Albert Zotkin

Reflections on Science

Self-similar Doppler shift: An example of correct derivation that Einstein Relativity was preventing us to break through

Posted by Albert Zotkin on September 5, 2016

In this short paper I present a simple but correct derivation of the complete Doppler shift effect. I will prove that Doppler effect of electromagnetic waves is a self-similar process, and therefore Special Relativity, that pretends to be complete for every inertial system, is excluded from that self-similarity of the Doppler effect.

I. The derivation: Successive stages towards self-similarity

Let a source of electromagnetic waves emit at a frequency f₀, and let a detector move away from it at a speed v, as messured in the reference frame of the source. If that speed v is close to c (the speed of light in a vacumm), then Special Relativity claims the meassured frequency in that detector is relativistic, so:

\displaystyle    f = f_0 \sqrt{\cfrac{1- \frac{v}{c}}{1+\frac{v}{c}}}    (1)

But, under classical mechanics assumptions, that Doppler shift is computed as follows:

\displaystyle   f = f_0 \left (1- \frac{v}{c}\right )     (2)
Let us split that speed into two, v = u + u, and place an antenna in the intermediate reference frame that moves at u with respect to the source. The detector is placed in a final third collinear reference frame, which is moving away at a speed u with respect to the second one. Then, this detector meassures a double Doppler effect, which in classical mechanics is:

\displaystyle   f_1 = f_0 \left (1- \frac{u}{c}\right )\left (1- \frac{u}{c}\right ) =f_0 \left (1- \frac{v}{2c}\right )^2    (3)
therefore, we observe that f₁ is not equal to f in [2], and that means the Doppler effect under classical mechanics assumption is not self-similar. so it is incomplete. But, let’s see now how the prediction under Special Relativity assumptions is for that double Doppler:

\displaystyle    f_2 = f_0 \sqrt{\cfrac{1- \frac{u}{c}}{1+\frac{u}{c}}} \sqrt{\cfrac{1- \frac{u}{c}}{1+\frac{u}{c}}} = f_0 \cfrac{1- \frac{v}{2c}}{1+\frac{v}{2c}}     (4)
We can see that f₂ is not equal to f in [1] either, so Special Relativity predicts a non self-similar Doppler, an incorrect one, since that theory pretends to be complete for every inertial system. A complete Doppler must be strictly self-similar. So, how can we design a model that can predict a complete Doppler? Let’s do the following:

Above, we have divided the speed v into two halves, u. Let’s now divide v into n parts, v = u + u + … + u. Then, we compound those $n$ parts, under classical mechanics, assuming there is an antenna in each intermediate reference frame that relays the signal to the next collinear one. Then, we have:

\displaystyle  f_n= f_0 \left (1- \frac{v}{n c}\right )^n    (5)

If we now compute the limit of f_n when n tends to infinity, we get:

\displaystyle    f = \lim_{n \to \infty} f_n= \lim_{n \to \infty} f_0 \left (1- \frac{v}{n c}\right )^n = f_0 \exp\left(-\frac{v}{c}\right)      (6)
and we can happily see that [6] does express a strictly self-similar Doppler effect, and therefore it is the correct formula for the complete Doppler. If someone still can’t see the point about the meaning of equation [6] being an expression of a strictly self-similiar Doppler, it means the speed v of the detector with respect to the source can be split into so many parts as you like, and each part will correspond to an intermediate antenna that relays the signal to the next collinear antenna, so the compound Doppler will match the simple one, that is directly observed between source and final detector.

II. A test of self-similarity for detecting false theories of relativity

In the above derivation, someone could argue that I did not take into account the Einstein addition of velocities when I claimed that Special Relativity failed the test of self-similarity for the Doppler effect. Actually, Special Relativity equation [1] is not self-similar if we apply a canonical sum of velocities, v = v₁ + v₂ , but if we apply the convention of Einstein addition of velocities we can achieve that equation [1] becomes self-similar. Why that amazing feat?. Actually, it is not any amazing feat at all. Any theory of relativity that owns a formula for Doppler effect of electromagnetic waves can be proclaimed as self-similar if we define a method of how the composition of velocities must be. Any theory of relativity exhibits the following generic Doppler formula:

\displaystyle     f = f_0 \exp \left (\mathrm{S}(\beta)\right )    (7)

where obviously, $latex\beta=\frac{v}{c}$ and \mathrm{S}(\beta) is a function of \beta. Since Special Relativity has the following equation for Doppler effect:

\displaystyle   f = f_0 \sqrt{\cfrac{1+\beta}{1-\beta}}    (8)

that means the function \mathrm{S}(\beta) must be:

