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Quantum effects in the Bertotti-Robinson metric
Volume 86A, number 2
PHYSICS LETTERS
2 November 1981
QUANTUM EFFECTS IN THE BERTOTTI—ROBINSON METRIC Varun SAHNI Department of Physics, Moscow State University, Moscow, USSR Received 21 July 1981
Scalar and spin 1/2 fields have been studied in the Bertotti—Robinson space—time and analytical solutions have been obtained for the Klein—Gordon equation and the Dirac equation. For large e particle creation takes place as in the pseudoeudidean metric under the influence of a constant external field.
1. introduction. The Bertotti—Robinson electromagnetic universe represents an empty; singularity-free nonasymptotically flat space—time endowed with a constant, homogeneous electric field [1] . Since Cikim = 0 and Fjk.l 0, we get R ikim = 0, i.e. the gravitational field is likewise homogeneous. The Bertotti—Robinson line elemerh also describes the geometry of an extreme Reissner—Nordström throat. It may be written as (1) ds2 = Q2(dr2 cos2r dz2 d~2 sin2~9dp~), •,~
—
—
—
where El = c2/~/~ Q. Q has the dimension of length and determines the distance at which the gravitational field makes its presence felt by substantially altering the space—time structure. ForE = 1 V/cm, Q 1027 cm 1o~ light years. The metric (1) possesses a high degree of symmetry [2] and is an appealing background for the study of pair creation and vacuum instabilities in the presence of a strong external gravitational and electromagnetic field. Its properties make it appropriate for suitable generalisation to the world of elementary particles. The metric (1) may be represented statically by the metric ds2
=
Q2(cosh2r dt2
—
dr2
—
sin2t~dp2
—
dt92),
(2)
by the coordinate transformations tanhz = tanhrsec
t,
sin r=coshrsin
(3)
t.
We shall confme ourselves, however, to the spatially homogeneous metric (1) as the latter is more suitable for dealing with nonstationary problems describing evolution. It should likewise be pointed out that metric (1) is regular at ‘r = ±ir/2and can be analytically continued through these hypersurfaces. 2. The scalar field. Since the scalar curvature R ten as (c =it = G = 1) —
=
0 for the metric (1), the Klein—Gordon equation may be writ-
ieA,~)~,/~jg’~’(a~, ieA~)4,(1+ m2 ~,1i= 0,
whereA~= (0,A~,0, 0),AZ
(4)
—
=
Q2E
5sin r. For the metric (1) this reduces to ‘a \2 a2 2 (~——ieA 2r Z ~ ~in2~ ap2 5)~ ar look for solutions ~ cosof eq. (5) of the general form ~‘(r,z, z~,~)= T(r) e~PzY We
2’ a’ I —~—tanr-~~-— a
(5) 1j~(t~, ~). Making the transformation
0 031-9163/81/0000—0000/$ 02.75 © 1981 North-Holland
87
Volume 86A, number 2
x
=
(1
cos(r
—
—
PHYSICS LETTERS
2 November 1981
ir/2) we obtain
x2).~__~ —
2
2x
—
[0~)
+(ieEQ2)2 —2ip(ieEQ2)x
—
1(1 + 1)
—
m2Q2
+ (eEQ2)2]
T= 0.
A general solution of eq. (6) can be obtained in terms of the hypergeometric functions. For p = ieEQ2 = ie’, k(k + 1) = 1(1 + 1) + m2Q2 (eEQ2)2]
?
(6)
eEQ2 [a= ip, b
—
T
2(l ~_X)(a_b)/2F(k + a
+
1, a
—
k;a
—
b
+
1; (1 —x)/2),
(7a)
1(x) = const1.(l +X)(a+b)/ 1;(1 —x)/2),
(7b)
2F(k÷b+1,bk;b_a+l;(l —x)/2), x)(b_a)/ x)_(b_a)/2F*(k + b + 1, b k; b a + 1; (1 x)/2),
(8a)
T
2F*(k+a+ l,a
—
k;a
—
b
+
2(x)—const2~(l+X)_(a+b)/2(l _x)_(0_~
and for eEQ2 ~p T
1(x)const3~(l +X)(b+a)12(l T
2(l
—
—
—
—
(8b)
2(x) = const4 (1 +X)_(~a)/ describe the two linearly independent solutions of eq. (6). Analysing the solutions asymptotically for p ~ eEQ2 we obtain in general C
2
1(l —x) i(p e)/2
+
c2(l _Xy(P_e)/
~
(C
2+
1B1
+ C2B)(l +X)1(P~-e’)/
(C
2.
