Four-force
Template:Short description In the special theory of relativity, four-force is a four-vector that replaces the classical force.
In special relativityEdit
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,:
<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>, the four-momentum is given by the relation <math>\mathbf{P} = m\mathbf{U}</math>, where <math>\mathbf{U}=\gamma(c,\mathbf{u})</math> is the four-velocity. In analogy to Newton's second law, we can also relate the four-force to the four-acceleration, <math>\mathbf{A}</math>, by equation:
<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; and <math>E</math> is the total energy of the particle.
Including thermodynamic interactionsEdit
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>Template:Cite journal</ref> Note that work and heat cannot be meaningfully separated, though, as they both carry inertia.<ref name=eckart1940>Template:Cite journal</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>Template:Cite journal</ref><ref name=truesdelletal1960 /> and which in the Newtonian limit becomes <math>h + \mathbf{f} \cdot \mathbf{u}</math>.
In general relativityEdit
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>Template:Cite book</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>
ExamplesEdit
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.