\displaystyle   \exp \left (\mathrm{S}(\beta)\right )= \sqrt{\cfrac{1+ \beta}{1-\beta}} (9)
\displaystyle   \mathrm{S}(\beta)=\ln \sqrt{\cfrac{1+ \beta}{1-\beta}} (10)
\displaystyle   \mathrm{S}(\beta)= \frac{1}{2} \ln \cfrac{1+ \beta}{1-\beta} (11)
\displaystyle   \mathrm{S}(\beta)=\mathrm{artanh}\ (\beta)    (12)
So, we can clearly see that when we apply the generic Doppler to the case of Special Relativity, we automatically attain a law of composition of velocities that makes it self-similar. In other words, let’s suppose we want to compound two distinct β, β₁ and β₂, then we have:

\displaystyle    \exp \left (\mathrm{S}(\beta_1)\right )\exp \left (\mathrm{S}(\beta_2)\right ) = \exp \left (\mathrm{S}(\beta)\right ) (13)
\displaystyle   \exp \left (\mathrm{S}(\beta_1)+ \mathrm{S}(\beta_2) \right )=\exp \left (\mathrm{S}(\beta)\right ) (14)
\displaystyle   \mathrm{artanh}\ (\beta_1) +\mathrm{artanh}\ (\beta_2) = \mathrm{artanh}\ (\beta)  (15)

so, we definitively get:

\displaystyle    \beta =\cfrac{\beta_1 +\beta_2}{1+ \beta_1\beta_2}    (16)

as the well-known Einstein’s addition of velocities.

Above, we have seen that Doppler equation under classical mechanics is

\displaystyle   f = f_0  \left (1+\beta \right )

\displaystyle   f = f_0  \left (1+\beta \right )      (17)
and we proved it is not self-smilar under a canonical sum of velocities. But curiously, we can transform it into a self-similar equation, as we did for Special Relativity, if we can find a suitable law of non canonical composition of velocities for it. Let’s see how we can do it:

\displaystyle    \left (1+\beta \right ) = \exp \left ( \mathrm{S}(\beta) \right ) (18)
\displaystyle   \mathrm{S}(\beta) =\ln \left (1+\beta \right )    (19)

So the sum of betas would be now:

\displaystyle  \mathrm{S}(\beta_1) +\mathrm{S}(\beta_2)=\mathrm{S}(\beta) (20)
\displaystyle  \ln \left (1+\beta_1 \right ) +\ln \left (1+\beta_2 \right ) = \ln \left (1+\beta \right ) (21)
\displaystyle  \left (1+\beta \right )=\left (1+\beta_1 \right )\left (1+\beta_2 \right ) (22)
\displaystyle  \beta =\left (1+\beta_1 \right )\left (1+\beta_2 \right )-1  (23)
\displaystyle  \beta =\beta_1 +\beta_2 +  \beta_1 \beta_2   (24)
This would be the law of composition of velocities under classical mechanics assumptions if we want a self-similar Doppler. Logically, if we have endowed classical mechanics with that law of composition, it becomes a different theory, so we would have to called it with a different name. Anyway, now it is easy to prove the only theory of relatvity that is self-similar with a canonical sum of velocities is {\bf the Extended Galilean Theory of Relativity}, which we already know exhibits the complete Doppler formula:

\displaystyle   f=f_0 \exp \left(\beta \right) (25)
\displaystyle   f=f_0 \exp \left (\frac{v}{c} \right) (26)


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Reflection of light from a moving mirror falsifies Special Relativity

Posted by Albert Zotkin on October 12, 2012

A source of light and an observer are at rest in the same inertial frame, where r is the vector distance between them. A mirror, moving at receding velocity u perpendicular to r through its midpoint, at height h, reflects light of source off to observer. We can know the component velocities v₁ and v₂ of velocity u, with respect to source and to observer, respectively. Those component velocities actually have the same magnitude v, so we can write

u = v_1 + v_2 \\  v = |v_1| = |v_2|

and the angle alpha between v₁ and v₂ as

\alpha = 2\tan^{-1}\left(\cfrac{|r|}{2h} \right)


|u| = 2 \cos \left ( \cfrac{\alpha}{2} \right )v \\ \\ \\  v = \cfrac{|u|}{ 2 \cos (\frac{\alpha}{2})}

The observer then detects the reflected light as if it were coming from the image behind the mirror. Since the mirror is moving with velocity v₁ wrt to source and v₂ wrt observer, it creates a virtual image of the source moving with receding velocity w = 2v along the observer line of sight,

w = 2v = \cfrac{|u|}{\cos(\frac{\alpha}{2})}

Then, for that virtual velocity w, which can even be a superluminal velocity, because it doesn’t correspond to any real motion between source and observer (recall source and observer are really at rest), we can predict an observed Doppler frequency red-shift of the original frequency f₀ emitted by source, as

f = f_0 \exp \left (-\cfrac{w}{c} \right ) = f_0 \exp \left (-\cfrac{|u|}{c\cos(\frac{\alpha}{2})}  \right )


