Relational Relativity
| Scientific Paper | |
|---|---|
| Title | Relational Relativity |
| Read in full | Link to paper |
| Author(s) | Amir M Abbassi |
| Keywords | Inertia, Mach's Principle, Relativity |
| Published | 2002 |
| Journal | Apeiron |
| Volume | 9 |
| Number | 2 |
| No. of pages | 18 |
Read the full paper here
Abstract
According to a simple model of inertia a Machianized theory of special and general relativity is presented.
Overview
This paper by Amir H. Abbassi (Tarbiat Modarres University) and Amir M Abbassi (Tehran University) attempts to remove absolute space from relativity itself rather than merely from Newtonian mechanics. Its starting point is a model of Inertia as a genuine two-body interaction, from which the authors derive a modified second law, a modified kinetic energy, and finally modified geodesic equations for general relativity. The goal is a formulation in which the so-called inertial frame is nothing but the frame attached to the centre of mass of the Universe, and in which the physical constants — including G — are determined by the global content of the world.
The dissident thrust is aimed at a claim usually made in passing by textbooks: that general relativity is Machian. The authors think it is not, and that the standard way of showing it — appealing to boundary conditions to explain why the field equations admit flat space Rμν = 0 for an empty universe — is an evasion. Their reconstruction is designed so that the empty universe gives the identity 0 = 0 instead, and a single-particle universe admits no motion at all. Both are stated as the tests a properly Machian theory should pass.
The argument
Mach's two conclusions
The authors take from Ernst Mach's critique of Newtonian mechanics exactly two propositions: (i) only the relative motion of a body with respect to other bodies is observable, not motion with regard to absolute space; and (ii) the inertial motion of a body is influenced by all the masses in the Universe. Everything else in the paper is an attempt to build these into the formalism rather than to satisfy them approximately. They locate their work alongside Assis's relational mechanics and Ghosh's extended Mach's principle, and against the background of the 1993 Tübingen conference volume From Newton's Bucket to Quantum Gravity.
Inertia as a two-body force
The model treats inertia as a real interaction. For two particles in an arbitrary non-rotating frame S, the force is proportional to the difference of their accelerations and to an "inertial charge" carried by each:
- Finertia = μ c1c2(a1 − a2)
with μ an inertial coupling constant. For N particles the total force on particle i sums over all others. The bridge to ordinary mechanics is the definition
- mi = μ ci Σj cj
summed over all particles in the Universe. Inertial mass therefore has two factors: the particle's own inertial charge, and a global effect of every other particle. The authors note this is consistent with observation, since local inhomogeneities are not seen to affect inertial mass — the mass is fixed by the global structure.
A modified second law
Rewriting the N-body force in terms of masses gives
- Fi = miai − mi (Σj mjaj) / (Σj mj)
i.e. Newton's second law with the acceleration of the Universe's centre of mass subtracted. This is invariant under a wider group than the Galilean, which the authors call generalized Galilean transformations, including a constant relative acceleration b as well as velocity and position offsets. Absolute space is thereby demoted: it "is just the frame attached to the center of mass of the Universe in which the Newtonian second law Fi = miai is recovered." Newton's third law follows automatically, with F1 = −F2 for two bodies and ΣFi = 0 for N.
Gravitation and the origin of G
Extending the model to gravity, the equivalence principle is read as the statement that inertia and gravitation have the same source, so the gravitational force between two particles is written μ2c1c2/r2. Comparison with Newton's law then yields
- G = μ / Σj mj
so the gravitational constant is itself a global effect to which every particle contributes, and its finiteness implies Σcj is finite. The authors take this as the signature of a good Machian model: "the so-called physical constants (including G) should be determined from global features of the Universe."
Lagrangian and the vanishing of centre-of-mass energy
Applying D'Alembert's principle to the modified law replaces the usual kinetic energy Σ½mivi2 with an expression built from relative velocities, μ Σij mimj|vi − vj|2 / 4Σmk. The difference is exactly the kinetic energy of the centre of mass, which is cancelled. The authors defend this as physically obligatory in cosmology: "Where we are dealing with the whole Universe, motion and kinetic energy of its center of mass have no physical meaning." A consequence they list as an advantage is that the Lagrangian, Hamiltonian, and hence energy become scalar invariants for a non-rotating observer. They credit Lynden-Bell with reaching a relational Newtonian mechanics by a different route.
They summarize five Machian features: no absolute space is needed; inertial mass is not a natural constant and may change when the total inertial charge changes (they instance a pair-production era); G likewise; energy is frame-independent; and an empty universe predicts no structure.
Relational special relativity
The authors first make a pointed observation about the standing of special relativity in a Machian scheme. From G = μ/Σmj one may set μ = 0 and obtain a world without inertia, but a world without gravitation is impossible — so "the subject of special relativity because of its ignorance of gravitation is under question and cannot be considered as a global theory from a Machian standpoint." They then construct one anyway.
