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. ∫ ∫ 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.

We are still left with an important question. What is the physical mech- anism or process responsible for the tired light? Finding such a mechanism

5 means also finding the mechanism responsible for Hubble’s law (eq. 1). Be- cause of that, Soares [2] called such a mechanism the “Hubble effect” and put forward a heuristic program aiming to its discovery. The experimental evidences of this program are the following [2, sec. A.3]:

(a) the effect depends on the flux of the source of radiation according to Hubble’s law, given by [2, eq. 2];

(b) the redshift does not depend — or, has a very weak dependence — on the wavelength of the radiation;

(c) the effect is quantized ([13], [14] and references therein);

(d) the effect does not cause light scattering (see section 1).

The physical mechanism that satisfies these observational constraints might or might not be compatible with the instrumental description given by eqs. 3 and 4 since the latter represent just a working hypothesis.

3 Final remarks

The concept of tired light arose as an alternative to the explanation of red- shifts as caused by the expansion of space — which is, incidentally, indistin- guishable from the Doppler effect for small redshifts [5, p. 311] [15]. But in the end, expanding space may well be considered as one of the acceptable physical processes for the explanation of the “fatigue” of light when traveling from the source to the observer on Earth. The British cosmologist Edward Harrison (1919-2007) [5, p. 315] cites in this context, the statement of the French astrophysicist Evry Schatzman (1920-2010), “For anyone who does not accept the expansion of the universe, the red shift of spectral lines re- mains an important but totally unexplained physical phenomenon.” and he comments:

We note that tired-light redshifts and expansion redshifts have much in common. Unlike the Doppler effect, both consist of a steady displacement toward the red end of the spectrum as light traverses vast regions of space. One might even with justice call the expansion redshift a theory of tired light. Light rays are pro- gressively robbed of energy and become fatigued by the expansion

6 of the universe. It is ironic that all the difficulties of the tired- light theory are banished by changing its name and attributing fatigue to expansion: we have fatigue but no scattering; (. . . )

Formally, there is nothing against, for example, that Friedmann’s critical model (eq. 2) be adopted as the physical mechanism for tired light. And if this is the case, as mentioned in the end of the previous section, it does not need to comply exactly with the instrumental description expressed by eq. 3. That is exactly what would happen, as illustrated in figure 1. In any case, this is not the searched physical process because what one wants is an alternative to expansion models, due to their inadequacy (cf. [6]).

References

[1] E.P. Hubble, A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae, 1929, Proceedings of the National Academy of Sciences 15, 168-173

[2] D. Soares, The Age of the Universe, the Hubble Constant, the Accelerated Expansion and the Hubble Effect, 2009a, arXiv:0908.1864v6 [physics.gen- ph]

[3] A.K.T. Assis, M.C.D. Neves, D.S.L. Soares, Hubble’s Cosmology: From a Finite Expanding Universe to a Static Endless Universe. In: Frank Pot- ter (ed.) 2nd Crisis in Cosmology Conference, Port Angeles, Washington, September 2008. ASP Conference Series, vol. 413 (Astronomical Society of the Pacific, San Francisco, 2009), pp. 255-267

[4] F. Zwicky, On the Red Shift of Spectral Lines through Interstellar Space, 1929, Proceedings of the National Academy of Sciences 15, 773-779

[5] E. Harrison, Cosmology – The Science of the Universe (Cambridge Uni- versity Press, Cambridge, 2000).

[6] D. Soares, A Stone of Stumbling to General Relativity Theory, 2009b (http://www.fisica.ufmg.br/ dsoares/ensino/grt-stn.pdf)

[7] E.P. Hubble, M.L. Humason, The Velocity-Distance Relation among Extra-Galactic Nebulae, 1931, ApJ 74, 43-80

7 [8] A. Viglioni, D. Soares, Note on the Classical Solutions of Friedmann’s Equation, 2010, http://arxiv.org/abs/1007.0598.

[9] B.W. Carroll, D.A. Ostlie, An Introduction to Modern Astrophysics (Addison-Wesley Publ. Co., Inc., Reading, 1996).

[10] M. L´opez-Corredoira, Alcock-Paczynski cosmological test, 2013, arXiv:1312.0003v2 [astro-ph.CO]

[11] W.L. Freedman et al., Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant, 2001, ApJ 553, 47-72

[12] A.K.T. Assis, M.C.D. Neves, The Redshift Revisited, 1995, Astrophysics and Space Science 227, 13-24

[13] W.G. Tifft, Redshift periodicities, The Galaxy-Quasar Connection, 2003, Astrophysics and Space Science 285, 429-449

[14] H. Arp, Seeing Red: Redshifts, Cosmology and Academy (Apeiron, Mon- treal, 1998).

[15] D. Soares, Universo relativista: expanso no espao ou do espao?, 2012, in Portuguese (http://www.fisica.ufmg.br/ dsoares/expn/expn.htm)

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