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

In special relativity

The four-force is defined as the rate of change in the four-momentum of a particle with respect to the particle's proper time. Hence,:

For a particle of constant invariant mass is the four-velocity. In analogy to Newton's second law, we can also relate the four-force to the four-acceleration, by equation:

Here

and

are 3-space is the total energy of the particle.

Including thermodynamic interactions

. This is only true in purely mechanical situations, when heat exchanges vanish or can be neglected.

In the full thermo-mechanical case, not only work, but also heat contributes to the change in energy, which is the time component of the energy–momentum covector. Note that work and heat cannot be meaningfully separated, though, as they both carry inertia. This fact extends also to contact forces, that is, to the stress–energy–momentum tensor.

Therefore, in thermo-mechanical situations the time component of the four-force is not but has a more generic expression, to be given case by case, which represents the supply of internal energy from the combination of work and heat,.

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. 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.

, relative to the other one, is obtained using a Lorentz transformation:

.

In general relativity, the expression for force becomes

with covariant derivative. The equation of motion becomes

is the Christoffel symbol. If there is no external force, this becomes the equation for geodesics in the curved space-time, we can use then the equivalence principle:

Examples

In special relativity, Lorentz four-force

where

• is the electromagnetic tensor,

• is the four-velocity, and

• is the electric charge.

See also

• four-vector

• four-velocity

• four-acceleration

• four-momentum

• four-gradient

References