Liénard–Wiechert potentials in even dimensions
Metin Gürses, and Özgür SarıoğluCitation: Journal of Mathematical Physics 44, 4672 (2003); doi: 10.1063/1.1613040 View online: http://dx.doi.org/10.1063/1.1613040
View Table of Contents: http://aip.scitation.org/toc/jmp/44/10
Lie´nard–Wiechert potentials in even dimensions
Metin Gu¨rsesa)
Department of Mathematics, Faculty of Sciences, Bilkent University, 06800, Ankara, Turkey
O¨ zgu¨r Sarıog˘lub)
Department of Physics, Faculty of Arts and Sciences, Middle East Technical University, 06531, Ankara, Turkey
共Received 29 March 2003; accepted 21 July 2003兲
The motion of point charged particles is considered in an even dimensional Minkowski space–time. The potential functions corresponding to the massless sca-lar and the Maxwell fields are derived algorithmically. It is shown that in all even dimensions particles lose energy due to acceleration. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1613040兴
I. INTRODUCTION
Recently Gal’tsov1 and Kazinski et al.2 have considered the Lorentz–Dirac equation for a radiating point charge in a Minkowski space–time of arbitrary dimension. They showed that the mass renormalization is possible only in three and four dimensions. In their discussion, they have also given the retarded Green’s functions of the D’Alembert equation in any dimensions which was in fact constructed rigorously a long time ago.3Motivated by these works, we are interested in the radiation problem of accelerated point charges in all even dimensions共for the reason why we did not consider odd dimensions, please see Appendix B兲. Here we find the Lie´nard–Wiechert potentials corresponding to the massless scalar and the Maxwell fields in all even dimensions. We then use these potentials to relate the radiation from an accelerated point particle to its motion and the geometry of its trajectory. We derive the energy flux for this radiation and show that acceler-ating point charged particles lose energy in all even dimensions.
In Sec. II, we develop the kinematics of a curve C in a D-dimensional Minkowski manifold
MD. In Sec. III we find the Lie´nard–Wiechert potentials of massless free scalar fields in an even dimensional Minkowski space. We calculate the energy radiated due to the acceleration. We show that in all even dimensions such particles lose energy, as can be expected. In Sec. IV, we determine the Lie´nard–Wiechert potentials for the Maxwell theory. We give a recursion relation between the vector potentials of the theory in two consecutive even dimensions. In Sec. IV, we also show that particles carrying electric charges lose energy in all even dimensions. We construct explicit solu-tions of the electromagnetic vector field due to the acceleration of charged particles in 4,6,8,10 dimensions. We then find the energy fluxes in 4,6,8 dimensions due to acceleration. In Appendix A, we give the Serret–Frenet equations in an arbitrary Minkowski space–time and also some auxiliary tools used in the calculation of the energy flux integrals. In Appendix B, we give a proof of the recursion relation introduced in Sec. IV.
II. CURVES IND-DIMENSIONAL MINKOWSKI SPACE
In our previous works,4 – 6 we developed a curve kinematics to be utilized in finding new solutions and in calculating energy fluxes due to the acceleration in the framework of Einstein’s general theory of relativity. Here we use the same approach to solve the scalar and Maxwell field
a兲Electronic mail: gurses@fen.bilkent.edu.tr b兲Electronic mail: sarioglu@metu.edu.tr
4672
equations in all even dimensions. For this purpose, we shall now give a summary of the geometry of a regular curve in MD, Minkowski space–time manifold of dimension D.
