# Gravitational field strength

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 the gravitational tensor. [1]

Since the gravitational field is a field, its strength ${\displaystyle ~\mathbf {\Gamma } }$ on the time and the coordinates of the point in space where the field strength is measured:

${\displaystyle ~\mathbf {\Gamma } =\mathbf {\Gamma } (x,y,z,t).}$

The gravitational field strength ${\displaystyle ~\mathbf {\Gamma } }$ and the gravitational torsion field ${\displaystyle ~\mathbf {\Omega } }$ describe gravitational field in the Lorentz-invariant theory of gravitation and obey the Maxwell-like gravitational equations.

In general relativity, the gravitational field strength is called the strength of the gravitoelectric field, and the torsion field corresponds to the gravitomagnetic field. In the weak gravitational field limit, the specified quantities are included in the equations of gravitoelectromagnetism.

The gravitational field strength in the international system of units is measured in meters per second squared [m/s2] or in Newtons per kilogram [N/kg].

## The gravitational field strength in Lorentz-invariant theory of gravitation

If we write the relations of the Lorentz-invariant theory of gravitation (LITG) in terms of 4-vectors and tensors, we find that the vector of gravitational field strength and the vector of torsion field make up the gravitational tensor and are part of the gravitational stress-energy tensor and of Lagrangian for a single particle in gravitational field, and the scalar and vector potentials of the gravitational field form the gravitational four-potential. [2] We can also calculate with ${\displaystyle ~\mathbf {\Gamma } }$  and ${\displaystyle ~\mathbf {\Omega } }$  the following: the vector of energy flux density of gravitational field or the Heaviside vector ${\displaystyle ~\mathbf {H} ,}$  the energy density of gravitational field ${\displaystyle ~u}$  and the vector of the momentum density of gravitational field ${\displaystyle ~\mathbf {P_{g}} }$ :

${\displaystyle ~\mathbf {H} =-{\frac {c_{g}^{2}}{4\pi G}}[\mathbf {\Gamma } \times \mathbf {\Omega } ],}$
${\displaystyle ~u=-{\frac {1}{8\pi G}}\left(\Gamma ^{2}+c_{g}^{2}\Omega ^{2}\right),}$
${\displaystyle ~\mathbf {P_{g}} ={\frac {1}{c_{g}^{2}}}\mathbf {H} ,}$

where ${\displaystyle ~c_{g}}$  is the propagation speed of gravitational effect (speed of gravity), ${\displaystyle ~G}$  is the gravitational constant.

### The gravitational force

The total force, at which the gravitational field acts on a test particle, is expressed by the following formula:

${\displaystyle ~\mathbf {F} =M\left(\mathbf {\Gamma } +\mathbf {V} \times \mathbf {\Omega } \right),}$

where: ${\displaystyle ~M}$  is the mass of the particle, ${\displaystyle ~\mathbf {V} }$  is the particle velocity, ${\displaystyle ~\mathbf {\Omega } }$  is the gravitational torsion field vector.

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 the velocity of the particle and on the torsion field acting on the particle. It is assumed that ${\displaystyle ~\mathbf {\Gamma } }$  and ${\displaystyle ~\mathbf {\Omega } }$  are the strength and the torsion field from the 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 the extended body, within the limits of which the strength and the torsion of the 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 the force vector ${\displaystyle ~\mathbf {f} }$ , understood as the gravitational force acting on a unit of moving volume, is part of the space-like component of the 4-vector of gravitational force density (see four-force). In the covariant theory of gravitation this 4-vector is given by:

${\displaystyle ~f^{\nu }=g^{\nu \lambda }\Phi _{\lambda \mu }J^{\mu }=-\nabla _{\mu }U^{\nu \mu },}$

where ${\displaystyle ~g^{\nu \lambda }}$  is the metric tensor, ${\displaystyle ~\Phi _{\lambda \mu }}$  is the gravitational tensor, ${\displaystyle ~J^{\mu }}$  is the 4-vector of mass current density , ${\displaystyle ~U^{\nu \mu }}$  is the gravitational stress-energy tensor.

The expression for the 4-vector of the gravitational force density in the Lorentz-invariant theory of gravitation can be represented through the gravitational field strength:

${\displaystyle ~f^{\nu }=({\frac {\mathbf {\Gamma } \cdot \mathbf {J} }{c_{g}}},\mathbf {f} ),}$

where ${\displaystyle ~\mathbf {J} }$  is the mass current density, the gravitational force density is given by ${\displaystyle ~\mathbf {f} =\gamma \rho _{0}(\mathbf {\Gamma } +\mathbf {V} \times \mathbf {\Omega } )=\rho \mathbf {\Gamma } +\mathbf {J} \times \mathbf {\Omega } ,}$

${\displaystyle ~\gamma ={\frac {1}{\sqrt {1-({\frac {v}{c_{g}}})^{2}}}}}$  is the Lorentz factor, ${\displaystyle ~\rho _{0}}$  is the mass density in the comoving reference frame.

