Acceleration field

Acceleration field is a two-component vector field, describing in a covariant way the 4-acceleration of individual particles and the 4-acceleration that occurs in systems with multiple closely interacting particles. The acceleration field is a component of the general field, which is represented in the Lagrangian and Hamiltonian of an arbitrary physical system by the term with the energy of particles’ motion and the term with the field energy. ^[1] The acceleration field enters into the equation of motion through the acceleration tensor and into the equation for the metric through the acceleration stress-energy tensor.

The acceleration field was presented by Sergey Fedosin within the framework of the metric theory of relativity and covariant theory of gravitation, and the equations of this field were obtained as a consequence of the principle of least action. ^[2] ^[3]

Mathematical description

The 4-potential of the acceleration field is expressed in terms of the scalar $~ \vartheta$ and vector $~ \mathbf {U}$ potentials:

~u_\mu = \left(\frac {\vartheta }{c},- \mathbf {U} \right) .

The antisymmetric acceleration tensor is calculated with the help of the 4-curl of the 4-potential:

~ u_{\mu \nu} = \nabla_\mu u_\nu - \nabla_\nu u_\mu = \partial_\mu u_\nu - \partial_\nu u_\mu .

The acceleration tensor components are the components of the field strength $~\mathbf {S}$ and the components of the solenoidal vector $~\mathbf {N}$ :

~ u_{\mu \nu}= \begin{vmatrix} 0 & \frac {S_x}{ c} & \frac {S_y}{ c} & \frac {S_z}{ c} \\ -\frac {S_x}{ c} & 0 & - N_{z} & N_{y} \\ -\frac {S_y}{ c} & N_{z} & 0 & -N_{x} \\ -\frac {S_z}{ c}& -N_{y} & N_{x} & 0 \end{vmatrix}.

We obtain the following:

~ \mathbf {S} = - \nabla \vartheta - \frac {\partial \mathbf {U}}{\partial t},\qquad\qquad \mathbf {N} = \nabla \times \mathbf {U}.\qquad\qquad (1)

Action, Lagrangian and energy

In the covariant theory of gravitation the 4-potential $~u_\mu$ of the acceleration field is part of the 4-potential of the general field $~ s_\mu$ , which is the sum of the 4-potentials of particular fields, such as the electromagnetic and gravitational fields, acceleration field, pressure field, dissipation field, strong interaction field, weak interaction field and other vector fields, acting on the matter and its particles. All of these fields are somehow represented in the matter, so that the 4-potential $~ s_\mu$ cannot consist of only one 4-potential $~u_\mu$ . The energy density of interaction of the general field and the matter is given by the product of the 4-potential of the general field and the mass 4-current: $~ s_\mu J^\mu$ . We obtain the general field tensor from the 4-potential of the general field, using the 4-curl:

~ s_{\mu \nu} =\nabla_\mu s_\nu - \nabla_\nu s_\mu.

The tensor invariant in the form $~ s_{\mu \nu} s^{\mu \nu}$ is up to a constant factor proportional to the energy density of the general field. As a result, the action function, which contains the scalar curvature $~R$ and the cosmological constant $~ \Lambda$ , is given by the expression: ^[1]

~S =\int {L dt}=\int (kR-2k \Lambda - \frac {1}{c}s_\mu J^\mu - \frac {c}{16 \pi \varpi} s_{\mu\nu}s^{\mu\nu} ) \sqrt {-g}d\Sigma,

where $~L$ is the Lagrange function or Lagrangian; $~dt$ is the time differential of the coordinate reference system; $~k$ and $~ \varpi$ are the constants to be determined; $~c$ is the speed of light as a measure of the propagation speed of the electromagnetic and gravitational interactions; $~\sqrt {-g}d\Sigma= \sqrt {-g} c dt dx^1 dx^2 dx^3$ is the invariant 4-volume expressed in terms of the differential of the time coordinate $~ dx^0=cdt$ , the product $~ dx^1 dx^2 dx^3$ of differentials of the space coordinates and the square root $~\sqrt {-g}$ of the determinant $~g$ of the metric tensor, taken with a negative sign.

The variation of the action function gives the general field equations, the four-dimensional equation of motion and the equation for the metric. Since the acceleration field is the general field component, then from the general field equations the corresponding equations of the acceleration field are derived.

