## Posts Tagged ‘**principle of least action**’

## Universe’s expansion may be understood without dark energy

**Least-time paths of light**

by **Arto Annila**

**ABSTRACT**

The variational principle in its original form á la Maupertuis is used to delineate paths of light through varying energy densities and to associate shifts in frequency and changes in momentum. The gravitational bending and Doppler shift are in this way found as mere manifestations of least-time energy dispersal. In particular, the general principle of least action due to Maupertuis accounts for the brightness of Type 1a supernovae versus redshift without introducing extraneous parameters or invoking conjectures such as dark energy. Likewise, the least-time principle explains the gravitational lensing without the involvement of additional ingredients such as dark matter. Moreover, time delays along curved geodesics relative to straight paths are obtained from the ratio of the local to global energy density. According to the principle of least action the Universe is expanding uniformly due to the irrevocable least-time consumption of diverse forms of bound energy to the lowest form of energy, i.e. the free electromagnetic radiation.

**1 INTRODUCTION**

A ray of light takes the path of least time. Thi well – known principle by Pierre de Fermat is a special form of the general principle of least action (De Maupertuis 1744; Tuisku, Pernu & Annila 2009).

Namely, light, as any other form of energy in motion, will naturally select the path of propagation that will maximize the dispersal of energy (Kaila & Annila 2008). The derivation of Snell’s law by the least-time principle is a familiar textbook example (Alonso & Finn 1983).

However, does the same variations principle govern also light’s passage through the expanding Universe from a high-density distant past to the low-density near-by present?

The answer will illuminate interpretation of supernovae data (Goldhaber & Perlmutter 1998; Garnavich et al., 1998) that seems to signal for a faster expansion than is expected on the basis of known forms of energy – possibly due to dark energy (Perlmutter 2003).

Moreover, light caught bending when passing by the Sun, is a famous proof of general relativity (Einstein 1911; Berry 2001).

However, does the least-time principle also govern light’s refraction when passing by all gravitating bodies?

The answer will explain galactic gravitational lensing (Blandford & Narayan 1992) that seems to be stronger than expected on the basis of luminous matter – possibly due to dark matter (Goldsmith 1991).

According to the principle of least action, light will follow the path where the integrand of variations is at a minimum (Feynman 1965).

Customarily the integrand is a Lagrangian which, as a conserved quantity, can be used to determine stationary paths of stationary-state systems. However, the expanding Universe is an evolutionary system where light must on its way adapt to changing circumstances. Likewise, light must adjust its energy to varying surroundings, when passing by a local variation in the universal energy density.

Enlightening light’s least-time paths through changing surroundings is the objective of this study. Therefore, rather than using the conserved Lagrangian form of the action principle (Kovner 1990), its original form á la Pierre Louis Moreau de Maupertuis will be used here. In the general form of the action principle kinetic energy is integrated over time, or quivalently momentum is integrated over the path.

This form has for long been shunned but recently it has been derived from the statistical physics of open systems (Kaila & Annila 2008; Sharma & Annila 2007; Annila 2010a). Subsequently it has been used to describe diverse evolutionary processes (Mäkelä & Annila 2010; Annila & Salthe 2009; Annila & Salthe 2010; Annila, 2010b).

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**6. DISCUSSION**

The principle of least time is known to be a powerful way of analyzing propagation of light through a varying energy density. However, the variations in the evolutionary trajectory to be minimized are given by Maupertuis action, whereas the commonly used Lagrangian integrand qualifies only to elucidate paths within stationary surroundings.

For this reason the results by Fermat’s principle presented here differ from those obtained via general relativity. More generally any formulation that complies with any one group

of symmetry, such as that of Poincaré, cannot break its norm which would be necessary to delineate least-time paths through varying energy densities.

When the spontaneous symmetry breaking is not understood as a nonunitary process, but the invariant form of the space-time curvature is insisted, the discrepancy between observations of evolutionary processes and predictions will be inevitable.

It will prompt one to save the unitary theory by invoking ad hoc explanations, most notably dark energy and dark matter or to propose impromptu expansions, most notably modified

gravity. In short We cannot solve problems by using the same kind of thinking we used when we created them (Calaprice 2005).

Obviously energy density gradients affect not only rays of light but also paths of bodies. To this end the principle of least action á la Maupertuis accounts also for galactic rotational curves and anomalous accelerations as well as for advancing perihelion precession (Koskela & Annila 2010; Annila 2009).

Thus the universal principle provides a holistic and self-consistent worldview in an elementary mathematical form (Annila 2010). The Universe is irrevocably processing from high-symmetry states of bound energy to states of lower and lower symmetry, eventually

attaining the lowest group U(1).

This free form of energy is electromagnetic radiation. In thermodynamic terms it makes

it makes sense to measure all bound forms of energy via

E = mc^{2}= m/μ_{ο}ε_{ο}

relative to the free space, the lowest state, characterized by permittivity εo and permeability μo. This is to say that the speed of light is dictated by the surrounding energy density of any kind, most notably, by that of free space. The cosmological principle, i.e., the high degree of

homogeneity at the largest scale is, according to the thermodynamic tenet, a mere consequence of maximal dispersal of energy. It is the combustion of bound forms of

energy to the free form of energy by stars, pulsars, black holes etc. that powers the expansion. This is to say, the Big Bang did not happen – it is still going on.

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Read also: **physorg.com**