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

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

where $\Gamma^\mu_{\nu\lambda}$ 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 $f^\alpha_f$ 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 $x^\mu$:

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

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.