Gravitation as an Inertial Process

Scientific Paper
Title	Gravitation as an Inertial Process
Read in full	Link to paper
Author(s)	David F Roscoe
Keywords	Gravitation, Inertial Process
Published	1988
Journal	Apeiron
Volume	1
Number	3
No. of pages	13
Pages	1-13

Read the full paper here

Abstract

According to our common experience all our useful concepts of space and time are absolutely correlated with the existence and evolution of matter. This statement represents a qualitative definition of the strong interpretation of Mach's Principle (SIMP), and the primary purpose of the presented paper is to derive a formal quantitative expression of it. This purpose is achieved by a careful consideration of the "inertial" concept, and the resulting formalism is a description of inertial processes given in terms of relations between our concepts of spacetime and our concepts of inertial matter. Thus, for example, the formalism suggests an understanding of the origin of "inertial forces", which is directly analogous to our understanding of the electromagnetic force exerted on charged particles by electromagnetic fields.

Overview

David F Roscoe, then at the University of Sheffield's Department of Applied and Computational Mathematics, sets out in this 1988 Apeiron paper to give Mach's Principle a formal mathematical expression rather than a philosophical one. He works from what he calls the strong interpretation (SIMP): "all our useful notions of space and time are absolutely correlated with the existence, and the evolution, of matter." The paper's programme is to convert that sentence into an equation, and then to see what kind of theory of gravitation the equation turns out to be.

The result is unusual. Roscoe arrives at a description in which the metric properties of spacetime are determined by a scalar "inertial radiation field" U satisfying a covariant wave equation. On this picture inertial matter stands to the inertial field exactly as charged matter stands to the electromagnetic field, so that "inertial forces" experienced by an accelerating body admit the same mediated-exchange description used for electromagnetic forces on charges. Because the inertial field is a real radiative medium filling spacetime, it also supplies the "material vacuum" that tired-light cosmologies require — and Roscoe closes by identifying the cosmic background radiation as the product of photons from discrete sources losing energy to that field. Where general relativity treats gravitation as geometry sourced by a stress-energy tensor, Roscoe represents "the conventional gravitational processes as pure radiation processes in a thermodynamically evolving Universe."

The argument

What a formal statement of Mach's Principle must look like

Roscoe first argues that any formal SIMP statement is bound to have a peculiar logical character. He reminds the reader how Newtonian mechanics is actually built: from the observation that in a collinear collision of two smooth balls the ratio of the velocity changes Δv₁/Δv₂ is constant, independent of the initial velocities. Defining that ratio as the inertial mass of one ball relative to the other, and momentum as mass times velocity, the third law and the whole of Newtonian mechanics follow from the conservation of a single quantity.

The lesson he draws is that "concepts of 'inertia' are fundamentally descriptions of certain particle properties expressed in terms of a particular model of spacetime." In Newtonian mechanics the inertia concept is inertial mass and the spacetime model is Galilean. Writing S for the spacetime model and I for the inertial model, we always have I ≡ I(S), so a formal statement of SIMP must take the form f₂[I(S)] ≡ f₁(S): a description of spacetime given in terms of spacetime measurements. Roscoe accepts openly that this "is ultimately tautological!" — and defends it. The equivalence statement itself merely records a relation intuitively understood "for at least two centuries, since Bishop Berkeley"; the physical content lies entirely in the explicit form of the operators f₁ and f₂.

From the light cone to a wave equation

To fill in those operators he needs concrete concepts of S and I. For S he takes Lorentzian spacetime, represented by a metric tensor, on the grounds that there is no reason to think it inadequate as a local model. For I he needs an inertial property that is relativistic with respect to that model, and he finds it by reformulating the everyday notion that "inertia is resistance to change in motion" as: inertia is the property which, within an inertial frame, confines a particle's possible states of motion to the interior of the local light cone. Formally, the worldline must satisfy a quadratic inequality in the metric — and Roscoe then makes his key move: he defines an "inertial point particle" as the set of all the particle's future possibilities originating from the point where it is defined, that is, by the light-cone condition itself. He stresses this is not a definition of inertial mass, which is a relative concept, but of the essential inertial properties of a single particle expressed through the spacetime model.

