Spohn H. Dynamics of Charged Particles and their Radiation Field
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7.2 Taylor expansion 83
equation
m
b
˙v(t) = e
E
ex
(q(t)) + v(t) × B
ex
(q(t))
+
e
2
12π R
2
ϕ
v(t − 2R
ϕ
) − v(t)
,
(7.14)
where we have reintroduced the external fields.
The memory equation (7.14) is of suggestive simplicity. To have a well-defined
dynamics one has to prescribe q(0) and v(t) for −2R
ϕ
≤ t ≤ 0asinitial data.
Of course, the coupled system determines these data completely. However, the
supporters of differential–difference equations regard (7.14) as the starting point
with no instruction for the choice of initial data. Their claim is that solutions to
(7.14) are not very sensitive to this choice. While there is some evidence on the
linearized level, the dependence on the initial data for the full nonlinear problem
remains to be studied.
7.2 Taylor expansion
We return to Eq. (7.5). As will be explained in section 7.3, one knows that there
exists a constant C, independent of ε for ε<ε
0
, such that
|¨q
ε
(t)|≤C , |
...
q
ε
(t)|≤C
1 + ε(ε +|t|)
−2
,
|
....
q
ε
(t)|≤C
1 + ε(ε +|t|)
−2
+ ε(ε +|t|)
−3
(7.15)
for all t, provided the total charge e is sufficiently small. This smallness condition
merely reflects the fact that at present we do not know how to do better mathemat-
ically. Physically we expect (7.15) to hold no matter how large e.
Note that in higher time derivatives the mismatch of the initial conditions be-
comes visible. Only if the charge is allowed to move for a time span of order ε
1/3
,
which is short on the macroscopic scale but long as
O(ε
−2/3
) on the microscopic
scale, do the derivatives become uniformly bounded.
Because of (7.15) we are allowed to Taylor-expand in (7.6). To simplify notation
we set v
ε
(t) = v and t − s = τ . Then
v
ε
(s) = v
ε
(t − τ) = v −˙vτ +
1
2
¨vτ
2
+ O(τ
3
), (7.16)
e
−ik·(q
ε
(t)−q
ε
(s))
= e
−ik·(q
ε
(t)−q
ε
(t−τ))
= e
−i(k·v)τ
1 +
1
2
τ
2
i(k ·˙v) −
1
6
τ
3
i(k ·¨v)
−
1
2
1
2
τ
2
(k ·˙v) −
1
6
τ
3
(k ·¨v)
2
+ O((|k|τ
2
)
3
)
. (7.17)
84 Self-force
Inserting in (7.6) and substituting s
= ε
−1
s, k
= εk yields
F
ε
self
(t) = e
2
ε
−1
t
0
dτε
−1
d
3
k|ϕ(k)|
2
e
−i(k·v)τ
(|k|
−1
sin |k|τ)ik
− (cos |k|τ)
v − ετ ˙v +
1
2
ε
2
τ
2
¨v
− (|k|
−1
sin |k|τ)
v × (ik × v)
− v × (ik × ετ ˙v) +
1
2
v × (ik × ε
2
τ
2
¨v)) +
1
2
ετ
2
i(k ·˙v)
×
(|k|
−1
sin |k|τ)ik−(cos |k|τ)(v−ετ ˙v)−(|k|
−1
sin |k|τ)(v×(ik×v)
− v × (ik × ετ ˙v))
+
−
1
6
ε
2
τ
3
i(k ·¨v) −
1
8
ε
2
τ
4
(k ·¨v)
2
×
(|k|
−1
sin |k|τ)ik − (cos |k|τ)v − (|k|
−1
sin |k|τ)(v × (ik × v))
