al GF
(GW) not only quantum ideas su
h as the inner produ
t of states [8, 9℄ or path independen
e
of the evolution of them but also their probability interpretation [8, 9℄.
In this work we use this su

essful quantization of the
ylindri
al GW as a basis for dis-

ussing the mentioned se
ond problem of (cid:28)nding the
onditions under whi
h the mentioned
probabilityin
reases andevenapproa
hes unity. In thequantum regime,where states
hange
with both time and spa
e [8, 9℄, there exists the known Zeno e(cid:27)e
t [22, 23, 24, 25℄ whi
h

auses these
hanging states to be
ome
onstant and (cid:28)xed and, therefore, to
ause their
probability, as in
lassi
al physi
s, to be
ome unity. This e(cid:27)e
t, whi
h were experimentally
validated [26, 27℄, has three di(cid:27)erent versions:
(1)Repeatinga largenumber of times, ina (cid:28)nitetotaltime, the same experiment of
he
king
the present state of some quantum system that has been prepared in an initial spe
i(cid:28)
state
so that in the limit of a very large number of repetitions in the same total time the initial
state is preserved in time [22, 26℄.
(2) Performing a large set of di(cid:27)erent experiments ea
h redu
ing the system to some state
(froma large number of di(cid:27)erent ones) thereby obtaininga largenumber of di(cid:27)erent possible
paths of states (Feynman paths [28, 29℄) so that in the limit of doing these experiments in
a (cid:16)dense(cid:17) manner one may (cid:16)realize(cid:17) [23, 24℄ any spe
i(cid:28)
path of states in the sense that the
probability to pro
eed along all its
onstituent states tends to unity.
(3) Simultaneously performing the same experiment in a large number of non-overlapping
spatial subregions all in
luded in a (cid:28)nite total region so that when these subregions in(cid:28)nites-
imally shrink, keeping the total region (cid:28)xed, one avoid any spatial shifting of the state [25℄.
This may be explained by the example of trying to lo
ate a very small parti
le-like obje
t
in the (cid:28)nite total spatial region whi
h be
ome easier when this region is divided into several
equal parts ea
h o

upying the same small obje
t and the sear
hing for it is done in ea
h of
these smaller regions. It is obvious that the smaller be
ome these regions in the (cid:28)nite total
region the probability to lo
ate the small obje
t in ea
h of them grows so that in the limit
INTRODUCTION 4
in whi
h they in(cid:28)nitesimaly shrink this probability tends to unity.
The (cid:28)rst two
ases (1)-(2) are, respe
tively, known [23, 24℄ as the stati
and dynami
Zeno
e(cid:27)e
ts while the third (3) is the the spa
e Zeno e(cid:27)e
t [25℄.
We note that whereas the quantum states are related to the ordinary intrinsi
time the
gravitational
ylindri
al states are
hara
terized by [10℄ an extrinsi
time variable whi
h is
related (and a
tually borrows [10℄ its name) to the extrinsi

urvature. That is, the grav-
itational state, represented by the probability that the related hypersurfa
e has
ylindri
al
geometry,
hanges in spa
etime with respe
t to extrinsi
time. Thus, sin
e this extrinsi
time, like the ordinary spatial variable, is [10℄
anoni
ally
onjugate to momentum (whi
h
is
onne
ted to extrinsi

urvature) the
orresponding gravitational Zeno e(cid:27)e
t should also
have spatial
hara
teristi
s as in the mentioned spa
e Zeno e(cid:27)e
t.
We dire
tly show in this work, using spa
e Zeno terms [25℄, that one may avoid any spa
e
shifting of the
ylindri
al GF (GW) thereby (cid:28)xing its
ylindri
al geometry and
ausing its
probability to approa
h unity. In the following we pre
ede this Zeno demonstration with
a dis
ussion wi
h shows that the
ylindri
al GF may be represented as a large number of

onstituent parts in some (cid:28)nite region of spa
e so that it may be dis
ussed in spatial Zeno
terms [25℄. The appropriate representation whi
h enables one to do so is the o

upation
number one [8, 9℄. Thus, we dis
uss here, in detail, this representation [8, 9℄ in relation to
the
ylindri
al GF. We note in this respe
t that although this representation in the
ontext
of
ylindri
al GF is mentioned in the literature [10, 30℄ no expli
it, as far as we know, and
detailed expressions of it exists so far
As known from quantum me
hani
s in the o

