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Vacuum polarization by a massive scalar field in Schwarzschild spacetime
Volume 115B, number 5
PHYSICS LETTERS
16 September 1982
VACUUM POLARIZATION BY A MASSIVE SCALAR FIELD IN SCHWARZSCHILD SPACETIME V.P. FROLOV and A.I. ZEL'NIKOV
P.N. Lebedev Physical Institute, Leninski Prospect 53, Moscow 117924, USSR Received 23 March 1982
The vacuum polarization by massive scalar particles in the gravitational field of the Schwarzschild black hole is discussed. The explicit expression for the vacuum energy-momentum tensor is obtained in the case when the Compton length hm = ~[me of the massive particle is much less than the gravitational radius of a black hole.
e n e r g y - m o m e n t u m tensor in the Hartle-Hawking vacuum IH), which corresponds to the black-body radiation [with temperature TBH = (8~rM) -1 at infinity] in equilibrium with a black hole. The advantage of the choice of the Hartle-Hawking vacuum is that the corresponding propagator G H (x ,x') = i (HI T(g~(x), ~(x'))[H) coincides with the analytic continuation o f the euclidean Green's function G E : GH(X,X' ) = iGE(X,x')lz=it , def'med as a regular decrease at the infinity solution o f the equation
The aim o f this paper is to obtain the vacuum average of the e n e r g y - m o m e n t u m tensor of a massive scalar field ~ satisfying the equation (El - ~R - m 2 ) ~ = 0
(1)
in the given Schwarzschild background geometry: ds 2 = - ( 1 - 2M/r)dt 2 + (1 - 2M/r)-I dr 2 + r2dco 2, dco 2 = d02 + sin20 d~02.
(2)
(Equation (1) is conformally invariant when m = 0, = 1/6.) This problem is essential when studying the backreaction of the vacuum polarization effects on the spacetime of a black hole. We consider the case when the Compton length Xm = ~/mc of the field is much less than the gravitational radius rg = 2GM/c 2 of the black hole:
e =- (m2/mM) 2 ~ 1,
([2 E - ~R E - m 2) GE(X,X' ) = --6(x,x')/X/~E
on the regular euclidean section o f the black-hole mettic (2), z being the angle coordinate with a period 8rak/ [3]. It means that (Tuu)H can be obtained by the analytic continuation of the euclidean energy-momenturn tensor (T~v)E after all the necessary calculations (including the renormalization procedure) have been done in the euclidean section. Using the DeWitt's effective action approach [4,5] and applying Schwinger's regttlarization prescription [ 5 - 8 ] one gets
(3)
and construct the expansion for the vacuum e n e r g y momentum tensor in powers of the ratio of k m to the radius o f the spacetime curvature ,1. (Here mp1 = (hc/ G)I/2 is the Planek mass.) The inequality (3) guarantees that M >> mpl , and hence the approximation of a given static background metric is justified. We begin by considering the average value of the
wE
,1 The opposite case (m = 0) was considered in great detail by Candelas [ 1]. The generalization of Candelas' results for the charged rotating black hole is discussed in ref. [2]. 372
(4)
=
I ~ 3 (n - 3)! f4x 32.2 -d
(5)
where an(X ) are local scalar invariants constructed from the curvature tensor and its covariant derivatives up to order 2(n - 1) [4]. These quantities are uni-
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
Volume 115B, number 5
formly bounded laE(x)l < CnM-2n on the euclidean section of the black hole and eq. (5) can be considered as the expansion of
(6)
Here (and later on) the dots denote the omitted terms which do not contribute to (TUV)ren in a Ricci-flat space and
W1 _
1 96 × 7!rr2m 2
W/6g.v
then the non-zero components of 6 for the background metric (9) can be expressed in terms of the functional derivatives o f W[~,, ~2] b y means of the following relations
W/6gAB = n - 2 ~W[')', e]/~')'AB' guv6W[Sgt~v
C at~'r~ }.