1B2
(9) + C2B)(1 +X)_i(P+e)/
By use of the WKB approximation it can be shown that forx -÷ 1(1 _x)~(P~’)/2 and (1 _x)—i(P—e’)/2, and for —1(1 +x)—i(P+e’)/2 and (1 +x)i(P~’)I2correspond to positive and negative frequency solutions of eq. (6), respectively. Hence eq. (9) describes particle creation in the metric (1). Using the normalisation condition —*
T*T?
—
TT~1= —i/(l —x2),
(10)
we obtain B 2. sin2irk+sinh2irp ,I2B I~l sinh2lTp sinh2ire’ In the case of eEQ2 ~ p
2_ SIfl2lTk+siflh2lle
sinh2irp
—
—
‘
B1
2
—
lB 21. 2_
sinh27re’
C
2—~C D (1 +xy(e+P)/2 + C 2, 1(l _X)_i(e —P~/ x—÷—1 1 1 1D2(l +X)_i(e~)/ C (1 _X)1(e’_P)/2 —~~c D~(1÷X) i(e+p)/2 +C 12,
(lla) (lib)
2D~(1+Xy(e+P)
describe the two linearly independent solutions of eq. (6). For normalised coefficients we obtain
D 2_ coshir(p +y)coshir~—‘) 11 sinhir(e’+p)siithir(e’—p) 2 sinhir(e’+p)sinhn(e’—p)’ 2Q2] For large e wherey = e’ (2e’)l [1(1+ 1) + m 1D 2 ~ 1 + exp(_ir 1q÷ +m), ID 11 2 12 ~~exp(_ir ~ ~ +~n). 1D
2_
cosh lr(e’ +y)coshir(e’ —y)
—
12
‘
.
—
~
(13)
The above equation is identical to the one obtained by Nikishov [3] in his study of pair production by a constant electric field in the pseudo-eudidean metric. 3. Equation for spin ~ particles. In curvilinear coordinates the Dirac equation takes the form [4]
88
[i@(Wxk
- ITk - ieA,)
where {yi, yi} = 28,
Simplifying
ax _=a7
{?‘,yjj
(14)
= 2g%renti, sin 7,
for the metric (l),
r3 = - +y2y3cog9.
eq. (14) we get
(- -
70% a cos 7
- m] $ = 0,
l?I = - f~&
IT2 = r0 = 0,
2 November 1981
PHYSICS LETTERS
Volume 86A, number 2
az
ieEQ2sin r
)
(15)
x - TIZx - i?umQx,
where x = (Q~COS T sin 9)lj2 JI and 1 is the eigenvalue of the operator i = ~1~,,~2 a/W t ~I~0~3a/a~, ix = lx. As in the scalar case we look for solutions of eq. (15) of the general form ~(7, z,, 0, cp)= x1 (7) e-ipza(8, cp). Only the two matrices T,, and y1 remain explicitly in eq. (15) after the operator L has been replaced by the number 1; they can therefore be represented by 2 X 2 matrices and the time factor by a two-component spinor;
Eq. (15) thereby reduces to the system of equations
df= -~Tg-ieEQ2tanrg-lg-imef, *
dg G=
dr
(16)
-&f-ieEQ2tanrf+Iftim&.
