1 CHAPTER NINE

TRIPLE UNIFICATION: FLUID GRAVITATION

In this chapter the ECE2 theory is used to unify the field equations of fluid dynamics

and gravitation to produce the subject of fluid dynamics, which can be described as the effect

of the fluid vacuum or aether on gravitational theory. To start the chapter it is shown that

fluid dynamics can describe all the main features of a whirlpool galaxy, so is preferred both

to the Newton theory and the Einstein theory.

In fluid dynamics the acceleration due to gravity is defined as:

where E is the Kambe electric field of fluid dynamics, used in chapter eight:

where v is the velocity field of the spacetime, aether or vacuum, h is the enthalpy per unit

mass, and is the scalar potential defined by:

The vacuum magnetic field is the vorticity:

and the vacuum law:

follows from Eqs. ( ) to ( ). This is equivalent to the Faraday law of induction

of electrodynamics.

The Newtonian acceleration due to gravity is defined by:

where M is a gravitating mass, G is Newton’s constant and r the distance between M and an

orbiting mass m. The acceleration due to gravity g induces E in the vacuum, and

conversely the vacuum velocity field induces g in matter.

There is an exact analogy between the ECE2 gravitational field equations:

and the ECE2 field equations of fluid dynamics:

Both sets of equations are ECE2 covariant. Here is the gravitomagnetic field, g is the

gravitational field, is the mass density, is defined in terms of the spin

connection, is the current of mass density, is the scalar potential of ECE2

gravitation, and is the vector potential of ECE2 gravitation. In the ECE2 equations of

fluid dynamics, E is the fluid electric field, B is the fluid magnetic field, is the

fluid charge, J is the fluid current, and the assumed constant speed of sound.

It follows that:

and

and that:

From the gravitational field equation:

it follows that:

so material mass density is:

and originates in the vacuum charge:

In general:

so any spacetime velocity field gives rise to material mass density. Conversely any mass

density induces a spacetime velocity field.

The vacuum wave equation of chapter eight and UFT349 ff. is:

given the vacuum Lorenz condition:

which is a particular solution of vacuum continuity equation:

in which the vacuum current is:

The d’Alembertian in Eq. ( ) is :

The Newtonian solution is:

as in Note 358(3). This solution is discussed numerically and graphically later on in this

chapter. To exemplify the elegance of fluid gravitation consider as in Note 358(4) on

www.aias.us the constant vacuum angular momentum:

which can be defined for any central force between m and M. Here r is the position vector

and v the velocity field. The reduced mass is defined by:

of m orbiting M. The subscript F for any quantity denotes the fluid vacuum. For a planar

orbit:

and:

so the vacuum velocity field is:

where:

and:

This is a divergenceless velocity field:

The material gravitomagnetic field is then:

and is perpendicular to the plane of the orbit. In a whirlpool galaxy for example the

gravitomagnetic field is perpendicular to the plane of the galaxy and the gravitational field

between a star of mass m of the whirlpool galaxy and its central mass M is:

In Cartesian coordinates:

so:

Now assume that

and it follows that:

where:

Finally use:

to find an inverse cube law between m and M:

the force being:

From the vacuum Binet equation:

the orbit of m around M is the hyperbolic spiral:

In plane polar coordinates ( ) the velocity of a star in the whirlpool galaxy is:

From lagrangian theory:

so the velocity of the star is:

From Eqs. ( ) and ( ) it follows that:

so the vacuum potential is constant:

in a whirlpool galaxy. It follows from the wave equation ( ) that:

The vacuum charge of the whirlpool galaxy is:

so from Eq. ( ):

and there is a very large mass at the centre of the galaxy, as observed.

The spacetime current ( ) that gives rise to a whirlpool galaxy is:

if E is time independent. Therefore with this assumption:

and J is proportional to v .

Numerical and graphical analysis of these characteristics are developed later in

this chapter.

In the subject of fluid gravitation. Newtonian gravitation produces a rich structure

in the fluid vacuum, a structure which can be illustrated with the velocity field, vorticity,

charge and static current using gnuplot graphics. A new law of planar orbital theory can be

inferred and the Newtonian acceleration due to gravity becomes the convective derivative of

the orbital linear velocity.

