Four-force

In the special theory of relativity, four-force is a four-vector that replaces the classical force.

In special relativity

The four-force is the four-vector defined as the change in four-momentum over the particle's own time:

${\mathbf {F}}={d{\mathbf {P}} \over d\tau }$.

For a particle of constant invariant mass $m>0$, ${\mathbf {P}}=m{\mathbf {U}}\,$ where ${\mathbf {U}}=\gamma (c,{\mathbf {u}})\,$ is the four-velocity, so we can relate the four-force with the four-acceleration $\mathbf {A}$ as in Newton's second law:

${\mathbf {F}}=m{\mathbf {A}}=\left(\gamma {{\mathbf {f}}\cdot {\mathbf {u}} \over c},\gamma {{\mathbf f}}\right)$.

Here

${{\mathbf f}}={d \over dt}\left(\gamma m{{\mathbf u}}\right)={d{\mathbf {p}} \over dt}$

and

${{\mathbf {f}}\cdot {\mathbf {u}}}={d \over dt}\left(\gamma mc^{2}\right)={dE \over dt}$.

where $\mathbf {u}$, $\mathbf {p}$ and $\mathbf {f}$ are 3-vectors describing the velocity and the momentum of the particle and the force acting on it respectively.

In General Relativity

In general relativity the relation between four-force, and four-acceleration remains the same, but the elements of the four-force are related to the elements of the four-momentum through a covariant derivative with respect to proper time.

$F^{\lambda }:={\frac {DP^{\lambda }}{d\tau }}={\frac {dP^{\lambda }}{d\tau }}+\Gamma ^{\lambda }{}_{{\mu \nu }}U^{\mu }P^{\nu }$

In addition, we can formulate force using the concept of coordinate transformations between different coordinate systems. Assume that we know the correct expression for force in a coordinate system at which the particle is momentarily at rest. Then we can perform a transformation to another system to get the corresponding expression of force.^[1] In special relativity the transformation will be a Lorentz transformation between coordinate systems moving with a relative constant velocity whereas in general relativity it will be a general coordinate transformation.

Consider the four-force $F^{\mu }=(F^{0},{\mathbf {F}})$ acting on a particle of mass $m$ which is momentarily at rest in a coordinate system. The relativistic force $f^{\mu }$ in another coordinate system moving with constant velocity $v$, relative to the other one, is obtained using a Lorentz transformation:

${{\mathbf {f}}}={{\mathbf {F}}}+(\gamma -1){{\mathbf {v}}}{{{\mathbf {v}}}\cdot {{\mathbf {F}}} \over v^{2}},$

$f^{0}=\gamma {\boldsymbol {\beta }}\cdot {\mathbf {F}}={\boldsymbol {\beta }}\cdot {\mathbf {f}}.$

where ${\boldsymbol {\beta }}={\mathbf {v}}/c$.

In general relativity, the expression for force becomes

$f^{\mu }=m{DU^{\mu } \over d\tau }$

with covariant derivative $D/d\tau$. The equation of motion becomes

$m{d^{2}x^{\mu } \over d\tau ^{2}}=f^{\mu }-m\Gamma _{{\nu \lambda }}^{\mu }{dx^{\nu } \over d\tau }{dx^{\lambda } \over d\tau },$

where $\Gamma _{{\nu \lambda }}^{\mu }$ is the Christoffel symbol. If there is no external force, this becomes the equation for geodesics in the curved space-time. The second term in the above equation, plays the role of a gravitational force. If $f_{f}^{\alpha }$ is the correct expression for force in a freely falling frame $\xi ^{\alpha }$, we can use the then the equivalence principle to write the four-force in an arbitrary coordinate $x^\mu$:

$f^{\mu }={\partial x^{\mu } \over \partial \xi ^{\alpha }}f_{f}^{\alpha }.$

Examples

In special relativity, Lorentz 4-force (4-force acting to charged particle situated in electromagnetic field) can be expressed as:

$F_{\mu }=qF_{{\mu \nu }}U^{\nu }$,

where

• $F_{\mu\nu}$ is electromagnetic tensor,
• $U^{\nu }$ is 4-velocity, and
• $q$ - electric charge.

See also

• four-vector
• four-velocity
• four-acceleration
• four-momentum

References

1. Steven, Weinberg (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley & Sons, Inc. ISBN 0-471-92567-5.

• Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford: Oxford University Press. ISBN 0-19-853953-3.