John Bruce Davies

Dept. of Physics (Retd.)

University of Colorado

Boulder, CO 80309


Essay written for Gravity Research Foundation 2010 Awards for Essays on Gravitation

Submitted: March 31, 2010


Recent observation of acceleration of the expansion of the Universe has caused a crisis in the accepted model with the postulation of Dark Energy as a cause. We propose that returning to Einstein’s original equation governing the spatial and time dependence of the gravitational potential can remove this crisis without invoking phantom energies. Using the Newtonian limit approximation and only radial dependence, we derive a Klein-Gordon semi-linear equation governing the gravitational potential valid in the time after decoupling of matter from radiation. This original equation has solutions that co-move with the expansion of the Universe. Certain of these localized-in-space traveling waves have time dependencies that are periodic, which are termed Breathers. To an observer located at our local co-moving coordinate system, the observable Universe will then appear to be oscillating, with deceleration changing to acceleration and so on.

© Copyright 2010 by John Bruce Davies All Rights Reserved


We exist and observe only along and in our own small region of spacetime. Analyses indicate that the Universe we observe is expanding with recent acceleration following deceleration. Thus, all our measurements are made in and based on a co-moving coordinate system that moves with the expansion and its changes. The recent acceleration of the expansion has been attributed to phantom phenomena termed Dark Energy, which has been linked with Einstein’s cosmological constant and the hypothesized vacuum energy. We propose that these observed changes in the Universe expansion can alternatively be explained by solutions of the governing Einstein/Newton equations relative to our co-moving coordinates.

We focus on the period of the Universe from the time of the decoupling of matter and radiation. We use the Newtonian approximation for Einstein’s equation, together with the assumptions of hydrostatic equilibrium and a polytropic equation of state. These reduce to a Klein-Gordon equation in the gravitational potential with a power law potential. The Green’s function solution for the spherically symmetric case incorporates oscillatory Bessel functions, with their property of phase differences between the various gradients. For a co-moving co-ordinate system, there exists a certain class of solutions that are localized traveling waves. We are familiar with the solitary wave or soliton solution that is localized in space and steady in time. A similar solution is a localized wave in space with a profile that changes periodically or aperiodically in time, mathematically and cogently termed Breathers. A breather is a wave-packet type of solution, a periodic traveling wave with an envelope that limits the wave to reside in a bounded region of space. Thus, the gravitational gradient governing the acceleration/deceleration will appear to the co-moving observer as oscillatory in the observable time back to the decoupling. Such a solution is more general than that under the usual additional assumption of a spatially constant density and Hubble velocity relation which gives a power law dependence in time of Universe expansion, Zeldovitch (1977).

Governing Equations

The fundamental relation connecting the metric gradients (g) and the symmetric curvature (G) is:

(1) 1/2g = -G

This is Einstein’s basic equation, which, on expansion, is equivalent to that derived by Weinberg (1972) in his classic text (eq. 7.2.6). His approach was based on the assumptions of maximally general metric structure, linearized fields, conservation of symmetric curvature and the nonrelativistic limit of the field equation giving Newton's Law. Davies (1988) has shown, in a more general derivation, that this relation is a consequence of the linearization of the Bianchi Identities in a more general U4 space-time.

We are only examining the Universe after the decoupling of matter from radiation, where densities are in the Newtonian regime. In order to incorporate expansion and its effect on the gravitational potential, the explicit time dependence of the gravitational field must be incorporated. It is thus necessary to use the relativistic limit of Einstein's field equation, which corresponds to the time-dependent Newton's Law, c.f. Weinberg (1972), p. 156, relating the gravitational potential, , gradients to mass energy density, , where we consider radial dependence only:

(2) 2/r2 + 2/r ./r – 1/c2 . 2/t2 = - 4G

In the simplest case, we assume that the equation of hydrostatic equilibrium is valid where pressure p is due to the gas:

(3) .p/r = ./r

The applicable equation of state that relates pressure and density in this system is taken to be polytropic:

(4) p = k n

where n is the usual adiabatic gas exponent and k is the appropriate constant.

On differentiating the equation of state and substituting into the hydrostatic equation, we obtain:

(5) nk (n-2)./r = /r

which, on integration, yields:

(6) nk/(n-1) . (n-1) = + C

where the constant of integration, C, is taken to be zero for the reference state.

Substituting equation (6) into (2) gives the fundamental time-dependent equation governing the gravitational potential:

(7) 2/r2 + 2/r ./r – 1/c2 . 2/t2 = - 4G [(n-1)/nk .]1/(n-1)

This is the standard form of a Klein-Gordon equation where the left-hand side is linear while the right-hand side changes for the various equations of state. In general, the rhs will have fractional or whole powers of the gravitational potential.

Klein-Gordon Solutions

Monatomic hydrogen, the gaseous component at the time of decoupling, has the value of polytropic index n = 5/3, so that the governing equation has the right-hand side to the power 1.5, i.e:

(8) 2/r2 + 2/r ./r – 1/c2 . 2/t2 = - 4G [2/5k .]3/2

We know of no exact solutions of this specific equation. The near-linear approximate solution of the non-linear Klein-Gordon equation is usually obtained by expanding the r.h.s. in a Taylor series. Rosales (2003) has shown that such solution methods are valid in the broad envelope regime and where the solution of the zeroth-order linearized equation governs the co-moving coordinate system. In our case, the system moves with expansion of the Universe at the velocity of light, i.e. the solutions are in terms of a co-moving variable (r-ct).

We expand the rhs term in a Taylor series about the initial radius R and time T of decoupling, where the gravitational potential has the value so that:

(9) .3/2 = 3/2 1/2 . + 3/4 -1/2.2 - 3/8 –3/2 .3 + O(4)

Rosales (2003) restricted his analysis to 1+1 spatial/time equations and found that Breather solutions will have the form where amplitude decays exponentially toward infinity relative to the co-moving coordinates and will have periodic oscillations. These solutions have existence conditions, which depend on the coefficients of the first and second orders in the Taylor series expansion. These conditions are quite broad so that we can expect similar conditions on this spherically symmetric equation and its resulting Breather solutions.

Whitham (1974) shows that only in the linear case is the dispersion relation independent of the amplitude of the wave. Thus it can be expected that amplitude dependence of the dispersion relation will apply to these nonlinear waves. Also, relative to the co-moving coordinate system, the frequency of the gravitational potential and its gradient’s oscillations will satisfy a, usually, non-linear dependence on amplitude and wave-number. This oscillation frequency will depend on the value of the gravitational potential at decoupling and the relevant polytropic constant k.


Our aim was to show that there are solutions to Einstein’s equation that can satisfy the observed changes in the expansion rate of the Universe without invoking Dark Energy. The atomic hydrogen gas that constitutes the matter after decoupling has an equation of state that produces a non-linear term in the governing Klein-Gordon equation. Solutions are given in the co-moving coordinate system that moves with the expanding Universe. These include traveling waves that are localized in space and oscillatory in time. As the gas component and its pressure effects are reduced in time as more gas becomes locked in stars, the operating equation of state can be expected to vary from that at the time of decoupling. However, we will still expect these Breather solutions though with probably changing amplitude and frequency dependence.


Davies, J.B. (1988), Foundations of Physics, 18, 563.

Rosales, R.R. (2003),

Weinberg, S. (1972), Gravitation and Cosmology, Wiley Press.

Whitham, G.B. (1974), Linear and Non-Linear Waves, Wiley Press.

Zeldovitch, Ya. B. (1977), Ann. Rev. Fluid Mech., 9, 215.