# 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 defined as the rate of change in the four-momentum of a particle with respect to the particle's proper time:

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

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

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

and

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

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

## Including thermodynamic interactions

From the formulae of the previous section it appears that the time component of the four-force is the power expended, $\mathbf {f} \cdot \mathbf {u}$ , apart from relativistic corrections $\gamma /c$ . This is only true in purely mechanical situations, when heat exchanges vanish or can be neglected.

If 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. The time component of the four-force includes in this case a heating rate $h$ , besides the power $\mathbf {f} \cdot \mathbf {u}$ . 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 proportional to the power $\mathbf {f} \cdot \mathbf {u}$  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, and which in the Newtonian limit becomes $h+\mathbf {f} \cdot \mathbf {u}$ .

## 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. 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 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 four-force (four-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 the electromagnetic tensor,
• $U^{\nu }$  is the four-velocity, and
• $q$  is the electric charge.