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

See also

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.