+
O(ε
2
). (7.18)
The terms proportional to ε
−1
cancel by symmetry. We sort all other terms,
F
ε
self
(t) = e
2
d
3
k |ϕ(k)|
2
− (v ·˙v)∇
v
+˙v(v ·∇
v
)
ε
−1
t
0
dτ e
−i(k·v)τ
(|k|
−1
sin |k|τ)
+
˙v +
1
2
v(˙v ·∇
v
)
ε
−1
t
0
dττe
−i(k·v)τ
(cos |k|τ) + ε
1
2
− (v
2
− 1)
× (˙v ·∇
v
)∇
v
+ v(v ·∇
v
)(˙v ·∇
v
) + (v ·¨v)∇
v
−¨v(v ·∇
v
)
+
1
6
− (1 − v
2
)(¨v ·∇
v
)∇
v
− v(v ·∇
v
)(¨v ·∇
v
) + 3(v ·˙v)(˙v ·∇
v
)∇
v
−3˙v(v ·∇
v
)(˙v ·∇
v
)
+
1
8
(v
2
− 1)(˙v ·∇
v
)
2
∇
v
−v(v ·∇
v
)(˙v ·∇
v
)
2
ε
−1
t
0
dττe
−i(k·v)τ
(|k|
−1
sin |k|τ)
+ ε
−¨v −
1
6
v(¨v ·∇
v
) + 3˙v(˙v ·∇
v
)
ε
−1
t
0
dττ
2
e
−i(k·v)τ
cos |k|τ
+
O(ε
2
). (7.19)
7.2 Taylor expansion 85
To take the limit ε → 0wegoback to position space and use the fundamental
solution of the wave equation. Then
lim
ε→0
ε
−1
t
0
dτ
d
3
k|ϕ(k)|
2
e
−i(k·v)τ
(|k|
−1
sin |k|τ)τ
p
=
∞
0
dt
d
3
x
d
3
y ϕ(x)ϕ(y)
1
4πt
δ(|x + vt − y|−t) t
p
=
d
3
k |ϕ(k)|
2
[k
2
− (k · v)
2
]
−1
for p = 0 ,
d
3
x ϕ(x)
d
3
yϕ(y)(γ
2
/4π) for p = 1 .
(7.20)
By the same method
lim
ε→0
ε
−1
t
0
dτ
d
3
k |ϕ(k)|
2
e
−i(k·v)τ
τ
1+ p
d
dτ
(|k|
−1
sin |k|τ)
=−
1 + p + (v ·∇
v
)
∞
0
dt
d
3
k |ϕ(k)|
2
e
−i(k·v)t
(|k|
−1
sin |k|t)t
p
=
−
d
3
k |ϕ(k)|
2
(k
2
+ (k · v)
2
)[k
2
− (k · v)
2
]
−2
for p = 0 ,
−
d
3
x ϕ(x)
d
3
yϕ(y)(2γ
4
/4π) for p = 1 .
(7.21)
Collecting all terms the final result reads
F
ε
self
(t) =−m
f
(v)˙v + ε(e
2
/6π)
γ
4
(v ·¨v)v + 3γ
6
(v ·˙v)
2
v
+3γ
4
(v ·˙v) ˙v + γ
2
¨v
+ O(ε
2
) (7.22)
for t > 0 with
m
f
(v) = m
e
|v|
−2
γ
2
(3 − v
2
) −|v|
−3
(3 + v
2
)arctanh|v|
v ⊗ v
+
−|v|
−2
+|v|
−3
(1 + v
2
) arctanh|v|
1l
. (7.23)
Note that m
f
(v) = d(P
s
− m
b
γ v)/dv asa3× 3 matrix.
Up to order ε, F
ε
self
(t) consists of two parts of a rather different character. The
term −m
f
(v)˙v is the contribution from the electromagnetic field to the change in
total momentum. We computed this term already in section 4.1 via a completely
different route. As emphasized there, since the Abraham model is semirelativistic,
the velocity dependence of m
f
has no reason to be of relativistic form and indeed
it is not. The term proportional to ε in (7.22) is the radiation reaction.Again there
86 Self-force
is no a priori reason to expect it to be relativistic, but in fact it is. Using the four-
vector notation of section 2.5, the radiation reaction can be rewritten as
ε(e
2
/6π)(
¨
u − (
˙
u ·
˙
u)u) = ε(e
2
/6π)(g + u ⊗ u) ·
¨
u . (7.24)
The space part is the term proportional to e
2
of (7.22), i.e. the radiation reaction
force, and the time part is the work done by this force per unit time.