The velocity field induced by the Newtonian acceleration due to

gravity g is defined by:

where:

Similarly, the static electric field strength in volts per metre induces vacuum structure in an

exactly analogous manner:

In general there are many solutions of Eq. ( ), one of which is developed as follows

from Note 359(6) on www.aias.us . :

where:

These components are graphed later on in this chapter using gnuplot, and are richly

structured. The above velocity field gives:

where:

The three vacuum charges are:

and sum to zero in this solution:

The vacuum charges exhibit a swirling motion as shown by gnuplot later on in this chapter.

The vacuum vorticities are:

and self consistently obey the equations:

These are also graphed later in this chapter using gnuplot, and also exhibit a swirling motion.

We therefore arrive at a new law of orbits the Newtonian acceleration due to

gravity

between m and M is the convective derivative of the orbital linear velocity.

By definition:

In ECE2 electromagnetism they become the homogeneous field equations:

The vacuum current is defined by:

where is the assumed constant speed of sound. Therefore the inhomogeneous field

equations of the vacuum are:

and

If it is assumed that:

then:

and using Eq. ( ):

which again exhibit a swirling motion.

The fundamental philosophy of fluid gravitation is:

so the familiar Newtonian g induces

in the vacuum. The converse is also true as developed in chapter eight.

In order to apply fluid gravitation to planar orbits consider the linear velocity in

plane polar coordinates:

where the unit vectors are:

and

In fluid gravitation there is a new and general relation between the orbital velocity v and the

Newtonian acceleration g:

From Eqs. ( ) and ( ) it follows that:

and

From Eqs. ( ) and ( ) it follows that:

therefore for any planar orbit:

and

For an elliptical planar orbit for example:

and:

where:

is the semi major axis, the half right latitude and the eccentricity. In plane

polar coordinates:

and in Cartesian coordinates:

where the semi minor axis is:

In the case of the ellipse:

and:

It follows that for the planar elliptical orbit:

and:

These properties are graphed and analyzed later in this section.

For circular orbits:

so:

Therefore a new and general theory of orbits has been inferred.

Using fluid gravitation it can be shown as follows that all observable orbits can be

expressed as a generally covariant inverse square law:

Therefore g is the convective derivative of v. For planar orbits, the inverse square law is:

and from Eq. ( ) the orbital velocity is:

In plane polar coordinates:

and for elliptical orbits:

so for the elliptical orbit:

and the acceleration due to gravity is:

Eq. ( ) shows that the acceleration due to gravity can be expressed as:

For all planar orbits:

and

In the case of the whirlpool galaxy:

and it follows that:

where the angular momentum L is a constant of motion. It follows as in Note 360(3) that:

and

For the precessing orbit in a plane in a simple model:

where:

It follows as in Note 360(4) that:

and

so X and Y can be graphed. They are illustrated later on in this chapter.

As shown in Note 360(5):

so for the precessing orbit:

ECE2 dynamics with the convective derivative can be developed as in UFT361 by

expressing the velocity field as:

where cylindrical polar coordinates have been used. In classical dynamics:

The material derivative is:

in which:

and:

is a property of the coordinate system. By construction:

so it follows that:

The second matrix has the antisymmetric structure of a rotation generator. The derivative

( ) is a special case of the Cartan derivative:

in which the spin connection in plane polar coordinates is:

Therefore:

The velocity vector in plane polar coordinates is:

so:

and:

where the angular velocity of the rotating frame is:

Therefore the spin connection components are:

and in terms of unit vectors:

Eq. ( ) is the following covariant derivative of Cartan geoemetry

So for any acceleration:

The individual covariant derivatives are:

and

Therefore:

and:

There are new accelerations:

which are absent from classical dynamics, in which:

On the right hand side of Eq. ( ) appear the Newtonian acceleration:

the centrifugal acceleration:

and the Coriolis accelerations:

The use of the convective derivative leads to the accelerations ( ) which

occur in addition to the fundamental accelerations of classical dynamics. They are the result

of replacing v(t) of classical dynamics by the velocity field:

These new accelerations are interpreted later on in this chapter and developed numerically.