Under SR, the prediction is as follows. Apply an Einstein’s addition of velocities,

w =  \cfrac{2v}{1 + \frac{v^2}{c^2}}

Now, apply a relativistic Doppler,

\displaystyle  f' = f_0 \sqrt{\cfrac{1 - \frac{w}{c}}{1 + \frac{w}{c}}}

and after some algebra, knowing that v = |u|/(2\cos(\alpha/2)) , it yields

\displaystyle  f' = - f_0  \cfrac{|u| - 2c\cos(\frac{\alpha}{2}) }{|u| + 2c\cos(\frac{\alpha}{2})}

Where is the misleading mistake in this SR derivation? There are two misconceptions that when acting cooperatively try to slightly compensate the wrong answer to a right one. The first error is to assume there must exist a relativistic addition of velocities as v = 2v(1 + v²/c²). This is nonsense, since w is a VIRTUAL velocity of source with respect to observer (they are actually at rest), not a real one (it can be even virtually superluminal), a relativistic addition of velocities must NOT be applied. If the virtual w is superluminal it means that, once the observer detects the reflected light as if it were coming from the image behind the mirror, the information is not FTL, because that information has travelled a larger path length than that of a straight line from source to observer, so actually that information travelled with the light speed, it is not FTL. This error of misconception is slightly corrected/compensated when the SR’s relativistic Doppler formula is applied to fₒ through that wrong w, yielding a predicted red-shifted frequency f‘ that is very close to the correct one f, provided above. Actually, f/fₒ and f‘/fₒ only start to differ from the third order term of their respective Taylor expansion series,

\displaystyle  \frac{f}{f_0}=  1-\frac{\text{Sec}\left[\frac{\alpha }{2}\right] |u|}{c}+\frac{\text{Sec}\left[\frac{\alpha }{2}\right]^2 |u|^2}{2 c^2}-\frac{\text{Sec}\left[\frac{\alpha }{2}\right]^3 |u|^3}{6 c^3}+\frac{\text{Sec}\left[\frac{\alpha }{2}\right]^4 |u|^4}{24 c^4}\dots\\ \\ \\ \\    \frac{f'}{f_0}= 1-\frac{\text{Sec}\left[\frac{\alpha }{2}\right] |u|}{c}+\frac{\text{Sec}\left[\frac{\alpha }{2}\right]^2 |u|^2}{2 c^2}-\frac{\text{Sec}\left[\frac{\alpha }{2}\right]^3 |u|^3}{4 c^3}+\frac{\text{Sec}\left[\frac{\alpha }{2}\right]^4 |u|^4}{8 c^4}\dots\


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Natural borders of the observable Universe

Posted by Albert Zotkin on September 10, 2012

1 Natural Units:

Let’s propose N as a natural large enough number, such that we can define natural lower and upper bounds:
These natural borders have to be seen as proper boundaries for material systems. We will see how relative interactions between two systems can be defined by assuming invariant the upper bounds, and doing the lower ones relatively variable.

Some initial assumptions follow:

\hbar is the reduced Planck constant (Dirac’s constant) \hbar = h / 2\pi , with h Planck constant.
G is the gravitational constant.
R_h is Hubble radius.
lp is Planck length.
N = R_h / l_p is Natural Computational Capacity.
k is Boltzmann constant.
\epsilon_0 is permittivity in a vacuum

\begin{tabular}{ || p{2.5cm} | p{2.5cm} | p{4.7cm} | p{4.5cm} || } \hline
\multicolumn{4}{|c|}{\textbf{NATURAL BORDERS of the OBSERVABLE UNIVERSE}} \\ \hline
\textbf{Magnitude} & \textbf{Dimension} & \textbf{Lower bound} & \textbf{Upper bound} \\ \hline

speed & LT\begin{math}^{-1}\end{math} & \textit{zero-point-speed}: \newline \begin{math}
v_0 =\sqrt[3]{\hbar G /R_h^2}
\end{math} & \textit{Speed of light in the vacuum}: \newline \begin{math}
c =\sqrt[3]{\hbar G /l_p^2}
\end{math} \newline \textit{In units of }\begin{math}v_0\end{math}: \newline \begin{math} c =\sqrt[3]{N^2} \end{math} \\ \hline

length & L & \textit{Planck-length}: \newline \begin{math}
l_p =\sqrt{\hbar G /c^3}
\end{math} & \textit{Hubble Radius}: \newline \begin{math}
R_h =\sqrt{\hbar G /v_0^3}
\end{math} \newline \textit{In units of }\begin{math}l_p\end{math}: \newline \begin{math} R_h =N \end{math} \\ \hline

