Machian Derivation of the Friedmann Equation

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Within GR however, Friedmann uni- III. MACHIAN PHYSICS verses in general posses a gravitational horizon, due to propagation of gravity at the speed of light in a universe The Machian approach due to Schrödinger [3] shares of finite age. We therefore assume a propagating gravi- features of Newtonian mechanics, however defined in a tational horizon at comoving distance χg. purely relational framework, by which it considers causal relations of all matter within the cosmological horizon. Such an explicit treatment, as pursued hereafter, discerns the Machian approach from both Newton’s and Einstein’s theory. In a Machian context mass inertia is a relational property which depends on the “distribution of matter”, and so inertia is likely to evolve in an expanding universe with a cosmological horizon. Like in GR, and following Schrödinger approach in his Machian derivation of the anomalous perihelion precession [3], we assume the influ- ence of the matter sources is through the gravitational potential. That is, we assume the inertia of a point particle to depend on the gravitational potentials of all other interacting point particles, in a one-on-one fashion. By integration of the elementary point particle relations one can derive expressions for the interaction of arbitrary fi- FIG. 1. Gravitational force F inside an empty spherical cavity nite bodies, like e.g. two solid spheres, or (as in our case) within the horizon χg. a small test particle relative to the surrounding sphere of cosmic matter, as explained below. How to actually determine (cosmic) gravitational energy is a matter still open to debate. But given the The whole argument of Newton’s derivation is based surprising success of Newtonian cosmology, we assume on the assumption that the mass distribution outside an a Newtonian potential. Though, we do allow the density empty spherical cavity of the universe is spherically sym- parameter ρ to include the various matter sources. We metric, so that the shell theorem applies and the field in- assume a homogeneous and isotropic universe. Different side the cavity is uniformly zero. This assumption holds from Newton’s derivation, we consider test particle m at in an infinite homogeneous isotropic universe. However, the center of the cosmic sphere of comoving radius χg, the presence of a gravitational horizon, centered at an thus regard all matter causally connected with m. The observer somewhere in the cavity, breaks this symmetry; infinitesimal Newtonian potential at the origin due to 3 due to the introduction of a horizon the field inside the the cosmic matter element d M at spherical coordinates cavity is no longer uniformly zero, as is clear from Fig.1; (χ,θ,φ) is the asymmetry causes a non-zero field directed away from 3 the center of the cavity. 3 −G d M 2 d ϕ(χ,θ,φ) = = −Gρa χ sin θ dφdθdχ. (3) aχ As a result the Newtonian derivation collapses for universes bounded by a gravitational horizon, which, ironi- Then the Newtonian cosmic potential due to all matter cally, is the case with general Friedmann universes. And within the horizon is actually, one would not expect differently, since due to χg π 2π 3 − 2 2 symmetry the gravitational field in the perfectly homoge- ϕ = d ϕ = 2πGρa χg. (4) neous isotropic universe is zero everywhere, with or with- Z0 Z0 Z0 out a horizon. This makes the concept of deceleration by This expression contrasts with Sciama [6], who derived, gravitational attraction questionable. For the same rea- by the gravitoelectromagnetic analog of Maxwell’s equa- son one must question the concept of a “repulsive force” tions, a cosmic potential equal to due to vacuum energy. 2 ϕu = −c . (5) With no forces acting, what then is left to even explain deceleration in a Newtonian sense? The zero field points Remarkably, the potential ϕu is independent of any cos- at constancy of the kinetic energy of recession; no energy mic parameter. But as Sciama noted, it has to be like is being exchanged. In a Newtonian context, where mass that, given that in our inertial frame Newton’s laws hold inertia is invariant, constant kinetic energy implies con- without any reference to the cosmic masses. It is tempt- stant recession velocities, i.e., a coasting universe, as in ing to assume both potentials to be equal, ϕ = ϕu, but Melia [4]. Hence, if Newtonian physics predicts anything, it is unnecessary for the present derivation of the Fried- then it is a coasting universe, rather than a decelerating mann equation to make such an assumption. universe; a conclusion earlier arrived at e.g. by Layzer Note that the Newtonian gravitational potential is [5]. genuinely Machian by its relational, frame independent 3 nature. Newtonian kinetic energy, on the other hand, where the effective potential is frame dependent, thus not relational, so clearly not 1 Machian. There is no established definition of Machian ϕeff = 3 ϕ (8) kinetic