7.3 How can the acceleration be bounded?
We return to the microscopic time scale. From the conservation of energy together
with condition (P),wehave
E
s
(v
0
) + eφ(εq
0
) = E (E
0
, B
0
, q
0
, v
0
) = E (E(t), B(t), q(t), v(t))
≥ m
b
γ(v(t)) + e
¯
φ (7.25)
and therefore
sup
t∈
R
|v(t)|≤ ¯v<1 . (7.26)
In (6.4) the external forces are of order ε. Superficially the self-force is of order
one. However for a Coulombic charge soliton field the self-force vanishes. Thus
if we could show that the deviations from the appropriate local soliton field are of
order ε, then the acceleration would satisfy
sup
t∈
R
|˙v(t)|≤C ε (7.27)
with C a suitable constant. This is what we want to prove. We will not keep track
of the constants, and the value of C changes from equation to equation. We make
sure, however, that the e-dependence is explicit and that C depends only on
v,
and thus is determined by the initial conditions. Of course, to justify the Taylor
expansion of section 7.2, we also need analogous estimates of higher derivatives,
which can be obtained with considerably more effort through the same scheme.
Here we want to explain how to get (7.27) and why we need e to be sufficiently
small, at least for the moment.
From the equations of motion one has
˙v = m
0
(v)
−1
εe
E
ex
(εq) + v × B
ex
(εq)
+ e
E
ϕ
(q) + v × B
ϕ
(q)
, (7.28)
where m
−1
0
(v) = (m
b
γ)
−1
(1l −v ⊗ v) is the matrix inverse of m
0
(v). Clearly by
(7.26) we have m
0
(v)
−1
≤C and, by condition (P), the first term is bounded as
ε
e
E
ex
(εq) + v × B
ex
(εq)
≤ C|e|ε. (7.29)
7.3 How can the acceleration be bounded? 87
On the other hand, the self-force looks to be of order one. To reduce it in order
we have to exploit the fact that E, B deviate only slightly from E
v
, B
v
close to the
charge distribution, i.e. we subtract zero and rewrite the self-force as
e
E
ϕ
(q) − E
vϕ
(q) + v × (B
ϕ
(q) − B
vϕ
(q))
. (7.30)
Our goal is to show that this difference is of order ε.
Let us define then
Z (x, t) =
E(x, t) − E
v(t)
(x − q(t))
B(x, t ) − B
v(t)
(x − q(t))
. (7.31)
Using Maxwell’s equations and the relations (v ·∇) E
v
=−∇×B
v
+ eϕv,
(v ·∇)B
v
=∇×E
v
one obtains
d
dt
Z (t) = AZ(t) − g (t), (7.32)
where A is defined in (2.18) and
g (x, t) =
(˙v(t) ·∇
v
)E
v
(x − q(t))
(˙v(t) ·∇
v
)B
v
(x − q(t))
. (7.33)
Therefore (7.32) has again the structure of the inhomogeneous Maxwell equations.
Since Z (0) = 0byour assumption on the initial data, one has
Z (t) =−
t
0
ds U (t − s)g (s). (7.34)
In terms of Z(t), using (7.28), (7.30), the acceleration is bounded through
|˙v(t)|≤C(ε +|e|)
d
3
xϕ(x)|Z
1
(x + q(t), t) + v(t) × Z
2
(x + q(t), t)| .
(7.35)
Let us set W (t, s) = U (t − s)g (s). Below we will prove that
|W
1
(t, s, q(t) + x)|+|W
2
(t, s, q(t) + x)|≤|e|C|˙v(s)|(1 + (t − s)
2
)
−1
(7.36)
for |x|≤R
ϕ
. Therefore inserting (7.36) in (7.35) one obtains
|˙v(t)|≤|e|C
ε +|e|
t
0
ds (1 + (t − s)
2
)
−1
|˙v(s)|
. (7.37)
88 Self-force
Let κ = sup
t≥0
|˙v(t)|. Then (7.37) reads
κ ≤|e| C
ε +|e|κ
∞
0
ds (1 + s
2
)
−1
,κ≤
|e| C
1 − e
2
C
ε. (7.38)
From the computation below we will see that C depends on ¯v (and on model
parameters like the form factor ϕ), but not on e. Thus taking |e| sufficiently small
one can ensure e
2
C < 1 and therefore κ ≤ Cε as claimed.