Given the gauge condition of the cosmological constant in the form

~ c k \Lambda = - s_\mu J^\mu ,

is met, the system energy does not depend on the term with the scalar curvature and is uniquely determined: ^[3]

~E = \int {( s_0 J^0 + \frac {c^2 }{16 \pi \varpi } s_{ \mu\nu} s^{ \mu\nu} ) \sqrt {-g} dx^1 dx^2 dx^3},

where $~ s_0$ and $~ J^0$ denote the time components of the 4-vectors $~ s_{\mu }$ and $~ J^{\mu }$ .

The system’s 4-momentum is given by the formula:

~p^\mu = \left( \frac {E}{c}{,} \mathbf {p}\right) = \left( \frac {E}{c}{,} \frac {E}{c^2}\mathbf {v} \right) ,

where $~ \mathbf {p}$ and $~ \mathbf {v}$ denote the system’s momentum and the velocity of the system’s center of mass.

Equations

The four-dimensional equations of the acceleration field are similar in their form to Maxwell equations and are as follows:

\nabla_\sigma u_{\mu \nu}+\nabla_\mu u_{\nu \sigma}+\nabla_\nu u_{\sigma \mu}=\frac{\partial u_{\mu \nu}}{\partial x^\sigma} + \frac{\partial u_{\nu \sigma}}{\partial x^\mu} + \frac{\partial u_{\sigma \mu}}{\partial x^\nu} = 0.

~ \nabla_\nu u^{\mu \nu} = - \frac{4 \pi \eta }{c^2} J^\mu,

where $J^\mu = \rho_{0} u^\mu$ is the mass 4-current, $\rho_{0}$ is the mass density in the co-moving reference frame, $u^\mu$ is the 4-velocity of the matter unit, $~ \eta$ is a constant, which is determined in each problem, and it is supposed that there is an equilibrium between all fields in the observed physical system.

The gauge condition of the 4-potential of the acceleration field:

~ \nabla^\mu u_{\mu} =0 .

In Minkowski space of the special theory of relativity the form of the acceleration field equations is simplified and they can be expressed in terms of the field strength $~\mathbf {S}$ and the solenoidal vector $~\mathbf {N}$ :

~ \nabla \cdot \mathbf{S} = 4 \pi \eta \gamma \rho_0, \qquad\qquad \nabla \times \mathbf{N} = \frac{1}{c^2} \left( 4 \pi \eta \mathbf{J} + \frac{\partial \mathbf{S}} {\partial t} \right),

~ \nabla \times \mathbf{S} = - \frac{\partial \mathbf{N} } {\partial t} , \qquad\qquad \nabla \cdot \mathbf{N} = 0 .

where $~ \gamma = \frac {1}{\sqrt{1 - {v^2 \over c^2}}}$ is the Lorentz factor, $~ \mathbf{J}= \gamma \rho_0 \mathbf{v }$ is the mass current density, $~ \mathbf{v }$ is the velocity of the matter unit.

Using as well the gauge condition in the form of $~ \partial^\mu u_{\mu} =0$ and relation (1), we can obtain from the field equations the wave equations for the acceleration field potentials:

~ \frac {1}{c^2}\frac{\partial^2 \vartheta }{\partial t^2 } -\Delta \vartheta = 4 \pi \eta \gamma \rho_0, \qquad\qquad (2)

~ \frac {1}{c^2}\frac{\partial^2 \mathbf{U} }{\partial t^2 } -\Delta \mathbf{U}= \frac {4 \pi \eta}{c^2} \mathbf{J}. \qquad\qquad (3)

The equation of motion of the matter unit in the general field is given by the formula:

~ s_{\mu \nu} J^\nu =0

Since $~ J^\nu = \rho_0 u^\nu$ , and the general field tensor is expressed in terms of the tensors of particular fields, then the equation of motion can be represented with the help of these tensors and the 4-acceleration $~ a_\mu$ :

~ - u_{\mu \nu} J^\nu = \rho_0 a_\mu =F_{\mu \nu} j^\nu + \Phi_{\mu \nu} J^\nu + f_{\mu \nu} J^\nu + h_{\mu \nu} J^\nu + \gamma_{\mu \nu} J^\nu + w_{\mu \nu} J^\nu .

Here $~ F_{\mu \nu}$ is the electromagnetic tensor, $~ j^\nu$ is the charge 4-current, $~ \Phi_{\mu \nu}$ is the gravitational tensor, $~ f_{\mu \nu}$ is the pressure field tensor, $~ h_{\mu \nu}$ is the dissipation field tensor, $~ \gamma_{\mu \nu}$ is the strong interaction field tensor, $~ w_{\mu \nu}$ is the weak interaction field tensor.