energy. Therefore, before considering Machian kinetic energy of the expanding universe, we will briefly in- serves as a normalization parameter which preserves con- troduce Machian physics according to the approach taken sistency with Newtonian inertia, as shown below. The 1 by Schrödinger in his Machian derivation of the anoma- reason that a factor 3 of the cosmic potential ϕ appears lous perihelion precession [3]: (see Schrödinger [3]) is that in a Machian sense only the The main conceptual difference with Newtonian radial component of motion counts, i.e., the two perpen- physics is that in Machian physics according Schrödinger dicular components of motion do not contribute. Hence, inertia and kinetic energy are not intrinsic properties of a in any peculiar motion effectively only one third of the particle, but are mutual properties which arise from the total cosmic potential contributes to the kinetic energy interaction of the particle with all other (causally con- between a mass m and the universe (see appendix A for nected) particles, just like this is true for potential energy details). and the force of gravity. For the pair of point particles Of course one can argue whether the above Machian mi and mj we consider a frame independent definition of definitions are correct. Obviously they are no subtitute their mutual Machian kinetic energy, according of GR. Yet, the underlying physical concepts (observ- ≡ 1 2 ability, causal connection of all matter within the cosmic Tij 2 µij r˙ij , (6) horizon, abandoning absolute space, mutuality of both inertia and kinetic energy) are ontologically as good as where r denotes the proper radial distance (separation) ij irrefutable. But also the precise form of the definitions of the particles. Crucial here is that from a relational Eqs. (6,7) appears sensible. Inertia between two point point of view only radial motion is meaningful in the particles is defined as their mutual potential energy and physical relationship of the two point particles. This so expresses the equivalence of mass and energy. From ontological notion (epistemological if you like) is due to these same definitions Schrödinger straightforwardly re- Bishop Berkeley [7], one of the earliest critics of Newton, produced the GR expression of the anomalous perihe- who first pointed out that any motion between two point lion precession. Since the Machian approach explicitly particles, other than their relative radial motion, in oth- considers the causal connection of all matter within the erwise empty space is unobservable, therefore physically gravitational cosmic horizon, it may provide a useful al- meaningless, or inexistent for that matter. Thus, two ternative in the physical interpretation of cosmology, like particles in circular orbit of each other have zero mutual in the derivation of the Friedmann equation hereafter. kinetic energy between them. Each particle does however have mutual kinetic energy relative to the surrounding cosmic particles, namely proportional to the (square of the) radial component of motion towards each cosmic IV. MACHIAN DERIVATION OF THE particle. Thus Machian physics is intimately connected FRIEDMANN EQUATION with all the cosmic matter. One can picture Machian kinetic energy between two Using the above elementary definitions [Eqs. (6,7,8)], particles as the energy that would be dissipated if one one can formulate the kinetic energy between a small would freeze the relative motion between the particles. mass m at the origin and all receding matter within the Fixing the separation of two spheres which are in elliptic gravitational horizon χg. We assume (only initially) par- orbit would definitely affect kinetic energy, but only as ticle m has a peculiar velocity v. This to evaluate consis- far as the radial component of motion is concerned (this tency of the Machian definitions with Newtonian physics. also explains why the anomalous perihelion precession For simplicity and without loss of generality we assume depends on the amount of eccentricity of the ellipse). To this peculiar velocity is in the polar direction (θ = 0). the contrary, fixing the distance between two spheres in The radial velocityr ˙ between m and cosmic matter ele- 3 perfect circular orbit would not affect energies at all; their ment d M at coordinates (χ,θ,φ) is thus mutual (Machian) kinetic energy is zero, just as Berkeley argued. So, if there is zero kinetic energy between spheres r˙ = χa˙ − v cos θ. (9) in perfect circular orbit, then the orbital kinetic energy of these two spheres must be entirely due to the presence of According Eq. (7) the mutual inertia between m and 3 cosmic matter. That is, due to the motion of each sphere matter element d M is, using Eqs. (3,4), in or from the direction of surrounding cosmic matter. Like kinetic energy Tij , Machian inertia µij is a re- 3 3 d ϕ(χ,θ,φ) 3χ sin θ lational and mutual property between any pair of point d µ = m 1 = m 2 dφdθdχ. (10) particles and is defined as 3 ϕ(χg,a) 2πχg

