Four-force

Four-force (4-force) is a four-vector, considered as a relativistic generalization of the classical 3-vector of force to the four-dimensional spacetime. As in classical mechanics, the 4-force can be defined in two ways. The first one measures the change in the energy and momentum of a particle per unit of proper time. The second method introduces force characteristics – strengths of field, and with their help in certain energy and momentum of the particle is calculated 4-force acting on the particle in the field. The equality of 4-forces produced by these methods, gives the equation of motion of the particle in the given force field.

In special relativity 4-force is the derivative of 4-momentum $~ p^\lambda$ with respect to the proper time $~ \tau$ of the particle: ^[1]

~F^\lambda = \frac{dp^\lambda }{d\tau}. \qquad\qquad (1)

For a particle with constant invariant mass m > 0, $~ p^\lambda = m U^\lambda$ , where $~ U^\lambda$ is 4-velocity. This allows connecting 4-force with 4-acceleration $~ A^\lambda$ similarly to Newton's second law:

~F^\lambda = m A^\lambda

Given $~ \mathbf{u}$ is the classic 3-vector of the particle velocity, $~ \gamma = \frac{1}{\sqrt{1-(\frac{u}{c})^2}}$

||F^\lambda ||=\left(\gamma \frac{\mathbf{f}\cdot \mathbf{u}}{c} , \gamma \mathbf{f} \right)

~{\mathbf {f}}={d \over dt} \left(\gamma m {\mathbf {u}} \right)={d\mathbf{p} \over dt}= \gamma m\left(\mathbf{a} +\gamma ^2 \frac{\left(\mathbf{u} \cdot \mathbf{a} \right)}{c^2}\mathbf{u}\right)= m \gamma^3 \left( \mathbf{a} + \frac {\mathbf{u} \times [ \mathbf{u} \times \mathbf {a}] } {c^2} \right)

is the 3-vector of force, ^[2]

$~ \mathbf{p}$ is the 3-vector of relativistic momentum, $~ \mathbf{a}= \frac {d \mathbf{u}}{dt}$ is the 3-acceleration,

~\mathbf{f}\cdot\mathbf{u}={d \over dt} \left(\gamma m c^2 \right)={dE \over dt}

$~ E$ is relativistic energy.

In general relativity, the 4-force is determined by the covariant derivative of 4-momentum with respect to the proper time: ^[3]

F^\lambda := \frac{Dp^\lambda }{d\tau} = \frac{dp^\lambda }{d\tau } + \Gamma^\lambda {}_{\mu \nu}U^\mu p^\nu

where $~ \Gamma^\lambda {}_{\mu \nu}$ are the Christoffel symbols.

Examples

4-force acting in the electromagnetic field on the particle with electric charge $~q$ , is expressed as follows:

~ F^\lambda = \frac{q}{c}g_{\mu \nu }U^\nu F^{\lambda \mu}

where $~F^{\lambda \mu}$ is the electromagnetic tensor

$||F_{\mu \nu}|| = \begin{bmatrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & cB_z & -cB_y \\ E_y & -cB_z & 0 & cB_x \\ E_z & cB_y & -cB_x & 0 \end{bmatrix}$

, $~U^\nu$ is 4-velocity.

The density of 4-force

To describe liquid and extended media, in which we must find forces in different points in space, instead of 4-vector of force 4-vector of force density is used, acting locally on a small volume unit of the medium:

~f^\lambda := \frac{dJ^\lambda }{d\tau}, \qquad\qquad (2)

where $~ J^{\lambda} = \rho_0 U^{\lambda}$ is the mass 4-current, $~ \rho_0$ is the mass density in the rest reference frame relative to the matter.

In the special theory of relativity, the relations hold:

~ ||U^\lambda || = \left(\gamma c, \gamma {\mathbf {u}}\right)

~||f^\lambda || = \begin{bmatrix} \frac{\gamma }{c} \frac{ d\varepsilon }{dt} \\ \gamma f_{R}^x \\ \gamma f_{R}^y \\ \gamma f_{R}^z \end{bmatrix}

where $~ f^\lambda = {d \over dt} \left(\gamma \rho_0 {u^\lambda } \right)={dJ^\lambda \over dt}$ is 3-vector of force density, $~ \mathbf{J}$ is 3-vector of mass current, $~ \varepsilon = \gamma \rho_0 c^2$ is the density of relativistic energy.

If we integrate (2) over the invariant volume of the matter unit, measured in the co-moving reference frame, we obtain the expression for 4-force (1):

~\int {f^\lambda dV_0}= F^\lambda = \int {\frac{d(\rho_0 U^\lambda ) }{d\tau} dV_0} = \frac {d}{ d\tau } \int {\rho_0 U^\lambda dV_0} =\frac {d}{ d\tau } \int { U^\lambda d m } =\frac{dp^\lambda }{d\tau}.

Four-force in CTG

If the particle is in the gravitational field, then according to the covariant theory of gravitation (CTG) gravitational 4-force equals:

~ F^\nu = m \Phi^{\nu \mu} U_\mu = \Phi^{\nu \mu} p_\mu

where $~\Phi^{\nu \mu}$ is the gravitational tensor, which is expressed through the gravitational field strength and the gravitational torsion field, $~p_\mu$ is 4-momentum with lower (covariant) index, and particle mass $~ m$ includes contributions from the mass-energy of fields associated with the matter of the particle.