Requiring the relation between S and I to be covariant under arbitrary coordinate transformations then yields, at a point, an expression setting the metric tensor proportional to the second partial derivatives of U with respect to the coordinates (a factor of one-half being absorbed into U). Roscoe reads this as "a statement of the essential pointwise equivalence between the concepts 'inertial mass' and 'inertial reference frame'."

The theory becomes complete when this local statement is promoted to a universal one. U generalises from a point particle to an inertial particle field, and the relation is rewritten in generally covariant form with the partial derivatives replaced by covariant derivatives involving the Christoffel symbols. Contracting with the inverse metric then gives, as Roscoe shows, the generally covariant scalar wave equation — which is why he says U "is more properly described as an 'inertial radiation field'". The summary is compact: the geometric properties of spacetime are described purely in terms of an inertial radiation field associated with the matter content, determined by a relativistic wave equation.

Meeting the standard tests

Roscoe then asks whether this qualifies as a viable gravitation theory by the standards of the day, which he takes from Clifford Will's Theory and Experiment in Gravitational Physics. He argues the Einstein Equivalence Principle is satisfied automatically: the light-cone definition of an inertial point particle can hold only if all possible individual trajectories are locally geodesic near the origin, so geodesic motion for test particles is built into the theory. Weak Mach's Principle — the metric coupled to matter — is satisfied a priori. These, he says, suffice for consistency with the classical tests.

The harder standard is the Strong Equivalence Principle, which he characterises as requiring that a viable theory consist of a metric description of spacetime coupled directly to a description of its matter content "with no other field of any kind being involved" — a criterion which, he notes, disqualifies virtually every historical rival of general relativity. Roscoe claims his theory passes, since its equations relate the metric directly to a representation of the matter content (its inertial properties) with no additional fields. He connects this to the binary pulsar observations of Taylor and McCulloch on PSR 1913+16, which on Will's multipole reading require the absence of dipole terms in gravitational radiation. Performing a multipole analysis of a general spherically symmetric field, Roscoe reports three results: the zero-order term describes a "heat death" Universe; the expansion to the monopole term gives the weak-field case and conforms to all the classical tests, with a line element of exactly the same form as the Eddington form of the Schwarzschild line element of general relativity to order 1/R³; and the dipole term is completely absent. He therefore claims the theory is "unique, with GR, in conforming to the requirements of the SEP."

Flat spacetime, the background radiation and tired light

The final section takes the limiting case. If spacetime is assumed globally flat, the covariant wave equation reduces to the flat-space wave equation, whose solution Roscoe writes as a retarded integral over all sources with homogeneous conditions at infinity. Since every point of spacetime is then a source of equal strength, U is perfectly homogeneous and in radiative equilibrium. Quantised with Bose statistics — appropriate to a scalar field — and subject to the ordinary laws of thermodynamics, it must have a perfect black-body spectrum.

Roscoe then asks what matter distribution such a field could correspond to, and answers by looking at local experience: the locally flat spacetime of a free-fall observer is correlated with an electromagnetic background that is approximately black-body and, though Doppler-shifted differently in different frames, isotropic in one particular frame. A scalar field derived from it inherits that isotropy, and being scalar is isotropic in all inertial frames. If the CBR is an artifact of thermodynamic evolution — as it would be in any tired-light theory — then the associated scalar matter field must ultimately evolve to a perfect black-body, isotropic state, which is exactly the character of the inertial radiation field of the limiting-case Universe. The limit case is therefore identified with the "heat death" Universe.

If that identification is unique, a further consequence follows: a thermodynamically active Universe cannot be globally flat, so heat death and global flatness must be approached together, and thermodynamic evolution must proceed through interaction with the inertial radiation field. That field then plays the part of the "material vacuum" demanded by tired-light theories — a role Roscoe explicitly likens to Dirac's 1951 proposal, and places in the line running through Zwicky (1929), Finlay-Freundlich (1954), and Pecker, Roberts and Vigier (1972, 1986). The CBR is then the product of a tired-light effect: photons from discrete sources interact with the inertial field, lose energy — producing a redshift — and drive a general redistribution of inertial and electromagnetic energy toward the black-body state.

The paper ends with an unusually personal note: "for me, Newtonian mechanics represents the perfect example for the development of physical theory — observations, concepts, formalism, theory."

