CC BY
22
UDC 530.1; 539.1
Renormalization of static self-potential A. A. Popov
institute of Mathematics and Mechanics, Kazan Federal University, 18 Kremlyovskaya St., Kazan, 420008, Russia
E-mail: apopov@ksu.ru
A method is presented which allows for the renormalization of the self-potential of a scalar point charge at rest in static curved spacetime. The method is based on the local expansion of the self-potential for a scalar point charge at rest in general static spacetimes.
Keywords: self-force, renormalization.
1 Introduction
Renormalization
It is known that a charged particle interacts with the field, the source of which is this particle [1-6]. A discussion of the self-force in detail may be found in reviews [7-9].
Calculating the self-force one must evaluate the field that the point charge induces at the position of the charge. This field diverges and must be renormalized. There are different methods of such type of renormalization. Some of them are reviewed in recent papers [10,11].
Note also the zeta function method [12] and ’’the massive field approach” for the calculation of the selfforce [13,14]. In the ultrastatic spacetimes the renormalization of the field of static charge can be realize by the subtraction of the first terms from DeWitt-Schwinger asymptotic expansion of a three dimensional Euclidean Green’s function [15-18].
In this paper similar approach expands to the case of static spacetimes. In framework of suggested procedure one subtracts some terms of expansion of the corresponding Green function of a massive scalar field with arbitrary coupling to the scalar curvature from the divergent expression obtained. The quantities of the terms to be subtracted are defined by simple rule
- they no longer vanish as the field’s mass goes to the infinity.
Such approach is similar to renormalization introduced in the context of the quantum field theory in curved spacetime [19, 20]. The Bunch and Parker method [21] is used for expansion of the corresponding Green’s function of a scalar field.
Our conventions are those of Misner, Thorne, and Wheeler [22]. Throughout this paper, we use units c = G = 1.
Let us consider equation for scalar massive field with source
^m^-(m2 + £R) = —4nqj ¿(4)(x—xo(r))—j
dr
—g(4)' (1)
where £ is a coupling of the scalar field with mass m to the scalar curvature R, g(4) is the determinant of the metric g^v, q is the scalar charge and r is its proper time. The world line of the charge is given by Xq(t). The metric of static spacetime can be presented as follows
ds2 = —gtt(xl>dt2 + gjk (x®>dxj dxk,
(2)
where i,j, k = 1, 2, 3. This means that one can write the field equation in the following way
1
d
d j i vWgvfc x0>
fgtty g(3) dxj V dxk
— (m2 + £R(x> ) ^"(x1; xQ) = —
4nq5(3) (x1, xQ>
v/g(3) ,
(3)
where m is the mass of scalar field, g(3) = det gj and we take into account that dr/dt = for the particle at rest. In the case
m » 1/L,
(4)
where L is the characteristic curvature scale of the background geometry, it is possible to construct the iterative procedure of the solution of Eq. (3) with small parameter 1/(mL) [19-21]. This expansion can be used in the regularization procedure of Rosenthal [13]
f’“<(xo) = q lim (lim d ( x0>-*»(*xo))
^ | d^0 3'X^
| qm2nM(xo> | qmaM(xo> ) ^
+ 2 + 2 p {5)
because this procedure demands the calculation of the expansion of ^m(x;x0) in terms of xg — xg mid 1/m accurate to order O ( (x — x0)2) + O (1/m) only. In the expression (5) ^(x; x0) is the massless field induced byscalar charge q and x is a point near the charge’s word line x0(r), defined as follow s. At x0 we construct a unit spatial vector ng, which is perpendicular to the object’s
x0
ngng = 1, nMwM = 0). In the direction of this vector we construct a geodesic, which extends out an invariant length S to the point x(x0,ng, S); throughout this manuscript u^d ag denote the object’s four-velocity
x0
To construct the expansion of ^m(x; x0), let us consider the equation for the three-dimensional Green’s function GE(x®,x0)
that G(y:) satisfies the equation
d 2G dy:dyj
g«,i g«,k 2gtt2
2^ , ailgii,* dG - mG+^ 2g: dj
+
g«,ik
2g«
k dG + R j yV d2G
y n j + R k i 0 r-, : ^ j
dyj 3 ay:ayj
+ (f - = -¿(3)(y).
Let us present G(yi) = G0(yi) + G1(yi) + G2 (yi) + . . . :
(11)
(12)
where Ga(y®) has a geometrical coefficient involving a derivatives of the metric at point y® = 0. Then these functions satisfy the equations
+
gjk dgtt dGE(x:,x0) dxj dxk
¿(3) (x:, x0)
d2Go
dy:dyj
d2Gi
dy:dyj
- m2Go = -¿(3)(y),
- m2G1 +
gti,j dGo 2gtt dyj
(6)
and introduce the Riemann normal coordinates y® in 3D space with origin at the point x0 [23]. In these coordinates one has
(yi) = % - 3-Rikji|y=o ykyi + O fLy-3
d2.G2 . - m2 G2 + ^
dy:dyj 2gu dyj
dGo dyj
|R : j yky1 d2Go , f R \ G
(7) +jR ki~ M3 - eR)Go = °.
