Profound conceptual incompatibilities exist between General Relativity (GR) and quantum mechanics (QM) as currently formulated. For instance; 1) whereas in GR, time is just another covariantly transforming coordinate of an arbitrarily differentiable spacetime manifold, in QM it is an external, Newtonian parameter and properties are discrete, and 2) the respective macroscopic and microscopic domains of applicability remain as disconnected as ever. The paper proposes perspective at as a possible means of unification. A simple mathematical principle is then invoked as a natural limit on and the implied existence of a “maximum physical reality” therefore means that effects that are clearly perspectival at must become “more real” (effectively observer-independent) with each increment. (The apparently microscopic thus becomes the effectively microscopic, and 3-space scale extremes are juxtaposed.) Characteristics of the relationship between the earth- bound observer and the horizon are applied to, suggesting perspectival interpretations of space, universal expansion, radiation, entanglement, gravitation, mass, dimensional reduction and measurement (as projection rather than amplification). The difference between apparent flatness of the 2-surface and actual global closure corresponds to the boson of the gauge interpretation (i.e., the observer’s height, projected from the local, macroscopic frame to the remote, effectively microscopic frame, where it appears as a “collapse vector”). Although horizons therefore appear to radiate, a Hawking photon is instead thought of as a “unit of projection” (of the local frame to the remote), thus challenging the notion that photons “travel through space”.

The paper is structured as follows. Section 2 considers the natural limit on and its implications. The main geometrical characteristics of are outlined in Section 3, including a duality of the fundamental constants of GR and QM (i.e., and). Section 4 interprets perspective as gauge, and Section 5 then interprets the Standard Model gauge group as perspective, with gravity included by way of charge- mass duality. Characteristics from Section 3 are then applied to black holes in Section 6 using an Alice-and-Bob scenario. A scheme for quantum gravity (QG) is outlined, together with an account of black hole entropy and an interpretation of the information paradox as a perspectival effect. Finally, Section 7 is a discussion and brief

summary of the main ideas. In terms of notation, the case in which a surface is spherically closed and embedded in is denoted typically as.

References and are given (Figure 1), and at each iteration, a mid-point or difference is specified between the references, using the previously-established difference as the new reference. (After the second iteration, the line of references approaches the difference asymptotically.) After three iterations (the dashed line) the difference is completely detached from the original references.

If differences are now expressed as derivatives, then the difference between two spacetime positions and is a change of position or velocity, and the difference between two velocities is an acceleration, etc. At each iteration, the previously established difference is used as a new reference, and the de-

taching again occurs between iterations 3 and 4. Thus, at iteration the original references are “superseded” (the original reference only appears inside brackets after this level). Assuming the applicability of this simple principle, the obvious claims would be 1) that the Universe in effect “runs out of references” by which to distinguish differences (i.e., that the Universe only differentiates twice with respect to time), 2) that is effectively indistinguishable beyond D6 and 3) that physical observables are therefore also limited. This also implies that there is a “maximum physical reality” at D6, that effects that are clearly perspectival at D3 become “more real” (effectively observer-independent) with each increment, that D7 is nonphysical (i.e., mathematical) and that D8 may be associated with observer objectivity.

Since the observer also possesses limited freedoms, perspectival effects at (e.g., geometry, horizons) become increasingly observer-independent with each increment with respect to the observer; i.e., the

observer possesses diminishing freedoms with respect to physical phenomena. Therefore we define dimensionality in terms of freedoms. Whereas in the observer possesses the freedom to orientate in any superposition

of axes, in the observer has limited freedom to orientate in 4-space in axes ¹, and in

, freedom is further curtailed; a body now has mass (i.e., is subject to spacetime curvature). Similarly for horizons; at the planetary horizon is global (remote), surrounds the observer, and is completely observer-dependent, moving as the observer changes 2-position (i.e., is totally “permeable”). At the cosmological horizon is also global (remote) and surrounds the observer, but is less observer-dependent, moving as the observer changes 3-position (i.e., with the observer’s velocity) but being causally “semi-permeable”. Finally, at, massive bodies are now subject to local (multiple) black hole horizons which do not surround the observer (rather, it is as though a remote, global horizon has been brought into the local frame, in some sense). The event horizon is now a strictly one-way membrane with unclear observer-dependent status; one would expect it to accelerate with the observer. (The Rindler horizon, with similar thermal properties, appears at constant 3-distance for constant acceleration.)