We still have to establish (7.36). U (t) is given in Eqs. (2.12), (2.13). Since
∇·g
1
(s) = 0 =∇·g
2
(s), the term proportional to k ⊗ k drops out. In real space
|k|
−1
sin |k|t becomes G
t
from (2.15) and cos |k|t becomes ∂
t
G
t
. Therefore
W
1
(t, s, x) =
1
4π(t − s)
2
d
3
y δ(|x − y|−(t − s))
×
(t − s)∇×g
2
(y, s) + g
1
(y, s) − (x − y) ·∇g
1
(y, s)
,
W
2
(t, s, x) =
1
4π(t − s)
2
d
3
y δ(|x − y|−(t − s))
×
− (t − s)∇×g
1
(y, s) + g
2
(y, s) − (x − y) ·∇g
2
(y, s)
.
(7.39)
We insert g from (7.33). E
v
and B
v
are first-order derivatives of the function φ
vϕ
which according to (4.7) is given by
φ
vϕ
(x) = e
d
3
yϕ(x − y)(4π)
−1
(1 − v
2
)y
2
+ (v · y)
2
−1/2
. (7.40)
Using (4.5) one has componentwise
|∇
v
E
v
(x)|+|∇
v
B
v
(x)|≤C ( |∇φ
v
(x)|+|∇∇
v
φ
v
(x)|),
|∇∇
v
E
v
(x)|+|∇∇
v
B
v
(x)|≤C ( |∇∇
v
φ
v
(x)|+|∇∇∇
v
φ
v
(x)|) (7.41)
and taking successive derivatives in (7.40) one obtains the bounds
|∇φ
v
(x)|+|∇∇
v
φ
v
(x)|≤eC(1 +|x|)
−2
,
|∇∇φ
v
(x)|+|∇∇∇
v
φ
v
(x)|≤eC(1 +|x|)
−3
, (7.42)
which imply
|g
1
(x, s)|+|g
2
(x, s)|≤eC|˙v(s)|(1 +|x − q(s)|
2
)
−1
,
|∇g
1
(x, s)|+|∇g
2
(x, s)|≤eC|˙v(s)|(1 +|x − q(s)|
3
)
−1
. (7.43)
Notes and references 89
We insert the bound (7.43) in (7.39) which results in an upper bound on
W (t, s, q(t) + x). Using the condition that |x|≤R
ϕ
and |q(t) − q(s)|≤ ¯v|t − s|
finally yields (7.36).
We summarize our findings as
Theorem 7.1 (Bounds on the velocity and its derivatives).For the Abraham
model satisfying conditions (C), (P), and (I ) there exist constants C, depending
through ¯v only on the initial conditions, and ¯e such that
|v(t)|≤ ¯v<1 , |˙v(t)|≤Cε, |¨v(t)|≤C
ε
2
+ ε(1 +|t|)
−2
,
|
...
v (t )|≤C
ε
3
+ ε
2
(1 +|t|)
−2
+ ε(1 +|t|)
−3
(7.44)
for all t on the microscopic time scale, provided the charge is sufficiently small,
i.e. |e| < ¯e.
By keeping track of the constant C, one could get a bound on the charge ad-
missible in Theorem 7.1. Since we believe this restriction to be an artifact of the
method anyhow, there is no point in the effort.
Notes and references
Section 7.1
Sommerfeld (1904a, 1905) systematically uses memory equations. In fact he con-
siders the Abraham model with the kinetic energy m
b
v
2
/2 for the particle and
wants to understand what happens when v(0)>c.Heargues that the particle
rapidly loses its energy to become slower than c by emitting what we now call
ˇ
Cerenkov radiation. The differential–difference equation (7.14) is derived by Page
(1918) and its relativistic generalization by Caldirola (1956). For reviews we refer
to Erber (1961) and Pearle (1982). Moniz and Sharp (1974, 1977) supply a linear
stability analysis and show that the solutions to (7.14) are stable provided R
ϕ
is not
too small. For that reason Rohrlich (1997) regards (7.14) and its relativistic sister
as the fundamental starting point for the classical dynamics of extended charges.
We take the Abraham model as the basic dynamical theory. Memory equations are
a useful tool in analyzing its properties.
Section 7.2
The Taylor expansion is taken from Kunze and Spohn (2000a). Such an expansion
was already used in Sommerfeld (1904a, 1905), to be repeated in various disguises.
The traditional expansion parameter is the size of the charge distribution, which in
90 Self-force
our context is replaced by the scaling parameter ε controlling the variation of the
potentials.