time & T & \textit{Planck time}:\newline \begin{math}
t_p =\sqrt{\hbar G /c^5}
\end{math} & \textit{Cassini time}: \newline \begin{math}
t_c=H_0^{-1}=\sqrt{\hbar G /v_0^5}
\end{math} \newline \textit{In units of }\begin{math}t_p\end{math}: \newline \begin{math} t_c =\sqrt[3]{N^5} \end{math} \\ \hline

acceleration & LT\begin{math}^{-2}\end{math} & \textit{zero-point-acceleration}: \newline \begin{math}
\end{math} & \textit{Cassini acceleration}: \newline \begin{math}
\end{math} \newline \textit{In units of }\begin{math}a_0\end{math}: \newline \begin{math} A_c=N \end{math} \\ \hline

mass & M & \textit{zero-point-mass}: \newline \begin{math}
m_0=\sqrt{\hbar v_0 /G}
\end{math} & \textit{Planck mass}: \newline \begin{math}
m_p=\sqrt{\hbar c /G}
\end{math} \newline \textit{In units of }\begin{math}m_0\end{math}: \newline \begin{math} m_p =\sqrt[3]{N} \end{math} \\ \hline

density& ML\begin{math}^{-3}\end{math} & \textit{zero-point-density}: \newline \begin{math}
\rho_0=v_0^5/(\hbar G^2)
\end{math} & \textit{Planck density}: \newline \begin{math}
\rho_p=c^5/(\hbar G^2)
\end{math} \newline \textit{In units of }\begin{math}\rho_0\end{math}: \newline \begin{math} \rho_p =\sqrt[3]{N^{10}} \end{math} \\ \hline

energy & ML\begin{math}^{2}\end{math}T\begin{math}^{2}\end{math} & \textit{zero-poin-energy}:\newline \begin{math}
E_0=\sqrt{\hbar v_0^5 /G}
\end{math} & \textit{Planck energy}: \newline \begin{math}
E_p=\sqrt{\hbar c^5 /G}
\end{math} \newline \textit{In units of }\begin{math}E_0\end{math}: \newline \begin{math} E_p =\sqrt[3]{N^5} \end{math} \\ \hline

temperature & (T) & \textit{zero-point-temperature}: \newline \begin{math}
T_0=\sqrt{\hbar v_0^5 /Gk^2}
\end{math} & \textit{Planck temperature}: \newline\begin{math}
T_p=\sqrt{\hbar c^5 /Gk^2}
\end{math} \newline \textit{In units of }\begin{math}T_0\end{math}: \newline \begin{math} T_p =\sqrt[3]{N^5} \end{math} \\ \hline

action & ML\begin{math}^{2}\end{math}T\begin{math}^{-1}\end{math} & \textit{Reduced Planck Constant}: \begin{math}
\end{math} & \textit{Cassini Constant}: \newline \begin{math}
H_c=\hbar \sqrt{c^5 /v_0^5}
\end{math} \newline \textit{In units of }\begin{math}\hbar\end{math}: \newline \begin{math} H_c =\sqrt[3]{N^5} \end{math}\\ \hline

gravity & M\begin{math}^{-1}\end{math}L\begin{math}^{3}\end{math}T\begin{math}^{-2}\end{math} & \textit{zero-point-gravity}: \newline \begin{math}
\end{math} & \textit{Gravitational Constant}: \newline \begin{math}
\end{math} \newline \textit{In units of }\begin{math}G_0\end{math}: \newline \begin{math} G =N^2 \end{math} \\ \hline

Charge& Q & \textit{electron charge}: \newline \begin{math}
e_0=\sqrt{4\pi \epsilon_0 \hbar c/\ln{N}}
\end{math} & \textit{Planck charge}: \newline \begin{math}
q_p=\sqrt{4\pi \epsilon_0 \hbar c}
\end{math} \newline \textit{In units of }\begin{math}e_0\end{math}: \newline \begin{math} q_p =\sqrt{\ln(N)} \end{math} \\ \hline

computational\newline structure& dimensionless & \textit{fine-structure-constant}: \newline \begin{math}
\alpha= e_0^2 / (\hbar c 4\pi \epsilon_0) \newline \alpha= 1/\ln(N)
\end{math} & \textit{Natural Computational Capacity}: \newline \begin{math}
N =\exp(1/\alpha)
\end{math} \\ \hline

2 Discussion:

Absolute zero of temperature (0° K, -273.15°C) is unattainable in practice, and that means there must be a real achievable lower bound, rather than 0° K. Here I’m going to deduce that lower bound for temperatures is T_0 . It is reasonable to think that if Planck temperature, T_p = \sqrt{c^5/Gk^2}, is the upper bound for temperatures, then there should exist a lower bound explicitly expressed when substituting c with v_0 , that is T_0 = \sqrt{v_0^5/Gk^2}. From this reasonable assumption we deduce a lower bound for speeds, and that leads us to the zero-point energy,

E_0 = \sqrt{\cfrac{v_0^5}{G}}

that is a non zero value.