Perspective in 3-space is that effect whereby a receding object appears to shrink to an apparent point. Perspective plus curvature (e.g., on the closed surface of Earth) means that the size of the object does not go to zero before the object disappears over the horizon. At, using metres as units, Alice stands with 3-height above the 2-surface, which is closed with constant radius of curvature. More formally,

we specify a manifold that is closed with constant radius of curvature and embedded in according to the Nash theorems. At we define a 3-height and a tangent space bounded at a constant distance from x by, where. The manifold is bounded by, where at a distance along such that, where is coincident with the “top” of

. The tangent space is defined as the perception of due to in the frame of an observer located at x with. We note the following characteristics.

1) Assuming, the 2-surface looks flat to good approximation (Figure 2(a)).

2) There is an increasing divergence between apparent flatness and actual global closure (Figure 2(b)).

3) Parallel lines on the surface (e.g., railway lines) appear to converge to an apparent D0 point or singularity on the horizon (at a distance).

4) With increasing distance from Alice, objects appear to contract and space appears to expand, and Alice reasons that these rates must be exactly equivalent;.

5) There is a dimensional reduction from Alice’s 3-height to an apparent D0 point on.

6) There are two such points; A, where Alice (naively) believes she is looking, and B, where she is actually looking. In 3-space, these are clearly the same point; A and B are identified.

7) If Alice gains sufficient 3-height, she will see the D0 point singularity transform or open up to become a D1 line singularity (i.e., there is a non-zero separation, , between railway lines as they disappear over the horizon). This forced transition, , acts as a gate that reveals the embedding of in (i.e., the case).

8) The difference between apparent flatness and actual global closure is just Alice’s local 3-height, projected (and rotated through) to the remote, horizon. (Alice’s apparent 3-height, as projected to the remote frame and seen from the local frame, is less than the thickness of the average human hair;.)

9) The angle subtended at Alice’s eye between the true horizontal line of sight (i.e., tangential at a radius of) and a line of sight to the actual horizon is necessarily equal to that subtended at the centre of the Earth between radial lines to the locations of Alice and the horizon, so the relationship (and transition) between points A and B is proportional to and associated with those between the local observer and the remote horizon.

10) The transition is also that from triangle (1) with, to triangle (2), with, so the transition is associated with the loss of the term.

11) Observer-dependency of the S¹ horizon can be expressed as the fact that 2-distance to the S¹ horizon is a constant in every frame.

12) The of an object that disappears over does not cease to exist but is “conserved” in 3-space.

At, Alice’s 4-height is taken to be velocity (rather than the Minkowski spacetime interval,). Assuming, the 3-surface “looks flat”, but due to the existence of, the apparent velocity of Bob, receding, “contracts” with 3-distance to some irreducible value, interpreted here as a minimum energy,. However, when Bob is at, his velocity relative to Alice is, and so the curvature limits are the characteristic GR and QM constants, and. At

the linear expansion of spacetime with increasing 3-distance from Alice corresponds to universal expansion with constant velocity [1] [2] . We also note that observer-dependency of the S² (cosmological) horizon can be expressed in terms of the fact that 3-distance to the S² horizon is a constant in every frame.

At, Alice’s 5-height is acceleration. Assuming, the 4-surface “feels flat”; i.e., implying a physical basis for, as required by the Equivalence Principle (EP), but due to the existence of, the apparent acceleration of Bob, falling towards the black hole event horizon, decreases with 4-distance to some irreducible value, interpreted here as a minimum acceleration,. At the same time, Bob’s acceleration relative to Alice is, where is an effective maximum beyond which the Unruh or Hawking radiation wavelength is (i.e., a firewall) and so the curvature limits are and. At, the second-order expansion of spacetime with increasing 3-distance corresponds to accelerating universal expansion [3] [4] . We also note that observer-dependency of the S³ (black hole) horizon can be expressed as the fact that 4-distance (v = c) to the S³ horizon is a constant in every frame. Thus the fixed 4-distance to theblack hole event horizon in the local, macroscopic frame is the quantum of energy in the remote, effectively microscopic frame. In this sense, then, ².

