Latest revision 12-12-2022
index
$
\def \hieruit {\quad \Longrightarrow \quad}
\def \slechts {\quad \Longleftrightarrow \quad}
\def \SP {\quad ; \quad}
\def \MET {\quad \mbox{with} \quad}
\def \half {\frac{1}{2}}
\def \artanh {\operatorname{artanh}}
$
Relativity Special
With Unified Alternative Cosmology (UAC), there is Length Contraction
and Time Dilation, two notions which are commonly associated with
Special Relativity (SR). This suggests that there could
be sort of a Lorentz transformation associated with UAC as well.
The Lorentz transformation of Special Relativity (one-dimensional in space) is:
$$
x' = \frac{x-(v/c)\,ct}{\sqrt{1-(v/c)^2}} \\
ct' = \frac{ct-(v/c)\,x}{\sqrt{1-(v/c)^2}}
$$
where $x=$ space, $t=$ time, $v=$ speed of observer, $c=$ speed of light in vacuum.
The Hyperbolic Connection: some mathematics about
Hyperbolic functions is required in order to understand
what follows.
Define the parameter $\,p\,$ by $\,\tanh(p) = (v/c)\,$. Then:
$$
\frac{1}{\sqrt{1-(v/c)^2}} = \frac{1}{\sqrt{1-\tanh^2(p)}} =
\frac{1}{\sqrt{[\cosh^2(p)-\sinh^2(p)]/\cosh^2(p)}} = \cosh(p) \\
\frac{v/c}{\sqrt{1-(v/c)^2}} = \tanh(p)\cosh(p) = \sinh(p)
$$
Herewith the Lorentz transformation becomes:
$$
\begin{cases}
x' = + \cosh(p)\,x - \sinh(p)\,ct \\
ct' = -\sinh(p)\,x + \cosh(p)\,ct
\end{cases}
\slechts \begin{bmatrix} x' \\ ct' \end{bmatrix} =
\begin{bmatrix} +\cosh(p) & -\sinh(p) \\ -\sinh(p) & +\cosh(p) \end{bmatrix}
\begin{bmatrix} x \\ ct \end{bmatrix}
$$
The determinant of the transformation matrix is: $\,\cosh^2(p)-\sinh^2(p) = 1\,$. The same transformation matrix is found in
What does shear mean?
near the bottom lines of the answer.
But let's take a further look to the formula that defines the parameter $\,p\,$ and the inverse function of it.
With help of Wikipedia:
$$
\artanh(x) = \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right) \qquad x \lt 1 \hieruit \\
p = \artanh\left(\frac{v}{c}\right) = \ln\left(\sqrt{\frac{1+(v/c)}{1-(v/c)}}\right)
$$
Between parentheses the Relativistic longitudinal Doppler effect is recognized. This effect may be held responsible for
Redshift $\,z\,$ with motion completely
in the radial or line-of-sight direction.
$$
1+z = \frac{\lambda_r}{\lambda_s} = \frac{f_s}{f_r} = \sqrt{\frac{1+\beta}{1-\beta}} \MET \beta = \frac{v_\parallel}{c}
$$
where $\lambda=$ wavelength, $f=$ frequency. With Hubble Parameter then we have,
intrinsic redshift assumed:
$$
p = \ln(1+z_i) = \frac{H_i\,D}{c_0} = \ln\left(\frac{m_0}{m}\right)
$$
And so:
$$
x' = + \frac{e^{+p}+e^{-p}}{2}x - \frac{e^{+p}-e^{-p}}{2}ct = \frac{1}{2}\left[+\frac{m_0}{m}(x-ct) + \frac{m}{m_0}(x+ct)\right] \\
ct' = - \frac{e^{+p}-e^{-p}}{2}x + \frac{e^{+p}+e^{-p}}{2}ct = \frac{1}{2}\left[-\frac{m_0}{m}(x-ct) + \frac{m}{m_0}(x+ct)\right] \\
\hieruit x'-ct' = \frac{m_0}{m}(x-ct) \SP x'+ct' = \frac{m}{m_0}(x+ct)
$$
In order to find a neat correspondence with UAC theory, a simplified form of the wave equation for light is
considered:
$$
\frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0
$$
References by my own concerning it are the following, in reverse order of publication.