(7)
C ~ C ~ r Ce.~~~ = ½Ca~, ~ C7~ e~C e ~ '
X (3C~IS7~C 7~ erCer a - 2R~ Ca.~a C3"r~ + ...).
6~2-3~2:A:A,
(11)
(12)
where d6o 2 is the metric on a unit sphere and ~2 is a scalar function on a two-dimensional space P with a metric dP 2 = 7A~ dx A dx ~, (A, B = 0,1). Any local scalar invariant w [g] of the metric and its derivatives being restricted on the spacetime (9) can be considered as a local invariant ~ [7, ~ ] of TAB and ~2. If we denote
W1-
(10) ~ = ~4w[g] [g=(7,s2),
(13)
1
/'d2 ¢r~
96 X 7!zr2m 2a ×
{1_!~[1 + 3(1 - 6~)1K3~2 -2
- [36 + 112(1 - 6~)]K2~2-3~2:A:A}.
(14)
After performing the necessary variations, using eq. (11) and substituting ~2 = r, K = 6M[r we arrive at the following expression for the non-zero components of r uv in the Schwarzschild background (2): r~ = A ( r ) [ - 2 8 5 + 626M/r + 168(1 - 6 ~ ) ( - 5 + 11M/r)], rrr =A(r)[105 - 154M/r + 168(1 - 6~)(2 - 3M/r)], 0 7"0
=f d4x x/-S'gw[gl ,
= I 1.20--493 3..... ~
Here K - 1 is the gaussian curvature of the two-dimensional metric dF 2 and ( ):A denotes the covariant derivative with respect to dF 2. Eqs. (12), (13) allow us to write the expression for the restriction W1 of the action (7) on the metric (9) in the form
(8)
(9)
C~78e
Ca#~/a C'r8 et Ce~'a# = ~ K3 ~2-6.
Now we note that the general spherically symmetric line element can be presented in the form
~[7,n] =Jd2xv"c'4~,[7,nl,
n~[~,nll~n.
The calculation of the first term r ~ = 2 ( - g ) - l / 2 f i W1/ fig~v of the (TUV)ren expansion [(TUV)ten = r ~v + O ( m - 4 ) ] in the Schwarzschild spacetime is greatly simplified if we use these relations and note that for the metric (9) the scalar curvature is
076e
To arrive at this expression the following relations containing the Weyl tensor C ~ 7 ~ were used:
W[gl
'
and the Weyl tensor satisfies the following relations
X ( C ~ r~ C ~:8e~.C~" ~ + 50R~ C 8 e C ts'r~e
ds 2 = ~22(dr 2 + d6o2),
=
R = 2~2-2K -
f d4x
+ [ - 8 + 14(1 - 6~)]RC ~
16 September 1982
PHYSICS LETTERS
=,/Ao
= A ( r ) [ - 3 1 5 + 734M/r + 168(1 - 6~) ( - 6 + 14M/r)], (15)
.2 We use the MTW sign conventions of ref. [10]. It should be noted that the definitions of the curvature tensor in Gilkey's paper [9] differs by the sign from the MTW definition.
where A(r) = (10080rr2) -1 ~3G2c-SM2[(m2rS). The components r pu in the Schwarzschild coordinates remain finite at the event horizon. When ~ = 1/6 the 373
Volume 115B, number 5
PHYSICS LETTERS
energy density - 4 is positive at large distances. It vanishes at the surface r = 1 ~ss rg lying outside the horizon and remains negative on and under the horizon. The obtained vacuum e n e r g y - m o m e n t u m tensor does not indicate any peculiarity near r = rg which might be connected with the conjectural high concentration of real massive particles near the event horizon by Zel'dovich [11 ]. The higher order corrections contain the additional factor ~e(rg/r) 3 . It means that the vacuum e n e r g y - m o m e n t u m tensor (15) is uniformly valid for r >>el/3rg. The comparison of the obtained expression (15) with Candelas results [ 1 ] for the vacuum polarization by a massless scalar field near the bifurcation point of the horizons shows that the vacuum e n e r g y - m o m e n t u m tensor in the massless case is larger by the factor e -1 . Up to now we consider only the average value o f the e n e r g y - m o m e n t u m tensor in the Hartle-Hawking vacuum
PHYSICS LETTERS
16 September 1982
VACUUM POLARIZATION BY A MASSIVE SCALAR FIELD IN SCHWARZSCHILD SPACETIME V.P. FROLOV and A.I. ZEL'NIKOV
P.N. Lebedev Physical Institute, Leninski Prospect 53, Moscow 117924, USSR Received 23 March 1982
The vacuum polarization by massive scalar particles in the gravitational field of the Schwarzschild black hole is discussed. The explicit expression for the vacuum energy-momentum tensor is obtained in the case when the Compton length hm = ~[me of the massive particle is much less than the gravitational radius of a black hole.
