Fermi–Walker transport is a process in
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
used to define a
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
or
reference frame
In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system, whose origin, orientation, and scale have been specified in physical space. It is based on a set of reference points, defined as geometric ...
such that all
curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
in the frame is due to the presence of mass/energy density and not due to arbitrary spin or rotation of the frame. It was discovered by Fermi in 1921 and rediscovered by Walker in 1932.
Fermi–Walker differentiation
In the theory of
Lorentzian manifold
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ...
s, Fermi–Walker differentiation is a generalization of
covariant differentiation. In general relativity, Fermi–Walker derivatives of the
spacelike
In mathematical physics, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold.
Lorentzian manifolds can be classified according to the types of causal structures they admit (''ca ...
vector fields in a frame field, taken with respect to the
timelike
In mathematical physics, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold.
Lorentzian manifolds can be classified according to the types of causal structures they admit (''ca ...
unit vector field in the frame field, are used to define non-inertial and non-rotating frames, by stipulating that the Fermi–Walker derivatives should vanish. In the special case of
inertial frame
In classical physics and special relativity, an inertial frame of reference (also called an inertial space or a Galilean reference frame) is a frame of reference in which objects exhibit inertia: they remain at rest or in uniform motion relative ...
s, the Fermi–Walker derivatives reduce to covariant derivatives.
With a
sign convention, this is defined for a vector field ''X'' along a curve
:
:
where is four-velocity, is the covariant derivative, and
is the scalar product. If
:
then the vector field is Fermi–Walker transported along the curve. Vectors perpendicular to the space of
four-velocities in
Minkowski spacetime
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a s ...
, e.g., polarization vectors, under Fermi–Walker transport experience
Thomas precession
In physics, the Thomas precession, named after Llewellyn Thomas, is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope. It relates the angular velocity of the spin of a par ...
.
Using the Fermi derivative, the
Bargmann–Michel–Telegdi equation for spin precession of electron in an external electromagnetic field can be written as follows:
:
where
and
are polarization four-vector and
magnetic moment
In electromagnetism, the magnetic moment or magnetic dipole moment is the combination of strength and orientation of a magnet or other object or system that exerts a magnetic field. The magnetic dipole moment of an object determines the magnitude ...
,
is four-velocity of electron,
,
, and
is the
electromagnetic field strength tensor. The right side describes
Larmor precession
Sir Joseph Larmor (; 11 July 1857 – 19 May 1942) was an Irish mathematician and physicist who made breakthroughs in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter. His most influential work was ...
.
Co-moving coordinate systems
A coordinate system co-moving with a particle can be defined. If we take the unit vector
as defining an axis in the co-moving coordinate system, then any system transforming with proper time is said to be undergoing Fermi–Walker transport.
Generalised Fermi–Walker differentiation
Fermi–Walker differentiation can be extended for any
where
(that is, not a
light-like
In mathematical physics, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold.
Lorentzian manifolds can be classified according to the types of causal structures they admit (''ca ...
vector). This is defined for a vector field
along a curve
:
:
Except for the last term, which is new, and basically caused by the possibility that
is not constant, it can be derived by taking the previous equation, and dividing each
by
.
If
, then we recover the Fermi–Walker differentiation:
and
See also
*
Basic introduction to the mathematics of curved spacetime
*
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian and naturalized American physicist, renowned for being the creator of the world's first artificial nuclear reactor, the Chicago Pile-1, and a member of the Manhattan Project ...
*
Arthur Geoffrey Walker
Arthur Geoffrey Walker FRS FRSE (17 July 1909 in Watford, Hertfordshire, England – 31 March 2001) was a British mathematician and professor of the University of Sheffield who made important contributions to physical cosmology. Although he ...
*
Transition from Newtonian mechanics to general relativity
Notes
References
*.
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{{DEFAULTSORT:Fermi-Walker transport
Mathematical methods in general relativity