As in the scalar case a general solution of eq. (16) has been obtained in terms of the hypergeometric = cos(7 - n/2)] : fi(x)=constI*(l
+x)(~+~‘+I/~)/~(~
-x)(“-b’+1’2)‘2F(ktu+
b’=b-3,
f,(x)=const*f;(x),
l,a-k;a-
b’t
functions
[x
I;(1 -x)/2),
k(kt1)=12+m2Q2-e’2-~,
(17a)
and gI (x) = const2*(l
_x)’ a - b~+1’2)~2F(k+.t1,.-k;.-b”t1;(1-x)/2),
t x) (a+b”+WID(l
g2(x) = const *g:(x),
(17b)
b” = b t 4,
describe the two linearly independent solutions of eq. (16) corresponding to f(x) andg(x), respectively, for the case a > b. Interchanging b’ and CI,and b” and a in eq. (17a) and (17b), respectively, we get the solutions of eq: (16) for the case b > a. Analysing the solution asymptotically for the case b > a we find that C1(l -x)-@-a)‘2 C2(l _ x)(b-aY2
x-+--i CIB1(l
tx)(b+a)/2
t C1B2(1 tx)-(b+a)/2, (lga)
x-_i
C2BI(1 t x)-@+a)‘2
t C2Bi(l
t x)@+a)l2,
- C,B,(l
tx)-(**a~2,
and ~~(1
-,)-U-Y2
C2(l _x)(b-a)/2
-..-+ CIB1(l x~_l xz
C2B31
+x)(b+al’z +x)-
@+a)/2 _ C2B;(l
(lgb)
+ x)(b+a)/2,
describe the two linearly independent solutions of eqs. (17a, b). Using the normalisation X $T’$J = l/Q we obtain for the coefficients B1, B2, C,, C2
conditionjo
= fi
89
Volume 86A, number 2
lB 1
2
=
1/4 + (e’
IC
2
PHYSICS LETTERS
2 sinh n(e’ + — p)2 p) sm1i ~
=~
~
— —
÷~
y) sinh 1T(e n(e’ ++ y) 1’) ~th P) y
=
lB2
12
=
2 November 1981
1/4 p)2 coshir(y—p) 1/4 + + (e’ (e’ + p)2 coth ir(e + p) cosh cosh lTIy ir(e +p) p) —
—
e’ — (2e1)_1(12 +m2Q2).
(19)
1 12 = 1C21 For large e, lB
2) lB 2/Q2+ m2) lB 2 = 1, 1 12 1— exp(_ir-~L~ m 2 2 ~exp(_7T 1 1 2 + 1B21 which again coincides with the results obtained by Nikishov [3J. Similarly we obtain for a ~
2,
C
1(1 —x) (a—b)/2 X-~— ~ C1D1(1 ÷X)(a+b)12+ C1D2(1 ÷xy~”~’-’~I
C 2(1 _X)(~~_b)I2 —~C2D~(l+X)_(~z+b)I2+ and
—i 2 C~D
C1(l _X)_(~~_b)I
1(1+X)(a+b)I2
—
b (2la)
c2D(l ÷xya+b)12, C1D2(1 ÷X)(a+b)I2, 2,
c2(1
(20)
(2lb)
C2D~(l+X)_(a+b)12 C2D(l +X)(a+b)/ which describe the solutions forf(x) andg(x), respectively, and _X)(a
b)/2
___..÷
—
ID 1 I2_ 1/4+(p—e’)2cosh27Tp—cos2nk 1/4 + (p + e’)2 cosh2np + siith2ire’ For e
=
ID 2
2 cosh2np cosh2ne’—sin2irk 1/4 + (p + e’)2 + sinh2iTe’ 2_1/4+(p~)
m = 0, ID
1 2 = 1, ID2 12 ly flat metrics [5].
=
0, which corresponds to the fact that neutrinos cannot be created in conformal-
The author is grateful to A.A. Starobinsky for many helpful discussions and advice. References [1] B. Bertotti,Phys. Rev. 116 (1959) 1331; I. Robinson, Bull. Acad. Pol. Sd. 7 (1959)
351. [2] G. Denardo and E. Spallucci, Nuovo Cimento 47B (1978) 25; 49B (1979) 162. [31A.I. Nildshov, Zh. Eksp. Teor. Fiz. 57 (1969) 1210. [4] D.R. Brill and J.A. Wheeler, Rev. Mod. Phys. 29 (1957) 465. [5] YaB. Zeldovich and A.A. Starobinsky, Zh. Eksp. Teor. Fiz. 61(1971) 2161.