e n e r g y - m o m e n t u m tensor in the Hartle-Hawking vacuum IH), which corresponds to the black-body radiation [with temperature TBH = (8~rM) -1 at infinity] in equilibrium with a black hole. The advantage of the choice of the Hartle-Hawking vacuum is that the corresponding propagator G H (x ,x') = i (HI T(g~(x), ~(x'))[H) coincides with the analytic continuation o f the euclidean Green's function G E : GH(X,X' ) = iGE(X,x')lz=it , def'med as a regular decrease at the infinity solution o f the equation
The aim o f this paper is to obtain the vacuum average of the e n e r g y - m o m e n t u m tensor of a massive scalar field ~ satisfying the equation (El - ~R - m 2 ) ~ = 0
(1)
in the given Schwarzschild background geometry: ds 2 = - ( 1 - 2M/r)dt 2 + (1 - 2M/r)-I dr 2 + r2dco 2, dco 2 = d02 + sin20 d~02.
(2)
(Equation (1) is conformally invariant when m = 0, = 1/6.) This problem is essential when studying the backreaction of the vacuum polarization effects on the spacetime of a black hole. We consider the case when the Compton length Xm = ~/mc of the field is much less than the gravitational radius rg = 2GM/c 2 of the black hole:
e =- (m2/mM) 2 ~ 1,
([2 E - ~R E - m 2) GE(X,X' ) = --6(x,x')/X/~E
on the regular euclidean section o f the black-hole mettic (2), z being the angle coordinate with a period 8rak/ [3]. It means that (Tuu)H can be obtained by the analytic continuation of the euclidean energy-momenturn tensor (T~v)E after all the necessary calculations (including the renormalization procedure) have been done in the euclidean section. Using the DeWitt's effective action approach [4,5] and applying Schwinger's regttlarization prescription [ 5 - 8 ] one gets
(3)
and construct the expansion for the vacuum e n e r g y momentum tensor in powers of the ratio of k m to the radius o f the spacetime curvature ,1. (Here mp1 = (hc/ G)I/2 is the Planek mass.) The inequality (3) guarantees that M >> mpl , and hence the approximation of a given static background metric is justified. We begin by considering the average value of the
wE
,1 The opposite case (m = 0) was considered in great detail by Candelas [ 1]. The generalization of Candelas' results for the charged rotating black hole is discussed in ref. [2]. 372
(4)
=
I ~ 3 (n - 3)! f4x 32.2 -d
(5)
where an(X ) are local scalar invariants constructed from the curvature tensor and its covariant derivatives up to order 2(n - 1) [4]. These quantities are uni-
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
Volume 115B, number 5
formly bounded laE(x)l < CnM-2n on the euclidean section of the black hole and eq. (5) can be considered as the expansion of
(6)
Here (and later on) the dots denote the omitted terms which do not contribute to (TUV)ren in a Ricci-flat space and
W1 _
1 96 × 7!rr2m 2
W/6g.v
then the non-zero components of 6 for the background metric (9) can be expressed in terms of the functional derivatives o f W[~,, ~2] b y means of the following relations
W/6gAB = n - 2 ~W[')', e]/~')'AB' guv6W[Sgt~v
C at~'r~ }.
