The fields of a charged particle in hyperbolic motion

The entire xy-plane first “sees” the charge at time t = 0. If this seems surprising, refer to Appendix  A.

The fields of a point charge in hyperbolic motion were first considered by

M.

 

Born

, “,”  , – ().

For the early history of the problem see

W.

 

Pauli

, (reprint by , , ), Section 32(γ).

For a comprehensive history see

E.

 

Eriksen

and

Ø.

 

Grøn

, “,”  , – ().

See also

S.

 

Lyle

, (, , ).

See, for example,

D. J.

 

Griffiths

, , 4th ed. (, , ), Eq. (10.72.)

This field was first obtained by

G. A.

 

Schott

, (, , ), pp. –.

E. M.

 

Purcell

and

D. J.

 

Morin

, , 3rd ed. (, , ), Section 5.6.

J. M.

 

Aguirregabiria

,

A.

 

Hernández

, and

M.

 

Rivas

, “,”  , – (), Eq. (50).

The fields of a massless point charge are considered also in

J. D.

 

Jackson

, , 3rd ed. (, , ),

Prob. 11.18,

W.

 

Thirring

  (, , ), pp. – and

M. V.

 

Kozyulin

and

Z. K.

 

Silagadze

, “,”  , – (), Eq. (20).

H.

 

Bondi

and

T.

 

Gold

, “,”  , – ().

D. G.

 

Boulware

, “,”  , – () obtained Eq. (21) using the truncated hyperbolic model.

The latter was also explored by

W.

 

Thirring

 , 2nd ed. (, , ), p. . See also Lyle, ref. 2, Section 15.9.

Reference 3, Eqs. (10.46) and (10.47).

See Appendix  B for details of these calculations. Equation (29) reduces to Eq. (9), of course, when t = 0 and s = x.

R.

 

Jackiw

,

D.

 

Kabat

, and

M.

 

Ortiz

, “,”  , – ().

Potentials (30) and (31) satisfy the Lorenz gauge condition, as does Eq. (37), but Eq. (36) does not.

The delta-function terms in the potentials were first obtained by

T.

 

Fulton

and

F.

 

Rohrlich

, “,”  , – ().

Reference 3, Eq. (10.19). Since all we're doing is searching for a missing term, we may as well concentrate on V, and set t = 0.

Reference 7, quoted in Eriksen and Grøn, Ref. 2.

Reference 12, quoted in Eriksen and Grøn, Ref. 2.

Lyle (Ref. 2, page 216) thinks it “likely” that the extra terms could in fact be obtained at the level of the Liénard-Wiechert potentials “if we were more careful about the step function,” but he offers no justification for this conjecture.

A.

 

Zangwill

, (, , ), Sec. 20.3.

C.

 

LaMont

, “,” Reed College senior thesis, .

The sign of the radical is enforced by the condition $T>Tr$⁠.

Any reader with lingering doubts is invited to check that these fields satisfy all of Maxwell's equations. Note the critical role of the delta functions in Gauss's law and the Ampère-Maxwell law.

The factor $(1−r·v/rc)$ accounts for the fact that the rate at which energy leaves a (moving) charge is not the same as the rate at which it (later) crosses a patch of area on the sphere. See Ref. 3, page 485.

The fact that a charged particle in hyperbolic motion radiates has interesting implications for the equivalence principle—in fact, it is this aspect of the problem that has attracted the attention of most of the authors cited here. Incidentally, the particle experiences no radiation reaction force—see

R.

 

Peierls

, (, , , Chapter 8.)

Reference 3, Eq. (11.73).