In CTG gravitational tensor with covariant indices $~ \Phi_ {rs}$ is determined directly, and for transition to the tensor with contravariant indices in the usual way the metric tensor is used which is in general a function of time and coordinates:

~ \Phi^{\nu \mu}= g^{\nu r} g^{s \mu } \Phi_{rs} .

Therefore the 4-force $~ F^\nu$ , which depends on the metric tensor through $~ \Phi^{\nu \mu}$ , also becomes a function of the metric. At the same time, the definition of 4-force with covariant index does not require knowledge of the metric:

~ F_\mu = m \Phi_{\mu \nu} U^\nu = \Phi_{\mu \nu} p^\nu.

In the covariant theory of gravitation 4-vector of force density is described with the help of acceleration field: ^[4]

~f^\nu = - g^{\nu \mu } u_{\mu \lambda } J^\lambda = \nabla_\mu B^{\nu \mu}= \rho_0 \frac{ DU^\nu } {D \tau }= \rho_0 U^\mu \nabla_\mu U^\nu =\rho_0 \frac{ dU^\nu } {d \tau }+\rho_0 \Gamma^\nu _{\mu \lambda} U^\mu U^\lambda = \rho_0 a^\nu , \qquad (3)

where $~ B^{\nu \mu}$ is the acceleration stress-energy tensor, $~ u_{\mu \lambda }$ is acceleration tensor, $~ a^\nu$ is the 4-acceleration.

In the above expression the operator of proper-time-derivative $~\frac{ D } {d \tau }= U^\mu \nabla_\mu$ is used, which generalizes the material derivative (substantial derivative) to the curved spacetime. ^[2]

If there are only gravitational and electromagnetic forces and pressure force, then the following expression is valid:

~f^\nu = g^{\nu \lambda }\left(\Phi_{\lambda \mu } J^\mu + F_{\lambda \mu } j^\mu + f_{\lambda \mu } J^\mu \right) = -\nabla_\mu \left(U^{\nu \mu }+ W^{\nu \mu } + P^{\nu \mu } \right), \qquad\qquad (4)

where $~ g^{\nu \lambda}$ is the metric tensor, $~ j^\mu = \rho_{0q} U^\mu$ is the 4-vector of electromagnetic current density (4-current), $~\rho_{0q}$ is the density of electric charge of the matter unit in its rest reference frame, $~ f_{\lambda \mu }$ is the pressure field tensor, $~ U^{\nu \mu}$ is the gravitational stress-energy tensor, $~ W^{\nu \mu}$ is the electromagnetic stress-energy tensor, $~ P^{\nu \mu}$ is the pressure stress-energy tensor.

In some cases, instead of the mass 4-current the quantity $~ h^\lambda = \rho U^\lambda$ is used, where $~ \rho$ is the density of the moving matter in an arbitrary reference frame. The quantity $~ h^\lambda$ is not a 4-vector, since the mass density is not an invariant quantity in coordinate transformations. However, integrating over the moving volume of the matter unit due to the relation $~ dm= \rho_0 dV_0=\rho dV$ again we obtain the 4-force:

~ F^\lambda = \int {\frac {dh^\lambda}{ d\tau } dV}= \int {\frac{d(\rho U^\lambda ) }{d\tau} dV} = \frac {d}{ d\tau } \int {\rho U^\lambda dV} =\frac {d}{ d\tau } \int { U^\lambda d m } =\frac{dp^\lambda }{d\tau}.

In general relativity, it is believed that the stress-energy tensor of matter is determined by the expression $~ T^{\nu \lambda }= J^\nu U^\lambda$ , and for it $~ h^{\lambda} = \frac {T^{0 \lambda }}{c}$ , that is the quantity $~ h^\lambda = \rho U^\lambda$ consists of four timelike components of this tensor. The integral of these components over the moving volume gives respectively the energy (up to the constant, equal to $~ c$ ) and the momentum of the matter unit.

Instead of it, in the covariant theory of gravitation 4-momentum containing the energy and momentum is derived by using of Hamiltonian and not from the stress-energy tensors.

Components of 4-force density

The expression (4) for 4-force density can be divided into two parts, one of which will describe the bulk density of energy capacity, and the other describe total force density of available fields. We assume that speed of gravity is equal to the speed of light. In order do not depend on the metric tensor, we can write (4) with the lower, covariant index:

~ f_\lambda = \Phi_{\lambda \mu } J^\mu + F_{\lambda \mu } j^\mu + f_{\lambda \mu } J^\mu .

In this relation we make a transformation:

~ J^\mu = \rho_0 U^\mu = \rho_0 \frac {cdt}{ds} \frac {dx^\mu }{dt} = \rho \frac {dx^\mu }{dt} ,

where $~ ds$ denotes interval, $~ dt$ is the differential of coordinate time, $~ \rho= \rho_0 \frac {cdt}{ds}$ is the mass density of moving matter, four-dimensional quantity $~ \frac {dx^\mu }{dt}=(c, \mathbf{v} )$ consists of the time component equal to the speed of light $~ c$ , and the spatial component in the form of particle 3-velocity vector $~ \mathbf{v}$ .

