The tired-light paradigm Domingos Soares Departamento de F´ısica,Universidade Federal de Minas Gerais C.P. 702, 30123-970, Belo Horizonte, Brazil February 24, 2014 Abstract I discuss some features of the so-called \tired-light paradigm", which constitutes one of the possible explanations for the dependence of the spectral redshift of a distant cosmic source with its distance to the observer. The most popular phenomenological representation of the paradigm is presented in some detail. Furthermore, since the physical process responsible for the hypothetical phenomenon is still unknown, I suggest also guidelines for its discovery. 1 Introduction In 1929, the 15th volume of the Proceedings of the National Academy of Sciences published two articles whose results resound until present days. In the first one, on page 168 [1], Edwin Hubble (1889-1953) presents the relation which later became known as \Hubble's law", i.e., the linear relation between velocities and distances of distant galaxies [2]. This work is a fruitful source of discussions (e.g. [3]). On page 773 [4], appears the second one, an article by the astronomer Fritz Zwicky (1898-1974). Bulgarian at birth, Swiss citizen and who lived in the United States from 1925 until his death, Zwicky was known to be a strong and controversial personality. In the mentioned work, already in the begin- ning, he cites Hubble's article and presents an alternative to the expanding universe ideas that had been raised as an explanation for Hubble's law. He invents the hypothesis that became known as the tired-light paradigm. 1 In his article, Hubble obtains, in fact, a relation between redshifts z and distances. Hubble provisionally converts redshifts into velocities using the mathematical expression of the Doppler effect v = cz, where c is the speed of light in vacuum. Zwicky argues properly in terms of redshifts not velocities. He puts forward the hypothesis that when light \travels" from the distant source (a galaxy, for example) progressively looses energy. In other words, light \tires". Light is characterized by a frequency ν and a wavelength λ = c/ν. Its quantum of energy, the photon, has energy E = hν, being h Planck's constant. If the energy diminishes because of the fatigue of the voyage, this means a diminishing in ν and, consequently, an increasing in its wavelength λ because λ is inversely proportional to ν. Therefore, the greater the distance to the source the greater the increasing in λ, that is, the greater the redshift. One must recall that, in the visible spectrum, that runs from the violet to the red, the violet has the smaller wavelength and the red the larger. Hence the nomenclature \redshift" in order to refer to an increasing in wavelength, being in the visible range or not. Zwicky goes beyond putting forward the hypothesis (the paradigm) and proposes also a physical mechanism for the fatigue of light in its voyage. The suggested mechanism is, in his words, \a sort of gravitational analogue of the Compton effect" [4, p. 773]. He assumes that the photon looses energy due to the gravitational drag (dynamical friction) during its voyage from the source to the observer; the drag is caused by the existing matter throughout the trajectory. Zwicky's mechanism has nowadays only historical interest. A fundamental feature in the search for the physical mechanism of the paradigm is given by the constraint that the proposed mechanism does not cause light scattering. In general, light scattering is selective, i.e., wavelength- dependent, and in this case images of distant extended sources would appear \blurred", as if they were out of focus, which is not observed (e.g. [5, p. 312]). The importance of Zwicky's contribution is the idea behind the paradigm. Light loses energy (by a physical process still unknown) during its trajectory from the distant source to the observer. Therefore, the greater the distance to the light source the larger the redshift and hence such a possibility could also explain Hubble's law. Consequently, the model of an expanding universe could be dismissed, which would be convenient since this model has serious difficulties to establish itself (see, for example, [6]). I present, in the next section, a very popular mathematical expression for the tired-light paradigm. It must be pointed out that I shall only discuss the 2 mathematical expression and not the possible mechanism responsible for it. I discuss also the prescription put forward by Soares [2] for the discovery of such a mechanism. In the last section, I make some additional remarks. 