upation number representation [8, 9, 31℄,
one may, theoreti
ally, prepare any quantum state by merely applying the relevant
reation
and destru
tion operators [8, 9℄ any required number of times upon some initial basis state.
One also knows from the
anoni
al formalism [6, 13, 14℄ of general relativity that it is
possible to prepare the geometry of some spa
etime hypersurfa
e by
ontrolling the form of
the lapse and shift fun
tions [6, 13, 14℄. That is, any spe
i(cid:28)
evolution of spa
etime should
INTRODUCTION 5
be pre
eded by determining beforehand these fun
tions so that one
an be sure (with a
unity probability) that the related spa
etime is developed along the spe
i(cid:28)ed route. This
operation,
hara
terized by the determination of the lapse and shift fun
tions, should be
related, when dis
ussing the quantum properties of the GF, to the
orresponding operation
of the mentioned quantum
reation and destru
tion operators upon some initial basis state.
We derive here in detail the appropriate expressions whi
h a

ordingly relate the
ylindri
al
lapse and shift fun
tions to the
reation and destru
tion operators.
When the mentioned probability of the
ylindri
al GW approa
h unity (also in
onse-
quen
e of the spatial Zeno e(cid:27)e
t) the related GW produ
e
ertain e(cid:27)e
ts upon the neigh-
bouring spa
etime through whi
h it pro
eeds su
h as implanting its
ylindri
al geometry
upon it and giving rise to some trapped surfa
e [32, 33, 34, 35℄. We follow the develope-
ment of this
ylindri
al GW, on
e its probabilisti

han
es were greatly in
reased, and (cid:28)nd
its properties in the transverse-tra
eless (TT) gauge and also
al
ulate the geometry of the
generated trapped surfa
e [32, 33, 34, 35℄.
In Se
tion II we introdu
e the prin
ipal expressions [10℄ related to the Einstein-Rosen

ylindri
al GW [36℄ as, espe
ially, represented in [10, 30℄. In Se
tion III we dis
uss the
ylin-
dri
al GF (GW) in the
ommutation number representation [8, 9℄ so that it may be thought
of as
omposed of a very large number of gravitational quanta whi
h, like the quantum ones
[8, 9℄, are
reated and destroyed by the
orresponding gravitational
reation and destru
-
tion operators. In Se
tion IV we relate the
ylindri
al lapse and shift fun
tions [6, 13, 14℄
whi
h determine the geometry of spa
etime hypersurfa
es to the mentioned gravitational

reation and destru
tion operators whi
h, likewise, determine this geometry through
reat-
ing or (and) destroying the quantum
omponents of the generating GW. We note that the

