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:
.
For a particle of constant invariant mass m > 0, where
is the four-velocity, so we can relate the four-force with the four-acceleration
as in Newton's second law:
.
Here
and
.
where ,
and
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.
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 acting on a particle of mass
which is momentarily at rest in a coordinate system. The relativistic force
in another coordinate system moving with constant velocity
, relative to the other one, is obtained using a Lorentz transformation:
where .
In general relativity, the expression for force becomes
with covariant derivative . The equation of motion becomes
where 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
is the correct expression for force in a freely falling frame
, we can use the then the equivalence principle to write the four-force in an arbitrary coordinate
:
Examples
In special relativity, Lorentz 4-force (4-force acting to charged particle situated in electromagnetic field) can be expressed as:
,
where
is electromagnetic tensor,
is 4-velocity, and
- electric charge.
See also
References
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.