Let z() describe a smooth curve C in MD, whereis the arclength parameter of the curve. From an arbitrary point x outside the curve, there are two null lines intersecting the curve C. These points are called the retarded and the advanced times. Let ⌽ be the distance 共world func-tion兲 between the points x and z(), then by definition it is given by
⌽⫽1
2共x⫺z共兲兲 共x⫺z共兲兲, 共1兲
where⫽diag(⫺1,1,...,1). Hence ⌽ vanishes at the retarded,0, and advanced,1, times. In this work we shall focus on the retarded case only. The Green’s function for the vector potential chooses this point on the curve C.7,8By differentiating⌽ with respect to xand letting⫽0, we get ⬅,⫽ x⫺z共0兲 R , R⬅z˙ 共 0兲 共x⫺z共0兲兲, 共2兲 where R is the retarded distance,is a null vector, and a dot over a letter denotes differentiation with respect to0. The derivatives of R and, using共2兲, are given by
,⫽ 1 R关⫺z˙⫺z˙⫺共A⫺⑀兲 兴, 共3兲 R,⫽共A⫺⑀兲 ⫹z˙, 共4兲 where A⫽z¨共x⫺z兲, z˙˙z⫽⑀⫽0,⫾1. 共5兲
Here⑀⫽0,⫺1 for null and time-like curves, respectively. Furthermore, we have
˙z⫽1, R,⫽1. 共6兲
Letting a⫽ A/R, it is easy to prove that
a,⫽0. 共7兲
Similarly, other scalars (a1,a2,...), satisfying the same property共7兲 obeyed by a can be defined
ak⬅
dk¨z
d0k , k⫽1,2,...,n. 共8兲
Moreover one has
ak,␣␣⫽0, 共9兲
for all k (k⫽0 is also included if we let a0⫽a). For a more detailed discussion, please refer to Ref. 4. Here n is a positive integer which depends on the dimension D of the manifold MD. An analysis using Serret–Frenet frames shows that the scalars (a, ak) are related to the curvature
scalars of the curve C in MD. The number of such scalars is D⫺1. 9
Hence we let n⫽D⫺1. 4673 J. Math. Phys., Vol. 44, No. 10, October 2003 Lie´nard–Wiechert potentials in even dimensions
III. MASSLESS SCALAR FIELD
Letdescribe a massless scalar field satisfying the free field equation
2
xx⫽0. 共10兲
Let D be a positive even integer, and (D) and(D⫹2) denote the retarded solutions共Lie´nard– Wiechert potentials兲 of the massless scalar field in D and D⫹2 dimensions, respectively. Then
(D⫹2)⫽1 R
d d
(D). 共11兲
In this recursion relation we emphasize that the expressions on the right-hand side are those of D-dimensions. Take the solution(D) in D-dimensions, take itsderivative and divide this by the R of D-dimensions. The result is the solution (D⫹2) of D⫹2-dimensions. For the proof of relation 共11兲 see Appendix B. In the following we explicitly give these solutions for D
⫽4,6,8,10: (4)⫽c R, 共12兲 (6)⫽ 1 R2关c˙⫺pc兴, 共13兲 (8)⫽ 1 R3关c¨⫺3pc˙⫹共⫺a1⫹3p 2兲c兴, 共14兲 (10)⫽1 R4
冋
d3c d3⫺6pc¨⫹共15p 2⫺4a 1兲c˙⫹冉
⫺a2⫹10pa1⫺15p3⫹ 1 R˙z␣ d3z␣ d3冊
c册
, 共15兲 where c⫽c() is the共time dependent兲 scalar charge and p⬅a⫺⑀/R.The flux of massless scalar field energy is then given by共see Refs. 7 and 11 for this definition and also for the integration surface S)
dE⫽⫺
冕
S
z
˙TdS, 共16兲
where T ⫽ ⫺14(␣␣ ) is the energy momentum tensor of the massless
scalar field . The surface element dS on S is given by
dS⫽nRD⫺3dd⍀, 共17兲 where n is orthogonal to the velocity vector field z˙ which is defined through
⫽⑀z˙⫹⑀1 n
R , n
n
⫽⫺⑀R2. 共18兲
Here ⑀1⫽⫾1. For the remaining part of this work we shall assume ⑀⫽⫺1 (C is a time-like curve兲. One can consider S in the rest frame as a sphere of radius R. Here d⍀ is the solid angle. Letting dE/d⫽N, we have
N(D)⫽⫺
冕
SD⫺2
z
where SD⫺2 is the (D⫺2)-dimensional sphere centered at⫽0 on the curve C. At very large values of R the energy flux is given by
N(D)⫽⫺
冕
SD⫺2
d⍀ RD⫺3共z˙␣␣兲共n兲.
It turns out that the energy flux expression has a fixed sign for all D. The energy flux of the massless scalar fieldas R→⬁ is given by
N(D)⫽⫺⑀1
冕
SD⫺2关
(D)兴2d⍀, where we obtain R independent functions共for each D) (D) from
(D)⫽ lim
R→⬁
关RD/2(D⫹2)兴.