The stress-energy tensor

The acceleration stress-energy tensor is calculated with the help of the acceleration tensor:

~ B^{ik} = \frac{c^2} {4 \pi \eta }\left( -g^{im} u_{n m} u^{n k}+ \frac{1} {4} g^{ik} u_{m r} u^{m r}\right)

We find as part of the tensor $~ B^{ik}$ the 3-vector of the energy-momentum flux $~\mathbf {K}$ , which is similar in its meaning to the Poynting vector and the Heaviside vector. The vector $~\mathbf {K}$ can be represented through the vector product of the field strength $~ \mathbf {S}$ and the solenoidal vector $~ \mathbf {N}$ :

~ \mathbf {K}=c B^{0i} = \frac {c^2}{4 \pi \eta }[\mathbf {S}\times \mathbf {N}],

here the index is $~ i=1,2,3.$

The covariant derivative of the stress-energy tensor of the acceleration field specifies the 4-force density:

~ f^\alpha = \nabla_\beta B^{\alpha \beta} = - u^{ \alpha}_{k} J^k = - \rho_0 u^{ \alpha}_{k} u^k = \rho_0 a^\alpha = \rho_0 \frac {Du^\alpha }{D \tau},\qquad \qquad (4)

where $~ D \tau$ denotes the proper time differential in the curved spacetime.

The stress-energy tensor of the acceleration field is part of the stress-energy tensor of the general field $~ T^{ik}$ , but in the general case the tensor $~ T^{ik}$ contains also the cross-terms with the products of strengths and solenoidal vectors of particular fields:

~ T^{ik}= k_1W^{ik}+ k_2U^{ik}+ k_3B^{ik}+ k_4P^{ik} + k_5Q^{ik}+ k_6 L^{ik}+ k_7A^{ik}+ cross \quad terms,

where $~ k_1{,} k_2{,} k_3{,} k_4{,} k_5{,} k_6{,} k_7$ are certain coefficients, $~ W^{ik}$ is the electromagnetic stress–energy tensor, $~ U^{ik}$ is the gravitational stress-energy tensor, $~ P^{ik}$ is the pressure stress-energy tensor, $~ Q^{ik}$ is the dissipation stress-energy tensor, $~ L^{ik}$ is the strong interaction stress-energy tensor, $~ A^{ik}$ is the weak interaction stress-energy tensor.

Through the tensor $~ T^{ik}$ the stress-energy tensor of the acceleration field enters into the equation for the metric:

~ R^{ik} - \frac{1} {4 }g^{ik}R = \frac{8 \pi G \beta }{ c^4} T^{ik},

where $~ R^{ik}$ is the Ricci tensor, $~ G$ is the gravitational constant, $~ \beta$ is a certain constant, and the gauge condition of the cosmological constant is used.

Specific solutions for the acceleration field functions

One particle

The four-potential of any vector field for a single particle can be represented as: ^[2]

~ L_\mu = \frac { k_f \varepsilon_p }{\rho_0 c^2} u_\mu ,

where $~ k_f = \frac {\rho_0}{\rho_{0q}}$ for electromagnetic field and $~ k_f = 1$ for other fields, $~ \rho_{0}$ and $~\rho_{0q}$ are the mass density and accordingly charge density in comoving reference frame, $~ \varepsilon_p$ is the field energy density of the particle, $~ u_\mu$ is the covariant four-velocity.

For the acceleration field $~ \varepsilon_p = p_0 c^2$ , $~ k_f = 1$ , and according to the definition, for the four-potential of the acceleration field of one particle we have the following:

~ L_\mu = u_\mu,

i. e. for the single particle the field 4-potential is the 4-velocity with a covariant index. In the special relativity (SR) we can write:

~u_\mu = \left( \frac {\vartheta }{c},- \mathbf {U} \right) = \left(\gamma c, - \gamma \mathbf {v} \right).