(7)
C ~ C ~ r Ce.~~~ = ½Ca~, ~ C7~ e~C e ~ '
X (3C~IS7~C 7~ erCer a - 2R~ Ca.~a C3"r~ + ...).
6~2-3~2:A:A,
(11)
(12)
where d6o 2 is the metric on a unit sphere and ~2 is a scalar function on a two-dimensional space P with a metric dP 2 = 7A~ dx A dx ~, (A, B = 0,1). Any local scalar invariant w [g] of the metric and its derivatives being restricted on the spacetime (9) can be considered as a local invariant ~ [7, ~ ] of TAB and ~2. If we denote
W1-
(10) ~ = ~4w[g] [g=(7,s2),
(13)
1
/'d2 ¢r~
96 X 7!zr2m 2a ×
{1_!~[1 + 3(1 - 6~)1K3~2 -2
- [36 + 112(1 - 6~)]K2~2-3~2:A:A}.
(14)
After performing the necessary variations, using eq. (11) and substituting ~2 = r, K = 6M[r we arrive at the following expression for the non-zero components of r uv in the Schwarzschild background (2): r~ = A ( r ) [ - 2 8 5 + 626M/r + 168(1 - 6 ~ ) ( - 5 + 11M/r)], rrr =A(r)[105 - 154M/r + 168(1 - 6~)(2 - 3M/r)], 0 7"0
=f d4x x/-S'gw[gl ,
= I 1.20--493 3..... ~
Here K - 1 is the gaussian curvature of the two-dimensional metric dF 2 and ( ):A denotes the covariant derivative with respect to dF 2. Eqs. (12), (13) allow us to write the expression for the restriction W1 of the action (7) on the metric (9) in the form
(8)
(9)
C~78e
Ca#~/a C'r8 et Ce~'a# = ~ K3 ~2-6.
Now we note that the general spherically symmetric line element can be presented in the form
~[7,n] =Jd2xv"c'4~,[7,nl,
n~[~,nll~n.
The calculation of the first term r ~ = 2 ( - g ) - l / 2 f i W1/ fig~v of the (TUV)ren expansion [(TUV)ten = r ~v + O ( m - 4 ) ] in the Schwarzschild spacetime is greatly simplified if we use these relations and note that for the metric (9) the scalar curvature is
076e
To arrive at this expression the following relations containing the Weyl tensor C ~ 7 ~ were used:
W[gl
'
and the Weyl tensor satisfies the following relations
X ( C ~ r~ C ~:8e~.C~" ~ + 50R~ C 8 e C ts'r~e
ds 2 = ~22(dr 2 + d6o2),
=
R = 2~2-2K -
f d4x
+ [ - 8 + 14(1 - 6~)]RC ~
16 September 1982
PHYSICS LETTERS
=,/Ao
= A ( r ) [ - 3 1 5 + 734M/r + 168(1 - 6~) ( - 6 + 14M/r)], (15)
.2 We use the MTW sign conventions of ref. [10]. It should be noted that the definitions of the curvature tensor in Gilkey's paper [9] differs by the sign from the MTW definition.
