The Heaviside vector is a vector
of energy flux density of gravitational field, which is a part of the gravitational stressenergy tensor in the Lorentzinvariant theory of gravitation.
The Heaviside vector can be determined by the cross product of two
vectors: ^{[1]}
where is
the vector of gravitational field strength
or gravitational acceleration, is
the gravitational constant, is
the gravitational torsion field or
torsion of the field, is
the speed of gravity.
The Heaviside vector magnitude is
equal to the amount of gravitational energy transferred through the unit area
which is normal to the energy flux per unit time. The minus sign in the
definition of means that the energy is transferred in the
direction opposite to the vector.
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To determine the vector of
momentum density of
gravitational field we must divide the Heaviside vector by the square of the
speed of gravitation propagation:
The
vector is a part of the gravitational
stressenergy tensor in
the form of three timelike components, when the indices of the tensor are i = 0, k = 1,2,3. To determine
the momentum of the gravitational field, we must integrate the vector over the moving space volume, occupied by the field,
taking into account the Lorentz contraction of this volume.
From the law of conservation of
energy and momentum of matter in a gravitational field in the Lorentzinvariant
theory of gravitation should Heaviside theorem:
where is
the mass current density.
According to this theorem, the
gravitational energy flowing into a certain volume in the form of the energy
flux density is
spent to increase the energy of the field
in this volume and to carry out the gravitational work as the product of
field strength and the mass current density .
Maxwelllike
gravitational equations, in the form of which the equations of
Lorentzinvariant theory of gravitation are presented, allow us to determine
the properties of plane gravitational waves from any point sources of field. In
a plane wave the vectors and are perpendicular to each other and to the
direction of the wave propagation, and the relation holds for the amplitudes.
If we assume that the wave
propagates in one direction, for the field strengths it can be written:
where and are
the angular frequency and the wave vector.
Then for the gravitational energy
flux it will be:
The average value over time and
space of the squared cosine is equal to ½, so:
In practice, it should be noted
that the pattern of waves in a gravitationally bound system of bodies has
rather quadrupole than dipole character, since in case of emission we should
take into account the contributions of all field sources. According to the
superposition principle we must first sum up at each point of space all the
existing fields and , find them as functions of coordinates
and time, and only then calculate with the obtained total magnitudes the energy
flux in the form of the Heaviside vector.
Suppose that there is a
gravitational energy flux falling on some unit material area absorbing all the
energy. The energy flux propagates at the speed
and transfers the momentum density of the
field
Then the maximum possible
gravitational pressure is:
where is the mean Heaviside vector and is the amplitude of the
gravitational field strength vector of incident plane gravitational wave. The
formula for the maximum pressure can be understood from the definition of
pressure as force , applied to the area , the definition of
force as the momentum of field during the time , provided that ; ; volume, absorbing
field momentum ; average density of
gravitational momentum :
Since the gravitational energy flux
passes through bodies with low absorption in them, to calculate the pressure it
is necessary to take the difference between the incident and outgoing energy
fluxes.
Representation of the
gravitational energy flux first appeared in the works by Oliver
Heaviside. ^{[2]} Previously the Umov vector for
the energy flux in substance (1874) and the Poynting vector for the
electromagnetic energy flux (1884) had been determined.
The Heaviside vector is in agreement with that used by Krumm and Bedford, ^{[3]} by
Fedosin, ^{[4]} by H. Behera
and P. C. Naik.^{[5]}