2 An instrumental description of tired light Before discussing tired light, I shall present two alternative descriptions for the dependence of the redshift z with the distance r, which will be useful to demarcate the location of tired light in the space of relevant parameters z and r. The first one is the observational relation known as \Hubble's law". Hub- ble's [1] figure 1 shows the linear relation velocity-distance for the extragalac- tic nebulae (nowadays called galaxies). Such a relation is mathematically ex- pressed as v = H◦r, where v is the velocity assigned to the galaxy (by means of v = cz), r is its distance and H◦ is the so-called \Hubble's constant". In fact, what Hubble measures is the redshift z. The transformation v = cz im- plicitly assumes the Doppler effect as an explanation of the redshifts. Hubble is aware of this, as he and Humason state, two years later, in another place [7, p. 73]: \The quantities actually observed in the present investigation are redshifts and apparent magnitudes. (...) The fact that the red- shifts are expressed on a scale of velocities is incidental; for the present purpose they might as well be expressed as dλ/λ." (Bold is mine; note that dλ/λ is the very definition of redshift z. Incidentally, Hubble and Humason's paper [7] has the title The Velocity-Distance Relation among Extra-Galactic Nebulae whereas Hubble's 1929 paper [1] is entitled A Relation between Distance and Radial Velocity among Extra-Galactic Nebu- lae, that is, both investigate the same problem; the paper with Humason is a more detailed version besides presenting new observations which enlarge the range of distances.) The relation v = H◦r plus the \incidental interpretation" v = cz (cf. Hubble and Humason cited above), lead to the following expression for the function z(r): H◦ z = r : (1) c The second alternative description is a theoretical relation from one of the solutions of the General Relativity Theory, namely, Friedmann's solutions [8, 3 sec. 4.2]. The Standard Model of Cosmology is based on these solutions (see details in [8]). There are three classical Friedmann's models, depending on the matter density of the model universe. If the universe density is equal to the critical Friedmann's density one has the critical model, which has Euclidian spatial geometry. This model is, out of the three, the one that has the most simple mathematical description (see, for example, [8, eq. 2]). The function z(r) for the critical Friedmann's model is given by ([9, eq. 27.89], [10, eq. A8]): ( )−2 H◦ z = 1 − r − 1 ; (2) 2c where r is the comoving distance of the source. This expression reduces to −1 eq. 1 for H◦r=2c ≪ 1 (or r ≪ 8 Gpc, for H◦ = 72 km/s Mpc , cf. [2] and [11]), with the aid of the power series of Newton's binomial theorem (1 + x)n = 1 + nx=1! + n(n − 1)x2=2! + :::, keeping only the first two terms. In order to get z(r) for the tired light, one can use an instrumental def- inition of the paradigm, considering the lack of knowledge of its physical mechanism. The most popular is ([10, app. A-6], [12, eq. 2 with dt =dr=c]): dE H◦ = − dr ; (3) E c which states that, in infinitesimal terms, the relative decreasing of the ra- diation energy is proportional to the distance traveled. This equation must be integrated to get the energy E◦ of the observed photon, which has been emitted with energy E by the source located at r = 0. Z Z E◦ dE H◦ r = − dr E E c 0 E◦ H◦ ln = − r E c Writing the energy in terms of the radiation wavelength λ and with the definition of the redshift z, one gets the mathematical expression of z(r) corresponding to the instrumental description given by eq. 3: c E = hν = h λ 4 ∆λ λ◦ − λ λ◦ z = = = − 1 λ λ λ H◦ r z = e c − 1 ; (4) which, again, reduces to eq. 1 for H◦r=c ≪ 1 (or r ≪ 4 Gpc), with the aid of the power series expansion of the exponential function ex = 1 + x=1! + x2=2! + :::, keeping only the first two terms. Eq. 4 is the same as eq. A24 of [10]. Figure 1 shows the three functions z(r) given by eqs. 1, 2 and 4. The coordinate axes are in arbitrary scales. It is worthwhile stress that the tired- light model sits exactly between Hubble's law and the model of an expanding space. 6 Friedmann Tired light z = (Ho/c)r 5 4 3 redshift (z) 2 1 0 0 0.5 1 1.5 2 distance (r) Figure 1: The function z(r) for the models represented by eqs. 1, 2 e 4. The model \Friedmann" (critical or flat model; see [8]) is also known as \Einstein-de Sitter" model. All of them converge to a linear function for r ≪ 1. The scale in the abscissa axis r is in units of c/H◦ = 4:2 Gpc.