# Physics/Essays/Fedosin/Gravitational field strength

The gravitational field strength is a vector physical quantity which characterizes gravitational field at a given point and is numerically equal to the ratio of gravitational force $~\mathbf {F}$ acting on a stationary test particle, placed at a given point of the field, to gravitational mass $~M$ of this particle:
    $~\mathbf {\Gamma } ={\frac {\mathbf {F} }{M}}.$
This reduces the strength to the gravitational force acting on a unit mass. There is another definition, where the field strength is found by space and time derivatives of the gravitational field potentials or by the components of gravitational tensor.
Since the gravitational field is a vector field, its strength $~\mathbf {\Gamma }$ depends on time and coordinates of a point in space where the field strength is measured:
    $~\mathbf {\Gamma } =\mathbf {\Gamma } (x,y,z,t).$
The gravitational field strength $~\mathbf {\Gamma }$ and gravitational torsion field $~\mathbf {\Omega }$ describe gravitational field in Lorentz-invariant theory of gravitation and obey the Maxwell-like gravitational equations.
In general relativity, the gravitational field strength is called the strength of gravitoelectric field, and the torsion field corresponds to the gravitomagnetic field. In the weak gravitational field limit, the specified quantities are included in equations of gravitoelectromagnetism.
The gravitational field strength in the international system of units is measured in meters per second squared [m/s$^2$] or in Newtons per kilogram [N/kg].

## Gravitational field strength in Lorentz-invariant theory of gravitation

If we write relations of Lorentz-invariant theory of gravitation (LITG) in terms of 4-vectors and tensors, we find that the vector of gravitational field strength $~\mathbf {\Gamma }$ and the vector of torsion field $~\mathbf {\Omega }$ make up gravitational tensor and are part of gravitational stress-energy tensor and of Lagrangian for a single particle in gravitational field, and the scalar and vector potentials of gravitational field form the gravitational four-potential. We can also calculate with $~\mathbf {\Gamma }$ and $~\mathbf {\Omega }$ the energy density of gravitational field $~u$ and the vector of energy flux density of gravitational field or the Heaviside vector $~\mathbf {H}$:
    $~u=-{\frac {1}{8\pi G}}\left(\Gamma ^{2}+c^{2}\Omega ^{2}\right),$
    $~\mathbf {H} =-{\frac {c^{2}}{4\pi G}}[\mathbf {\Gamma } \times \mathbf {\Omega } ],$
where $~c$ is the speed of light, $~G$ is the gravitational constant.

### Gravitational force

The total force, at which gravitational field acts on a test particle, is expressed by the following formula:
    $~\mathbf {F} =M\left(\mathbf {\Gamma } +\mathbf {V} \times \mathbf {\Omega } \right),$
where $~M$ is the mass of particle, $~\mathbf {V}$ is the particle velocity.
In this formula, the first term of the force is proportional to the gravitational field strength, and the second term of the force depends on velocity of the particle and on torsion field acting on the particle. It is assumed that $~\mathbf {\Gamma }$ and $~\mathbf {\Omega }$ are the strength and the torsion field from external gravitational field, averaged over the volume of the particle, and the proper field of the particle can be neglected due to its smallness.
To calculate the total force acting on extended body, within the limits of which the strength and the torsion of gravitational field change on a significant scale, we perform partition of the body into small parts and calculate for each part their force and then make the vector summation of all these forces.
The density of force vector $~\mathbf {f}$, understood as gravitational force acting on a unit of moving volume, is part of space-like component of 4-vector of gravitational force density (see four-force). In covariant theory of gravitation this 4-vector is given by:
    $~f^{\nu }=g^{\nu \lambda }\Phi _{\lambda \mu }J^{\mu }=-\nabla _{\mu }U^{\nu \mu },$
where $~g^{\nu \lambda }$ is the metric tensor, $~\Phi _{\lambda \mu }$ is the gravitational tensor, $~J^{\mu }$ is the 4-vector of mass current density , $~U^{\nu \mu }$ is the gravitational stress-energy tensor.
The expression for 4-vector of gravitational force density in the Lorentz-invariant theory of gravitation can be represented through the gravitational field strength:
    $~f^{\nu }=({\frac {\mathbf {\Gamma } \cdot \mathbf {J} }{c}},\mathbf {f} ),$
where $~\mathbf {J}$ is mass current density, gravitational force density is given by $~\mathbf {f} =\gamma \rho _{0}(\mathbf {\Gamma } +\mathbf {V} \times \mathbf {\Omega } )=\rho \mathbf {\Gamma } +\mathbf {J} \times \mathbf {\Omega } ,$
$~\gamma ={\frac {1}{\sqrt {1-({\frac {v}{c}})^{2}}}}$ is the Lorentz factor, $~\rho _{0}$ is the mass density in the comoving reference frame.
The formula shows that the product $~\mathbf {\Gamma } \cdot \mathbf {J}$ is equal to power of work done by the gravitational force per unit volume, and torsion field is not included in this product and does not perform work on matter.