As an example let D⫽4. We take (6) from共13兲, multiply it by R2 and let R→⬁ 共then p→a), and finally we obtain(4). The explicit expressions of(D) are as follows:
(4)⫽c˙⫺ac, 共20兲 (6)⫽c¨⫺3ac˙⫹共⫺a 1⫹3a2兲c, 共21兲 (8)⫽d 3c d3⫺6ac¨⫹共15a 2⫺4a
1兲c˙⫹共⫺a2⫹10aa1⫺15a3兲c, 共22兲
(10)⫽d 4c d4⫺10a d3c d3⫹共45a 2⫺10a
1兲c¨⫺共5a2⫺60aa1⫹105a3兲c˙
⫺共a3⫺15aa2⫺10a1 2⫹105a
1a
2⫺105a4兲 c. 共23兲
Hence we have共assuming c⫽constant)
N(4)⫽⫺⑀1
冉
4 3冊
c 2 1 2 , 共24兲 N(6)⫽⫺⑀1冉
82 105冊
c 2关20 1 4⫹7 ˙1 2⫹7 1 2 2 2兴, 共25兲 N(8)⫽⫺⑀1冉
163 10 395冊
c 2兵99关共¨ 1⫺41 3⫺ 12 2兲2⫹共2˙ 12⫹1˙2兲2⫹1 2 2 2 3 2兴 ⫹1 2 关9001 4⫹1100 1 2 2 2⫹3597 ˙12兴其. 共26兲IV. ELECTROMAGNETIC FIELD
In the Lorentz gauge (A⫽0), the Maxwell equations reduce to the wave equation for the vector potential A, A␣⫽0. By using the curve C, we can construct divergence free 共Lorentz gauge兲 vector fields A␣ satisfying the wave equation outside the curve C in any even
dimension D. Similar to the case of the massless scalar field, such vectors obey the following recursion relation A(D⫹2)⫽1 R d dA (D). 共27兲 4675 J. Math. Phys., Vol. 44, No. 10, October 2003 Lie´nard–Wiechert potentials in even dimensions
In the recursion relation above A(D)is the electromagnetic vector potential in even D-dimensions, with⫽0,1,...,D⫺1. On the right-hand side of the recursion relation all operations are done in D-dimensions, just like the scalar case. However the result is to be considered as the electromag-netic vector potential of D⫹2-dimensions, with ⫽0,1,...,D⫹1 on the left-hand side. As an example we have A(4)⫽z˙/R as the electromagnetic vector potential of four dimensions.8Here z˙ is the four velocity, R andare, respectively, the retarded distance and time in four dimensions. Using the recursion relation 共27兲 the right-hand side becomes
z¨⫺az˙ R2 ⫹⑀
z˙ R3.
We then regard this expression as the solution A(6) of the Maxwell field equations in dimensions. Indeed it satisfies both the Lorentz condition and the field equations of six-dimensions, as can be verified separately. Starting from D⫽4, we can generate all even dimen-sional vector potentials satisfying the Maxwell equations. For instance, the vector potentials for D⫽4,6,8,10 are explicitly given by
A(4)⫽z˙ R , 共28兲 A(6)⫽ 1 R2关z¨⫺pz˙兴, 共29兲 A(8)⫽ 1 R3
冋
d3z d3 ⫺3pz¨⫹共⫺a1⫹3p 2兲z˙ 册
, 共30兲 A(10)⫽1 R4冋
d4z d4 ⫺6p d3z d3 ⫹共15p 2⫺4a 1兲z¨⫹冉
⫺a2⫹10pa1⫺15p3⫹ 1 R˙z␣ d3z␣ d3冊
˙z册
. 共31兲The flux of electromagnetic energy is then given by7共the integration surface S is also given in this reference兲 dE⫽⫺
冕
S z ˙TedS, 共32兲 where Te ⫽F␣F␣⫺14F 2 is the Maxwell energy momentum tensor, F⫽A,⫺A, is
the electromagnetic field tensor and F2⬅F␣F␣. Letting dE/d⫽Ne,10we have
Ne(D)⫽⫺
冕
SD⫺2
z
˙TenRD⫺3d⍀. 共33兲
At very large values of R, for all even D, we get
Ne(D)⫽⫺⑀1
冕
SD⫺2 (D) (D)d⍀, 共34兲 where (D)⫽ lim R→⬁ 关A(D⫹2)RD/2兴, 共35兲Here we have two remarks. The first one is on the gauge dependence of共35兲. The only gauge freedom left in our solutions is A→A⫹, wheresatisfies the scalar wave equation共10兲. However we have already found the solutions of the scalar wave equation for all even dimensions. It can be shown that the contribution of such scalar functions to the norm of(D) is zero in the limit R→⬁. Our second remark is on the sign of Ne(D) in 共34兲. The vectors (D) in all even dimensions are orthogonal to the null vector, hence they must be either共i兲 space-like vectors,