The acceleration tensor components according to (1) will equal:

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

Since in the equation of motion for the 4-acceleration with a covariant index $~ a_\mu$ the relation holds

~ \rho_0 a_\mu = \rho_0 \frac {Du_\mu }{D \tau}= - u_{\mu \nu} J^\nu = - \rho_0 u_{\mu \nu} u^\nu,

then in SR we obtain the following:

~ \frac {Du_\mu }{D \tau}=\gamma \frac {du_\mu }{dt}, \qquad\qquad u^\nu =\left(\gamma c, \gamma \mathbf {v} \right),

and the equations for the Lorentz factor $~ \gamma$ and for the 3-acceleration $~ a= \frac {d \mathbf { v }}{dt}$ :

~ \frac {d \gamma }{dt}= - \frac {1 }{c^2} \mathbf {S}\cdot \mathbf { v }, \qquad (5) \qquad \frac {d (\gamma \mathbf { v })}{dt}= \gamma \mathbf { a }+ \frac {d \gamma}{dt}\mathbf { v } = - \mathbf {S}- [\mathbf { v }\times \mathbf {N}]. \qquad (6)

Substituting the quantity $~ \frac {d \gamma }{dt}$ from equation (5) to (6), multiplying equation (6) by the velocity $~ \mathbf { v },$ taking into account relation $~\gamma^{-2}=1 - {v^2 \over c^2},$ we find the well-known expression for the derivative of the Lorentz factor using the scalar product of the velocity and acceleration in SR:

~ - \mathbf {S}\cdot \mathbf { v }=\gamma^3 \mathbf {v}\cdot \mathbf { a }=c^2 \frac {d \gamma }{dt}.

We can prove the validity of equation (6) by substituting in it the expression for the strength and solenoidal vector:

~ \frac {d (\gamma \mathbf { v })}{dt}= c^2 \nabla \gamma + \frac {\partial (\gamma \mathbf { v })}{\partial t} - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] . \qquad\qquad (7)

Indeed, the use of the material derivative gives the following:

~ \frac {d (\gamma \mathbf { v })}{dt}= \frac {\partial (\gamma \mathbf { v })}{\partial t} + (\mathbf { v } \cdot \nabla) (\gamma \mathbf { v }) = \frac {\partial (\gamma \mathbf { v })}{\partial t}+\gamma (\mathbf { v } \cdot \nabla) \mathbf { v } + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .

In addition

~ - \mathbf { v }\times [ \nabla \times (\gamma \mathbf { v }) ] = - \gamma \mathbf { v }\times [ \nabla \times \mathbf { v } ] - \mathbf { v }\times [ \nabla \gamma \times \mathbf { v }] = -\frac {\gamma }{2} \nabla v^2 + \gamma (\mathbf { v } \cdot \nabla) \mathbf { v } - v^2 \nabla \gamma + \mathbf { v } (\mathbf { v } \cdot \nabla\gamma) .

Substituting these relations in (7), taking into account the expression $~ \gamma^{-2}=1 - {v^2 \over c^2},$ we obtain the identity:

~ c^2 \nabla \gamma - \frac {\gamma }{2} \nabla v^2 - v^2 \nabla \gamma =0 .

If the components of the particle velocity are the functions of time and they do not directly depend on the space coordinates, then the solenoidal vector $~ \mathbf { N }$ vanishes in such a motion.

The system of particles

Due to interaction of a number of particles with each other by means of various fields, including interaction at a distance without direct contact, the acceleration field in the matter changes and is different from the acceleration field of individual particles at the observation point. The acceleration field in the system of particles is given by the strength and the solenoidal vector, which represent the typical average characteristics of the matter motion. For example, in a gravitationally bound system there is a radial gradient of the vector $~ \mathbf { S },$ and if the system is moving or rotating, there is a vector $~ \mathbf { N }.$ From (4) there follows the general expression for the 4-acceleration with covariant index:

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

where $~ ds$ denotes a four-dimensional space-time interval. For a stationary case, when the potentials of the acceleration field are independent of time, under the assumption that $~ \vartheta = \gamma c^2,$ wave equation (2) for the scalar potential is transformed into the equation:

~ \Delta \gamma= - \frac {4 \pi \eta \gamma \rho_0}{c^2}.

The solution of this equation for a fixed sphere with the particles randomly moving in it has the form: ^[4]

~\gamma= \frac {c \gamma_c }{r \sqrt {4 \pi \eta \rho_0} } \sin \left(\frac {r}{c}\sqrt {4 \pi \eta \rho_0} \right) \approx \gamma_c - \frac {2 \pi \eta \rho_0 r^2 \gamma_c }{3 c^2}.

where $~ \gamma_c = \frac {1}{\sqrt{1 - {v^2_c \over c^2}}}$ is the Lorentz factor for the velocities $~ v_c$ of the particles in the center of the sphere, and due to the smallness of the argument the sine is expanded to the second order terms. From the formula it follows that the average velocities of the particles are maximal in the center and decrease when approaching the surface of the sphere.