Zero-point energy is deduced as result of boundary conditions, noting that the lower speed is

v_0 = \cfrac{c}{\sqrt[3]{N^2}}

It is a mystery how the fine-structure constant, \alpha,  matches the inverse of the natural logarithm of N, but it does!

The natural number N, defined as the Natural Computational Capacity of a material system with respect to itself, or proper N, results to be the radius of the observable universe (Hubble radius R_h ), expressed in units of Planck length, l_p .
Its value has been estimated to be about N=1.2\;\mathrm{x}\; 10^{62}. And the fine-structure-constant is then:

\alpha = \cfrac{1}{\ln(R_h/l_p)}


N = \exp \left ( \cfrac{1}{\alpha}\right )

It seems that Feynman[1] was right, the fine-structure constant is related to the base of natural logarithms!

Now, suppose we define the number P as:

P = \alpha N = \cfrac{N}{\ln(N)}

then, it is roughly the number of prime numbers lower or equal to N. So, P(n) is roughly the Prime Counting Function for any n<N . Thus, the question is, would Quantum Gravity be related to prime numbers?
It seems to be so, as N is defined as the proper Computational Capacity of a material system. But two sytems, A and B, would exhibit relative N’s, such that

N_{\text{AB}} < N \\ \\ N_{\text{AB}} > N
N_{\text{AB}} = N

The conjeture is now ready to be served:

N_{\text{AB}} must always be a prime number”.

Relativistic boosts would be regarded as relative shifts of the lower boundaries towards its upper ones, regarded as invariant. But those relative shifts of the lower boundaries might be constrained to values, such that the ratio between upper bound and lower bound, would be a prime number, if the physical magnitude being considered is the intersystemic distance. More clearly:
Let A and B be two inertial systems a distance r apart, and with an intersystemic speed v. Then, both systems must produce relativistic boosts such that N_{AB} must be a prime number. As they are inertial systems, their coordinate N’s must be symmetrical:

N_{\text{AB}}= N_{\text{BA}}

with N_{\text{AB}}<N, iff r >0 or v>0

It is now clear that the fine-structure constant is a relativistic variable, when considered from a system with respect to other, and not from the proper system itself. That relativistic a, measured in a system A with respect to other B, would be then:

\alpha_{\text{AB}} = \cfrac{1}{\ln \left (N_{\text{AB}} \right )}

“Quantum Gravity is the art of computing N_{\text{AB}}


[1] S-I. Tomonaga, J. Schwinger and R. P. Feynman,  

[2] Feynman, Richard (1985). QED: The strange theory of light and matter. Princeton University Press. ISBN 0-691-08388-6.

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Prime numbers: Symmetry patterns in Ulam Spiral

Posted by Albert Zotkin on September 7, 2012

The aim of this article is not, in principle, to account for what Ulam spiral is and how we could build it, but an attempt for seeking patterns and
symmetries, and if possible, a satisfactory explanation to the mysterious phenomenon of how prime numbers get distributed over certain diagonals in that spiral. Anyway, if you still know nothing at all about Ulam spiral, here I give you a short explanation of how you can build one:

On a blank gridded paper, look for a central square, and from that point spirally counterclockwise count squares, so when your reach to a prime number you mark that square with a cross.

When you had completed enough squares you will be able to see how prime numbers are being concentrated in certain diagonals, and you can glimpse some patterns in that strange distribution. Actually, there is not a satisfactory explanation for those strange scribbles, although I hope that after you had read my article that strange phenomenon will be less mysterious than before.

This squared spiral, where prime numbers are highlighted over black background, was accidentally discovered by nuclear phisicist Stanislaw Ulam while he was getting bored in a conference, as he later told.

The Ulam Spiral is just the limit to infinity of a family of spirals with increasing degrees. So, the first spiral of that family,that with degree n=1, is built forming the succession of all natural numbers that are not divisible by 2, and number 2 is also included. In mathematics, it happens very often that limits tendind to infinity are hardly comprehensible objects, it is saying they are irrational. The same issue occur with prime numbers. The simple definition of prime number is the main obstacle to comprehend the structure of that kind of numbers. Anyway, things can be simpler if, rather than considering prime number from the beginning, we use numbers that are very similar to primes, and the they become to identy with them when the limit of considered degree is infinity.