共ii兲 proportional to , or 共iii兲 zero vectors.11They are zero only when the curve C is a straight
line which leads to no radiation. They cannot be proportional to the null vectoreither, because this again leads to the trivial case of zero radiation. In the first three cases共4, 6, 8 dimensions兲 it can be easily observed that zero radiation implies that (D) is a zero vector. Hence (D) is a space-like vector in all even dimensions. Therefore the sign of the right-hand side of共34兲 is the same in all dimensions. These vectors are explicitly given as follows:
(4)⫽z¨⫺az˙, 共36兲
(6)⫽
d3z
d3 ⫺3az¨⫹共⫺a1⫹3a 2兲z˙ , 共37兲 (8)⫽ d4z d4 ⫺6a d3z d3 ⫹共15a 2⫺4a
1兲z¨⫹共⫺a2⫹10aa1⫺15a3兲 z˙, 共38兲
(10)⫽ d5z d5 ⫺10a d4z d4 ⫹共45a 2⫺10a 1兲 d3z
d3 ⫹共⫺5a2⫹60aa1⫺105a 3兲z¨
⫹共⫺a3⫹15aa2⫹10a1 2⫺105a
1a2⫹105a4兲z˙. 共39兲
These lead to the following energy flux expressions:
Ne(4)⫽⫺⑀1 8 3 1 2 , 共40兲 Ne(6)⫽⫺⑀1 322 15
冉
˙1 2⫹ 1 2 2 2⫹9 71 4冊
, 共41兲 Ne (8)⫽⫺⑀ 1 323 10395再
297冋冉
¨1⫺ 4 31 3⫺ 12 2冊
2 ⫹共2˙12⫹1˙2兲2⫹1 2 2 2 3 2册
⫹41 2关300 1 4⫹506 1 2 2 2⫹825 ˙12兴冎
. 共42兲To be compatible with the classical results,7,8one should take ⑀1⫽⫺1.
V. CONCLUSION
In this work we have considered radiation of scalar and vector fields due to acceleration of point charged particles. We first examined the geometric properties of their paths in an even dimensional Minkowski space MD. By using the curve kinematics we developed, we have first found the retarded solutions of the scalar field equations in MD. These solutions describe the potentials of the accelerated scalar charges and we have examined the energy loss due to such a radiation. We have shown that in all even dimensions such scalar point particles lose energy. We have given explicit examples for D⫽4,6,8,10. We then found the retarded solutions of the Max-well field equations that describe the point particles carrying electric charges. Again, using the curve kinematics we developed an algorithm to calculate the vector potential A in 4677 J. Math. Phys., Vol. 44, No. 10, October 2003 Lie´nard–Wiechert potentials in even dimensions
D⫹2-dimensions from the one in D-dimensions. We have given explicit examples for D ⫽4,6,8. We have calculated the energy flux in each case, and we have shown that particles lose
energy due to acceleration in all even dimensions.
ACKNOWLEDGMENTS
This work is partially supported by the Scientific and Technical Research Council of Turkey and by the Turkish Academy of Sciences.
APPENDIX A: SERRET–FRENET FRAMES
In this appendix, we first give the Serret–Frenet frame in D dimensions. Here we shall assume that the curve C described in Sec. II is time-like and has the tangent vector T⫽z˙. Starting from this unit tangent vector, by repeated differentiation with respect to the arclength parameter0, one can generate an orthonormal frame兵T,N1,N2,...,ND⫺1其, the Serret–Frenet frame:
T˙⫽1N1, 共A1兲 N˙1⫽1T⫺2N2, 共A2兲 N˙2⫽2N1⫺3N3, 共A3兲 ¯ N˙D⫺2⫽D⫺2ND⫺3⫺D⫺1ND⫺1, 共A4兲 N˙D⫺1⫽D⫺1ND⫺2. 共A5兲
Here i (i⫽1,2,...,D⫺1) are the curvatures of the curve C at the point z(0). The normal vectors Ni(i⫽1,2,...,D⫺1) are space-like unit vectors. Hence at the point z(0) on the curve we have an orthonormal frame which can be used as a basis of the tangent space共of MD) at this point. In Sec. II, we have defined some scalars
ak⫽ dkz¨ d0k , where ⫽⑀T⫹⑀ 1 n R .