In such a system, the scalar potential $~ \vartheta$ becomes the function of the radius, and the vector potential $~ \mathbf {U}$ and the solenoidal vector $~ \mathbf { N }$ are equal to zero. The acceleration field strength $~\mathbf {S}$ is found with the help of (1). Then we can calculate all the functions of the acceleration field, including the energy of particles in this field and the energy of the acceleration field itself. ^[5] For cosmic bodies the main contribution to the 4-acceleration in the matter makes the gravitational force and the pressure field. At the same time the relativistic rest energy of the system is automatically derived, taking into account the motion of particles inside the sphere. For the system of particles with the acceleration field, pressure field, gravitational and electromagnetic fields the given approach allowed solving the 4/3 problem and showed where and in what form the energy of the system is contained. The relation for the acceleration field constant in this problem was found:

~\eta = 3G- \frac {3q^2}{4 \pi \varepsilon_0 m^2 },

where $~ \varepsilon_0$ is the vacuum permittivity, $~q$ and $~m$ are the total charge and mass of the system.

The solution of the wave equation for the acceleration field within the system results in temperature distribution according to the formula: ^[4]

~ T=T_c - \frac {\eta M_p M(r)}{3kr} ,

where $~ T_c$ is the temperature in the center, $~ M_p$ is the mass of the particle, for which the mass of the proton is taken (for systems which are based on hydrogen or nucleons in atomic nuclei), $~ M(r)$ is the mass of the system within the current radius $~ r$ , $~ k$ is the Boltzmann constant.

This dependence is well satisfied for a variety of space objects, including gas clouds and Bok globules, the Earth, the Sun and neutron stars.

The wave equation (3) for the vector potential of the acceleration field was used to represent the relativistic equation of the fluid’s motion in the form of the Navier–Stokes equations in hydrodynamics and to describe the motion of the viscous compressible and charged fluid. ^[6]

Other approaches

Studying the Lorentz covariance of the 4-force, Friedman and Scarr found incomplete covariance of the expression for the 4-force in the form $~ F^\mu = \frac {d p^\mu }{d \tau } .$ ^[7] This led them to conclude that the 4-acceleration in SR must be expressed with the help of a certain antisymmetric tensor $~ A^\mu_\nu$ :

~c \frac { d u^\mu }{d \tau } = A^\mu_\nu u^\nu .

Based on the analysis of various types of motion, they estimated the required values of the acceleration tensor components, thereby giving indirect definition to this tensor. From comparison with (4) it follows that the tensor $~ A^\mu_\nu$ is up to a sign and a constant factor equal to the acceleration tensor $~ u^{ \alpha}_{k}$ with mixed indices.

Mashhoon and Muench considered transformation of inertial reference frames, co-moving with the accelerated reference frame, and obtained the relation: ^[8]

~c \frac { d \lambda_\alpha }{d \tau } = \Phi_\alpha^\beta \lambda_\beta.

The tensor $~ \Phi_\alpha^\beta$ has the same properties as the acceleration tensor $~ u_\alpha^\beta.$

References

1 2 Fedosin S.G. The Concept of the General Force Vector Field. OALib Journal, Vol. 3, P. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459.
1 2 Fedosin S.G. The procedure of finding the stress-energy tensor and vector field equations of any form. Advanced Studies in Theoretical Physics, Vol. 8, No. 18, 771 - 779 (2014). http://dx.doi.org/10.12988/astp.2014.47101.
1 2 Fedosin S.G. About the cosmological constant, acceleration field, pressure field and energy. vixra.org, 5 Mar 2014.
1 2 Fedosin S.G. The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field. American Journal of Modern Physics. Vol. 3, No. 4, P. 152-167 (2014). http://dx.doi.org/10.11648/j.ajmp.20140304.12.
↑ Fedosin S.G. Relativistic Energy and Mass in the Weak Field Limit. Jordan Journal of Physics. Vol. 8, No. 1, P. 1-16 (2015).
↑ Fedosin S.G. Four-Dimensional Equation of Motion for Viscous Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure Field and Dissipation Field. International Journal of Thermodynamics. Vol. 18, No. 1, P. 13-24 (2015). http://dx.doi.org/10.5541/ijot.5000034003.
↑ Yaakov Friedman and Tzvi Scarr. Covariant Uniform Acceleration. Journal of Physics: Conference Series Vol. 437 (2013) 012009 doi:10.1088/1742-6596/437/1/012009.
↑ Bahram Mashhoon and Uwe Muench. Length measurement in accelerated systems. Annalen der Physik. Vol. 11, Issue 7, P. 532–547, 2002.

External links

Acceleration field in Russian

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