Expansion by Proxy

Effective Universality of Laws

The universality assumption underpins our descriptive physical Laws: that their aptness is neither peculiar to our locale or the eon in which we evolved to ascribe them to observations. Put another way, although we constitute a particular inertial observer with (potentially) a peculiar set of only locally appropriate measuring "metrics' with which to probe, we interpolate and generalise on the basis of data patterns and then formulate extrapolated laws, trusting their appropriateness in the beyonds of the very small and big.

Phenomenologically we see this in our descriptive measures and dynamical laws constructed to have particular domains of applicability. Thermodynamic Pressure and Temperature are "effective" coarse grained measures of the state of gases and liquids containing orders of magnitude of moles of atoms. Newton's inertial equations of motion are the "effective" field equations of Schrodinger's wave equations, as such appropriately describing the motion of a large collection of atoms called a billiard ball. Newton's inverse square law, NG of a conservative gravitational force,

Inverse Square Law of Gravitation (1)

is the "effective" field theory of the non-conservative field equations Einstein's General theory of Relativity GR,

Contracted Curvature tensors of space-time geometry equated to Energy-Momentum Sources (2)

Where a prior model such as NG is subsumed by a theory with broader explanatory power such as GR, we seek consistency such that GR reduces to NG in the (far from source) weak field limit which we can observe directly. Accordingly we fix the measure of coupling strength, 𝜅 of energy-momentum to the geometry of space-time according to 𝜅 =8πG/c^4 for Newton's gravitational field strength, G.

Einstein's formulation of gravitation, GR expresses the dynamics of a "metric" ruler of intervals. GR constrains a gauge freedom that would otherwise enable absolute statements about lengths to be made. Wetterich in a 2013 paper,  (Universe without expansion) respects the possible extra degree of freedom or gauge ambiguity that an extended geometry allows one to entertain, arguing that  meter-rulers that represent our laboratory based absorption spectra are simply shrinking giving the impression that the universe is expanding. We explore this novel interpretation in the following.

Conservative forces and Integrability

NG describes a "conservative" force in that it can be equivalently formulated as the gradient (spatial differential) of a "gauge" potential, 𝝓

Force as gradient of Potential (3)

We are relating here a verb (l.h.s.), the "force" to the gradient of a gauge field, a noun on the r.h.s. Using Newton's Second Law, F=ma, rearranging (assuming the equivalence of inertial and gravitational mass) and focusing rather on the kinematical acceleration "interpolating-Faraday-field" g=F/m we note the equivalence between the accelerated motion of a test body to the contoured gauge potential. Observing that lifting a mass through the gravitational field, g and then returning it to its original position, or in moving a mass around a loop contour in the field we do no work, we describe this by a (closed contour) Integral for a displacement vector, r as

Work Done against a conservative Force around closed contour (4)

Indeed equation (3) follows from assuming (4) by the Fundamental theorem of Calculus, if we define the function 𝝓(x) as an integral:

The gravitational "gauge" potential (5)

In this way NG, (eq 1) are the "integrability conditions" for the scalar gauge potential, 𝝓(x). They are a description of a conservative force since mechanical energy is not dissipated as an ideal test body moves through it.  A body's potential to do work is dependent only on its position in the field and independent of path taken to arrive at that point. It retains no memory of its journey through the field.

Gauss's field formulation (GF) of Newton's equation of gravitation (1) equates the divergence of the field, g to a central attracting massive source object of density, 𝝔. The formulation is more explanatory in that it emphasises the primacy of the gravitational field, g in making it explicitly a noun (it exists even if no test mass is there to feel it) and teases out the wave character of the interpolating gauge potential 𝝓(x) 

Gauss's formulation of Gravitational attraction (6)

Substituting (6) into a g-field version of (3),

we obtain Poisson's equation,

Poisson Equation as Integrability condition for Gauss-Faraday-field (7)

which reduces to Laplace's wave equation for 𝝔=0 and we begin to see this central body conservative field generator formulation as an effective Schrodinger-type wave field equation in an indirectly observed field 𝝓(x). Moreover we see that Poisson's equations represent the integrability conditions for the path-independent Gauss-Faraday-gauge field, 𝝓(x).

Weyl's Gauge "Scale" Invariance  

The Universe in most interpretations is expanding: our measured value of the Hubble's constant (that relates recessional velocity of galaxies to their distance from us), Ho is argued to be representative of all other observers in the universe. Through the inverse of Hubble, 1/Ho we derive the proper age of our universe as 13.8 billion years. There are highly suggestive reasons, including detailed evidence from laboratory experiments, for interpreting redshifts of distant Galaxies' emission spectrum as an increase in the ratio of measured cosmological distances to those on atomic scales.

Dirac encourages us to entertain two scales. One "atomic unit" measure appropriate to scales of order of the Bohr radius in which Plank's proportionality measure, h is a constant and one of an astronomical scale, a "Cosmological unit", in which Newton's measure of gravitational strength, G is a constant. That we cannot make an absolute statement of size only a relative one based on ratios reflects the presence of an inherent "gauge" freedom left in such a theory.

