The Pioneer Anomaly
The solution to the Pioneer Anomaly can be derived directly from the Hypergeometrical Universe topology. It is our current understanding that only matter curves spacetime, this means that locally the curvature is quite well defined by the inexistence of localized matter and equal to zero, that is, without the obvious occurrence of matter we believe that we live in a locally flat space. The addition of another spatial dimension in the Hypergeometrical Universe theory changes things, since there are now many curvatures to talk about.
Let's see what happens when one measures the speed of light within our standard paradigm. The measure time delay between shooting the laser pulse and measuring the reflection is equal to twice the distance divided by the speed of light.
For the reflected radiation bouncing back from the mirror at rest,
Now if we place the mirror in motion we arrive at the standard derivation of the Doppler Effect:
The Doppler shift is given by:
The derivation is different if one allows v to vary:
Resulting:
The derivation is different still if one allows both c and v to vary:
Resulting:
This means that both accelerations on the speed of light or the actual speed of the Pioneer spacecraft would contribute to the Doppler Shift. This also means that if you are not considering that the effective speed of light vary with distance, you will be missing the contribution
.
Under the current paradigm, the time it takes for light to bounce back from a mirror is given by 2L/c and doesn't change with the distance L. This considers that light and the mirror are traveling or existing in the same dimension (3D) which is not the Hypergeometrical Universe Paradigm.
There, light (a spatial modulation of the source of a dilaton field) travels within the 5D spacetime (4D spatial manifold). Anything else just flows within the 3D Hyperspherical Universe, which itself travels through the 4D spatial manifold.
If you consider our topology, it is clear that there is a curvature and that is equal to 1/R0 and it is independent of the existence of matter locally. Remember that a 5D Spacetime has more than one radius of curvature. The RX radius of curvature is R0, and so it is the ФX radius of curvature. The situation changes one consider the τX (τ is the local or proper time). The τX radius of curvature will depend upon which laws do you use to describe your Universe (Gauss Electrostatic Law, Newton's Gravitational Law). If you were to use the Quantum Lagrangian Principle to describe dilator dynamics, then you could use a Cartesian reference system for the local proper time. The usage of Newton's, Gauss' or similar laws to describe dilator dynamics makes it necessary to make local proper spacetime metric to be Lorentzian.
Now let's get down to the Pioneer Anomaly and how do we measure how long the Universe is alive from the observed Pioneer Spacecraft deceleration.
The first thing to consider is what would be the perceived speed of light deceleration resulting from the geometry. To do this, we need to consider how much light has to travel back as a function of the farthest distance traveled by the spacecraft.
This figure shows what would be the path traversed by each pulse emitted by the Pioneer spacecraft. The angle of the light ray is always 45 degrees. The leftmost radial line represents Earth position. This means that when Pioneer is far from Earth by 45 degrees or PI/4*R0, light will never come back (the light ray will be parallel to the Earth radial line. Two parallel lines never meet.
Let's say that at time zero Pioneer is at R0 from the 4D Center of the Universe and L from Earth and let's derive the deceleration from the simple geometry.
Let R(t) be the Radius of the Universe at the time t:
Where time is being measured after a light pulse is emitted from the Pioneer while it is at distance from the 4D Center of the Universe equal to R0.
The Cosmological Angle associated with Pioneer at time t given by:
for calculating the deceleration of the speed of light
And
for calculating the deceleration of the speed of the Pioneer spacecraft
The angle alpha is called Cosmological Angle and it is measured from the actual Center of the Universe. The time that measures the expansion of the Universe is the Cosmological Time φ, but for quasi-relaxed fabric of space (low velocities), the Cosmological Time is a reasonable approximation to the proper time.
The equation for the photon trajectory leaving the Spacecraft and reaching Earth later is given by:
Where y(t,t') is the horizontal distance represented in the figure above of the reflected light beam as a function of time t', which starts counting at the reflection moment. The time t refers to the t governing the expansion of the Universe from some given initial condition (R0, L).
Equating this equation to zero and solving for t', one obtains the time it takes for light to come from Pioneer to Earth at any given time or Pioneer position.
The perceived distance traversed is given by x=c.t':
Taking the second derivative, expanding in Taylor Series and simplifying to obtain the acceleration as:
The observed acceleration is 8.75e-10 m/s2, the speed of light = 299792458 m / s. This yields a value for R0= 2.05E+26 meters. Since the Pioneer velocity is much smaller than c and the second term can be neglected.
Since the 4D observed velocity is 
m this means that the Universe is 
years old or 1.5364E+10 years or
15.364 Billion Years Old…J
PerceiveAge=15.364 Billion Years Old…:) (using the Hubble Constant as a measure of the age of the Universe).
The important result is that the Universe is actually older that the age perceived by inspecting the Hubble Constant:
RealAge=15.36*sqrt(2)= 21.65 Billion Years Old:)
One salient point is that the Hypergeometrical Lightspeed Expanding Shockwave Universe expands at the observed speed of light, this means that the material world travels slower than the dilaton field, by a factor of cos(45) or 
.
This conclusion is very important because it breaks the symmetry between dilators and dilatons, in the sense that they do not travel at the same speed. On the other hand, this velocity relationship is strictly necessary for the Quantum Lagrangian Principle to make all dilators to surf the surrounding dilaton field.
Dilators interact with other dilators through retarded potentials or retarded dilaton fields.
The Beginning of Times
At time zero, the initial macroscopic metric fluctuation (the one that followed the dimensional transition) decayed into a huge number of dilators. Many recombined and released Gamma Rays. The initial radiation blast accelerated the extremely small Universe both radially and tangentially with the same strength. Dilators flowed with the retarded dilaton field in all directions of the 4D spatial manifold, creating the 3D Lightspeed Expanding Shockwave Universe. The 4D Big Bang also synchronized all dilators' spinning. Only dilators that were flush (in perfect overlap) with the 3D Hypersphere at time zero, were accelerated into c.
Up to now, I considered that dilators and dilatons always traveled at the only allowed speed (c).
Of course, I noticed that dilators have inertia with respect to lateral motion but I thought that the radial motion was the natural motion of a metric deformation, thus requiring no initial push to be achieved.
If I accept that the dilaton speed or light speed is actually larger than the observed speed of light then the picture associated with the initial Big Bang changes from a simple decaying process into a tangential explosion which set the 3D Hypersphere to travel at the observed speed of light.
It is clear that a dilaton field cannot accelerate dilators to its speed (sqr(2)*c) because interaction has to occur through RETARDED POTENTIALS. Retarded Potentials always propagate at c (Lightspeed) due to the initial geometry.
The interaction of dilators with the retarded dilaton field from other dilators produces a distribution of tangentially dilators traveling at c which is also a distribution of radially traveling dilators at that same speed. In other words, a lightspeed traveling Hyperspherical Universe and an sqr(2)*c dilaton-dilator interaction form a self-consistent solution to the problem.
This salient point can be summarized as:
The speed of light is SQR(2) times C or light is faster than the speed of light..J
This is due to motion be part of a coherence that was placed in motion at the beginning of times through the interaction of near field dilatons. The interference pattern of near field radially propagating homogeneous dilatons travels along 45 degrees. This is another very salient point.
Cheers,
MP