ylindri
al lapse and shift fun
tions were related in [10℄ to the Einstein-Rosen parameters
(see Eqs (29)-(30) in [10℄). The detailed
al
ulation relating the
ylindri
al lapse and shift
A
fun
tions to the
reation and destru
tion operators are shown in Appendix . In Se
tion
V we show that, beginning from a gravitational
ylindri
al geometry in some hypersurfa
e,
THE EINSTEIN-ROSEN CYLINDRICAL GRAVITATIONAL WAVE 6
the probability to (cid:28)nd the same geometry upon this hypersurfa
e tends to unity in the limit
of the spa
e Zeno e(cid:27)e
t [25℄. As mentioned, we rather dis
uss the spa
e Zeno e(cid:27)e
t and
not the (intrinsi
) time analogue of it be
ause in the
ylindri
al geometry one dis
usses [10℄
the extrinsi
time variable whi
h is
anoni
aly
onjugate to momentum [10℄ just as is any
spatial variable. We note that it has been shown [37℄, using the examples of the quantum
bubble and open-oyster pro
esses [31, 38℄, that the mentioned stati
and dynami
quantum
Zeno e(cid:27)e
ts are also valid in quantum (cid:28)eld theory [31, 38℄. We also note in this respe
t
that the quantum Zeno e(cid:27)e
t were dis
ussed [39℄ in the framework of gravitomagnetism [40℄.
B
The detailed
al
ulations of the appropriate probability is shown in Appendix . In Se
tion
VI we graphi
ally
orroborate our theoreti
al results so that one may see how the probabil-
ity approa
h unity in the Zeno limit. In Se
tion VII we dis
uss the
ylindri
al GW in the
transverse-tra
eless (TT) gauge whi
h is
hara
terized by a very simpli(cid:28)ed formalism [13℄
in whi
h, for example, the number of independent
omponents of the related GW is mini-
mal [13℄. In Se
tion VIII we dis
uss, using the method in [32℄, the related trapped surfa
e
[13, 32, 33, 34, 35℄ resulting from the passing
ylindri
al GW. In Se
tion IX we summarize
our dis
ussion in a Con
luding Remarks Se
tion.
2 The Einstein-Rosen Cylindri
al gravitational wave
A spa
etime is
onsidered to be
ylindri
ally symmetri
[10℄ if and only if one
an show that
(t, r, φ, z) −∞ < t < +∞ ∞ > r ≥ 0 2π > φ ≥ 0
there sxists a
oordinate system , , , ,
−∞ < z < +∞
in whi
h the line element be
omes
ds2 = −(N2 −e(ψ−γ)N2)dt2 +2N dtdr+e(γ−ψ)dr2 +R2e−γdφ2 +eγdz2,
1 1
(1)
R ≥ 0 γ, ψ, N, N t r
1
where and are fun
tions of and . The former dependen
e of the
g ,g ,g ,g ,g γ,ψ,R,N,N
11 22 33 00 01 1
nonzero metri
tensor
omponents upon the fun
tions is,
espe
ially,designed[10℄tosuittheADM[6℄
anoni
alformulationofgeneralrelativity. Thus,
THE EINSTEIN-ROSEN CYLINDRICAL GRAVITATIONAL WAVE 7
N N
1
the and are, respe
tively, the known ADM lapse and radial shift fun
tions [10, 13℄.
φ z
The
oordinates and are, essentially, (cid:28)xed ex
ept for a possible trivial transformation of
φ¯= ±φ+φ z¯= az+z t r
0 0
and whereas and may be subje
t, without
hanging the form of
the line element from (1), to the more general transformation
t¯= t¯(t,r), r¯= r¯(t,r)
(2)
(t, r, φ, z)
One may show [10℄, using the Killing ve
tor formalism in the system, that
R γ
the fun
tions and are s
alars. Thus, sin
e the metri
tensor
oe(cid:30)
ients depends, as
t r (t,r)
mentioned, only upon and one may write [10℄ the part of the line element (1) in the
following
onformally (cid:29)at form
ds2 = e(γ¯−ψ)(−dt¯2 +dr¯2)+R2e−ψdφ2 +eψdz2,
(3)
R ψ φ z
where the , , and are not barred due to their mentioned essential invarian
y. Now,
as emphasized in [10℄, if one writes the Einstein (cid:28)eld equations for the line element (3) one
R ∂2R − ∂2R = 0
may realize that must be a harmoni
fun
tion whi
h satis(cid:28)es ∂t¯2 ∂r¯2 . Thus, one
R T
may assume [10℄ to be a new radial
oordinate and the time
oordinate
orresponding
to it. That is, as emphasized in [10℄, the Einstein-Rosen
oordinates
an be uniquely and
rigorously de(cid:28)ned by invariant pres
riptions so that the line element (3) may be written as
ds2 = e(Γ−ψ)(−dT2 +dR2)+R2e−ψdφ2 +eψdz2
(4)
In su
h
ase the Einstein va
uum equations are
onsiderably simpli(cid:28)ed and redu
e to the
following three equations
∂2ψ ∂2ψ ∂ψ
− −R−1 = 0
∂T2 ∂R2 ∂R (5)
THE EINSTEIN-ROSEN CYLINDRICAL GRAVITATIONAL WAVE 8
∂Γ 1 ∂ψ ∂ψ
= R ( )2 +( )2
∂R 2 (cid:18) ∂T ∂R (cid:19) (6)
∂Γ ∂ψ ∂ψ
= R
∂T ∂T ∂R (7)
Asemphasized in[10℄, Eq(5)hasexa
tly thesameformasthewave equationofthe
ylin-
ψ
dri
ally symmetri
massless s
alar (cid:28)eld advan
ing in a Minkowskian spa
etime whereas
Eqs (6) and (7)are, respe
tively, the energy density and the radialmomentum density of this
(cid:28)eld in
ylindri
al
oordinates. The solution of Eq (5) is obtained by using the separation of
k
variables method [30℄ so that the resulting wave fun
tion for a parti
ular wave number is
ψ (R,T) = J (kR) A(k)e(ikT) +A∗(k)e−(ikT) ,
k 0
(8)
(cid:0) (cid:1)
j (kR) A(k), A∗(k)
0
where is the bessel fun
tion of order zero [41℄ and are the amplitude and
its
omplex
onjugate of the solution to the time part of Eq (5). Note that here we assume,
c = h¯ = 1 w = k = p w, k, p
as generally done in the relevant literature, that so that where
k
are respe
tively the frequen
y, wave number and momentum of some mode. Sin
e is a