### Heaviside's equations

The Lorentz-covariant equations of gravitation in inertial reference frames can be found in works by Oliver Heaviside. They are four vector differential equations, three of which include the vector of gravitational field strength:
    $~\nabla \cdot \mathbf {\Gamma } =-4\pi G\rho ,$
    $~\nabla \cdot \mathbf {\Omega } =0,$
    $~\nabla \times \mathbf {\Gamma } =-{\frac {\partial \mathbf {\Omega } }{\partial t}},$
    $~\nabla \times \mathbf {\Omega } ={\frac {1}{c^{2}}}\left(-4\pi G\mathbf {J} +{\frac {\partial \mathbf {\Gamma } }{\partial t}}\right),$
where: $~\mathbf {J} =\rho \mathbf {V}$ is the mass current density, $~\rho =\gamma \rho _{0}$ is the density of moving mass, $~\mathbf {V}$ is velocity of the mass flux creating the gravitational field and the torsion field.
These four equations fully describe gravitational field for the cases when the field is not large enough to affect the propagation of electromagnetic waves, their speed and frequency. In these equations, the sources of gravitational field are the mass density and the mass currents, and the formula for gravitational force, in turn, shows how the field acts on matter.
If gravitational field is large in size, its influence on electromagnetic processes leads to gravitational redshift, time dilation, deviation of motion of electromagnetic waves near the sources of gravitational field, and other effects. Since the time and space measurements are carried out by electromagnetic waves, then in a gravitational field the body sizes could be smaller for remote observer, and the rate of time could slow down. Similar effects are taken into account by introducing the spacetime metric which depends on coordinates and time. Therefore, in case of strong gravitational field more general equations of covariant theory of gravitation are used instead of the above equations, or the equations of general relativity, in which there is the metric tensor.
If we take gradient of the first Heaviside equation and the partial derivative with respect to time of the fourth equation, as a result we can obtain inhomogeneous wave equation for gravitational field strength:
    $~\nabla ^{2}\mathbf {\Gamma } -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {\Gamma } }{\partial t^{2}}}=-4\pi G\nabla \rho -{\frac {4\pi G}{c^{2}}}{\frac {\partial \mathbf {J} }{\partial t}}.$
Repeating the same actions for the second and third equations, we obtain the wave equation for torsion field:
    $~\nabla ^{2}\mathbf {\Omega } -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {\Omega } }{\partial t^{2}}}={\frac {4\pi G}{c^{2}}}\nabla \times \mathbf {J} .$
The presence of wave equations suggests that strength and torsion of gravitational field at any point can be found as the sums (integrals) of a set of separate simple waves, making their contribution to the total field, where each contribution should be calculated taking into account the delay of field sources influence due to limited speed of gravitational propagation.
The third Heaviside equation leads to possibility of gravitational induction, when a time-varying torsion field passing through some circuit, or a change of the circuit area at constant torsion field, generate circular gravitational field strength along the circumference of this circuit.

### Gravitational field potentials

The gravitational field strength can be expressed through scalar potential $~\psi$ as well as through vector potential of gravitational field $~\mathbf {D}$ as follows:
    $~\mathbf {\Gamma } =-\nabla \psi -{\frac {\partial \mathbf {D} }{\partial t}}.$
The torsion field depends only on the vector potential, since:
    $~\mathbf {\Omega } =\nabla \times \mathbf {D} .$

