Four-force explained

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,:

$$\backslash mathbf\; =\; .$$

m > 0 , the four-momentum is given by the relation P = m U , where U = \gamma(c, u ) is the four-velocity. In analogy to Newton's second law, we can also relate the four-force to the four-acceleration, A , by equation:

$$\backslash mathbf\; =\; m\backslash mathbf\; =\; \backslash left(\backslash gamma,\backslash gamma\backslash right).$$

Here

$$=\; \backslash left(\backslash gamma\; m\; \backslash right)=$$

and

$$=\; \backslash left(\backslash gamma\; mc^2\; \backslash right)=\; .$$

where

u , p and 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,

f ⋅ 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 f ⋅ 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]

Therefore, in thermo-mechanical situations the time component of the four-force is not proportional to the power

f ⋅ 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,^[4] and which in the Newtonian limit becomes h + f ⋅ 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.

$$F^\backslash lambda\; :=\; \backslash frac\; =\; \backslash frac\; +\; \backslash Gamma^\backslash lambda\; \_U^\backslash mu\; P^\backslash nu$$

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,} 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:

where

\boldsymbol{\beta} = v /c .

In general relativity, the expression for force becomes

$$f^\backslash mu\; =\; m$$

D/d\tau . The equation of motion becomes

$$m\; =\; f^\backslash mu\; -\; m\; \backslash Gamma^\backslash mu\_,$$

where

\Gamma ^{\mu} _{\nu λ} 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 then the equivalence principle to write the four-force in an arbitrary coordinate x ^{\mu} :

$$f^\backslash mu\; =\; f^\backslash alpha\_f.$$

Examples

In special relativity, Lorentz four-force (four-force acting on a charged particle situated in an electromagnetic field) can be expressed as:$$f\_\backslash mu\; =\; q\; F\_\; U^\backslash nu,$$

where

F _{\mu\nu} is the electromagnetic tensor,U ^{\nu} is the four-velocity, andq is the electric charge.

See also

• four-vector
• four-velocity
• four-acceleration
• four-momentum
• four-gradient

References

• Book: Rindler, Wolfgang . Introduction to Special Relativity . registration . 2nd . Oxford . Oxford University Press . 1991 . 0-19-853953-3.

Notes and References

1. Grot. Richard A.. Eringen. A. Cemal. Relativistic continuum mechanics: Part I – Mechanics and thermodynamics. 1966. Int. J. Engng Sci.. 4. 6. 611–638, 664. 10.1016/0020-7225(66)90008-5.
2. Eckart. Carl. The Thermodynamics of Irreversible Processes. III. Relativistic Theory of the Simple Fluid. 1940. Phys. Rev.. 58. 10. 919–924. 10.1103/PhysRev.58.919. 1940PhRv...58..919E.
3. 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.. On the covariant equations of the relativistic electrodynamics of continua. I. General equations. 1978. J. Math. Phys.. 19. 5. 1198–1205. 10.1063/1.523785. 1978JMP....19.1198M.
5. Book: Steven. Weinberg. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. 1972. John Wiley & Sons, Inc.. 0-471-92567-5. registration.