90
PHYSICS LETTERS
2 November 1981
QUANTUM EFFECTS IN THE BERTOTTI—ROBINSON METRIC Varun SAHNI Department of Physics, Moscow State University, Moscow, USSR Received 21 July 1981
Scalar and spin 1/2 fields have been studied in the Bertotti—Robinson space—time and analytical solutions have been obtained for the Klein—Gordon equation and the Dirac equation. For large e particle creation takes place as in the pseudoeudidean metric under the influence of a constant external field.
1. introduction. The Bertotti—Robinson electromagnetic universe represents an empty; singularity-free nonasymptotically flat space—time endowed with a constant, homogeneous electric field [1] . Since Cikim = 0 and Fjk.l 0, we get R ikim = 0, i.e. the gravitational field is likewise homogeneous. The Bertotti—Robinson line elemerh also describes the geometry of an extreme Reissner—Nordström throat. It may be written as (1) ds2 = Q2(dr2 cos2r dz2 d~2 sin2~9dp~), •,~
—
—
—
where El = c2/~/~ Q. Q has the dimension of length and determines the distance at which the gravitational field makes its presence felt by substantially altering the space—time structure. ForE = 1 V/cm, Q 1027 cm 1o~ light years. The metric (1) possesses a high degree of symmetry [2] and is an appealing background for the study of pair creation and vacuum instabilities in the presence of a strong external gravitational and electromagnetic field. Its properties make it appropriate for suitable generalisation to the world of elementary particles. The metric (1) may be represented statically by the metric ds2
=
Q2(cosh2r dt2
—
dr2
—
sin2t~dp2
—
dt92),
(2)
by the coordinate transformations tanhz = tanhrsec
t,
sin r=coshrsin
(3)
t.
We shall confme ourselves, however, to the spatially homogeneous metric (1) as the latter is more suitable for dealing with nonstationary problems describing evolution. It should likewise be pointed out that metric (1) is regular at ‘r = ±ir/2and can be analytically continued through these hypersurfaces. 2. The scalar field. Since the scalar curvature R ten as (c =it = G = 1) —
=
0 for the metric (1), the Klein—Gordon equation may be writ-
ieA,~)~,/~jg’~’(a~, ieA~)4,(1+ m2 ~,1i= 0,
whereA~= (0,A~,0, 0),AZ
(4)
—
=
Q2E
5sin r. For the metric (1) this reduces to ‘a \2 a2 2 (~——ieA 2r Z ~ ~in2~ ap2 5)~ ar look for solutions ~ cosof eq. (5) of the general form ~‘(r,z, z~,~)= T(r) e~PzY We
2’ a’ I —~—tanr-~~-— a
(5) 1j~(t~, ~). Making the transformation
0 031-9163/81/0000—0000/$ 02.75 © 1981 North-Holland
87
Volume 86A, number 2
x
=
(1
cos(r
—
—
PHYSICS LETTERS
2 November 1981
ir/2) we obtain
x2).~__~ —
2
2x
—
[0~)
+(ieEQ2)2 —2ip(ieEQ2)x
—
1(1 + 1)
—
m2Q2
+ (eEQ2)2]
T= 0.
A general solution of eq. (6) can be obtained in terms of the hypergeometric functions. For p = ieEQ2 = ie’, k(k + 1) = 1(1 + 1) + m2Q2 (eEQ2)2]
?
(6)
eEQ2 [a= ip, b
—
T
2(l ~_X)(a_b)/2F(k + a
+
1, a
—
k;a
—
b
+
1; (1 —x)/2),
(7a)
1(x) = const1.(l +X)(a+b)/ 1;(1 —x)/2),
(7b)
2F(k÷b+1,bk;b_a+l;(l —x)/2), x)(b_a)/ x)_(b_a)/2F*(k + b + 1, b k; b a + 1; (1 x)/2),
(8a)
T
2F*(k+a+ l,a
—
k;a
—
b
+
2(x)—const2~(l+X)_(a+b)/2(l _x)_(0_~
and for eEQ2 ~p T
1(x)const3~(l +X)(b+a)12(l T
2(l
—
—
—
—
(8b)
2(x) = const4 (1 +X)_(~a)/ describe the two linearly independent solutions of eq. (6). Analysing the solutions asymptotically for p ~ eEQ2 we obtain in general C
2
1(l —x) i(p e)/2
+
c2(l _Xy(P_e)/
~
(C
2+
1B1
+ C2B)(l +X)1(P~-e’)/
(C
2.