ϕj (rij ) ϕi(rij ) −Gmimj By deﬁnition Eq. (6) the kinetic energy between particle µij ≡ mi = mj = , (7) 3 ϕeff ϕeff ϕeff rij m and the cosmic matter element d M is 4

V. THE IDENTITY OF MACHIAN TOTAL ENERGY DENSITY 3 1 3 2 d T (χ,θ,φ) = 2 d µ r˙

3χ sin θ − 2 The Machian energy equation Eq. (13) is nearly iden- = m 2 (˙aχ v cos θ) dφdθdχ. tical to the Newtonian version Eq. (2), however with a 4πχg (11) notable difference: χs is an arbitrary fixed comoving ra- Then the total Machian kinetic energy between the parti- dius, while χg is (in general) an evolving gravitational cle m and all the cosmic matter within the horizon follows horizon, χg = χg(a). Thus in the Newtonian case total energy E appears as curvature energy density always, with some math (see appendix A) from the integral ∝ −2 −2 i.e., ρk a χs , while in the Machian case total energy density in general also evolves with the horizon, ∝ −2 −2 χg π 2π ρE a χg (a). Therefore, quite intriguingly, ρE may 3 m − 2 T = 4π χ2 χ sin θ(˙aχ v cos θ) dφdθdχ take identities different from curvature energy, depending g Z Z Z 0 0 0 (12) on the particular evolution of the horizon. For instance, in the de Sitter universe, and so in the late phase of = 3 ma˙ 2χ2 + 1 mv2. 4 g 2 the ΛCDM model, the proper distance to the event hori- −1 zon is constant, hence χg ∝ a . This translates total Thus one obtains two distinct energies: recessional and energy density to constant vacuum energy density, i.e., peculiar kinetic energy. The cross term of peculiar mo- ρE = ρΛ = const. Thus Machian cosmology potentially tion and recession vanishes due to symmetry. We see gives physical interpretation to the origin of the cosmo- that the peculiar part of the kinetic energy satisfies the logical constant. Newtonian definition and that the Newtonian inertia (m) is being retrieved. This means that the equivalence principle is maintained in peculiar motion, regard- less of cosmic evolution. Hence, the Machian principle remains unnoticed in the local frame. This is consistent with the Hughes-Drever experiments [8] (a null result of VI. CONCLUSION anisotropy of inertia) and in agreement with both Dicke [9] and Sciama [6]. Indeed, Newtonian physics makes no reference to cosmological parameters, even though it We argued that application of the shell theorem fails derives here from a cosmological context. with universes bounded by a gravitational horizon, like Confining ourselves again to recessional motion only, general Friedmann universes are. Consequently the New- we consider a unit test mass m at rest in the Hubble tonian derivation of the Friedmann equation collapses in flow (v =0), and drop the symbol m for simplicity. The the presence of a horizon, while the GR derivation holds potential energy of the particle is V = ϕ [Eq. (4)], thus for general Friedmann universes. We showed that, al- 3 2 2 ternatively, the Friedmann equation can be derived from together with recessional kinetic energy T = 4 χga˙ we have the Machian energy equation Machian arguments: for arbitrary homogeneous isotropic universes, either with or without a horizon, the Fried- 3 2 2 2 2 mann equation follows from conservation of a Machian χ a˙ − 2πGρa χ = E. (13) 4 g g definition of energy, without invoking the shell theorem. We further showed that, depending on the evolution of Similar to the translation of total energy E to curvature the horizon, the Machian total energy density takes dif- energy density ρk in the Newtonian case, we introduce ferent identities. In particular in a de Sitter universe, the (Machian) total energy density parameter Machian total energy density appears as constant vac- 2 2 uum energy density, thus provides interpretation to the ρE = E/2πGa χg, (14) cosmological constant. which, interestingly, is not necessarily curvature energy density, as χg in general evolves with the scale factor a (see next section). Eq. (13) can now be rewritten

3 2 2 2 2 Appendix A: 4 χga˙ = 2πG(ρ + ρE)a χg. (15)

2 Eliminating the common factor χg, we obtain the Fried- Calculation details of the integral Eq. (12), i.e., of the mann equation Machian kinetic energy between a test mass m at the origin and all receding matter within the cosmic horizon a˙ 2 8 = πG(ρ + ρ ). (16) χg, where m is assumed to have a peculiar velocity v in a2 3 E the polar direction θ = 0: 5

Since only the radial component of motion (r ˙ =aχ ˙ − v cos θ) counts, the potential due to the cosmic matter el- χg π 2π 3 m − 2 ement at coordinates (χ,θ,φ) contributes to the peculiar T = 4π 2 χ sin θ(˙aχ v cos θ) dφdθdχ χg Z0 Z0 Z0 (i.e., Newtonian) kinetic energy of the object m only by a 2 χg π fraction cos θ. On average, i.e., in the spherical integral 3 m 2 3 − 2 = 2 2 a˙ χ sin θ 2˙aχ v sin θ cos θ Eq. (A1), this fraction gives rise to a factor χg Z0 Z0 + χv2 sin θ cos2 θ dθdχ χ m g π = 3 − cos θa˙ 2χ3 +0 − 1 χv2 cos3 θ dχ 2 2 3 0 π χg Z0 2 0 sin θ cos θ dθ 1 χg π = . (A2) 3 m 2 3 2 2 R sin θ dθ 3 = 2 2 2˙a χ + 3 χv dχ 0 χg Z0 R

3 m 1 2 4 1 2 2 χg = 2 2 2 a˙ χ + 3 χ v 0 χg 3 2 2 1 2 = 4 ma˙ χg + 2 mv . Thus the eﬀective contribution of the cosmic potential to 1 (A1) the peculiar inertia is only ϕeff = 3 ϕ [Eq. (8)].

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