ontinuous parameter one may obtain the general solution to Eq (5) by integrating over all
k
the modes . Thus, the relevant general wave fun
tion is
∞
ψ(R,T) = dkJ (kR) A(k)e(ikT) +A∗(k)e−(ikT)
0
Z (9)
0 (cid:0) (cid:1)
π (T,R)
ψ
The
anoni
al
onjugate momentum may be obtained [30℄ by using the Hamilton
equation
∂ψ
= {ψ,H},
∂t (10)
ψ
where is given by Eq (9), the
urly bra
kets at the right denote the Poisson bra
kets and
THE EINSTEIN-ROSEN CYLINDRICAL GRAVITATIONAL WAVE 9
H
the Hamilton fun
tion is
∞
H = dr N˜H˜ +N˜1H˜
1
Z (cid:18) (cid:19) (11)
0
H˜ H˜
1
The quantities and are respe
tively the res
aled superHamiltonian and supermomen-
tum whi
h where shown in [10℄ (see Eqs (93)-(97) and (106)-(108) in [10℄) to be
1 1
H˜ = R Π +T Π + R−1π2 + Rψ2
,r T ,r R 2 ψ 2 ,r (12)
H˜ = T Π +R Π +ψ π ,
1 ,r T ,r R ,r ψ
r Π , Π
T R
where the su(cid:30)xed apostroph denote di(cid:27)erentiation with respe
t to and are the
T R N˜ N˜
1
respe
tive momenta
anoni
ally
onjugate to and . The quantities and respe
tively
N N π (T,R)
1 ψ
denote the res
aled lapse and shift fun
tion , (see Eq (96) in [10℄). Thus, were
shown [30℄ to have the form
∞
π (T,R) = iRR dkkJ (kR) A(k)e(ikT) −A∗(k)e−(ikT)
ψ ,r 0
Z (13)
0 (cid:0) (cid:1)
∞
−RT dkkJ (kR) A(k)e(ikT) +A∗(k)e−(ikT) ,
,r 1
Z
0 (cid:0) (cid:1)
j (kR)
1
where is the (cid:28)rst order Bessel fun
tion [41℄ whi
h may be obtained by di(cid:27)erentiating
j (kR) R j (kR) = −kj (kR) ψ(T,R) π (T,R)
0 0 ,R 1 ψ
with respe
t to as . The initialdata for and
T = 0 R = r Q(r) P(r)
are
al
ulatedfor and andare, respe
tively, denotedby and asfollows
∞
Q(r) = ψ (r) = ψ(T,R)| = dkJ (kr) A(k)+A∗(k)
0 T=0,R=r 0
Z (14)
0 (cid:0) (cid:1)
∞
P(r) = π (r) = π (T,R)| = ir dkkJ (kr) A(k)−A∗(k)
ψ0 ψ T=0,R=r Z 0 (15)
0 (cid:0) (cid:1)
THE EINSTEIN-ROSEN CYLINDRICAL GRAVITATIONAL WAVE 10
A(k) A∗(k)
Solving the last two equations for and one obtains
1 ∞
A(k) = drJ (kr) krQ(r)−iP(r)
0
2 Z (16)
0 (cid:0) (cid:1)
1 ∞
A∗(k) = drJ (kr) krQ(r)+iP(r)
0
2 Z (17)
0 (cid:0) (cid:1)
A(k), A∗(k)
One may show, using Eqs (16)-(17), that the variables satisfy the following
Poisson bra
kets
∞ δ(A(k))δ(A∗(k′)) δ(A(k))δ(A∗(k)) iδ(k −k′)
{A(k),A∗(k′)} = dr − =
Z (cid:20)δ(Q(r)) δ(P(r)) δ(P(r)) δ(Q(r)) (cid:21) 2
0
{A(k),A(k′)} = {A∗(k),A∗(k′)} = 0,
(18)
where use was made of the relation [30℄
∞ 1
drrj (kr)j (k′r) = δ(k −k′), n = 0,1,2,3....
n n
Z k′
0
Q(r)
In a similar manner, using Eqs (14)-(15), it is possible to show that the variables and
P(r)
satisfy the following Poisson bra
kets
∞ δ(Q(r)) δ(P(r′)) δ(Q(r)) δ(P(r′))
{Q(r),P(r′)} = dk − = −2iδ(r −r′)
Z (cid:20)δ(A(k))δ(A∗(k)) δ(A∗(k)) δ(A(k))(cid:21)
0
{Q(r),Q(r′)} = {P(r),P(r′)} = 0,
(19)
where use was made of the relation [30℄
∞ δ(r −r′)
dkkJ (kr)J (kr′) = , n = 0, 1,2,...
n n
Z r′
0
Using Eqs (6), (9) and the unnumbered relation written just after Eq (18) one may obtain