The gravitational field strength is a vector physical quantity which
characterizes the gravitational field at a given point and is numerically equal
to the ratio of the gravitational force acting on a stationary test particle, placed at a given point of the field,
to the gravitational mass of this particle:
This definition reduces the field 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 vector field, its strength
depends on the time and the coordinates of the point in space where the
field strength is measured:
The gravitational field strength and the gravitational torsion field describe gravitational field in the Lorentzinvariant
theory of gravitation and obey the Maxwelllike
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 the
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].
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If we write the relations of the
Lorentzinvariant theory of gravitation (LITG) in terms of 4vectors 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
stressenergy tensor and of Lagrangian
for a single particle in gravitational field, and the scalar and vector
potentials of the gravitational field form the gravitational fourpotential. ^{[2]} We can also calculate with and the following: the vector of energy flux density of gravitational field or
the Heaviside vector the energy density of gravitational field
and the vector of the momentum density of
gravitational field :
where is the propagation speed of
gravitational effect (speed of gravity), is the gravitational
constant.
The total force, at which the
gravitational field acts on a test particle, is expressed by the following
formula:
where is the mass of the particle, is the particle velocity, 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 particle and on the torsion
field acting on the particle. It is assumed that and 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 , understood as the gravitational force
acting on a unit of moving volume, is part of the spacelike component of the
4vector of gravitational force density (see fourforce).
In the covariant theory of gravitation
this 4vector is given by:
where is
the metric tensor, is the gravitational
tensor, is the 4vector of mass current density , is
the gravitational stressenergy tensor.
The expression for the 4vector
of the gravitational force density in the Lorentzinvariant theory of
gravitation can be represented through the gravitational field strength:
where is
the mass current density, the gravitational force density is given by
is
the Lorentz factor, is
the mass density in the comoving reference frame.
The formula shows that the
product 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.
The Lorentzcovariant 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:
where: is the mass current density, is the
density of moving mass, 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 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 gravitational field, and other effects. Since the
time and space measurements are carried out by electromagnetic waves, then in
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:
Repeating the same actions for
the second and third equations, we obtain the wave equation for the torsion
field:
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 the field sources influence due to limited
speed of gravitational propagation.
The third Heaviside equation
leads to the possibility of gravitational
induction, when the timevarying 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 strength
can be expressed through the scalar potential
as well as through the vector potential of
the gravitational field as follows:
The torsion field depends only on
the vector potential, since:
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 gravistatics the vector potential of gravitational field is neglected due to the absence or smallness of
translational or rotational motion of masses, creating the field, because 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:
where is
called the gravistatic potential to emphasize the static case of the
gravitational field. In gravistatics 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 the gravitational field strength does not
depend on time, the vector potential and the torsion field are
zero, in the Heaviside's equations only one equation is left:
If in (1) we use the
relation then we obtain the equation that has the form
of the Poisson equation:
Outside the bodies the mass
density at rest is zero, and the equation for the gravistatic
potential becomes the Laplace equation:
Poisson and Laplace 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]}
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:
where 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 outside the integral sign and then to
integrate only the surface area. In particular, the area of the spherical
surface ,
and for the field strength at the distance 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 )
we obtain:
This formula remains valid
regardless of the radius of the body of the spherical shape, as long as this
radius does not exceed ,
that is, when the field strength is sought outside of the body. For the point particle with mass we
can assume that the distance is measured from this particle.
In case when the divergence
theorem is applied to the spherical surface inside the body with 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 inside the spherical
surface with the radius :
For a sphere with uniform mass
density the mass is , which gives for the field strength:
In the center of the sphere,
where the field strength is zero, and with the radius ,
where is
the radius of the sphere, the strength reaches the maximum amplitude.
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 . If the source of the gravitational field
is the uniform spherical body with the gravitational mass , then according to the Newton's law of
universal gravitation outside the body:
where: is
the radius vector from the center of the body to the point in space, where the
gravitational field strength is
determined, and the minus sign indicates that the force and the field strength are directed opposite to the radius vector .
In the classical theory, the
scalar potential of gravitational field outside a spherical body is:
Using the formula we find the gravitational field strength in the vector form:
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:
i.e. the gravitational field
strength is equal in number (and in size) to the free fall acceleration of
the test particle in this field.