Assessment

The paper's most attractive feature is its methodological discipline. Roscoe does not begin with a favoured mechanism and hunt for support; he begins by asking what kind of statement a formalised Mach's Principle could possibly be, notices that it must be relational and in a sense tautological, and identifies precisely where the physical content can enter — in the explicit form of the operators connecting spacetime to inertia. The Newtonian collision analysis is a genuinely illuminating piece of exposition, and the definition of an inertial point particle as the totality of its future possibilities, bounded by the light cone, is an elegant way to make an inertial concept manifestly Lorentz-invariant. It is also to his credit that he tests the resulting theory against the strictest observational standard then available, the binary pulsar, rather than only against the classical solar-system tests, and that the theory's cosmological consequences fall out of the formalism rather than being bolted on.

The difficulties begin where the derivation compresses. The step from the covariance requirement to the relation between the metric and the second derivatives of U is asserted — "we can easily see that" — with no demonstration that this is the unique covariant relation available, and it is a strong claim, since it makes the entire ten-component metric a function of a single scalar. That is the theory's central weakness: a scalar field has far too few degrees of freedom to reproduce general relativity. Concretely, a theory whose metric is generated by one scalar cannot support the two transverse-traceless polarisation states of gravitational radiation, and its lowest radiative multipole is generally the monopole, not the quadrupole. Roscoe's claim to consistency with PSR 1913+16 rests on the absence of a dipole term, but the binary pulsar's measured orbital decay agrees with the general-relativistic quadrupole formula to a fraction of a percent; showing that a dipole is absent is necessary, not sufficient, and the paper never computes a radiated power to compare against Taylor's number. The multipole results are stated as things which "can be readily demonstrated" without the demonstration. Similarly, the claim that the weak-field line element matches the Eddington form of Schwarzschild "at O(1/R³)" is exactly the regime in which scalar and tensor theories are hardest to tell apart; the discriminating measurement is light deflection, where scalar gravitation famously gives the wrong coefficient, and the paper does not address it.

The claim to satisfy the Strong Equivalence Principle is also weaker than it appears. Roscoe argues that no field other than the metric and matter is involved because U is a description of the matter's inertial properties. But U is a dynamical field with its own wave equation and its own retarded solutions, and in the cosmological section it acquires an independent thermodynamic life, exchanging energy with photons. A field that can absorb energy from starlight is not merely a re-description of the matter content; treating it as one is what lets the SEP test be declared passed.

The cosmological section is the most speculative and the most exposed. The chain runs: assume global flatness, assume homogeneous conditions at infinity suffice, quantise a scalar field with Bose statistics, assume ordinary thermodynamics, and obtain a black-body spectrum — then assume the CBR is an artifact of thermodynamic evolution, and note that the two black-body isotropic fields resemble one another. The resemblance is then read back as an identification. Each step is plausible in isolation, but nothing here is derived from the observed background, and no temperature is predicted. More seriously, a tired-light interpretation of the redshift, of the kind Roscoe adopts from Pecker and Vigier, must contend with measurements that were not decisive in 1988 but are now: the (1+z) stretching of Type Ia supernova light curves, which follows from cosmological expansion but has no natural cause in photon energy loss; the (1+z)⁴ surface-brightness dimming of galaxies (the Tolman test); and the exquisite black-body quality of the background itself, since a scattering interaction strong enough to degrade photon energies over gigaparsec paths would be expected to blur images and distort the spectrum. Roscoe cannot be faulted for not answering objections that postdate him, but they bear directly on the paper's closing identification, and they are the reason the tired-light programme it belongs to has not prospered. Taken narrowly — as a formal expression of the strong Machian intuition, and as a demonstration that such an expression leads naturally to a radiative rather than a purely geometric account of inertia — the paper is a serious and careful piece of work. Taken as a rival to general relativity, it does not carry out the calculations that would let the comparison be made.

See also

• David F Roscoe
• Mach's Principle
• Inertia
• Tired Light
• Cosmic Microwave Background
• Redshift
• General Relativity
• Equivalence Principle
• Gravitational Waves
• Fritz Zwicky
• Jean-Pierre Vigier
• Paul Dirac
• Vacuum
• Apeiron