In the
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
of
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, the Abraham–Lorentz force (also known as the Lorentz–Abraham force) is the reaction force on an accelerating
charged particle
In physics, a charged particle is a particle with an electric charge. For example, some elementary particles, like the electron or quarks are charged. Some composite particles like protons are charged particles. An ion, such as a molecule or atom ...
caused by the particle emitting
electromagnetic radiation
In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength ...
by self-interaction. It is also called the radiation reaction force, the radiation damping force,
or the self-force. It is named after the physicists
Max Abraham and
Hendrik Lorentz
Hendrik Antoon Lorentz ( ; ; 18 July 1853 – 4 February 1928) was a Dutch theoretical physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for their discovery and theoretical explanation of the Zeeman effect. He derive ...
.
The formula, although predating the theory of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity,
"On the Ele ...
, was initially calculated for non-relativistic velocity approximations. It was extended to arbitrary velocities by
Max Abraham and was shown to be physically consistent by
George Adolphus Schott. The non-relativistic form is called Lorentz self-force while the relativistic version is called the Lorentz–Dirac force or collectively known as Abraham–Lorentz–Dirac force.
The equations are in the domain of
classical physics
Classical physics refers to physics theories that are non-quantum or both non-quantum and non-relativistic, depending on the context. In historical discussions, ''classical physics'' refers to pre-1900 physics, while '' modern physics'' refers to ...
, not
quantum physics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, and therefore may not be valid at distances of roughly the
Compton wavelength
The Compton wavelength is a quantum mechanical property of a particle, defined as the wavelength of a photon whose energy is the same as the rest energy of that particle (see mass–energy equivalence). It was introduced by Arthur Compton in 1 ...
or below.
[Fritz Rohrlich: ''The dynamics of a charged sphere and the electron'', Am. J. Phys. 65 (11) p. 1051 (1997)]
"The dynamics of point charges is an excellent example of the importance of obeying the validity limits of a physical theory. When these limits are exceeded the predictions of the theory may be incorrect or even patently absurd. In the present case, the classical equations of motion have their validity limits where quantum mechanics becomes important: they can no longer be trusted at distances of the order of (or below) the Compton wavelength. ... Only when all distances involved are in the classical domain is classical dynamics acceptable for electrons." There are, however, two analogs of the formula that are both fully quantum and relativistic: one is called the "Abraham–Lorentz–Dirac–Langevin equation", the other is the self-force on a moving mirror.
The force is proportional to the square of the object's
charge
Charge or charged may refer to:
Arts, entertainment, and media Films
* ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary
Music
* ''Charge'' (David Ford album)
* ''Charge'' (Machel Montano album)
* '' Charge!!'', an album by The Aqu ...
, multiplied by the
jerk that it is experiencing. (Jerk is the rate of change of
acceleration
In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
.) The force points in the direction of the jerk. For example, in a
cyclotron
A cyclotron is a type of particle accelerator invented by Ernest Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: Januar ...
, where the jerk points opposite to the velocity, the radiation reaction is directed opposite to the velocity of the particle, providing a braking action. The Abraham–Lorentz force is the source of the
radiation resistance
Radiation resistance is that part of an antenna's feedpoint electrical resistance caused by the emission of radio waves from the antenna. A radio transmitter applies a radio frequency alternating current to an antenna, which radiates the energy ...
of a radio
antenna radiating
radio wave
Radio waves (formerly called Hertzian waves) are a type of electromagnetic radiation with the lowest frequencies and the longest wavelengths in the electromagnetic spectrum, typically with frequencies below 300 gigahertz (GHz) and wavelengths g ...
s.
There are pathological solutions of the Abraham–Lorentz–Dirac equation in which a particle accelerates ''in advance'' of the application of a force, so-called ''pre-acceleration'' solutions. Since this would represent an effect occurring before its cause (
retrocausality
Retrocausality, or backwards causation, is a concept of cause and effect in which an effect precedes its cause in time and so a later event affects an earlier one. In quantum physics, the distinction between cause and effect is not made at the mos ...
), some theories have speculated that the equation allows signals to travel backward in time, thus challenging the physical principle of
causality. One resolution of this problem was discussed by
Arthur D. Yaghjian[
] and was further discussed by
Fritz Rohrlich and Rodrigo Medina.
Furthermore, some authors argue that a radiation reaction force is unnecessary, introducing a corresponding stress-energy tensor that naturally conserves energy and momentum in Minkowski space and other suitable spacetimes.
Definition and description
The Lorentz self-force derived for non-relativistic velocity approximation
, is given in
SI units
The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official st ...
by:
or in
Gaussian units
Gaussian units constitute a metric system of units of measurement. This system is the most common of the several electromagnetic unit systems based on the centimetre–gram–second system of units (CGS). It is also called the Gaussian unit syst ...
by
where
is the force,
is the derivative of
acceleration
In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
, or the third derivative of
displacement
Displacement may refer to:
Physical sciences
Mathematics and physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
, also called
jerk, ''μ''
0 is the
magnetic constant
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum'', ''magnetic constant'') is the magnetic permeability in a classical vacuum. It is a physical constant, conventionall ...
, ''ε''
0 is the
electric constant, ''c'' is the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
in
free space
A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
, and ''q'' is the
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
of the particle.
Physically, an accelerating charge emits radiation (according to the
Larmor formula
In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.
When any charge ...
), which carries
momentum
In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
away from the charge. Since momentum is conserved, the charge is pushed in the direction opposite the direction of the emitted radiation. In fact the formula above for radiation force can be ''derived'' from the Larmor formula, as shown
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
* Ernst von Below (1863–1955), German World War I general
* Fred Belo ...
.
The Abraham–Lorentz force, a generalization of Lorentz self-force for arbitrary velocities is given by:
Where
is the Lorentz factor associated with
, the velocity of particle. The formula is consistent with special relativity and reduces to Lorentz's self-force expression for low velocity limit.
The covariant form of radiation reaction deduced by Dirac for arbitrary shape of elementary charges is found to be: