Fermi–Walker transport

Fermi-Walker transport is a process in general relativity used to define a coordinate system or reference frame such that all curvature in the frame is due to the presence of mass/energy density and not to arbitrary spin or rotation of the frame.

Acceleration is perpendicular to velocity in spacetime

Diagram 1. Changing views of spacetime along the world line of a rapidly accelerating observer. In this animation, the dashed line is the spacetime trajectory ("world line") of a particle. The balls are placed at regular intervals of proper time along the world line. The solid diagonal lines are the light cones for the observer's current event, and intersect at that event. The small dots are other arbitrary events in the spacetime. For the observer's current instantaneous inertial frame of reference, the vertical direction indicates the time and the horizontal direction indicates distance. The slope of the world line (deviation from being vertical) is the velocity of the particle on that section of the world line. So at a bend in the world line the particle is being accelerated. Note how the view of spacetime changes when the observer accelerates, changing the instantaneous inertial frame of reference. These changes are governed by the Lorentz transformations. Also note that:
• the balls on the world line before/after future/past accelerations are more spaced out due to time dilation.
• events which were simultaneous before an acceleration are at different times afterwards (due to the relativity of simultaneity),
• events pass through the light cone lines due to the progression of proper time, but not due to the change of views caused by the accelerations,and
• the world line always remains within the future and past light cones of the current event.

World lines for two particles at different speeds.

In Special relativity the location of a particle in 4 dimensional spacetime is given by its world line

$x^{\mu }=(ct,\mathbf {x} )$

where $\mathbf {x} =\mathbf {v} t$ is the position in space of the particle and $\mathbf {v}$ is the velocity in space.

The "length" of the vector is given by

$x_{\mu }x^{\mu }=\eta _{\mu \nu }x^{\mu }x^{\nu }=(ct)^{2}-\mathbf {x} \cdot \mathbf {x} \ {\stackrel {\mathrm {def} }{=}}\ \tau ^{2}$

where $\tau$ is c times the proper time as measured by a clock in the rest frame of the particle and the metric is given by

$\eta ^{\mu \nu }=\eta _{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}$.

The velocity in spacetime is defined as

$v^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {dx^{\mu } \over d\tau }=\left(c{dt \over d\tau },{dt \over d\tau }{d\mathbf {x} \over dt}\right)=\left(\gamma ,\gamma {\mathbf {v} \over c}\right)$

The velocity vectors in spacetime for a particle at two different speeds. In relativity an acceleration is equivalent to a rotation in spacetime

where

$\gamma \ {\stackrel {\mathrm {def} }{=}}\ {1 \over {\sqrt {1-{{\mathbf {v} \cdot \mathbf {v} } \over c^{2}}}}}$.

The magnitude of the 4-velocity is one,

$v_{\mu }v^{\mu }=1$.

The 4-velocity is therefore, not only a representation of the velocity in spacetime, it is also a unit vector in the direction of the position of the particle in spacetime.

The 4-acceleration is given by

$a^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {dv^{\mu } \over d\tau }\,\,$. (Equation 1)

The 4-acceleration is always perpendicular to the 4-velocity

$0={1 \over 2}{d \over d\tau }\left(v_{\mu }v^{\mu }\right)={dv_{\mu } \over d\tau }v^{\mu }=a_{\mu }v^{\mu }$.

Therefore, we can regard acceleration in spacetime as simply a rotation of the 4-velocity.

Gravitational forces

Define the 4-acceleration due to a gravitational force as $f^{\mu }$. Then the portion of this force parallel to the 4-velocity will have no effect on the 4-velocity. That portion of the 4-acceleration can be written

$\ v^{\mu }v_{\nu }f^{\nu }$.

The portion perpendicular to the 4-velocity is then

$f^{\mu }-v^{\mu }v_{\nu }f^{\nu }=\left(v_{\nu }v^{\nu }\right)f^{\mu }-v^{\mu }\left(v_{\nu }f^{\nu }\right)$

and the change in 4-velocity due to gravitational forces is

Equation 1:

${dv^{\mu } \over d\tau }=\left(v_{\nu }v^{\nu }\right)f^{\mu }-v^{\mu }\left(v_{\nu }f^{\nu }\right)$.

for a time-like metric.

Fermi derivative

This is defined for a vector field ${\mathbf {x}}$ along a curve $\gamma (s)$:

${\frac {D_{F}X}{\partial s}}={\frac {DX}{\partial s}}-g(X,{\frac {DV}{\partial s}})V+g(X,V){\frac {DV}{\partial s}}.$

When ${\frac {D_{F}X}{\partial s}}=0$, the vector is Fermi-Walker transported along the curve (See Hawking and Ellis, pag. 80).

Co-moving coordinate systems

A coordinate system co-moving with the particle can be defined. If we take the unit vector $v^{\mu }$ as defining an axis in the co-moving coordinate system, then any system transforming with proper time as Equation 1. is said to be undergoing Fermi Walker transport.

See also

• Albert Einstein
• Basic introduction to the mathematics of curved spacetime
• Enrico Fermi
• Lorentz scalar
• Transition from Newtonian mechanics to General relativity

References

• Einstein, A. (1961). Relativity: The Special and General Theory. New York: Crown. ISBN 0-517-02961-8.
• Landau, L. D. and Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0-08-018176-7.{{cite book}}: CS1 maint: multiple names: authors list (link)