Four-force

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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,:

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

For a particle of constant invariant mass m > 0 {\displaystyle m>0} {\displaystyle m>0}, the four-momentum is given by the relation P = m U {\displaystyle \mathbf {P} =m\mathbf {U} } {\displaystyle \mathbf {P} =m\mathbf {U} }, where U = γ ( c , u ) {\displaystyle \mathbf {U} =\gamma (c,\mathbf {u} )} {\displaystyle \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, A {\displaystyle \mathbf {A} } {\displaystyle \mathbf {A} }, by equation:

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

Here

f = d d t ( γ m u ) = d p d t {\displaystyle {\mathbf {f} }={\mathrm {d} \over \mathrm {d} t}\left(\gamma m{\mathbf {u} }\right)={\mathrm {d} \mathbf {p} \over \mathrm {d} t}} {\displaystyle {\mathbf {f} }={\mathrm {d} \over \mathrm {d} t}\left(\gamma m{\mathbf {u} }\right)={\mathrm {d} \mathbf {p} \over \mathrm {d} t}}

and

f ⋅ u = d d t ( γ m c 2 ) = d E d t . {\displaystyle {\mathbf {f} \cdot \mathbf {u} }={\mathrm {d} \over \mathrm {d} t}\left(\gamma mc^{2}\right)={\mathrm {d} E \over \mathrm {d} t}.} {\displaystyle {\mathbf {f} \cdot \mathbf {u} }={\mathrm {d} \over \mathrm {d} t}\left(\gamma mc^{2}\right)={\mathrm {d} E \over \mathrm {d} t}.}

where u {\displaystyle \mathbf {u} } {\displaystyle \mathbf {u} }, p {\displaystyle \mathbf {p} } {\displaystyle \mathbf {p} } and f {\displaystyle \mathbf {f} } {\displaystyle \mathbf {f} } are 3-space vectors describing the velocity, the momentum of the particle and the force acting on it respectively; and E {\displaystyle E} {\displaystyle 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 {\displaystyle \mathbf {f} \cdot \mathbf {u} } {\displaystyle \mathbf {f} \cdot \mathbf {u} }, apart from relativistic corrections γ / c {\displaystyle \gamma /c} {\displaystyle \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 {\displaystyle h} {\displaystyle h}, besides the power f ⋅ u {\displaystyle \mathbf {f} \cdot \mathbf {u} } {\displaystyle \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 f ⋅ u {\displaystyle \mathbf {f} \cdot \mathbf {u} } {\displaystyle \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 + f ⋅ u {\displaystyle h+\mathbf {f} \cdot \mathbf {u} } {\displaystyle 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 λ := D P λ d τ = d P λ d τ + Γ λ μ ν U μ P ν {\displaystyle F^{\lambda }:={\frac {DP^{\lambda }}{d\tau }}={\frac {dP^{\lambda }}{d\tau }}+\Gamma ^{\lambda }{}_{\mu \nu }U^{\mu }P^{\nu }} {\displaystyle 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 μ = ( F 0 , F ) {\displaystyle F^{\mu }=(F^{0},\mathbf {F} )} {\displaystyle F^{\mu }=(F^{0},\mathbf {F} )} acting on a particle of mass m {\displaystyle m} {\displaystyle m} which is momentarily at rest in a coordinate system. The relativistic force f μ {\displaystyle f^{\mu }} {\displaystyle f^{\mu }} in another coordinate system moving with constant velocity v {\displaystyle v} {\displaystyle v}, relative to the other one, is obtained using a Lorentz transformation:

f = F + ( γ − 1 ) v v ⋅ F v 2 , f 0 = γ β ⋅ F = β ⋅ f . {\displaystyle {\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}}} {\displaystyle {\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 β = v / c {\displaystyle {\boldsymbol {\beta }}=\mathbf {v} /c} {\displaystyle {\boldsymbol {\beta }}=\mathbf {v} /c}.

In general relativity, the expression for force becomes

f μ = m D U μ d τ {\displaystyle f^{\mu }=m{DU^{\mu } \over d\tau }} {\displaystyle f^{\mu }=m{DU^{\mu } \over d\tau }}

with covariant derivative D / d τ {\displaystyle D/d\tau } {\displaystyle D/d\tau }. The equation of motion becomes

m d 2 x μ d τ 2 = f μ − m Γ ν λ μ d x ν d τ d x λ d τ , {\displaystyle 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 },} {\displaystyle 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 Γ ν λ μ {\displaystyle \Gamma _{\nu \lambda }^{\mu }} {\displaystyle \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 α {\displaystyle f_{f}^{\alpha }} {\displaystyle f_{f}^{\alpha }} is the correct expression for force in a freely falling frame ξ α {\displaystyle \xi ^{\alpha }} {\displaystyle \xi ^{\alpha }}, we can use then the equivalence principle to write the four-force in an arbitrary coordinate x μ {\displaystyle x^{\mu }} {\displaystyle x^{\mu }}:

f μ = ∂ x μ ∂ ξ α f f α . {\displaystyle f^{\mu }={\partial x^{\mu } \over \partial \xi ^{\alpha }}f_{f}^{\alpha }.} {\displaystyle 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 μ = q F μ ν U ν , {\displaystyle f_{\mu }=qF_{\mu \nu }U^{\nu },} {\displaystyle f_{\mu }=qF_{\mu \nu }U^{\nu },}

where

See also

References

  1. 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.
  2. 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.
  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. (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.