where A(r) = (10080rr2) -1 ~3G2c-SM2[(m2rS). The components r pu in the Schwarzschild coordinates remain finite at the event horizon. When ~ = 1/6 the 373
Volume 115B, number 5
PHYSICS LETTERS
energy density - 4 is positive at large distances. It vanishes at the surface r = 1 ~ss rg lying outside the horizon and remains negative on and under the horizon. The obtained vacuum e n e r g y - m o m e n t u m tensor does not indicate any peculiarity near r = rg which might be connected with the conjectural high concentration of real massive particles near the event horizon by Zel'dovich [11 ]. The higher order corrections contain the additional factor ~e(rg/r) 3 . It means that the vacuum e n e r g y - m o m e n t u m tensor (15) is uniformly valid for r >>el/3rg. The comparison of the obtained expression (15) with Candelas results [ 1 ] for the vacuum polarization by a massless scalar field near the bifurcation point of the horizons shows that the vacuum e n e r g y - m o m e n t u m tensor in the massless case is larger by the factor e -1 . Up to now we consider only the average value o f the e n e r g y - m o m e n t u m tensor in the Hartle-Hawking vacuum
> rg) the differences between it and the average values in the Boulware (BITy~ IB)ren and Uuruh (UI TU IU)ren rac1) uum states are of the order o f exp (-m/TBri) exp (-81re-I/2) and they have to be neglected in our approximation. (BI TvUIB)ren ~ (1 - 2M/r) -2 as r --> 2M in the same manner as in the massless case [1]. For the t-r-components o f ~ in the Unruh vacuum one has (A, B = t, r) (UI TA IU)re n (~t +/3r-2(1 -
2M/r) -1
k/~r-2(1 - 2M/r) - 2
-~r-2 -~r-2(1 _
2M/r)-l]' (16)
374
16 September 1982
where fl is the quantity proportional to the Hawking flux of the massive particles into the black hole:/3 exp (--m[TBH). The additional terms proportional to/~ become essential only at the exponentially small (~/3) distances from the horizon. They make (UIT~IU)ren singular at the past horizon. In the coordinates covering the future horizon (UI T~I U)re n remains finite and the additional terms are uniformly small. It is a pleasure to acknowledge fruitful discussions with G.A. Vilkovisky.
References [1] P. Candelas, Phys. D21 (1980) 2185. [2] V.P. Frolov, Vacuum polarization near the event horizon of a charged rotating black hole, prepdnt N. 172, P.N. Lebedev Physical Institute (Moscow, 1981); V.P. Frolov, Phys. Rev. D (1982), to be published. [3] J.B. Hartle and S.W. Hawking, Phys. Rev. D13 (1976) 2188. [4] B.S. DeWitt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965). [5] B.S. DeWitt, Phys. Rep. 19C (1975) 295. [6] J. Schwinger, Phys. Rev. 82 (1951) 664. [7] S.M. Chdstensen, Phys. Rev. D14 (1976) 2490. [8] S.M. Christensen, Phys. Rev. D17 (1978) 946. [9] P.B. Gilkey, J. Diff. Geom. 10 (1975) 601. [10] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973). [ 11 ] Ya.B. Zel'dovich, Thermodynamical equilibrium and heavy particles near a black hole, preprint N. 121, Institute of Applied Mathematics (Moscow, 1975).
2M/r) -1
k/~r-2(1 - 2M/r) - 2
-~r-2 -~r-2(1 _
2M/r)-l]' (16)
374
16 September 1982
where fl is the quantity proportional to the Hawking flux of the massive particles into the black hole:/3 exp (--m[TBH). The additional terms proportional to/~ become essential only at the exponentially small (~/3) distances from the horizon. They make (UIT~IU)ren singular at the past horizon. In the coordinates covering the future horizon (UI T~I U)re n remains finite and the additional terms are uniformly small. It is a pleasure to acknowledge fruitful discussions with G.A. Vilkovisky.
References [1] P. Candelas, Phys. D21 (1980) 2185. [2] V.P. Frolov, Vacuum polarization near the event horizon of a charged rotating black hole, prepdnt N. 172, P.N. Lebedev Physical Institute (Moscow, 1981); V.P. Frolov, Phys. Rev. D (1982), to be published. [3] J.B. Hartle and S.W. Hawking, Phys. Rev. D13 (1976) 2188. [4] B.S. DeWitt, Dynamical theory of groups and fields (Gordon and Breach, New York, 1965). [5] B.S. DeWitt, Phys. Rep. 19C (1975) 295. [6] J. Schwinger, Phys. Rev. 82 (1951) 664. [7] S.M. Chdstensen, Phys. Rev. D14 (1976) 2490. [8] S.M. Christensen, Phys. Rev. D17 (1978) 946. [9] P.B. Gilkey, J. Diff. Geom. 10 (1975) 601. [10] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation (Freeman, San Francisco, 1973). [ 11 ] Ya.B. Zel'dovich, Thermodynamical equilibrium and heavy particles near a black hole, preprint N. 121, Institute of Applied Mathematics (Moscow, 1975).