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.

Eminem as an Immaterial Boy

Lee Smolin argues for the novelty of mathematical insight, that Mathematics is but a conjured set of rules, as materially real as the physicals of Nature they happen to describe. Not an ill-fitting off-the-rack selection from a Platonic warehouse, rather some bespoke made-to-measure cut by the latest scientific tailor.

That the mathematical code existed prior to its conjuring is akin to arguing for the pre-existence of ourselves as a conscious being, encoded as we are with successive precursors' DNA. After all, even primed with such knowledge of the fateful meetings of that multiplicity of forefathers and mothers, loaded as they were with their unique gubbings of code and code readers, we would strain to argue for our predestiny.

Is this immutable Platonic universe of ready-made mathematical machinery the object of a wilful lack of thinking; what we term a "belief" (see right). We need to be wary in drawing such a conclusion. Any sense of comfort drawn from an apparent comprehension of some Quantum chicanery is usually rendered cold not long after, sympathy for the Platonic view espoused by Penrose soon returns. Smolin asserts (fqxi.org) that the:
"Bulk of mathematics consists of elaborations of three core concepts grasped merely from looking at the world":

1-Geometry captures that objects take up space and form shapes;

2-Number captures that the world contains countable distinguishable objects;
3-Algebra captures that objects and number can be transformed, by processes carried out in time.

Applying then:

4-Logic as the distillation of our reasoning about the first three concepts,

we make deductions (and predictions) of future observations from past observations.

We not only infer and interpolate from spartan data, we extrapolate and generalise. The sciences and physics in particular is but pattern recognition from sparse observation with mathematics connecting the dots. We will follow an argument of each of these in the following as framed by googlePlus Philosophy forum.

On geometry: Einstein's Pragmatic View of geometry

Einstein deployed the trajectory of the photon as his pencil-line: "..light is propagated in a straight line, and indeed in a straight line in the sense of "practical geometry [as opposed to axiomatic, and as such a branch of the natural sciences].." His views on the nature of geometry from [Einstein on geometry and Experiment] is best summarised by the quote he gives of the French giant, H Poincaré whose shoulders he stood on with clod boots:

"Euclidean geometry is distinguished above all other imaginable axiomatic geometries by its simplicity..."

From all possible geometries we reveal with our light sensitive observations there is but a small subset of practical geometries that are descriptive of our reality. As Penrose conjures it right, [from The Large, the Small and the Human mind, Cambridge] from the Platonic World of Euclidean points, and line elements and a whole lot more besides, our Physical World instantiates but part of a small subset. Penrose envisions a Triality: to the Dualist's view of co-existence of immaterial conscious mind and the material world it apprehends, he sets as distinct that mind's best descriptor. More on this MindMatter&Maths Trifecta bet later.

That we need to accommodate objects with spatial extension, being in relative motion be they linear or rotating and thus defining an orientation, Einstein notes further the need for locally flat hyperbolic geometries:

 "In a system of reference rotating relatively to an inert system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inert systems we must abandon Euclidean geometry...

...because under closer inspection the real solid bodies in nature are not rigid, because their geometrical behaviour, that is, their possibilities of relative disposition, depend upon temperature, external forces, etc. Thus the original, immediate relation between geometry and physical reality appears destroyed"

Such practical geometry is reminiscent of the following analogy taken from Sciama (Unity of the Universe): envisage a circular metal plate emblazoned by the sun but for a shadow cast over its inner core. A ruler dragged around its hotter radius will be lengthened relative to that ruler dragged over part of its journey traversing the diameter. We have then 𝝅<C/D so we deem the plate curved.

The secret to our better understanding of the universes' workings it seems though has been to go well beyond such apprehendable analogising. Abstraction after all involves extension of dimensionality to beyond our apparent three and number type beyond the countable naturals on our hand.

On the Number and Algebra 

Smolin, like Eugene Wigner (Unreasonable_effectiveness) before him describes the effectiveness of Mathematics in its ability to saliently describe Physics. Through its succinct encoding of ideas, both rich and broad in scope, layers of interconnectedness are revealed in its multitude of forms. Why he asks,

"do the different division algebras organise the classification of the possible symmetry groups of continuous geometries?"