The formula shows that the product ${\displaystyle ~\mathbf {\Gamma } \cdot \mathbf {J} }$  is equal to the power of work done by the gravitational force per unit volume, and the torsion field is not included in this product and does not perform work on the matter.

### Heaviside's equations

The Lorentz-covariant equations of gravitation in inertial reference frames can be found in the works by Oliver Heaviside. [3] They are four vector differential equations, three of which include the vector of gravitational field strength:

${\displaystyle ~\nabla \cdot \mathbf {\Gamma } =-4\pi G\rho ,}$
${\displaystyle ~\nabla \cdot \mathbf {\Omega } =0,}$
${\displaystyle ~\nabla \times \mathbf {\Gamma } =-{\frac {\partial \mathbf {\Omega } }{\partial t}},}$
${\displaystyle ~\nabla \times \mathbf {\Omega } ={\frac {1}{c_{g}^{2}}}\left(-4\pi G\mathbf {J} +{\frac {\partial \mathbf {\Gamma } }{\partial t}}\right),}$

where: ${\displaystyle ~\mathbf {J} =\rho \mathbf {V} }$  is the mass current density, ${\displaystyle ~\rho =\gamma \rho _{0}}$  is the density of moving mass, ${\displaystyle ~\mathbf {V} }$  is the velocity of the mass flux creating the gravitational field and the torsion.

These four equations fully describe the 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 the gravitational field are the mass density and the mass currents, and the formula for the gravitational force, in turn, shows how the field acts on matter.

If the 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 the 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 the observer, and the rate of time could slow down. Similar effects are taken into account by introducing the spacetime metric which depends on the coordinates and time. Therefore, in case of strong gravitational field more general equations of the 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 the 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 the gravitational field strength:

${\displaystyle ~\nabla ^{2}\mathbf {\Gamma } -{\frac {1}{c_{g}^{2}}}{\frac {\partial ^{2}\mathbf {\Gamma } }{\partial t^{2}}}=-4\pi G\nabla \rho -{\frac {4\pi G}{c_{g}^{2}}}{\frac {\partial \mathbf {J} }{\partial t}}.}$

Repeating the same actions for the second and third equations, we obtain the wave equation for the torsion field:

${\displaystyle ~\nabla ^{2}\mathbf {\Omega } -{\frac {1}{c_{g}^{2}}}{\frac {\partial ^{2}\mathbf {\Omega } }{\partial t^{2}}}={\frac {4\pi G}{c_{g}^{2}}}\nabla \times \mathbf {J} .}$

The presence of wave equations suggests that strength and torsion of the 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 the field sources influence due to limited speed of gravitational propagation.

The third Heaviside equation leads to the possibility of gravitational induction, when the time-varying torsion field passing through some circuit, or the change of the circuit area at the constant torsion field, generate circular gravitational field strength along the circumference of this circuit.

### The gravitational field potentials

The gravitational field strength can be expressed through the scalar potential ${\displaystyle ~\psi }$  as well as through the vector potential of the gravitational field ${\displaystyle ~\mathbf {D} }$  as follows:

${\displaystyle ~\mathbf {\Gamma } =-\nabla \psi -{\frac {\partial \mathbf {D} }{\partial t}}.}$

The torsion field depends only on the vector potential, since:

${\displaystyle ~\mathbf {\Omega } =\nabla \times \mathbf {D} .}$

### Gravistatics

The simplest case for studying the properties of gravitation is the case of interaction of bodies which are fixed or moving at low speed. In gravitates the vector potential ${\displaystyle ~\mathbf {D} }$  of gravitational field is neglected due to the absence or smallness of translational or rotational motion of masses, creating the field, because ${\displaystyle ~\mathbf {D} }$  is proportional to the 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:

${\displaystyle ~\mathbf {\Gamma } =-\nabla \psi ,}$

where ${\displaystyle ~\psi }$  is called the gravitational potential to emphasize the static case of the gravitational field. In gravitates the gravitational field strength is the potential vector field, that is, the field that depends only on the 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 ${\displaystyle ~\mathbf {J} =0,}$  the gravitational field strength does not depend on time, the vector potential ${\displaystyle ~\mathbf {D} }$  and the torsion field ${\displaystyle ~\mathbf {\Omega } }$  are zero, in the Heaviside's equations only one equation is left:

${\displaystyle ~\nabla \cdot \mathbf {\Gamma } =-4\pi G\rho _{0}.\qquad \qquad (1)}$

If in (1) we use the relation ${\displaystyle ~\mathbf {\Gamma } =-\nabla \psi ,}$  then we obtain the equation that has the form of the Poisson equation:

${\displaystyle ~\Delta \psi =4\pi G\rho _{0}.}$

Outside the bodies the mass density at rest is zero, ${\displaystyle ~\rho _{0}=0,}$  and the equation for the gravistatic potential becomes the Laplace equation:

${\displaystyle ~\Delta \psi =0.}$

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 the total potential and the strength of the total gravitational field at any point of the 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 the graviton fluxes in the matter on the distance covered. [4]

#### Application of the divergence theorem

Equation (1) can be integrated over arbitrary space volume and then we can apply the divergence theorem, which substitutes the integral of the divergence of the vector function over a certain volume with the integral of the flux of this vector function over a closed surface around the given volume:

${\displaystyle ~\oint \limits _{S}\mathbf {\Gamma } \cdot d\mathbf {S} =-4\pi GM,}$

where ${\displaystyle ~M}$  is the total mass of the 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 ${\displaystyle ~\mathbf {\Gamma } }$  outside the integral sign and then to integrate only the surface area. In particular, the area of the spherical surface ${\displaystyle ~S=4\pi R^{2}}$ , and for the field strength at the distance ${\displaystyle ~R}$  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 ${\displaystyle ~R}$  ) we obtain:

${\displaystyle ~\Gamma =-{\frac {4\pi GM}{S}}=-{\frac {GM}{R^{2}}}.}$

This formula remains valid regardless of the radius of the body of the spherical shape, as long as this radius does not exceed ${\displaystyle ~R}$ , that is, when the field strength ${\displaystyle ~\Gamma }$  is sought outside of the body. For the point particle with mass ${\displaystyle ~M}$  we can assume that the distance ${\displaystyle ~R}$  is measured from this article.

In case when the divergence theorem is applied to the spherical surface inside the body with the spherically symmetric arrangement of the mass, the theorem implies that the gravitational field strength inside the body depends only on the mass of the body ${\displaystyle ~M(r)}$  inside the spherical surface with the radius ${\displaystyle ~r}$ :

${\displaystyle ~\Gamma =-{\frac {GM(r)}{r^{2}}}.}$

For a sphere with uniform mass density the mass is ${\displaystyle ~M(r)={\frac {4\pi r^{3}\rho _{0}}{3}}}$ , which gives for the field strength:

${\displaystyle ~\Gamma =-{\frac {4\pi G\rho _{0}r}{3}}.}$

In the center of the sphere, where ${\displaystyle ~r=0,}$  the field strength is zero, and with the radius ${\displaystyle ~r=a}$ , where ${\displaystyle ~a}$  is the radius of the sphere, the strength reaches the maximum amplitude.

#### The classical theory of gravitation

The expression for the gravitational field strength of a point particle can also be obtained from the Newton law for the gravitational force acting on a test particle with the mass ${\displaystyle ~m}$ . If the source of the gravitational field is the uniform spherical body with the gravitational mass ${\displaystyle ~M}$ , then according to the Newton's law of universal gravitation outside the body:

${\displaystyle \Gamma ={\frac {F}{m}}=-{\frac {\frac {GMm}{R^{2}}}{m}}=-{\frac {GM}{R^{2}}},}$

where: ${\displaystyle ~R}$  is the radius vector from the centre of the body to the point in space, where the gravitational field strength ${\displaystyle ~\Gamma }$  is determined, and the minus sign indicates that the force ${\displaystyle ~F}$  and the field strength are directed opposite to the radius vector ${\displaystyle ~R}$ .

In the classical theory, the scalar potential of gravitational field outside a spherical body is:

${\displaystyle ~\psi =-{\frac {GM}{R}}.}$

Using the formula ${\displaystyle ~\mathbf {\Gamma } =-\nabla \psi }$  we find the gravitational field strength in the vector form:

${\displaystyle ~\mathbf {\Gamma } =-{\frac {GM}{R^{2}}}{\frac {\mathbf {R} }{R}}.}$

If we consider the equivalence principle in which gravitational mass of a test particle is equal to the inertial mass of this particle in the Newton second law to be valid, then we obtain the following:

${\displaystyle F=mg={\frac {GMm}{R^{2}}}\Rightarrow g={\frac {GM}{R^{2}}}=\Gamma ,}$

i.e. the gravitational field strength is equal in number (and in size) to the free-fall acceleration ${\displaystyle ~g}$  of the test particle in this field.