Here n is a space-like vector orthogonal to T. It can be expressed as a linear combination of the unit vectors Ni’s as
n⫽␣1N1⫹␣2N2⫹¯⫹␣D⫺1ND⫺1,
where ␣12⫹␣22⫹¯⫹␣2D⫺1⫽R2. One can choose the spherical angles ,1,...,D⫺4
苸(0,), D⫺3苸(0,2) such that
␣1⫽R cos, ␣2⫽R sincos1, ␣3⫽R sinsin1cos2, . . . ,
␣D⫺2⫽R sinsin1¯ sinD⫺4cosD⫺3,
Hence we can calculate the scalars ak in terms of the curvatures of the curve C and the angles
(,1,...,D⫺3) at the point z(0). We need these expressions in the evaluation of energy flux formulas. As an example we give a and a1:
a⫽⫺⑀⑀11cos, a1⫽1 2⫺⑀⑀
1˙1cos⫹⑀⑀112sincos1. 共A6兲 The rest of the scalars can be determined similarly. It is clear that these scalars, ak, depend on the
curvatures and the spherical angles, for all k.
APPENDIX B: THE PROOF OF THE RECURSION RELATIONS„11…AND„27…
Here we give the proof for the vector potential case. The same type of proof applies also for the scalar case. Using the recursion relation共27兲 successively we get
A(D)⫽
冉
1 R d d冊
共D/2兲 ⫺2˙z R . 共B1兲On the other hand, from Refs. 1 and 2, we have
A(D)⫽
冕
G共x⫺z共兲兲 z˙d, 共B2兲where is the parameter of the curve C. The integral here is carried on the range of
苸(⫺⬁,⬁). Here G(x⫺z()) is the retarded Green function given by
G共x⫺z共兲兲⫽共x0⫺z0兲␦共D/2兲 ⫺2共⌽兲. 共B3兲
Here ⌽ is the world function given by 共1兲, (x) is the Heaviside step function and ␦k(x)
⬅ (dk/dxk)␦(x). Here we assume that D is an even integer. 关When D is an odd integer, the
expression for the Green function in 共B3兲 contains the step function instead of the ␦-function. Hence the potentials in all odd dimensions remain nonlocal共integral expressions兲. This makes our curve kinematics ineffective.兴 The zeros of ⌽ denote the advanced and retarded proper times on the curve C, but the step function(x0) chooses the retarded one. Since the integration is over the curve parameterin共B2兲, it is better to transform the derivative of the delta function with respect to⌽ to the derivative with respect to. As a simple example consider the D⫽6 case
d d⌽␦共⌽兲⫽
冋
1 d⌽/d d d ␦共⌽兲册
⌽⫽0 . 共B4兲It is easy to show that d⌽/d⫽⫺R. The delta function␦共⌽兲 can be expressed as follows:
␦共⌽兲⫽␦共⫺0兲
R ⫹
␦共⫺1兲
R .
The second term will vanish identically due to the step function in共B3兲. Hence
A(6)⫽1 R d d z˙ R ,
or simply A(6)⫽(1/R)(d/d) A(4). This verifies our relation共27兲. For the general case, we need higher order derivatives of␦共⌽兲 at ⌽⫽0. We find such terms by using 共B4兲 and taking successive derivatives. In the general case, for all k⫽0,1,2,... we obtain 共when ⌽⫽0)
dk d⌽k␦共⌽兲⫽
冋冉
⫺1 R d d冊
k ␦共⌽兲册
. 共B5兲 4679 J. Math. Phys., Vol. 44, No. 10, October 2003 Lie´nard–Wiechert potentials in even dimensionsUsing this expression in the Green’s function 共B3兲 for k⫽ D/2 ⫺2, inserting it in the integral equation共B2兲, A(D)⫽
冕
共x0⫺z0兲␦D/2⫺2共⌽兲 z˙d 共B6兲 ⫽冕
共x0⫺z0兲冉
⫺1 R d d冊
D/2⫺2 ␦共⌽兲 z˙d, 共B7兲and integrating by parts, we obtain共B1兲.
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