Similarly, we write the charge 4-current through the charge density of moving matter $~ \rho_{q}= \rho_{0q} \frac {cdt}{ds}$ :

~ j^\mu = \rho_{0q} U^\mu = \rho_{0q} \frac {cdt}{ds} \frac {dx^\mu }{dt} = \rho_{q}\frac {dx^\mu }{dt}.

In addition, we express the tensors through their components, that is, the corresponding 3-vectors of the field strengths. Then the time component of the 4-force density with covariant index is:

~ f_0 = \frac {1}{ c }( \rho \mathbf{\Gamma} \cdot \mathbf{v}+ \rho_{q} \mathbf{E} \cdot \mathbf{v}+\rho \mathbf{C} \cdot \mathbf{v} ) ,

where $~ \mathbf{\Gamma}$ is the gravitational field strength, $~ \mathbf{E}$ is the electromagnetic field strength, $~ \mathbf{ C}$ is the pressure field strength.

The spatial component of covariant 4-force is the 3-vector $~ - \mathbf{f}$ , i.e. 4-force is as $~ f_\lambda = (f_0{,} -f_x{,}-f_y{,}-f_z),$

wherein the 3-force density is:

~ \mathbf{f}= \rho \mathbf{\Gamma}+ \rho [\mathbf{v} \times \mathbf{\Omega}] + \rho_{q}\mathbf{E}+ \rho_{q} [\mathbf{v} \times \mathbf{B}] + \rho \mathbf{C}+ \rho [\mathbf{v} \times \mathbf{I}],

where $~ \mathbf{\Omega}$ is the gravitational torsion field, $~ \mathbf{B}$ is the magnetic field, $~ \mathbf{ I }$ is the solenoidal vector of pressure field.

Expression for the covariant 4-force can be written in terms of the components of the acceleration tensor and covariant 4-acceleration. Similarly to (3) we have:

~f_\nu = \nabla^\mu B_{\nu \mu}= \rho_0 \frac{ Du_\nu } {D \tau }= \rho_0 u^\mu \nabla_\mu u_\nu =\rho_0 \frac{ du_\nu } {d \tau }- \rho_0 \Gamma^\lambda _{\mu \nu} u_\lambda u^\mu = - u_{\nu \lambda } J^\lambda =\rho_0 a_\nu ,

~f_0 = \rho_0 a_0 = - \frac {\rho }{ c } \mathbf{S} \cdot \mathbf{v},

~ \mathbf{f}= - \rho \mathbf{S} - \rho [\mathbf{v} \times \mathbf{N}] ,

where $~ a_0$ is the time component of 4-acceleration, $~ \mathbf{S}$ is the acceleration field strength, $~ \mathbf{ N }$ is the acceleration solenoidal vector.

Hence, the 4-acceleration with covariant index can be expressed through its scalar and vector components:

~ a_\nu = \frac {cdt}{ds}\left(-\frac {1}{c} \mathbf{S} \cdot \mathbf{v}{,} \qquad \mathbf{S}+[\mathbf{v} \times \mathbf{N}] \right).

In special relativity $~ \frac {cdt}{ds}= \gamma = \frac{1}{\sqrt{1-(\frac{v}{c})^2}},$ and substituting the vectors $~ \mathbf{S}$ and $~ \mathbf{ N }$ for a particle, for the covariant 4-acceleration we obtain the standard expression:

~ \mathbf {S} = - c^2 \nabla \gamma - \frac {\partial (\gamma \mathbf { v })}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times (\gamma \mathbf { v }).

~ a_\nu = \gamma \left( \frac {d(\gamma c)}{dt}{,} \qquad - \frac {d(\gamma \mathbf{v}) }{dt} \right).

For a body with a continuous distribution of matter vectors $~ \mathbf {S}$ and $~ \mathbf {N}$ are substantially different from the corresponding instantaneous vectors of specific particles in the vicinity of the observation point. These vectors represent the averaged value of 4-acceleration inside the bodies. In particular, within the bodies there is a 4-acceleration generated by the various forces in matter. A typical example are the space bodies, where the major forces are the force of gravity and the internal pressure generally oppositely directed. Upon rotation of the bodies the 4-force density, 4-acceleration, vectors $~ \mathbf {S}$ and $~ \mathbf {N}$ are functions not only of the radius, but the distance from the axis of rotation to the point of observation.

References

↑ Rindler, Wolfgang (1991). Introduction to Special Relativity (2nd). Oxford: Oxford University Press. ISBN 0-853971-853951-5.
1 2 Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0. (in Russian).
↑ Landau L.D., Lifshitz E.M. (1975). The Classical Theory of Fields. Vol. 2 (4th ed.). Butterworth-Heinemann. ISBN 978-0-750-62768-9.
↑ Fedosin S.G. About the cosmological constant, acceleration field, pressure field and energy. vixra.org, 5 Mar 2014.

External links

Four-force in Russian

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