The second degree spiral (n=2) is that built from the succession of all natural numbers that are not divisible neither by 2 nor 3. The degree n es then the n-th prime number in that succession of prime numbers. The curious thing about these sequences is that they have finite degree patterns or periods, it is saying, they are somewhat rational sequences, so therefore they can be expressed by means of a general element. More exactly, they are periodic sequences, or in this case semi-periodic. Let’s see the first sequences of finite degree in their first elements.

1.-Sequence of natural numbers not divisible by 2:

S_1 = \{1, 2, 3, 5, 7, 9, 11, 13, 15, 17, ...\} ,

logically, since they are natural numbers not divisible by 2, they turns out to be the sequence of odd numbers from 2 onwards. Let us find now the differences between consecutive elements

d_n = s_{n +1} - s_n \\ D_1 = \{1, 1, 2, 2, 2, 2, ...\}

We see that it is easy to appreciate the repetition pattern in D_1 . The 2 repeats indefinitely from the position n=3 . So we have two kinds of objects to define. One is the period P_1 =\{2\} , another is the residue of degree n = 1 \mathrm{,}\; R_1 = \{1,1\} . When we mark the numbers of S_1 on the spiral, we get this pattern, that is logically a checkerboard of black and white squares, except at its center because the first two numbers of S_1 are 1 and 2.

2. – Succession of natural numbers not divisible neither by 2 nor by 3:

S_2 = \{1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, ...\} \\ D_2 = \{1, 1, 2, 2, 4, 2, 4, 2, 4, ...\} \\  R_2 = \{1, 1, 2\}\mathrm{,}\; P_2 = \{2,4\}

We can observe certain features both in the residues and periods. The period P_2 is based on P_1. In fact it is a transformation of P_1. The last element of a residue is always the first element of the next lower grade period. The sequence of prime numbers is actually the residue corresponding to an infinite degree, and that means there is not a period for the sequence  prime numbers, that is, in a way, an irrational succession. Or put another way, the only period for such succession is its own residue.

3.-Sequence of natural numbers not divisible by 2, 3, and 5:

S_3 = \{1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, ...\} \\ D_3 = \{1, 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, ...\} \\  R_3 = \{1, 1, 2, 2\}\mathrm{,}\; P_3 = \{4,2,4,2,4,6,4,6,4,2,4,2,4,6,2,6,4,2,4,2,4,6,2,6\}

Now we can see more clearly that the residue corresponding to all the prime numbers is formed by the first elements of the successive periods of convergent (I call convergent of degree n to succession D_n). In fact, observing S_3 we can guess the next prime number to 5 must be 7, and the residue is R_4 = \{1, 1, 2, 2, 4\}, which means that the next prime to 7 must have a difference of 4. Thus observing S_n, R_n and P_n for some degree n, we are ready to know what will be the next two prime numbers, and these numbers are logically in S_n. In fact, this process is just the famous sieve of Eratosthenes, but expressed in a more systematic way, and writing sequences indefinitely, not limited to a certain number.

When we mark the numbers of S_3 in spiral, we get this pattern, which is more complex than the previous one, but is based on it. In fact, it is almost the same pattern. It can be built marking in black the pixels of the previous pattern that are divisible by 5 in S_2. Here we can already see how the pattern of symmetry is more complex. As we increase the degree n we see that the pattern will get more and more like the full Ulam spiral of primes.

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A mathematical formula for searching circumstellar habitable zones

Posted by Albert Zotkin on September 6, 2012

A circumstellar habitable zone (CHZ) is a zone around a stelar system where life, as we know it, (carbon based life) could exist. So, we search for circumstellar regions where there may be rocky planets with a mean temperature in their surface of about 0 Celsius degrees.

Let’s propose and deduce a mathematical equation for this task. This equation will be based in the Unruh effect (derived by William Unruh in 1976) that aclaims the effective temperature experienced by a uniformly accelerating detector in a vacuum field is given by:

T =\cfrac{\hbar a}{2\pi c k}

a is the local acceleration
k is Boltzmann constant
\hbar is reduced Planck constant
c is the speed of light in a vaccum

Now let’s identify that local acceleration with the acceleration of gravity caused by the mass of a star on a orbiting rocky planet, and appply the Newtonian gravity

a = g = \cfrac{GM}{d^2}

M is the mass of the star
d is the distance of the rocky planet to the star.

So, we attain:

T =\cfrac{\hbar GM}{2\pi c k d^2}

This final equation means that if we are searching for a CHZ, we must tune the temperature to about T=273.15ºK, so if we know the stelar mass M , we can estimate a distance d   from that star where there could be a habitable rocky planet with liquid water and suitable for the existence of carbon-based life:

d =\sqrt{\cfrac{\hbar GM}{546.3\;\pi c k}}

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The complete Doppler formula: Return to the origin

Posted by Albert Zotkin on September 1, 2012

A new Doppler formula that shows the flawed limits of applicability of Special Relativity.

