Four-force

Last updated October 18, 2024

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,:

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

For a particle of constant invariant mass $m>0$ , the four-momentum is given by the relation $\mathbf {P} =m\mathbf {U}$ , where $\mathbf {U} =\gamma (c,\mathbf {u} )$ is the four-velocity. In analogy to Newton's second law, we can also relate the four-force to the four-acceleration, $\mathbf {A}$ , by equation:

$\mathbf {F} =m\mathbf {A} =\left(\gamma {\mathbf {f} \cdot \mathbf {u} \over c},\gamma {\mathbf {f} }\right).$

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; and $E$ is the total energy of the particle.

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.

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. The time component of the four-force includes in this case a heating rate $h$ , besides the power $\mathbf {f} \cdot \mathbf {u}$ .^[1] Note that work and heat cannot be meaningfully separated, though, as they both carry inertia.^[2] This fact extends also to contact forces, that is, to the stress–energy–momentum tensor.^[3]^[2]

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,^[2]^[1]^[4]^[3] 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.^[5] 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:

${\begin{aligned}\mathbf {f} &=\mathbf {F} +(\gamma -1)\mathbf {v} {\mathbf {v} \cdot \mathbf {F} \over v^{2}},\\f^{0}&=\gamma {\boldsymbol {\beta }}\cdot \mathbf {F} ={\boldsymbol {\beta }}\cdot \mathbf {f} .\end{aligned}}$

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 on a charged particle situated in an 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.

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References

1 2 Grot, Richard A.; Eringen, A. Cemal (1966). "Relativistic continuum mechanics: Part I – Mechanics and thermodynamics". Int. J. Engng Sci. 4 (6): 611–638, 664. doi:10.1016/0020-7225(66)90008-5.
1 2 3 Eckart, Carl (1940). "The Thermodynamics of Irreversible Processes. III. Relativistic Theory of the Simple Fluid". Phys. Rev. 58 (10): 919–924. Bibcode:1940PhRv...58..919E. doi:10.1103/PhysRev.58.919.
1 2 C. A. Truesdell, R. A. Toupin: The Classical Field Theories (in S. Flügge (ed.): Encyclopedia of Physics, Vol. III-1, Springer 1960). §§152–154 and 288–289.
↑ Maugin, Gérard A. (1978). "On the covariant equations of the relativistic electrodynamics of continua. I. General equations". J. Math. Phys. 19 (5): 1198–1205. Bibcode:1978JMP....19.1198M. doi:10.1063/1.523785.
↑ 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.

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[grotetal1966-1] 1 2 Grot, Richard A.; Eringen, A. Cemal (1966). "Relativistic continuum mechanics: Part I – Mechanics and thermodynamics". Int. J. Engng Sci. 4 (6): 611–638, 664. doi:10.1016/0020-7225(66)90008-5.

[eckart1940-2] 1 2 3 Eckart, Carl (1940). "The Thermodynamics of Irreversible Processes. III. Relativistic Theory of the Simple Fluid". Phys. Rev. 58 (10): 919–924. Bibcode:1940PhRv...58..919E. doi:10.1103/PhysRev.58.919.

[truesdelletal1960-3] 1 2 C. A. Truesdell, R. A. Toupin: The Classical Field Theories (in S. Flügge (ed.): Encyclopedia of Physics, Vol. III-1, Springer 1960). §§152–154 and 288–289.

[4] Maugin, Gérard A. (1978). "On the covariant equations of the relativistic electrodynamics of continua. I. General equations". J. Math. Phys. 19 (5): 1198–1205. Bibcode:1978JMP....19.1198M. doi:10.1063/1.523785.

[5] Steven, Weinberg (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity . John Wiley & Sons, Inc. ISBN 0-471-92567-5.

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