Why is that the two sets of Fields of Real and Complex numbers together with the set of Rings of Quaternions and Octonion (Cayley) numbers map to those Groups of objects whose Representations are those realised by the fundamental constituents of matter? The reasoning for this would go something like the following. We observe the world at its most simple appearing symmetric in the sense that it is built up from indistinguishable irreducibles filling a space that is similarly imbued with this indistinguishability property. We extrapolate this observation to Cosmic scales arguing that across such very large scales, space seems coursely to have no preferred directionality. That is, that we deem our universe to be both isotropic as well as homogenous on small and large scales is largely by convenient appeal to some descriptive mathematics.

All this allows us to describe constituent matter according to it being invariant representations (Baryon_decuplet-see right) of the continuous Lie Group SU(3)xSU(2)xU(1) of the standard model.

For space-time, the SL(2,C) group captures (group endomorphisms) both a double-fold rotational symmetry and the local connected Lorentz invariance under the Connected part of Special Orthogonal Group (see below). According to the Elie Cartan multi-vector calculus, spatially extended field-objects and space itself are best described by a graded Algebra of Grassmann-valued (quaternion-valued) exterior forms.

In between the micro and macro scales things are little messy, we call this phenomena "complex" and set it aside until our next best effective theory can address the awkward presence in this regime of relatively cold interacting stuff that amongst other things, forms an awareness that makes some little sense of all the hot simple stuff.

These "group symmetries" are Eddington's catch that we gather from the casting of our mathematical net over the universe:

"He casts a net into the water and brings up a fishy assortment. Surveying his catch, he proceeds in the usual manner of a scientist to systematise what it reveals. 

He arrives at two generalisations: No sea-creature is less than two inches long. (2) All sea-creatures have gills. These are both true of his catch, and he assumes tentatively that they will remain true however often he repeats it."

That there are no Cayley numbers caught in the net need not detract from the marvel that quaternions abound.(europarl.sanctions).

We may note that in our identifications above, the notion of number (as Group, Ring or Field) type has a richer and older history than that of the geometric-led observation of the continuous Lie group. We see right, two of the four division algebras that map to the symmetries as the real, R and Complex, C fields. Quaternions, H as the non-commutative Ring with the Cayley number (Octonions) are the others, the latter in being non associative is the least instantiated of the algebras of real-world indistinguishability.

Smolin in arguing that these Numbers "organise" the group structure mispeaks, in part. "Organise" even not taken literally as an active verb, implies the Number system does the job of categorising. To be consistent he would have to argue that the observed Lie group structure in nature is most conveniently catalogued by these four number systems. Indeed even beyond arguing semantics it feels that to be such a Smolin Naturalist ('Physicalist") of no independent Platonic persuasion our causally wired minds are left unsatisfied: that a clear causal link to our novel invocation of the best of breed effective theory of the day can never be established.

At another level Smolin seems to contradict. Observationally we see things by their very distinguishability, rendering their categorising and counting useful. Invoking the distinguishability character of Number would get you nowhere in understanding the "invariant" objects of study as captured by the Standard Model.

The delineation of our methods of mathematical enquiry along the three core concepts of geometry, number and algebra seem not wholly mutually consistent. That is with the precept that we are "merely" identifying useful correlates in our descriptions of the world.

Yes, in scoping the Cosmos both big and small we invoke Einstein's practical Geometry, classifying Algebraically, Nature's time and space according to its homogenous (and isotropically) invariant form. But we are not invoking what Smolin claims is the characterising quality of Number: to enumerate distinguishables as we do to identify ourselves uniquely (math.feld).

On logic: arguments for supremacy of human awareness

Penrose further from his [The Large, the Small and the Human mind, Cambridge], on beginning to pin down what is "consciousness", which he admits he does not know how to define, refers first to suggestive words such as "understanding" or better "insight" and says:

"I am not going to define these terms either-I don't know what they mean. There are two other words I do not [sic] understand -awareness and intelligence. Well, why am I talking about things when when I do not know what they really mean? It is probably because I am a mathematician and mathematicians do not mind so much about that sort of thing. They do not need precise definitions of the things they are talking about provided they can say something about the connections between them. "

Penrose admits to having no "knowledge" of the words' meaning; whatever that would entail, but finesses the potential for expressing mere belief (see first venn diagram again) by invoking our fourth core concept: logical reasoning. Without reference to definitions Penrose argues that the mere (categorical) interrelations of these words justifies the uses of a logical syllogism:

• Intelligence requires understanding, 
• Understanding requires awareness, 

Hence using deductive reasoning we have:

• Intelligence requires awareness. 