A source of light is receding from an observer at a relative fixed speed v . So, if original frequency of light is f_0 , the observed frequency is f<f_0 , because of that receding speed. The aim of this experiment is to accelerate the observer towards the source until both source and observer were at rest. So that will happen when the speed of the observer was -v with respect to the frame where he is now at rest. In order to achieve this, we increase the observed frequency f a differential df in its rest frame, by increasing his rest with a differential speed, dv , and that differential Doppler is addressed by means of the classical first order approximation Doppler formula,

f+df = f \left(1 + \cfrac{dv}{c}\right)

It is saying, the measured frequency by the observer when he is at rest in his frame is f , and when he starts to move a differential speed dv towards the source, he achieves to measure a greater frequency f+df because of the Doppler effect. So, simplifying, we get,

f+df = f + \cfrac{f\;dv}{c} \\ \\  df = \cfrac{f\;dv}{c} \\ \\  \cfrac{df}{f} = \cfrac{dv}{c}

So now, we are ready to integrate this differential equation, and the solution is

\ln \left (\cfrac{f}{f_0} \right) = \cfrac{v}{c} \\ \\  f = f_0\exp \left(\cfrac{v}{c}\right) \\ \\

This notable solution means that when the observer achieves to be at rest in the rest frame of the source of light, the measured frequency will be f=f_0 . This  result is also notable for the following consideration: since we used the classical first order aprroximation Doppler fórmula for expressing a differential of speed, then when that equation is integrated, we attain the correct solution for any speed. small or large, showing that it is in blatant discrepancy with Special Relativy.

The hilarious part of all of this, is that if anyone wants to prove my formula wrong, he must perform an experiment for discriminating between the relativistic formula and mine. But, nobody on Earth is still able to perform that test  because of a little detail (the devil is  in the details):  the expansion series (Taylor series) of both formulas  are indistinguishable within the third order approximation of v/c , and that means there is no manmade technologies available nowadays that could achieve such accurate measurements for discriminating one formula from the other. Can you believe it?

\frac{f}{f_0} = \exp (\frac{v}{c})= 1+\frac{v}{c}+\frac{v^2}{2 c^2}+\frac{v^3}{6 c^3}+\frac{v^4}{24 c^4}+... \\ \\ \frac{f}{f_0} = \sqrt\frac{1+v/c}{1-v/c} = 1+\frac{v}{c}+\frac{v^2}{2 c^2}+\frac{v^3}{2 c^3}+\frac{3 v^4}{8 c^4}+...

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De Broglie Relations expressed through rapidity

Posted by Albert Zotkin on August 29, 2012

It is well-known that phase velocity, c_p, of a matter wave can be expressed as

c_p = \frac{E}{p}

where E is total energy, and p is momentum. Likewise, group velocity, c_g, of a matter waves can be expressed as the derivative of E wrt p,

c_g = \frac{dE}{dp}

and this latter can be identified with the relative velocity v of the body which has associated that matter wave, v=c_g.

We also know that a rapidity, r, is defined as

v = c\tanh(r)

Therefore, we can express

\frac{dE}{dp} = c\tanh(r)

and also

\frac{E}{p} =c  \coth (r)

which means that

E = mc^2 \cosh(r)


p = mc \sinh(r)

Thus, the energy-momentum relations is

E^2 = (mc^2)^2 + (pc)^2

From properties of hyperbolic functions, we know that

\cosh(x)= 2\ \sinh^2(\frac{x}{2}) + 1

so, Eq.(1) yields

  E = mc^2 \cosh(r)= m c^2 (2\ \sinh^2(\frac{r}{2}) + 1)
  E =  m c^2  + 2 mc^2 \  \sinh^2(\frac{r}{2})

Now, if we define

p' = mc\sinh(r/2)

as the momentum of a body of rest mass m travelling at rapidity

r'= r/2

we get

2\frac{p'^2 }{m} = 2 (mc^2)\  \sinh^2(r/2)


 E  = mc^2  + \frac{2\ p'^2}{m}

which means that

E_k = 2p'^2/m

would be the kinetic energy of a body with rest mass m travelling at rapidity r.

The relation between both momenta, p and p', clearly is attained from Eqs.(3)(4), and it is

\sqrt{(mc^2)^2 + (pc)^2} = mc^2 + \frac{2\ p'^2}{m}

This latter relation is the simple square root that Dirac, Schrödinger, Oskar Klein, Walter Gordon, and many others were looking for, but they all failed.

This is the greatest contribution to QM ever!

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The real gravitational secret of mass and energy

Posted by Albert Zotkin on August 29, 2012

The following idea doesn’t seem to be new, but it can be revamped,

“mass/energy ‘steals’ space to its surroundings”

Let’s consider an empty sphere of radius r. Its volume is

V = \frac{4 \pi r^3 }{ 3}

A test particle travelling inertially would intersect that sphere preserving its straight path and velocity.

