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:
<math display="block">\mathbf{F} = {\mathrm{d}\mathbf{P} \over \mathrm{d}\tau}.</math>
For a particle of constant invariant mass <math>m > 0</math>, <math>\mathbf{P} = m\mathbf{U}</math> where <math>\mathbf{U}=\gamma(c,\mathbf{u})</math> is the four-velocity, so we can relate the four-force with the four-acceleration <math>\mathbf{A}</math> as in Newton's second law:
<math display="block">\mathbf{F} = m\mathbf{A} = \left(\gamma {\mathbf{f}\cdot\mathbf{u} \over c},\gamma{\mathbf f}\right).</math>
Here
<math display="block">{\mathbf f}={\mathrm{d} \over \mathrm{d}t} \left(\gamma m {\mathbf u} \right)={\mathrm{d}\mathbf{p} \over \mathrm{d}t}</math>
and
<math display="block">{\mathbf{f}\cdot\mathbf{u}}={\mathrm{d} \over \mathrm{d}t} \left(\gamma mc^2 \right)={\mathrm{d}E \over \mathrm{d}t} .</math>
where <math>\mathbf{u}</math>, <math>\mathbf{p}</math> and <math>\mathbf{f}</math> 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, <math>\mathbf{f}\cdot\mathbf{u}</math>, apart from relativistic corrections <math>\gamma/c</math>. 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 <math>h</math>, besides the power <math>\mathbf{f}\cdot\mathbf{u}</math>.<ref name=grotetal1966>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.</ref> Note that work and heat cannot be meaningfully separated, though, as they both carry inertia.<ref name=eckart1940>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.</ref> This fact extends also to contact forces, that is, to the stress–energy–momentum tensor.<ref name=truesdelletal1960>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.</ref><ref name=eckart1940 />
Therefore, in thermo-mechanical situations the time component of the four-force is not proportional to the power <math>\mathbf{f}\cdot\mathbf{u}</math> 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,<ref name=eckart1940 /><ref name=grotetal1966 /><ref>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.</ref><ref name=truesdelletal1960 /> and which in the Newtonian limit becomes <math>h + \mathbf{f} \cdot \mathbf{u}</math>.
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.
<math display="block">F^\lambda := \frac{DP^\lambda }{d\tau} = \frac{dP^\lambda }{d\tau } + \Gamma^\lambda {}_{\mu \nu}U^\mu P^\nu </math>
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.<ref>Steven, Weinberg (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley & Sons, Inc. ISBN 0-471-92567-5.</ref> 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 <math>F^\mu=(F^0, \mathbf{F})</math> acting on a particle of mass <math>m</math> which is momentarily at rest in a coordinate system. The relativistic force <math>f^\mu </math> in another coordinate system moving with constant velocity <math>v</math>, relative to the other one, is obtained using a Lorentz transformation:
<math display="block">\begin{align}
\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{align}</math>
where <math>\boldsymbol{\beta} = \mathbf{v}/c</math>.
In general relativity, the expression for force becomes
<math display="block">f^\mu = m {DU^\mu\over d\tau}</math>
with covariant derivative <math>D/d\tau</math>. The equation of motion becomes
<math display="block">m {d^2 x^\mu\over d\tau^2} = f^\mu - m \Gamma^\mu_{\nu\lambda} {dx^\nu \over d\tau} {dx^\lambda \over d\tau},</math>
where <math> \Gamma^\mu_{\nu\lambda} </math> 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 <math> f^\alpha_f </math> is the correct expression for force in a freely falling frame <math> \xi^\alpha </math>, we can use then the equivalence principle to write the four-force in an arbitrary coordinate <math> x^\mu </math>:
<math display="block">f^\mu = {\partial x^\mu \over \partial\xi^\alpha} f^\alpha_f.</math>
Examples
In special relativity, Lorentz four-force (four-force acting on a charged particle situated in an electromagnetic field) can be expressed as: <math display="block">f_\mu = q F_{\mu\nu} U^\nu ,</math>
where
- <math>F_{\mu\nu}</math> is the electromagnetic tensor,
- <math>U^\nu</math> is the four-velocity, and
- <math>q</math> is the electric charge.
See also
References
- Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd ed.). Oxford: Oxford University Press. ISBN 0-19-853953-3.