### Gravistatics

The simplest case for studying properties of gravitation is the case of interaction of bodies which are fixed or moving at low speed. In gravistatics the vector potential $~\mathbf {D}$ of gravitational field is neglected due to absence or smallness of translational or rotational motion of masses, creating the field, because $~\mathbf {D}$ is proportional to velocity of the masses. As a result the torsion field also becomes small, which is calculated as the curl of the vector potential. In this approximation, we can write:
    $~\mathbf {\Gamma } =-\nabla \psi ,$
where $~\psi$ is called the gravitational potential to emphasize the static case of gravitational field. In gravistatics the gravitational field strength is a potential vector field, that is, the field that depends only on gradient of some function, in this case of the scalar potential.
Provided that in the system under consideration there are no mass currents and therefore $~\mathbf {J} =0,$ the gravitational field strength does not depend on time, the vector potential $~\mathbf {D}$ and the torsion field $~\mathbf {\Omega }$ are zero, in the Heaviside's equations only one equation is left:
    $~\nabla \cdot \mathbf {\Gamma } =-4\pi G\rho _{0}.\qquad \qquad (1)$
If in (1) we use the relation $~\mathbf {\Gamma } =-\nabla \psi ,$ then we obtain the equation that has the form of the Poisson equation:
    $~\Delta \psi =4\pi G\rho _{0}.$
Outside the bodies the mass density at rest is zero, $~\rho _{0}=0,$ and the equation for the gravistatic potential becomes the Laplace equation:
    $~\Delta \psi =0.$
Poisson and Laplace's equations are valid both for the potential of a point particle and for the sum of the potentials of the set of particles, which makes it possible to use the superposition principle to calculate total potential and strength of total gravitational field at any point of a system. However, it follows from the modernized Le Sage’s theory of gravitation that in strong fields the superposition principle is violated because of the exponential dependence of graviton fluxes in matter on the distance covered.

#### Application of divergence theorem

Equation (1) can be integrated over arbitrary space volume and then we can apply the divergence theorem, which substitutes the integral of divergence of vector function over a certain volume with the integral of the flux of this vector function over a closed surface around the given volume:
    $~\oint \limits _{S}\mathbf {\Gamma } \cdot d\mathbf {S} =-4\pi GM,$
where $~M$ is total mass of matter inside the surface. The resulting expression is often called Gauss's law for gravity.
In many cases, it turns out that the flux of gravitational field strength on the surface is constant, which allows us to move the field strength $~\mathbf {\Gamma }$ outside the integral sign and then to integrate only the surface area. In particular, the area of spherical surface $~S=4\pi R^{2}$, and for the field strength at the distance $~R$ from the center of the sphere (and from the center of the body of the spherical shape with the proper radius not more than the radius of the surface $~R$ ) we obtain:
    $~\Gamma =-{\frac {4\pi GM}{S}}=-{\frac {GM}{R^{2}}}.$
This formula remains valid regardless of radius of body of spherical shape, as long as this radius does not exceed $~R$, that is, when the field strength $~\Gamma$ is sought outside of the body. For the point particle with mass $~M$ we can assume that the distance $~R$ is measured from this particle.
In case when the divergence theorem is applied to a spherical surface inside a body with the spherically symmetric arrangement of mass, the theorem implies that the gravitational field strength inside the body depends only on the mass of the body $~M(r)$ inside the spherical surface with the radius $~r$:
    $~\Gamma =-{\frac {GM(r)}{r^{2}}}.$
For a sphere with uniform mass density the mass is $~M(r)={\frac {4\pi r^{3}\rho _{0}}{3}}$, which gives for the field strength:
    $~\Gamma =-{\frac {4\pi G\rho _{0}r}{3}}.$
In the center of the sphere, where $~r=0,$ the field strength is zero, and with the radius $~r=a$, where $~a$ is the radius of the sphere, the strength reaches the maximum amplitude.

#### Classical theory of gravitation

Expression for gravitational field strength of a point particle can also be obtained from the Newton law for gravitational force acting on a test particle with the mass $~m$. If the source of the gravitational field is uniform spherical body with gravitational mass $~M$, then according to the Newton's law of universal gravitation outside the body:
    $\Gamma ={\frac {F}{m}}=-{\frac {\frac {GMm}{R^{2}}}{m}}=-{\frac {GM}{R^{2}}},$
where: $~R$ is radius vector from the center of the body to point in space, where the gravitational field strength $~\Gamma$ is determined, and the minus sign indicates that the force $~F$ and the field strength are directed opposite to the radius vector $~R$.
In the classical theory, the scalar potential of gravitational field outside a spherical body is:
    $~\psi =-{\frac {GM}{R}}.$
Using the formula $~\mathbf {\Gamma } =-\nabla \psi$ we find the gravitational field strength in vector form:
    $~\mathbf {\Gamma } =-{\frac {GM}{R^{2}}}{\frac {\mathbf {R} }{R}}.$
If we consider the equivalence principle of gravitational and inertial forces to be valid, in which gravitational mass of a test particle is equal to inertial mass of this particle in Newton's second law, then we obtain the following:
    $F=mg={\frac {GMm}{R^{2}}}\Rightarrow g={\frac {GM}{R^{2}}}=\Gamma ,$
i.e. the gravitational field strength is equal to the free-fall acceleration $~g$ of test particle in this field.