Now, put a point-like mass M in the center of that sphere. The idea is that now, the spherical volume becomes

V'  <  \ V

The mass M has ‘stolen’ part of the original volume, so a test particle travelling inertially from infinity will intersect the new volume V’ in such a way that its path is no longer a straight line (if it is non-radial) and its velocity will not remain invariant. How can we quantify that “steal” of space?

We consider a radial path whose length is r. A test particle travelling inertially at speed v and without a mass M in the center of the sphere, would reach that center in time t = r/v. But, if there is a mass M in the center, that mass ‘steals’ space such that the new effective radial length becomes

r' = r \exp \  \(- \frac {GM }{rc^2}\)

so now, the test particle would reach the center in time

         t' = \frac{r'}{v}= \frac{ r}{v} \exp \ \(-\frac{ G M }{ rc^2}\)}
         t' = t \exp\ \(-\frac{G M }{ rc^2}\)
which is equivalent to claim that particle has increased its speed to

 v' = \frac{ r}{t'} = v \exp \ \(\frac{G M }{ rc^2}\)
Therefore, volume V’ is

 V' = \frac{4 \pi r'^3 }{ 3} = \frac{4 \pi r^3}{3} \exp \ \(- \frac{3 G M }{rc^2}\)

and we can say that mass M has “stolen” a volume V” as

V'' = V - \ V' = V \(1 - \frac {4 \pi r'^3 }{3} \) = V \{1 - \frac {4 \pi r^3 }{3} \exp \ \( -\frac{GM}{rc^2}      \) \}

We can solve eq. [1] for r, if r’ and M are known,

W\( \frac { G M }{r'c^2} \)= \frac{G M }{rc^2}

where W is Lambert W-function, so

 r = \frac{G M }{ c^2 W\(\frac{G M }{r'c^2}\)}

We can imagine that “steal” of space as a negative spatial interval s, such that when added to r,
it yields a sum

 r' = r + s


 s = r \{\exp \ \(- \frac{GM }{ rc^2} \) - 1\}

So, masses can be regarded as negative volumes.


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Relativistic Problems for the 21st Century: Concise refutation of Special Relativity

Posted by Albert Zotkin on August 29, 2012

The correct derivation of an electromagnetic Doppler shift starts with a differential equation:

c\;dt = t\; dv \quad \quad \quad \quad \small\text{eq.[1]}

which correctly integrated yields,

t = t_0\exp(\frac{v}{c})

and this can be expressed as frequencies

\cfrac{1}{t} = \cfrac{1}{t_0}\;\exp(-\frac{v}{c}) \\\\  f = f_0\;\exp(-\frac{v}{c})

Now, let us see how a similar derivation would be under Special Relativity assumptions:

c\;dt = \dfrac{t\; dv}{1 - \frac{v^2}{c^2}} \\ \\  \ln(\frac{t}{t_0})=\tanh^{-1}(\frac{v}{c}) \\ \\  \ln(\frac{t}{t_0})=\frac{1}{2}\ln \left\{\cfrac{1+\frac{v}{c}}{1-\frac{v}{c}}\right\}

t = t_0 \sqrt{\cfrac{1+\frac{v}{c}}{1-\frac{v}{c}}} \\ \\  \cfrac{1}{t} = \cfrac{1}{t_0} \sqrt{\cfrac{1-\frac{v}{c}}{1+\frac{v}{c}}} \\ \\  f = f_0 \sqrt{\cfrac{1-\frac{v}{c}}{1+\frac{v}{c}}}

The initial factor 1/(1 -v^2/c^2) comes from the absurdity of the 2nd postulate of SR. Actually, that factor (which is equal to Lorentz factor squared, \gamma^2 , works fine for accelerating charged particles in electric/magnetic fields, because it tells us that a charged particle can’t be accelerated by the field beyond c. But, for neutral particles/bodies, that factor is a false constraint, and eq.[1] should be applied instead.

PS: Cosmological connection of the correct Doppler equation f= f_0\exp (-v/c) . For distant receding galaxies, v is often interpreted as a recessional speed, so the observed frequency redshift z of a distant galaxy is

z+1=\exp\left (\dfrac{v}{c}\right )

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Complete Doppler formula

Posted by Albert Zotkin on November 29, 2011

Let f be the observed frequency of an electromagnetic wave emitted by a source that is moving at a relative speed with respect to the observer. We know that frequency f times λ gives c, the speed of light, where λ is the observed wavelength, thus

Now if we increase f with a differential df then c is increased with a differential dv as λ remains constant,

so we have

and dividing both sides by f it yields

So integrating we get

is the original frequency in the light source.

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