1B2
(9) + C2B)(1 +X)_i(P+e)/
By use of the WKB approximation it can be shown that forx -÷ 1(1 _x)~(P~’)/2 and (1 _x)—i(P—e’)/2, and for —1(1 +x)—i(P+e’)/2 and (1 +x)i(P~’)I2correspond to positive and negative frequency solutions of eq. (6), respectively. Hence eq. (9) describes particle creation in the metric (1). Using the normalisation condition —*
T*T?
—
TT~1= —i/(l —x2),
(10)
we obtain B 2. sin2irk+sinh2irp ,I2B I~l sinh2lTp sinh2ire’ In the case of eEQ2 ~ p
2_ SIfl2lTk+siflh2lle
sinh2irp
—
—
‘
B1
2
—
lB 21. 2_
sinh27re’
C
2—~C D (1 +xy(e+P)/2 + C 2, 1(l _X)_i(e —P~/ x—÷—1 1 1 1D2(l +X)_i(e~)/ C (1 _X)1(e’_P)/2 —~~c D~(1÷X) i(e+p)/2 +C 12,
(lla) (lib)
2D~(1+Xy(e+P)
describe the two linearly independent solutions of eq. (6). For normalised coefficients we obtain
D 2_ coshir(p +y)coshir~—‘) 11 sinhir(e’+p)siithir(e’—p) 2 sinhir(e’+p)sinhn(e’—p)’ 2Q2] For large e wherey = e’ (2e’)l [1(1+ 1) + m 1D 2 ~ 1 + exp(_ir 1q÷ +m), ID 11 2 12 ~~exp(_ir ~ ~ +~n). 1D
2_
cosh lr(e’ +y)coshir(e’ —y)
—
12
‘
.
—
~
(13)
The above equation is identical to the one obtained by Nikishov [3] in his study of pair production by a constant electric field in the pseudo-eudidean metric. 3. Equation for spin ~ particles. In curvilinear coordinates the Dirac equation takes the form [4]
88
[i@(Wxk
- ITk - ieA,)
where {yi, yi} = 28,
Simplifying
ax _=a7
{?‘,yjj
(14)
= 2g%renti, sin 7,
for the metric (l),
r3 = - +y2y3cog9.
eq. (14) we get
(- -
70% a cos 7
- m] $ = 0,
l?I = - f~&
IT2 = r0 = 0,
2 November 1981
PHYSICS LETTERS
Volume 86A, number 2
az
ieEQ2sin r
)
(15)
x - TIZx - i?umQx,
where x = (Q~COS T sin 9)lj2 JI and 1 is the eigenvalue of the operator i = ~1~,,~2 a/W t ~I~0~3a/a~, ix = lx. As in the scalar case we look for solutions of eq. (15) of the general form ~(7, z,, 0, cp)= x1 (7) e-ipza(8, cp). Only the two matrices T,, and y1 remain explicitly in eq. (15) after the operator L has been replaced by the number 1; they can therefore be represented by 2 X 2 matrices and the time factor by a two-component spinor;
Eq. (15) thereby reduces to the system of equations
df= -~Tg-ieEQ2tanrg-lg-imef, *
dg G=
dr
(16)
-&f-ieEQ2tanrf+Iftim&.
