The Hidden Geometry of Electromagnetism

by Yaron Hadad

In collaboration with!Eli Cohen, Tel-Aviv University!

Ido Kaminer, M.I.T!Avshalom Elitzur, I.Y.A.R

Field Lines

Michael Faraday

The Problem of Radiation-ReactionThe Lorentz Force (LF) Eq: mu↵ = �eF↵�

ext

u�

The rate at which energy is radiated away from the electron is

R = �m�0u�u�

�0 =23

e2

m= 6.24� 10�24 s

=) an accelerating charge loses energy.This effect is not included in the Lorentz Force equation. The rate at which energy-momentum is emitted by radiation:

dP↵

d⌧= Ru↵

Lorentz 1892

A Plentitude of Models…mu� = �eF�⇥u⇥ + m�0

�u� + u2u�

⇥

mu↵ = �eF↵�u� � e⌧0hF↵�,� u�u

� � e

m

�F↵�F��u

� � F ��F��u�u�u

↵�i

mu↵ = �eF↵�u� � e⌧0

d

d⌧(F↵�u�)� F ��u� u�u

↵

�

mu↵ = �eF↵�u� � e⌧0⇥F↵� u� + F �� u�u�u

↵⇤

m

⌧0

⇥u↵(⌧ � ⌧0)� u↵(⌧)u�(⌧)u

�(⌧ � ⌧0)⇤= �eF↵�(⌧)u�(⌧)

mu↵ = �eF↵�(⌧)u�(⌧)�m

⌧0

⇥u↵(⌧ � ⌧0)� u↵(⌧)u�(⌧)u

�(⌧ � ⌧0)⇤

Lorentz-Abraham-Dirac (1938)

Landau-Lifshitz (1952)

Eliezer (1948)

Mo-Papas (1971)

Caldirola (1979)

Caldirola-Yaghjian (1992)

At least 7 more models were proposed….

Radiation-Reaction in Modern Experiments

The dimensionless intensity of !

a laser beam: a0 =eA0

m

� ⇠ 942 nm

Radiation-reaction criterion:�0a

20 ⇠ 108

A0 = wave amplitude

v0, �0

Field Lines Curvature

Charge acceleration Field lines curvature=)

=)

Electric CurvatureThe curvature of a curve

For an electric field line:

~

Eself(~x) ~

E

ext

(~x)~

E(~x)= +{

Coulomb’s law in the electron’s rest frame

k =| ~E ⇥ ( ~E ·r) ~E|

| ~E|3

~�

k =|~�0 ⇥ ~�00|

|~�0|

Electromagnetic Geodesics

Charges travel along trajectories of least electromagnetic curvature

{Never singular, not even

for point-like particles

= 0 () The charge travels along an “electromagnetic geodesic”

(lines of zero curvature)

k(~x0

(t+�t)) ⇡ 3(�t)2

2q| ~E

ext

⇥ ~a

0

(t)|

k(~x) ⇡ 3

q

| ~Eext

(~x)⇥ (~x� ~x

0

(t))|

Grazie mille!

www.yaronhadad.com