Strictly within the framework of GR it is an open question as to whether cosmological distances are absolutely increasing or the atomic comparative scales of the laboratory are contracting. If my ruler happens to shrink because it atomically contracts upon cooling, but the object whose length I am trying to measure maintains a fixed density it will appear to have relatively expanded. Equivalently, I will measure the same "growth" in the object of study if it had expanded with space but my ruler had stayed locked in the same density configuration.

In Einstein's GR, mass-energy comprising fluid pressure, electromagnetic field, fermionic matter sources etc, non-linearly shape space in a way that is described by the non-flat Euclidean geometry of Riemann.

The rotated angle of a vector (per unit area) when parallel transported around a closed loop of infinitesimal space "measures" the local curvature at the center of the loop. The vanishing of curvature of Euclidean space upon which Newton mechanics plays out is marked by the integrability of such finite parallel displacement of direction. NG path-independence is as discussed above.

In Weyl's 'Gauge" Geometry, a spacetime not only has the above curvature of direction (stanford.-weyl) but also possesses a curvature of length such that we observe path-dependency, non-integrability of the scalar measure of length when traversing that space and arriving at a common point.

Dirac's Metric Domains of Applicability

Dirac  provides a Large Number Hypothesis (LNH) to motivate the potential varying of G with epochs:

and supposes further that the bracketed quantity is a fixed constant of proportionality, containing purely locally peculiar "atomic" information remaining as such fixed at all epochs,t so that G∝Ho.
Dirac pointed out, the constancy or otherwise of a physical quantity depends on the system of units used, such that the constancy of charge, masses electrons and protons, Plank and speed of light, e, me, mp, h, and c presupposes the use of fixed "atomic unit" measure.

If we denote the scale factor characterising the "Cosmological metric" Radius of the universe at any epoch t as R(t), and fixing its length such that at our present time, t= to  we have R(to)=1, then
Ho(t) ∝(1/R)dR/dt. If the scale factor, scales as some power law of time since the time when the Cosmological and Atomic measures were one and the same, R∝t", we therefore have that, G∝1/t, so that (1/G)dG/dt=-1/t.

This states that the rate of change of Gravitational field strength, G as a proportion of its ongoing value is asymptotically decreasing.

To reconcile the fixed value of G as measured astronomically with its variation in terms of atomic units Dirac ask us to assume that the ratio of the two measures changes with epoch. Our two "metrics", one "Atomic", and one "Gravitational", denoted respectively by dsA and dsG form a time varying ratio across epochs, β(t)=dsA/dsG.

According to Narkliar:

"Gravity has stood apart from other interactions both in the unification program and in attempts at quantization. This isolation leads one to suspect whether G is in fact a fundamental constant, or even a quantity of purely local significance."

Wetterich's Alternative Interpretation of Redshift measures

Wetterich (Universe without expansion) asks us to imagine that the masses of electrons and protons were smaller at the time of photon emission from a distant galaxy. Smaller that is than they are in our locale as we observe them extinguish in our measuring apparatus of today. Given that the characteristic light emitted by atoms is also governed by the masses of the atoms' electrons, if an atom were to decrease in mass, Plank and Einstein's laws E=hf=mc^2 tell us the photons would become less energetic. A lower frequency of emission and absorption would manifest as a shift towards the red part of the electromagnetic spectrum. 

When we look at distant galaxies, backwards in time, seeing them as they would have been when they emitted the light, if all masses were once lower, and had since been linearly increasing, the colour of old galaxies would look redshifted in comparison to current laboratory frequencies, in proportion to their distances from Earth. The observed redshift would make galaxies appear to be receding according according to our usual Doppler shift interpretation of spectral line shifts.

In Wetterich's picture, the Weyl-Dirac meter-rulers of our absorption spectra (galaxy spectra) are simply shrinking physics.stackexchange.com.

The Universe is not expanding but rather the mass of everything has been increasing. In this model the cosmological scale factor, R remains constant, only the Planck mass and the particle masses increase with time as the "atomic" metric grows. In this picture the Universe's composite stuff is getting denser as time passes. 

Two questions:

1. Less inertial mass in the past associated to 'stuff' means the slower ticking of clocks. Would not the apparent precision of periodicity of a pulsar clocks versus an atomic clock on earth be brought into question if this view is entertained?
2. Does Wetterich's view explain Olber's paradox satisfactorily? 

On the latter. That the night sky is not filled with starlight as our complete field of view is not consumed by light. Just as the the Sun consumes our field of view and so its apparent brightness is the same as its intrinsic brightness, we do not see a sky thermally radiating at 6000K like the sun. 

Rather we observe the universe mostly radiating in microwave regime at 2.7K.

A nice explanation following Sciama would be to attribute the excess curvature we see in the universe (that we presently attribute to dark matter) as explicable by the presence of this cold old matter. Being less dense and barely ticking off time it does not radiate. The voids in the sky are just old times in which time flowed too slowly for anything to reach us. The excess curvature observed comes from these cold shadowy parts. We see curvature because a circular disc encircling these parts will be bathed in a cold heat (as it were). The rulers strangely should be shorter in these regions and thus circular loops will have circumferences greater than 2𝝅R.