Alice and Bob are astronauts moving away from each other in empty space, and, for the purposes of this consideration, we assume that neither has any prior spatial understanding. Alice first notes that Bob appears to be shrinking. Since they disagree on who is shrinking, they each theorise the existence of an embedding 3-space that makes sense of the fact that, with increasing distance, objects appear to contract (CON) and space appears to expand (EXP). Now as observers on Earth (but still without any geometrical understanding) Alice and Bob stand with equal 3-height above the 2-surface, which is closed with constant radius of curvature. Alice is directly aware of the existence of the 2-space, but has to infer (e.g., using parallax) the existence of a higher, 3-space. Assuming, the 2-surface looks flat, so Alice does not recognise as an horizon, but merely as a “dimensional separation”. Therefore the significance of 3-height (or potential) and its relation to the horizon remains unacknowledged. In other words, Alice does not recognise Earth as an object.

However, from outside her frame, we can see that the difference in her frame between apparent flatness and actual global closure is just the potential, projected (and rotated through) to the remote horizon, represented by the “collapse” vector R in Figure 2(b). Now Alice and Bob disagree about shrinkage and phase, which vary from point to point. In gauge terms, the 2-surface is the base space, and the unacknowledged potential is the fibre, defining an internal space at each point on which together comprise a fibre bundle, , such that. The gauge field comprises gauge connections that correspond to the group of continuous symmetries of, and in that there is a canonical projection, submersion or surjective map, , then the phase is a continuous image of in. Thus, R is just the potential, projected from the local to the remote frame, that appears in Alice’s frame as a “minimum unit” (i.e., the gauge boson or field quantum) that is the coupling constant or interaction strength that translates between frames and restores local gauge

invariance. Alice also notes that as she and Bob stand at each other’s horizon, the values 2 and define curvature limits at, allowing this geometry to be applied to.

The Standard Model (SM) is the non-Abelian gauge group that describes particles and interactions in terms of a sequence of early-epoch phase transitions following a Big Bang at. In the perspective approach, the SM comprises two groups of sets of perspective reductions. The first is, where are derivatives,. These generate apparent Planck dimension perspectival convergence limits (with), which are interpreted as fermions. At each level, a height is associated with spherical closure of, forming a series of “onion layers”, where each closure generates -forms that populate space; (i.e., a string), etc, as depicted in Figure 3. The second group is, generating apparent D2 area singularities that live on the horizon separating, interpreted as the leptons, the 2-forms that in the SM are “cousins” to quarks.

Assuming at each level, each surface can be described as a state defined by n direc-

tions or generators, so each lower level can be thought of as a state-splitting or copying process producing families of generations; e.g., the Cabibbo-Kobayashi-Maskawa matrix describing expectation-values for transitions between families and generations of quarks (with the Cabibbo angle corresponding to probabilities of decay modes), and the Pontecorvo-Maki-Nakagawa-Sakata matrix summarising weightings for lepton-mixing. The superpositions of directions (generators) at higher are rotations that transform lower particles, as depicted in Figure 4.

Each inter-dimension or phase transition is a perspective reduction; at each level, a -form possesses a local height (a property defined by that internal space), from which there is an apparent diminishing of until it disappears over the associated horizon. Thus, colour bleaches

out of Quark-world as quarks “fall into Quark-world black holes” (i.e., inside-out hadrons), which is the process of baryogenesis, marking the global closure of D2, the formation of colour-neutral baryons and the conservation of colour. Likewise, charge bleaches out of Nucleus-world as “nuclei fall into Nucleus-world black holes” (i.e., electrons, or inside-out atoms), corresponding to the formation of charge-neutral atoms and the conservation of charge at global closure of D4. In the perspective model at, information bleaches out of the (D4) universe as an SM phase transition. In QED, the atomic nucleus containing positively charged protons is sur-

rounded and perfectly balanced by shells populated by negatively charged electrons;. The 5-height of

the nucleus connects via D(n − 3) perspective reduction to a Dirac spinor of the Penrose zigzag electron, [5] in the last electron shell. In the nucleus frame, assuming, the separation of equally spaced calibrations of the n-surface appears to decrease with increasing distance and so the exact location of the horizon is uncertain, and electron shells are thus interpreted as multiple horizons. The quantum state

of the atom is defined as a superposition of all shells, or, in which are the eigenstates of an observable (e.g., the principle quantum number, n) corresponding to eigenvalues.