As in the scalar case a general solution of eq. (16) has been obtained in terms of the hypergeometric = cos(7 - n/2)] : fi(x)=constI*(l
+x)(~+~‘+I/~)/~(~
-x)(“-b’+1’2)‘2F(ktu+
b’=b-3,
f,(x)=const*f;(x),
l,a-k;a-
b’t
functions
[x
I;(1 -x)/2),
k(kt1)=12+m2Q2-e’2-~,
(17a)
and gI (x) = const2*(l
_x)’ a - b~+1’2)~2F(k+.t1,.-k;.-b”t1;(1-x)/2),
t x) (a+b”+WID(l
g2(x) = const *g:(x),
(17b)
b” = b t 4,
describe the two linearly independent solutions of eq. (16) corresponding to f(x) andg(x), respectively, for the case a > b. Interchanging b’ and CI,and b” and a in eq. (17a) and (17b), respectively, we get the solutions of eq: (16) for the case b > a. Analysing the solution asymptotically for the case b > a we find that C1(l -x)-@-a)‘2 C2(l _ x)(b-aY2
x-+--i CIB1(l
tx)(b+a)/2
t C1B2(1 tx)-(b+a)/2, (lga)
x-_i
C2BI(1 t x)-@+a)‘2
t C2Bi(l
t x)@+a)l2,
- C,B,(l
tx)-(**a~2,
and ~~(1
-,)-U-Y2
C2(l _x)(b-a)/2
-..-+ CIB1(l x~_l xz
C2B31
+x)(b+al’z +x)-
@+a)/2 _ C2B;(l
(lgb)
+ x)(b+a)/2,
describe the two linearly independent solutions of eqs. (17a, b). Using the normalisation X $T’$J = l/Q we obtain for the coefficients B1, B2, C,, C2
conditionjo
= fi
89
Volume 86A, number 2
lB 1
2
=
1/4 + (e’
IC
2
PHYSICS LETTERS
2 sinh n(e’ + — p)2 p) sm1i ~
=~
~
— —
÷~
y) sinh 1T(e n(e’ ++ y) 1’) ~th P) y
=
lB2
12
=
2 November 1981
1/4 p)2 coshir(y—p) 1/4 + + (e’ (e’ + p)2 coth ir(e + p) cosh cosh lTIy ir(e +p) p) —
—
e’ — (2e1)_1(12 +m2Q2).
(19)
1 12 = 1C21 For large e, lB
2) lB 2/Q2+ m2) lB 2 = 1, 1 12 1— exp(_ir-~L~ m 2 2 ~exp(_7T 1 1 2 + 1B21 which again coincides with the results obtained by Nikishov [3J. Similarly we obtain for a ~
2,
C
1(1 —x) (a—b)/2 X-~— ~ C1D1(1 ÷X)(a+b)12+ C1D2(1 ÷xy~”~’-’~I
C 2(1 _X)(~~_b)I2 —~C2D~(l+X)_(~z+b)I2+ and
—i 2 C~D
C1(l _X)_(~~_b)I
1(1+X)(a+b)I2
—
b (2la)
c2D(l ÷xya+b)12, C1D2(1 ÷X)(a+b)I2, 2,
c2(1
(20)
(2lb)
C2D~(l+X)_(a+b)12 C2D(l +X)(a+b)/ which describe the solutions forf(x) andg(x), respectively, and _X)(a
b)/2
___..÷
—
ID 1 I2_ 1/4+(p—e’)2cosh27Tp—cos2nk 1/4 + (p + e’)2 cosh2np + siith2ire’ For e
=
ID 2
2 cosh2np cosh2ne’—sin2irk 1/4 + (p + e’)2 + sinh2iTe’ 2_1/4+(p~)
m = 0, ID
1 2 = 1, ID2 12 ly flat metrics [5].
=
0, which corresponds to the fact that neutrinos cannot be created in conformal-
The author is grateful to A.A. Starobinsky for many helpful discussions and advice. References [1] B. Bertotti,Phys. Rev. 116 (1959) 1331; I. Robinson, Bull. Acad. Pol. Sd. 7 (1959)
351. [2] G. Denardo and E. Spallucci, Nuovo Cimento 47B (1978) 25; 49B (1979) 162. [31A.I. Nildshov, Zh. Eksp. Teor. Fiz. 57 (1969) 1210. [4] D.R. Brill and J.A. Wheeler, Rev. Mod. Phys. 29 (1957) 465. [5] YaB. Zeldovich and A.A. Starobinsky, Zh. Eksp. Teor. Fiz. 61(1971) 2161.
90