I rjj A g«,ik g«i g«,kA k
- 2gtt2 j y
(13)
(14)
(15)
g(3)(y:)
1 3|y=oy y
(8)
where the coefficients here and below are evaluated at
y® = 0 (i.e. at the point x0), Sj denotes the metric of a flat three-dimensional Euclidean spacetime, and denote the components of Riemann and Ricci tensors of the three-dimensional spacetime with metric
R = R-
2g«
gti,i’
g«
|
4gtt2 g«,i 5^ 2gtt2
G(y:) = G (y:)
The function Go(y:) satisfies the condition R< j k i d2Go r: . dGo n
R k i y y - - jy =°,
dy:dyj
dy:
|
(9)
where git,® denotes the covariant derivative of a scalar function gti(yj) with respect to y® in 3D space with metric gj (yk) (gtt^j is the covariant derivative of a vector git i at point yk = 0 in 3D space, which coincides with partial derivative as rj = 0 at yk =0 in the Riemann normal coordinates). All indices are raised and lowered with Sj. Defining G(y®) by
- ^R Go = °.
(16)
since Go(y:) can ^e ^^e function only of y:yj. Therefore Eq. (15) may be rewritten
d2G!2(y:) - m2G2(y:) + ^^j1
dy:dyj 2gu dyj
(17)
Let us introduce the local momentum space associated with the point y® =0 by making the 3-dimensional Fourier transformation
G„(y:)
dfcid&2 d&3
(2n)3
exp(*kjy:)Ga(k:). (18)
(10)
and retaining in (6) only the terms with coefficients involving two derivatives of the metric or fewer one finds
It is not difficult to see that
Go(k:) = 1
k2 + m2 '
°
G1(ki) = i
(20)
G2(k:) =
(k2 + m2)2
k:kj¿:kW 2Rji + gttji 5gtt,j gtt,i
|
gtt
4gtt2
(k2 | m2)3
where k2 = k.kj. Substituting (18), (19), (20), (21)
in (12) and integrating leads to
Go(y:) + G1(y:) + G2(y:) =
exp(-my) 8n
gtt,i y. + ^gtt,
2gtt y m 4gtt
¿3 | 3^:jgii,j gii,j - cr
|
16gtt2
I — | I I gii,*3 I 5gtt,: g«,j
~6j \ 4g7 16g«2 6
Ri.
yV
where
y = y ¿¿j yV.
1
H— m
gtt,
12gtt
+
8n [ V2a 2gi^v/2a
5gtt,igtt,: /_ 1
48gtt
2gtt
+ m2 V2a +
gtt,
12gtt V 2a + ( +
5gtt,i gtt^ 48gtt2
13gtt : gtt -
6gtt
a/2^
48gtt2 1
+O
(mS)+o (
3/2
(24)
where we take into account that in the arbitrary coordinates of 3D space
y: ^ u:(xo)As = -a:,
(25)
with metric gj (xo) (gtt,i;з• is the covariant derivative of a vector gtt : at point xo in 3D space),
g:j (xo)
2
(26)
is one half the square of the distance between the points x0 and x® along the shortest geodesic connecting them and (see, e.g., [24,25])
(21)
= - (x: - ;
Ip - 2ij'k
d r:
(xk -; (xk -;
(x; — x0) + O ^(x — x0)^ , (27)
where the Christoflfel symbols rjk are calculated at the
x0
Now we can use the expansion of
, (22)
(23)
^m(x:; x*o) = 4nqGE(x:; x*o)
(28)
in the regularization procedure (5). But if we take the limits before the partial differentiation in (5), then the last two terms do not to appear in the expression for /gelf (x0). And in the considered case of a charge at rest in a static spacetime we can renormalize self-potential
Using the definition of G(y:) (10), expansion (8), and expressions (9) one finds
n t i ^ _ 1 J 2 , gtt,:a
Ge (x ; xo) = 7T~ { +------------------- 2m
as
^ren(x) = lim (^(x; xo) - ^DS(x; xo)),
Xo -^-x
where
V v^a dxo 4gtt(xo)v/20:
(29)
(30)
and ^(x; x0) is the solution of (1) in the case of arbitrary mass m (even m = 0). Finally the self-force acting on a static scalar charge is
/uei/(x) = -I
qd^re„(x) 2"
(31)
3 Conclusion
The considered approach gives the possibility to renormalize (29) the self-potential of scalar point charge q at rest in static spacetime (2) and to calculate the self-force (31) acting on this charge. Note that in
1/m
scalar field is much smaller than the characteristic scale L of curvature of the background gravitational field at
x
imated expression for the renormalized self-potential
u®(x0) is the unit tangent vector to the shortest
x0 x
lated at points x0 and directed from x0 to x, As is the distance between these points along the considered geodesic, gtt,® denotes the covariant derivative of a scalar function gtt(x0) with respect to x0 in 3D space
^re:
q
2m
,(x) = lim (^m(x; xo) - ^DS(x; xo))
Xo^X
gtt,
12gtt
I
5gtt,:gtt/
48gtt2
- lC- 6 1R
+O
i2L3
(32)
:j
a
k
j
j
a
o
o
o
k
j
j
o
o
y
m
m
Of course the order of this expression in l/(mL) is less Acknowledgments than the correspondent order of фгеп for the massless field (or field with mass m < l/L). However the expression (32) can be used for the verification of asymptotic This work was supported in part by grant 11-02-
behavior of фгеп in the limit m ^ to. 01162 from the Russian Foundation for Basic Research.
References
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Received 01.10.2012
А. А. Попов
ПЕРЕНОРМИРОВКА СТАТИЧЕСКОГО СОБСТВЕННОГО ПОТЕНЦИАЛА
В работе представлен метод перенормировки собственного потенциала скалярного поля, создаваемого покоящимся точечным зарядом, во внешнем статическом гравитационном поле. Метод основан на локальном разложении поля такого заряда в его окрестности.
Ключевые слова: собственный потенциал, перенормировка.
Попов A.A., кандидат физико-математических наук, доцент.
Казанский (Приволжский) федеральный университет.
Ул. Кремлевская, 18 , Казань, Россия, 420008.
E-mail: apopov@ksu.ru