- Characteristics of equations of the form $u_{xy}=f(u_x,u_y,u)$
- Explicit formula of the solution of $u_{tt}=au_{xx}$ and for which values of $a$ is this a "wave equation"?
- Indefinite double integral
There is a well known proof in there that the most general solution of the wave equation is given by:
$$
u(x,t) = F(x-ct) + G(x+ct)
$$
Interpreted as the superposition of a wave $\,F\,$ travelling forward and a wave $\,G\,$ travelling backward.
In view of the above, it follows that
$$
u(x',t') = F(x'-ct') + G(x'+ct') = F\left(\frac{m_0}{m}(x-ct)\right) + G\left(\frac{m}{m_0}(x+ct)\right)
$$
The wave travelling forward is intrinsically redshifted both in space and time and might come from a galaxy observed by us.
The wave travelling backward is intrinsically blueshifted both in space and time and is travelling away from us perhaps towards
some galaxy, but anyway it cannot be observed. So the formulas of Special Relativity are only in concordance with
the Length Contraction and Time Dilation in UAC for the forward travelling wave
with parameter $(x-ct)$. Minkowsky space-time, however, requires that time behaves like a one-dimensional space coordinate,
it has no "arrow", it can go back and forth. That's where the - limited - consistency of UAC with SR comes from:
$$
(x'-ct')(x'+ct') = \frac{m_0}{m}(x-ct)\frac{m}{m_0}(x+ct) \hieruit (x')^2-(ct')^2 = (x)^2-(ct)^2
$$
Let us consult that Wikipedia page for support of the "old-fashioned" idea that space and time are separate entities. And that
there exists absolute space instead of Einstein's warped space-time. Time is quite another story.
Quote: Isaac Newton's rotating bucket argument [ .. ] was designed to demonstrate that true rotational motion
[ .. ] can be defined only by reference to absolute space. Indeed the accompanying formula is in concordance with just that:
water height $\,h\,$ is height at the bucket axis plus half (angular velocity $\,\Omega\,$ times distance to axis $\,r\,$) squared,
divided by gravitational acceleration $\,g\,$. Formally:
$$
h(r) = h(0) + \frac{1}{2g}\left(\Omega\,r\right)^2
$$
According to Mach's principle,
some cosmological parameters should show up in Newton's formula, because then the problem is supposed to be equivalent
with the cosmos rotating around the bucket in reverse direction with the same angular velocity.
Nothing of the kind is observed though. I've been trying to look up some references concerning this:
Mach's Principle confirmed by high-speed rotor experiments?
Further on we are led to the idea that the whole thing seems to be related to
Frame-dragging (Wikipedia) /
Frame-dragging: meaning, myths, and misconceptions.
However, the more I look at it, the more I get the impression that frame-dragging is only a panacea for the fact that absolute
space is denied by General Relativity.
This impression is reinforced by the name "gravitomagnetic" in the abstract of the
arXiv article.
Let's assume for a moment that there is indeed kind of a magnetic effect associated with gravity:
Gravitoelectromagnetism.
Then frame-dragging by the Cosmos as a force experienced by our Bucket would be analogous to the electric force inside a giant
"gravitational" solenoid.
However, the magnetic component of "gravity" inside such a solenoid would be constant. And thus there would be no gradient
in the accompanying electric → gravitational "Lorentz force".
We conclude that the water in the bucket would not curve. Hence Newton is right, Mach and Einstein (GR) are wrong.