Section 7.3
The contraction argument appears in Komech, Kunze and Spohn (1999). The
bound on ˙v(t) is taken from Kunze and Spohn (2000a), where also higher deriva-
tives are discussed. It is claimed that |¨v(t )|≤Cε
2
and |
...
v (t )|≤Cε
3
.Inthe ar-
gument some initial terms are overlooked and the correct bounds are as given in
(7.44).
8
Comparison dynamics
The expansion of the self-force suggests that if we are willing to accept an error
of order ε
2
, the trajectory of the charged particle is governed by an autonomous
equation–asubstantial simplification of the hitherto coupled problem. An error of
order ε
2
in the equation does not imply an error of the same order in the solution.
This point must be discussed, but let us proceed for a while in good faith and
simply ignore the error in Eq. (7.22). Then we obtain the following approximate
equation for the motion of the charge,
˙q = v ,
m(v)˙v = e
E
ex
(q) + v × B
ex
(q)
+ ε(e
2
/6π)
γ
4
(v ·¨v)v
+3γ
6
(v ·˙v)
2
v + 3γ
4
(v ·˙v) ˙v + γ
2
¨v
. (8.1)
Here m(v) is the effective velocity-dependent mass. It is the sum of the bare mass
and the mass (7.23) induced by the field,
m(v) = m
b
(γ 1l + γ
3
v ⊗ v) + m
f
(v). (8.2)
As anticipated in section 4.1, via a distinct route, the leading contribution to
(8.1) is derived from the effective Lagrangian
L
eff
(q, ˙q) = T ( ˙q) − e
φ
ex
(q) −˙q · A
ex
(q)
, (8.3)
or equivalently from the Hamiltonian
H
eff
(q, p) = E
eff
p − e A
ex
(q)
+ eφ
ex
(q). (8.4)
For later purposes it is more convenient to work with the energy function
H(q, v) = E
s
(v) + eφ
ex
(q), (8.5)
which is conserved by the solutions to (8.1) with ε = 0; compare with (4.14).
91
92 Comparison dynamics
The term of order ε in (8.1) describes the radiation reaction. If included, the
energy of the particle fails to be conserved and the energy balance becomes
d
dt
H(q, v) −
d
dt
ε(e
2
/6π)γ
4
(v ·˙v) =−ε(e
2
/6π)
γ
4
˙v
2
+ γ
6
(v ·˙v)
2
. (8.6)
The term −ε(e
2
/6π)γ
4
(v ·˙v) = E
schott
(v, ˙v) is the Schott energy.Ithas no def-
inite sign. The Schott energy is stored in the near field and can be reversibly ex-
changed with the mechanical energy of the charge. The right-hand side of (8.6) is
the irreversible loss of energy through radiation; compare with section 8.4. Equa-
tion (8.6) is analogous to the balance equations in hydrodynamics and a familiar
way torewrite it is
ev · E
ex
(q) =
d
dt
E
s
(v) + E
schott
(v, ˙v)
+ ε(e
2
/6π)
γ
4
˙v
2
+ γ
6
(v ·˙v)
2
.
(8.7)
In other words, the work done by the external electric field acting on the charge is
divided up into the change in its kinetic energy, the change of the Schott energy,
and radiation.
If we set G
ε
= E
s
+ E
schott
, then G
ε
is decreasing in time, and integrating both
sides of (8.6) yields
−G
ε
(q(t), v(t), ˙v(t)) + G
ε
(q(0), v(0), ˙v(0))
= ε(e
2
/6π)
t
0
ds
γ
4
˙v(s)
2
+ γ
6
(v(s) ·˙v(s))
2
. (8.8)
The mechanical energy is bounded from below, but the Schott energy does not
have a definite sign. If (!) the Schott energy remains bounded in the course of time,
then
∞
0
dt
γ
4
˙v(t)
2
+ γ
6
(v(t) ·˙v(t))
2
< ∞, (8.9)
which implies
lim
t→∞
˙v(t) = 0 . (8.10)
The same conclusion was already reached for the Abraham model in Theorem
5.1, with no adiabatic limit there. Instead of (8.9) we used the bounded energy
dissipation (5.9). Since both the approximate and the true solutions have the same
long-time asymptotics, we expect no further time scale, i.e. higher corrections to
(8.1) should not change the qualitative behavior of solutions and merely increase