These concepts can be used as in UFT362 to investigate the effect of the vacuum on

orbital theory. In classical dynamics there is no such effect. Consider the convective

derivative of an vector field:

in plane polar coordinates. In Eq. ( ) the velocity field is:

In plane polar coordinates Eq. ( ) becomes:

where the Leibnitz Theorem has been used. For plane polar coordinates {1 -12}:

so the convective derivative ( ) is

where:

In component format:

so:

In classical dynamics:

and there is no functional dependence of F on r(t) and . In classical dynamics

therefore:

and this result is assumed implicitly in classical orbital theory and cosmology.

The assumption ( ) simplifies Eq. ( ) to:

which is the ECE2 Cartan derivative with spin connection:

which is the rotation generator of the axes of the plane polar system. In classical dynamics, if

represents the position vector r(t), then the orbital velocity is given by the Cartan

derivative:

In vector notation the orbital velocity is:

If F represents the time dependent velocity vector v(t) of classical dynamics then the orbital

acceleration is given by:

In vector notation:

In the UFT papers on www.aias.us it has been shown that the Coriolis accelerations vanish

for any planar orbit:

so the Leibnitz equation:

is inferred for any planar orbit.

In classical dynamics the vacuum is a “nothingness”, but in fluid gravitation it is

richly structured as argued earlier in this chapter. It follows that the velocity ( ) is

generalized to:

and that the acceleration ( ) is generalized to:

in an orbit influenced by the vacuum. Different types of spin connection components appear

in Eqs. ( ) and ( ). In Eq. ( ):

and in Eq. ( ):

For example the orbital velocity components of classical dynamics are generalized to:

and:

so that the velocity vector becomes:

and its square becomes:

Orbital precession can be explained straightforwardly as an effect of the fluid

vacuum.

Consider the position vector R of an element of a fluid:

it follows that the velocity field of the fluid is:

and in component format:

In plane polar coordinates:

so:

The relevant spin connection matrix is therefore:

with components:

The velocity field components are therefore:

and:

The hamiltonian and lagrangian are therefore:

and:

where U is the potential energy. It is known in the solar system that precession is a very tiny

effect, so:

In contrast to the above analysis, classical dynamics is defined by:

i.e. by the convective derivative of the position r(t) of a particle rather than the position

R ( r , t) of a fluid element. So in classical dynamics:

In component format Eq. ( ) is:

which gives the Coriolis velocity:

Q. E. D.

The Euler Lagrange equations of the system are:

and:

From Eq. ( ) a new force law can be found using the lagrangian:

and is:

In comparison, the force law of a conic section orbit is:

so the orbit is changed by and .

Assume for the sake of analytical tractability that:

and the lagrangian simplifies to:

The constant angular momentum can be found from the Euler Lagrange equation:

and is a constant of motion:

In the approximation ( ) the force law ( ) becomes:

Using the Binet variable:

it follows that:

and can be interpreted as the Binet equation of an orbit in a fluid aether or vacuum. It reduces

the Binet equation of classical dynamics when:

If the orbit is a conic section:

it follows from Eq. ( ) that its force law is:

where:

y =

It is known from UFT193 that the force law of a precessing ellipse modelled by

is:

from the Binet equation of classical dynamics

Therefore:

and

To contemporary precision in astronomy:

so:

Therefore orbital precession is due to the effect of the fluid dynamic function

, which is the rate of displacement of a posiiton element R ( ) of a fluid

dynamic background, or vacuum. The spin connection makes the orbit precess. The conic

section orbit ( ) used in the Binet equation of fluid dynamics ( ) is exactly

equivalent to the use of the precessing orbit ( ) in the Binet equation of classical

dynamics. In both cases the law of attraction is the sum of terms inverse squared and inverse

cubed in r. The obsolete Einstein theory incorrectly claims to give a precessing orbit with a

sum of inverse squared and inverse fourth power terms.