The construction turns on how the line element is defined. A distance measurement, they insist, is only ever made between two physical points — points carrying mass — not with respect to an arbitrary geometrical origin. So for a non-interacting N-particle system they define the line element of particle a by subtracting the mass-weighted mean displacement:
- dsa2 = ημν (dxaμ − Σmbdxbμ/Σmb)(dxaν − Σmbdxbν/Σmb)
The resulting equation of motion for the kth particle is duk/dp − ducm/dp = 0, reducing to the modified Newtonian form in the low-velocity limit. Since the Lagrangian involves no a priori absolute space, they call the theory built on it relational special relativity.
Relational general relativity
Carrying this to curved spacetime is not straightforward, because summing vectors located at different points requires parallel transport along specified paths. Rather than define a relativistic centre of mass — which they note is anti-Machian in itself, since it is a point where the system's total mass is imagined located, and which different observers place differently — the authors introduce a centre of inertial charge and, more importantly, a variational constraint. Defining δXμ ≡ Σn mngμλ(xn)δxnλ, they postulate as a Machian principle that δXμ always vanishes.
Imposing this on the standard matter action by undetermined Lagrange multipliers yields a modified geodesic equation in which the mass-weighted average of the geodesic expression over all particles is subtracted from each particle's own — the relativistic analogue of subtracting the centre-of-mass acceleration. They acknowledge the constraint is not covariant, but treat it as "at least a clue," and then propose a covariant version in which the averaged term is parallel-transported to the location of the particle in question by an operator U.
The two Machian tests
The paper closes with the two results it was built to obtain. First, in the standard treatment one must invoke boundary conditions to explain why an empty universe admits Rμν = 0 rather than nothing at all. Here, since G ∝ 1/Σmi and the coupling appears on the left-hand side as G−1, an empty universe with Σmi = 0 makes the field equations read 0 = 0 — "a perfectly Machian result." Second, for a single-particle world the modified geodesic equations also give 0 = 0, "an ideal result from a Machian point of view," since a lone particle should have no motion and no inertia. The authors thank J. Barbour for comments.
Assessment
The paper's virtue is that it takes a criterion usually applied rhetorically and turns it into two sharp, checkable requirements — the empty universe must yield an identity rather than a solution, and a single particle must have no equation of motion — and then builds a formalism that meets both. That is a real contribution to the Machian literature, and the mechanism is elegant: because G is made proportional to the reciprocal of the total mass, emptying the universe removes the coupling constant along with the source, so both sides of the field equations vanish together. The two-body form of inertia, with acceleration differences and inertial charges, also has the attractive property that Newton's third law and momentum conservation are automatic rather than imposed. The frank admission that special relativity has no proper standing in a Machian scheme, made before constructing a relational version of it anyway, is more candid than most treatments.
The difficulties are substantial and mostly concern what is asserted rather than derived. The Machian condition δXμ = 0 is a postulate, introduced for convenience, and the authors state plainly that it is not covariant. The covariant form they then propose in equation (44), with a parallel-transport operator inserted, is offered without derivation and without a demonstration that it reduces to the non-covariant version or that the transport is path-independent — which in a general curved spacetime it is not. Since parallel transport along different paths gives different results, "the parallel transportation operator from the location of the jth particle to the location of the nth" is not well defined until a path is specified, and none is. This is the load-bearing step of the general-relativistic section.
A second gap is that the whole scheme is developed for non-interacting particles. The Lagrangians of equations (19), (22) and the action (34) are free-particle sums; the potential V(rij) appears only in the Newtonian Lagrangian (11). No field equation is derived from the modified action, so it is not shown that the modified geodesic equations are consistent with a conserved stress-energy tensor, nor that the Bianchi identities survive the constraint.
Third, the physical consequences are asserted rather than confronted with data. If inertial mass and G both change "whenever the total inertial charge of the world undergoes any change (e.g. in pair production era)," then this is a varying-constants theory, and varying-constants theories face measurement. Lunar laser ranging bounds the fractional rate of change of G at the level of about 10−13 per year, and Big Bang nucleosynthesis and the acoustic peaks of the Cosmic Microwave Background both constrain G at early times. The paper does not estimate the size of its predicted variation, so it cannot be told whether the model is comfortably inside these bounds or excluded by them — and since the effect is tied to pair-production epochs, when Σm would change substantially, the question is not idle.
Finally, the cancellation of the centre-of-mass kinetic energy is defended on the grounds that the Universe's centre-of-mass motion is meaningless. This is persuasive when the system is the Universe, but the same subtraction is applied to laboratory-scale N-particle systems in equations (10) and (11), where the centre-of-mass kinetic energy of, say, a moving body is very much measurable. The authors do not say how the sum over "all particles" is truncated in practice, and the model's predictions for a subsystem depend entirely on where that truncation falls.
On its own terms, the paper is a coherent and carefully motivated attempt to make general relativity satisfy a criterion its founder wanted it to satisfy. It should be read as a programme sketch rather than a finished theory: the Newtonian sector is worked out, the special-relativistic sector is plausible, and the general-relativistic sector rests on a non-covariant postulate and an undefined transport.