Editing Four-force

{{Short description|4-dimensional analogue of force used in theories of relativity}}
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,:

<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 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 (thermodynamics)|work]], but also [[Heat (thermodynamics)|heat]] contributes to the change in energy, which is the time component of the [[four-momentum|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>{{cite journal|last1=Grot|first1=Richard A.|last2=Eringen|first2=A. Cemal| title=Relativistic continuum mechanics: Part I – Mechanics and thermodynamics|date=1966|journal=Int. J. Engng Sci.| volume=4|issue=6|pages=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>{{cite journal| last1=Eckart|first1=Carl| title=The Thermodynamics of Irreversible Processes. III. Relativistic Theory of the Simple Fluid|date=1940|journal=Phys. Rev.| volume=58| issue=10| pages=919–924| doi=10.1103/PhysRev.58.919| bibcode=1940PhRv...58..919E}}</ref> This fact extends also to contact forces, that is, to the [[stress–energy tensor|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>{{cite journal| last1=Maugin|first1=Gérard A.|title=On the covariant equations of the relativistic electrodynamics of continua. I. General equations|date=1978|journal=J. Math. Phys.| volume=19| issue=5| pages=1198–1205| doi=10.1063/1.523785| bibcode=1978JMP....19.1198M}}</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 transformation]]s 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>{{cite book|last1=Steven|first1=Weinberg|title=Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity|date=1972|publisher=John Wiley & Sons, Inc.|isbn=0-471-92567-5|url-access=registration|url=https://archive.org/details/gravitationcosmo00stev_0}}</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 [[geodesic]]s 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 force|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 ==
* [[four-vector]]
* [[four-velocity]]
* [[four-acceleration]]
* [[four-momentum]]
* [[four-gradient]]

== References ==
{{Reflist}}

* {{cite book | author = Rindler, Wolfgang | title=Introduction to Special Relativity | url = https://archive.org/details/introductiontosp0000rind | url-access = registration | edition=2nd | location= Oxford | publisher=Oxford University Press | year=1991 | isbn=0-19-853953-3}}

[[Category:Four-vectors]]
[[Category:Force]]