Given the correspondence between at the macroscopic level and at the microscopic level, we now ask if there is a precedent for the notion of charge at the macroscopic scale. We note that mass tends towards horizons; either expanding (state; EXP) as part of the Hubble flow towards the cosmological horizon, or contracting (state; CON) towards black hole event horizons (Figure 5). In Alice’s frame, the rates

must be exactly equal;. Such labelling is arbitrary and therefore cannot be defined in any absolute way. In gauge-theoretic terms, just as the photon expresses the fact that we cannot define charge in an absolute sense in the Coulomb gauge, so the graviton would be the gauge particle mediating between EXP and CON “charges”³.

Generalising Newton’s third law, any derivative of action-force applied to a massive body (regardless of mass or composition) will elicit an instantaneous, exactly equivalent and perfectly counter-di- rected reaction-force,. Therefore, the picture suggested is of a massive 3-object (a CON

charge) surrounded by “derivative shells” populated by black holes. The of the 3-object is perfectly balanced by the of the derivative shells, and so the QG definition of freefall would be the superposition of all derivative shells. Therefore mass at the macroscopic level is charge at the microscopic level. In the perspective model, projection of height in the local, macroscopic frame to the remote (effectively) microscopic frame becomes more real, until, at the level of maximum physical reality, it is effectively real. That means that the observer is effectively projected into the microscopic domain, implying a thorough mixing of macroscopic and microscopic scales, and, somewhat outrageously, that electrons are black holes.

Black hole-electron duality (as suggested in [8] - [10] ) opens up many further correspondences. For instance, QED vacuum polarisation, in which the bare electron charge is dressed by slight displacement of virtual pairs away from and towards the charge, respectively, would correspond to GR vacuum polarisation, in

which the bare black hole “EXP” charge is dressed by displacement of virtual charges, dark energy and dark matter, respectively. Also, QED renormalisation, in which the cancellation of divergent inter-

grals may be achieved by removing the electron interior region (where is the classical radius) would suggest that gravitational renormalisation would be likewise achieved by removal of the black hole interior region,.

We might also note that the perspective projection maps the local, macroscopic (black hole) frame to the SM photon epoch following nucleosynthesis but before ions gained electrons. At this time, the densities of relativistic radiation (photons) and non-relativistic matter (atomic nuclei) were equal. The subsequent decoupling of matter and radiation and the formation of charge-neutral atoms corresponds in the perspective model to the global closure of D4. At recombination (the surface of last scattering), the photon mean path length became effectively infinite, and the Universe thus became transparent. At the macroscopic scale, the coincidence (or “why now?”) problem asks why the energy densities of dark energy (i.e., the cosmological constant)

and matter should be comparable. This suggests a perspective map, from the

coincidence problem at the macroscopic scale to the pre-decoupling radiation-matter densities of the SM. In the perspective approach, gravitons at the macroscopic scale are photons at the microscopic scale, in which case, the experimental absence of gravitons would indicate that the Universe is not “yet” gravitationally transparent. We next consider black holes, the firewall question and the information paradox.

One would like to explore a number of further implications of the perspective model, including;

1) that the “geometry of time” (i.e., a potential above a spherically closed spacetime surface) is common to GR (classical spacetime at) and QM (quantum spacetime at),

2) that apparent flatness and collapse correspond to the U and R processes of QM,

3) that at, assuming, quantum superposed alternatives are multiple -collapse thresholds (horizons) at one-graviton spacings, and that Schrödinger cat states collapse when associated spacetime curvature reaches a threshold of [26] .

4) that if black holes correspond to paths, impacts on the second screen of the double-slit set-up should replicate the distribution of black holes in that direction,

5) that Newton’s third law may be applied to EPR entanglement, in that the outcome of a measurement of spin along the z-axis of one entangled particle is perfectly correlated by the opposite outcome of a measurement of spin along the z-axis of the spacelike separated particle,

6) that, as at, apparent 3-height is maximum in the local frame and projected to an apparent minimum in the remote frame, so at, the local “rate at which time passes” (i.e., maximised proper time,) is projected to the remote frame and juxtaposed with “cosmic” time,

7) that the projection of local (macroscopic) values into the remote (effectively microscopic) frame explains apparent fine-tuning, naturalness, hierarchy and large number problems.

To summarise, the two principle reductions (and) and the two scale extremes (i.e., macroscopic and microscopic) define four sectors (Figure 6).

I am extremely grateful to Arkady Bolotin (Ben Gurion University of the Negev) for his comments, advice and patience.

RobLangley, (2016) Why Geometry Radiates: Quantum Gravity as Perspective. Journal of Modern Physics,07,489-500. doi: 10.4236/jmp.2016.76051