FLRW metric
The Friedmann-Lemaître-Robertson-Walker metric shows a predominant presence in nowadays cosmology. Therefore
I think it's wise to pay at least some attention to it. According to
[Fahr 2016] we have for its
Weltlinien element (i.e. infinitesimal space-time distance) $\,ds\,$ the following expression.
$$
ds^2 = c^2\,dt^2 - a^2(t)\left[\frac{dr^2}{1-kr^2} + r^2\,d\theta^2 + r^2 \sin^2(\theta)\,d\phi^2\right]
$$
Here $c=$ lightspeed, $t=$ time, $a=$ cosmological Scale factor, $k=$ space-time curvature, $(r,\theta,\phi)=$ common spherical coordinates. On page 11
of the same reference it is stated that zudem das Weltall als ungekrümmt entsprechend $\,k=0\,$
angesehen wird. Exactly as (erroneously?)
claimed with the $\,R_h=ct\,$ Universe. Why don't they say that immediately? Because now we have a much more traditional expression
between the square brackets.
$$
ds^2 = c^2\,dt^2 - a^2(t)\left[dr^2 + r^2\,d\theta^2 + r^2 \sin^2(\theta)\,d\phi^2\right]
$$
What follows is for my own understanding. Spherical coordinates are defined with Cartesian coordinates $(x,y,z)$ as:
$$
\begin{cases}
x = r\sin(\theta)\cos(\phi) \\
y = r\sin(\theta)\sin(\phi) \\
z = r\cos(\theta)
\end{cases}
$$
With accompanying infinitesimals:
$$
\begin{cases}
dx = dr\,\sin(\theta)\cos(\phi) + r\cos(\theta)\cos(\phi)\,d\theta - r\sin(\theta)\sin(\phi)\,d\phi \\
dy = dr\,\sin(\theta)\sin(\phi) + r\cos(\theta)\sin(\phi)\,d\theta + r\sin(\theta)\cos(\phi)\,d\phi \\
dz = dr\,\cos(\theta) - r\sin(\theta)\,d\theta
\end{cases}
$$
When putting this in matrix form, care must be taken that an orthogonal matrix is indeed emerging:
each row must be perpendicular to each other row (i.e. inner product $=0$) and the length of each row
must be $=1$.
$$
\begin{bmatrix} dx \\ dy \\ dz \end{bmatrix} =
\begin{bmatrix} \sin(\theta)\cos(\phi) & \cos(\theta)\cos(\phi) & -\sin(\phi) \\
\sin(\theta)\sin(\phi) & \cos(\theta)\sin(\phi) & +\cos(\phi) \\
\cos(\theta) & -\sin(\theta) & 0 \end{bmatrix}
\begin{bmatrix} dr \\ r\,d\theta \\ r\sin(\theta)\,d\phi \end{bmatrix}
$$
Then what we have is an orthogonal transformation and so:
$$
dx^2+dy^2+dz^2 = dr^2 + (r\,d\theta)^2 + (r\sin(\theta)\,d\phi)^2
$$
Which is exactly the space part (between the square brackets) of the FLRW space-time element.
Now almost the same infinitesimal space-time distance squared is found in the article by Jayant Narlikar and Halton Arp:
FLAT SPACETIME COSMOLOGY: A UNIFIED FRAMEWORK FOR EXTRAGALACTIC REDSHIFTS. It is formula (6), which must be combined
then with the "variable mass hypothesis" - Narlikar's Law - formulated as equation (7) in the article.
$$
ds^2 = c_0^2\,dt^2 - \left[dr^2 + r^2\,d\theta^2 + r^2 \sin^2(\theta)\,d\phi^2\right]
$$
For the time-dependent term, according to C-decay Theories, it is noticed that
$$
ds^2 = c^2\,dT^2 = c_0^2\left(\frac{T_0-A}{T-A}\right)^2\,dt^2\left(\frac{T-A}{T_0-A}\right)^2 = c_0^2\,dt^2
$$
So we have the same form for that part of the expression, for orbital time as well as for atomic time.
The space part of the expression must be preceded by the cosmic scale factor squared $\,a^2\,$ if we think
of expanding empty space instead of increasing VPM.
Our own interpretation of the whole thing would be that it is the metric of common Euclidean empty space,
expanding with a cosmic scale factor, expressed in orbital time $\,T\,$ or atomic time $\,t\,$ according to
Hubble Parameter:
$$
\frac{a(t)}{a_0} = e^{H_i(t-t_0)} = \left(\frac{T-A}{T_0-A}\right)^2
$$
We conclude that General Relativity is not at all required for a better understanding of this world.