Latest revision 08-10-2022

## Narlikar's Law

$\def \hieruit {\quad \Longrightarrow \quad} \def \slechts {\quad \Longleftrightarrow \quad} \def \SP {\quad ; \quad} \def \OF {\quad \mbox{or} \quad}$ Despite all the effort spent by established Big Bang science, quite another picture of the universe is emerging after Six decades of cosmology, as is the case with our Unified Alternative Cosmology.
An abundance of alternative arguments is found on the internet. Quotes from these internet sources will be displayed as Seeing Red (: click) in what follows. Bold font is emphasis by me (most of the time). However, most of the argumentation is already found in: FLAT SPACETIME COSMOLOGY: A UNIFIED FRAMEWORK FOR EXTRAGALACTIC REDSHIFTS.
1. On the Hoyle-Narlikar Theory of Gravitation (S. W. Hawking): It is shown that the direct-particle action-principle from which Hoyle & Narlikar derive their new theory of gravitation not only yields the Einstein field-equations in the 'smooth-fluid' approximation, but also implies that the 'm'-field be given by the sum of half the retarded field and half the advanced field calculated from the world-lines of the particles. This is in effect a boundary condition for the Einstein equations, and it appears that it is incompatible with an expanding universe [ .. ]
2. Non-standard Models and the Sociology of Cosmology (Martín López Corredoira). At slide 7/39 we read after the headers Alternative models and 1) Quasi steady-state Cosmology.
Total redshift: $$(1+z)=(1+z_{expansion})(1+z_{intrinsic})$$ Followed by a header Variable mass hypothesis: The intrinsic redshift does not indicate the distance but the age.
Hoyle & Narlikar (1964) developed a new theory of gravitation based on Mach's principle whose simple solution is the Minkowski metric and particle masses depending on time according to $m \sim t^2$. This brings about $$(1+z_{intrinsic}) = m_{observer}/m_{source} = t_0^2/(t_0-r.c)^2 \quad [ \mbox{ rest deleted } ]$$ [ .. ] (Narlikar 1977, Narlikar & Arp 1993).
3. A Different View of the Universe (Answers in Genesis): [ .. ] Arp presents his alternative to big bang cosmology. He uses the Narlikar version of general relativity (GR). This retains the possibility that particle masses can vary in space, and not (as is usually done) be eliminated at an early stage. Arp explains that this formulation implies that curved space-time is not needed, i.e. space[-time] is Euclidean.
4. There are NO massive black holes in the center of galaxies. Rather the idea is that new matter emerges into our universe in active galactic nuclei, where Arp suggests there may be white holes rather than black holes. This does [ not ] appear to be creatio ex nihilo in the biblical sense [ .. ] The postulated new matter has zero mass and very high redshift. It is then ejected, and increases in mass and decreases in redshift. Thus providing an possible explanation for the paradox of youth: Compared to theory, there is an overabundance of young stars [ .. ] in the Galactic Center (Wikipedia).
5. Disclaimer: the mere fact that I'm quote mining from "Answers in Genesis" does not mean that I am a Young Earth Creationist. It only means that some theories adhered by YEC people (and others) have drawn my attention.
As a next step, we shall be getting more to the point at hand of the following references:
1. From the paper Is Physics Slowly Changing? (Halton Arp) the following is quoted. What Narlikar showed is that the rigorous solution of the field equations (which in flat space are simply conservation of energy/momentum) requires the elementary particles to gain mass as $m = t^2$.
2. The UNIFIED THEORY : A Complete Paradigm Shift in Physics and Cosmology (Rati Ram Sharma) page 262: For the explanation, Arp [14] with Narlikar [15] believes that the mass $$m=at^2$$ of an elementary particle increases as direct square ($t^2$) of the cosmic time $t$. Narlikar [17] had deduced the above relation from the General Relativity. But Arp [16] arrives at it as below. [ i.e. heuristically ]
3. Intrinsic Redshifts in Quasars and Galaxies (Halton Arp). At page 34 it is read:
The core of the assumption is that elementary particle masses are born near zero mass and gain mass as t (time) squared.
4. From High Redshift Galaxies to the Blue Pacific (Halton Arp). Search for "Note":
Note: Flat space, no curves, no expansion.
The general solution of energy/momentum conservation (relativistic field equations) which Narlikar made with $m = t^2$ gives a Euclidean, three dimensional, uncurvedspace. The usual assumption that particle masses are constant in time only projects our local, snapshot view onto the rest of the universe.
5. Research With Fred (Halton Arp) Search for "Hubble":
And in the mathematical equations he presented he derived a Hubble law solely from a non-velocity, particle mass growth as a function of $t^2$.
Theory as has been developed by Narlikar rests on General Relativity. Consequently, Narlikar's "cosmic" time cannot be anything else than gravitational time $T$. Furthermore, linear relationships are easier tot manage than quadratic ones. This makes that the following Ansatz will be proposed instead of the vague $m\sim t^2$: $$\sqrt{\frac{m}{m_0}} = a.T+b$$ Here $T=$ orbital time, $m=$ elementary particle rest mass, $m_0=$ reference mass (here and now), $a$ and $b$ constants to be determined. There are two boundary conditions for the elementary particle rest mass. At $T=T_0$, the gravitational reference time, which is the "nowadays" timestamp, we have that the rest mass is the reference rest mass (here and now): $m=m_0$. Let $A$ be the gravitational timestamp corresponding with a beginning (Alpha), at the time when mass is created out of nothing. Then we have for $T=A$ that the elementary particle (rest) mass is zero: $m=0$. Formally: $$1 = a.T_0+b \SP 0 = a.A+b \hieruit a=\frac{1}{T_0-A} \SP b = \frac{-A}{T_0-A}$$ Therefore the following should hold. And it shall be memorized as Narlikar's Law: $$\large \boxed{\frac{m}{m_0} =\left(\frac{T-A}{T_0-A}\right)^2}$$ Okay, once again. With $T=$ orbital time, $m=$ elementary particle rest mass, $m_0=$ reference mass (here and now), $T_0=$ gravitational reference time, which is the "nowadays" timestamp, $A=$ gravitational timestamp corresponding with a beginning (Alpha), at the time when mass is created out of nothing. It should be noted that, here and now, Alpha is the only unknown quantity in Narlikar's Law. Hopefully we will find means to determine it later on. As a perspective, together with results from Length Contraction we have for an accompanying (intrinsic) redshift $\,z_i\,$: $$1+z_i = \frac{m_0}{m} = \left(\frac{T_0-A}{T-A}\right)^2$$ This formula is somehow in agreement with the one at page 144 of [Fahr 2016], where it is read that: $$(1+z_1)/(1+z_2) = t_2^2/t_1^2$$ But the author doesn't mention any two clocks, so we shall feel free to replace his "atomic" time by orbital time: $\,t\to(T-A)\,$.
Yet the author makes a good point - as others have done too. It has turned out that our first (boxed) formulation of Narlikar's law is not satisfactory in every respect. So let's go back to the basics and formulate in plain language what the core of the matter is. Let VPM stand for Variable (elementary) Particle (rest) Mass. Then $\,m\sim t^2\,$ can also be worded as: $$\frac{\mbox{VPM}_1}{\mbox{VPM}_2} = \left(\frac{\mbox{Age}_1}{\mbox{Age}_2}\right)^2 \OF \boxed{\large \frac{m_1}{m_2} = \left(\frac{T_0-A_1}{T_0-A_2}\right)^2}$$ Where $m=$ VPM, $T_0=$ here and now timestamp, $A=$ moment of creation. This may be called the second formulation of Narlikar's law and should be memorized as well.