Penrose's viewpoint on awareness (see image psychologieogy )and the relationship between conscious thinking and computation is then (not the "strong"-all thinking is computation, or less strong "computational simulation cannot of itself evoke awareness) but:

"Appropriate physical action of the brain evokes awareness, but this physical action cannot even be properly simulated computationally"

If one were to extend his line of logic explicitly to arrive at his viewpoint on the computational content of thinking, the predicate structure would follow:

• awareness requires more than computation, 
• therefore intelligence is more than computation.

The argument is thus that there is more to our intelligence than pure computation. Just because we encode our latest Ansatz or schema for describing the universe in a logical mathematical form, that does not mean that our present, rather evolved state of its appreciation could have been fashioned by some super-silicon-cubit crunching "designed" machine.

Living as Eminem and M theory

We need ask lastly whether to view Mathematics as outside the physical world and/or as distinct from an immaterial mind? Eugene Wigner argues that conscious systems, unlike inert systems cannot reside in the purgatory of the mere material world, deterministically evolving as superpositions of quantum states until we deign to take notice of them.

When a material system is in a superposition state it evolves "Unitarily" according to deterministic Schrodinger-or better- relativistic Dirac equations, into more complex superposed states. A measurement of location of an electron (say) via Einstein's photon probes will provide, by the collapse of wave function that electron's determinant place, if not its full state such is the cost of a complimentary indeterminate momentum measurement.

How does this viewpoint provide for an interpretation of the Double-Slit experiment performed with singly emitted monochromatic laser photons, as such strictly the purvey of Quantum Mechanics? Penrose explains the configuration of a source S, two slits A and B and a screen beyond upon which a point P on that screen marks where a photon's energy is noted as discharged.

The photons arrive at the screen as individual events and they are detected separately just as if they were ordinary [classical] particles. If only slit A were open (when source S has deemed to emit?) there would be a certain set of places on the screen that the photon could possibly reach. I choose carefully now a position P on the screen within that set and close slit A.

With only slit B open I may again find that the photon reaches that same chosen spot, P on the screen. But if I now open both A and B I may now find (if my point was chosen carefully enough) that the photon cannot reach that spot. That is to say that even though it would have hit P if either of A or B were separately open, somehow the two things which the photon might have done have cancelled eachother out.

In Classical physics, which informs much of our logical deductions according to Smolin, IF one scenario does not take place the alternative (OR) does. In this Quantum regime, Quantum logic applies; from a contrivance of two possible outcomes that "might" have happened the system conspires for neither to happen, as in an exclusive Or, (EOR) (quora.com):

The photon en route is in a Complex (probability field-weighted), w or z wave of alternative routes (SLP or SRP) taken, a superposition of w*SLP + z*SRP states. Only when the photon is observed at some P is the linear set of superposed plausibles reduced. The (states') wave function "collapses", upon "measurement" by virtue of being localised at the screen annihilation point. At the moment of this "State Vector Reduction" the square of the moduli of the complex valued z and w, non-deterministically, (that is, probabilistically) weight the relative likelihood that one path over the other was taken.

Wigner subsequently argues that such a "system" comprising an intermediary experimentalist, her observational apparatus and the double slit set-up cannot all be in the purgatory of a superposed state. The material screen, extinguishing photon and classically sized measuring apparatus do for a time persist in some multiple suspended state of superposition. Her conscious immaterial mind that envisioned the flash is, at a lapsed interval later in a definite state. There exists a shared reality of her and ourselves as passive (ex-system) onlookers at the time of (the still later) communication of her observation of the flash. I as removed from the measurement merely ask her mind what she saw and we share her one observed reality.




That she needed to invoke a level of mathematical and physical insight in the course of this interval suggests the Trivecta bet may be lost. Off the peg after all (moderngentlemanmagazine/)?

Would you bet on there being Trivecta of Mind, Matter and Mathematics?