It's typical that in A quasi-steady state cosmological model with creation of matter (F. Hoyle, G. Burbidge, J.V. Narlikar) it is stated (in the ABSTRACT) that newly created particles have a [ Planck ] mass $(3\hbar c/4\pi G)^{1/2}$ - which is quite a large mass for elementary particles. Quoting from the Wikipedia article: $$m_P = \sqrt{\frac{\hbar\,c}{G}} \approx 21.7651\,\mu g$$ Unlike all other Planck base units and most Planck derived units, the Planck mass has a scale more or less conceivable to humans. It is traditionally said to be about the mass of a flea, but more accurately it is about the mass of a flea egg. All this is in sharp contrast with Halton Arp's Theory, where newly created particles - at orbital time $A$ - have initial zero mass. It seems that Narlikar actually has developed two different theories, one for Hoyle and one for Arp, as becomes clear from ANOMALOUS REDSHIFTS AND THE VARIABLE MASS HYPOTHESIS. Why? Didn't Hoyle and Arp communicate? Needless to say that our formulation of Narlikar's Law is in support of Halton Arp's point of view, at least for the moment being.

Is there any empirical evidence for Narkilar's Law? Let's take a look at a reference, which is just one out of the few, found with googling on "Shrinking Kilogram Bewilders Physicists" (the original seems to be lost). Most relevant pieces of the text are quoted:

Physicist Richard Davis of the International Bureau of Weights and Measures, sits next to a copy of a 118-year-old cylinder that has been the international prototype for the metric mass, in his office in Sevres, southwest of Paris, Wednesday, Sept. 12, 2007. Davis said the reference kilo appears to have lost 50 micrograms compared to the average of dozens of copies. [ .. ]
The kilogram's fluctuation shows how technological progress is leaving science's most basic measurements in its dust. The cylinder was high-tech for its day in 1889 [ .. ]
With Narlikar's Formula, we need the Age of the Earth's material, which according to Wikipedia is $\,T_0-A = 4.54 \pm 0.05$ billion years. Furthermore we have $\,T_0 = 1889\,$ and $\,T = 2007\,$, resulting, indeed, in a $\,\delta = T-T_0 = 118$-year-old cylinder. So we have (because $\,\delta\,$ is small): $$\frac{m}{m_0} = \left(\frac{T-A}{T_0-A}\right)^2 = \left[\frac{(T_0-A)+(T-T_0)}{T_0-A}\right]^2 = \left(1+\frac{\delta}{T_0-A}\right)^2 \approx 1 + 2 \frac{\delta}{T_0-A} \\ \hieruit \frac{m-m_0}{m_0} \approx \frac{2 \times 118}{4.54 \times 10^9} \approx 52 \times 10^{-9}$$ Which is indeed approximately 50 micrograms when compared with the standard Kilogram. Coincidence or not? Let's quote from an email received about the subject.
P.S. I was at the BIPM meetings in the late 2000s when they discussed the problems with the 120-year old kilogram in Paris. Impurities in the platinum reacting with pollution in the atmosphere of Paris were thought to be the cause for the change of its mass. But one could not analyze the impurities in the kilogram without taking a sample from it, which would have changed its mass! Not being able to identify those complex chemical reactions was the source of bewilderment of the scientists, not the change of mass of the reference kilogram.