Growing entropy as an alternative to dark energy

*[email protected] 1 Growing entropy as an alternative to dark energy Wolfgang Salm* Abstract To avoid the dark energy hypothesis, we look at a frame of reference that is at rest relative to the dipole-corrected cosmic microwave background radiation, and in it a metric in which each observer assigns a time dilatation to all other bodies depending on their distance. Based on our metric we establish the Einstein field equations for weak fields. The radial equation can be interpreted as Fokker-Planck equation of entropy; thereby, we link entropy to the length of the time interval available for a transmitted unit of information. The growth of entropy leads to a dissipating force related to the change in Helmholtz free energy, which drives the accelerated expansion of the universe. In our approach, the light of recently observed massive galaxies with is emitted not 500 – 700 Myr after the big bang, but later than half the age of the universe. 1. Introduction The currently preferred cosmological models are based on the RW–metric, wich leads to the Friedmann field equations. The prevalent model in cosmology, the ΛCDM model, is based on these equations. Nevertheless, it runs into severe difficulties: Taking into account the accelerated expansion of the universe, one has to introduce a repulsive component, called dark energy, whose physical nature is enigmatic. A number of ideas have been proposed to avoid the dark energy hypothesis. As in our approach, Easson et al. proposed entropy accelerating the universe; but there, underlying the RW–metric, entropy is associated with the holographic information storage on the surface screen placed at the horizon of the universe. Here, we work without this concept. Farnes proposed a modified ΛCDM model, where continuously created negative masses can explain both, dark energy as well as dark matter phenomena. Intensive work over 12 years, based on the RW–metric, has been done in the TRR 33 colaboration without a conclusive breakthrough. Hauret and Magain proposed a modification of the RW–metric by introducing a cosmological time in order not to have to rely on the hypothesis of dark energy. We feel that their idea is pioneering. However, the cosmological time itself is linked to the curvature of spacetime thus exceeding the scope of the underlying RW– metric. Using the RW–metric, the redshift of the radiation emitted from distant sources correlates with the growth of the cosmic scale factor , while the time measured in an locally inertial moving reference frame flows uniformly in the universe. In our approach however, we are dropping the hypothesis of a uniform time flow thereby leaving the framework of comoving coordinate systems. We assume, that any observer in space possesses only its own, local time scale. Due to the principle of cosmology, the time scales of observers should equal each others. Isotropy of space demands that the time flow does not depend on the direction in space; so, the redshift of the radiation which is emitted by distant sources and absorbed at the position of an observer has to be isotrop. This isotropy is guaranteed if we choose a frame of reference which is at rest relative to the dipole corrected cosmic microwave background (cmb) radiation; the origin as position of the observer is freely selectable. Independent of cosmological models, the baryonic acoustic oscillations of the cmb show that the spatial curvature of space-time is indistinguisible from zero; we take this into account by putting the radial scale factor of our metric to be one. Whereas usually, ´mass grips spacetime, telling it how to curve´ , here, we will not start with a given stress-energy tensor, but propose our metric in sec. 2 and examine the properties of the corresponding stress-energy tensor in sec. 7. Underlying our metric, we establish the Einstein field equations for slowly varying variables as expected for the present and future universe in sec. 3. In sec. 4, we propose a solution of the field equations, which includes a distancedependent time dilation which is ascribed to an object at the distance by an observer at his position . We may consider this solution as an extension of the well known effect in special relativity, that any observer in his local inertial frame of reference ascribes a time dilatation to all other driveless moving objects, even reciprocally. The equations of the geodesic lines are evaluated in sec. 5 and illustrated in sec. 6, showing, that any observer notices a escape drift of all bodies away from his own position and a temporal growing redshift of the cmb. At greater distances, the timescale depends weaker on the distance than in the immediate vicinity of the observer; so, one does not expect the redshift to follow Hubble's law. In sec. 8, we notice, that the radial equation has the shape of the flux equation of entropy , the Fokker–Planck equation, if 2 is interpreted as the period of time which is disponible for one unit of information to be transmitted. In sec. 9, we discuss free Helmholtz entropy determining the expansion rate of the universe. The empirical redshift-distance law of the SNIa radiation is qualitatively interpreted in sec. 10. In our approach, the light of recently observed massive galaxies with is emitted not only 500 – 700 Myr after the big bang, but later than half of the age of the universe. 2. The metric In order to motivate our metric, we start with a gaussian geodesic metric with line element , thus omitting angular coordinates. In the case of the spatially flat RW–metric, we can identify with the cosmic radial scale factor , with the time coordinate and with the distance . In our metric however, we swap the roles of and ; we are entitled to do so because an extension of the wavelength of an observed radiation cannot be distinguished from a reduction in frequency. We use the line element . (1) Here, is the number of time units passing in the distance from an observer at , if he measures one time unit for himself. In his present at time he may set . As shown in sec. 5, can be selected freely in our approach. Whereas, due to the principle of cosmology, depends only on time t, may vary with time und distance . Before the detection of the cosmic microwave background (cmb) radiation, the RW–metric was the only conceivable frame of reference. In our approach, the frame of reference has to be at rest relative to the dipole corrected cmb; thus, the isotropy of space is guaranteed. Our metric has to describe two opposite evolutional tendencies of the universe: the contraction due to the gravitation and the accelerated expansion which we will ascribe to growing entropy in sec. 8. So, (2) is factorised in two parts. Regarding the future of the universe, where gravitation is expected to loose importance, we assume, that the first factor , with attracting gravitational potential hardly deviates from one. Until section 9, we will consider only the second part, . 3. The Einstein field equations The metric (1) determines the formulas for the Einstein field equations. Here, we consider only terms varying weakly with and , as expected in the future universe; so, the early universe is not considered. Thus, we start with the Ricci– tensor in linear approximation . (3) The tensor elements in the field equations are given in Tab. 1. While can be time-dependent, the tensor elements contain only derivatives with respect to the distance ; we mark them with dashes. The spherical stress-energy tensor with energy density and pressure using metric (1) is . (4) It will be considered in sec. 7. Ricci tensorelement Ricci tensorelement Ricci tensorelement 0 Riemannscalar Einstein tensorelement Einstein tensorelement Einstein tensorelement Tab. 1: Tensor elements of the Einstein field equations for weak fields, based on the metric eq. (1) with . Herewith, the field equations are calculated: , = constant of gravitation) or (5) or (6) or (7) 4. Solutions of the field equations Assuming that the universe is homogenous, we seek solutions with spatially constant p und ρ for very large distances r . In this limit, in eqs. (5) and (6) should not depend on . The metric with scale factor must be time-dependent; so, the solution of the eqs. (5) ,(6), (7) is (8) 3 with constant . So, at all fixed locations , the unit of time increases and thus the frequency of radiation emitting sources decreases with growing distance . At the position of an observer at , , independent of the time t. With , we can write eq. (8) in the diffusion form . The Riemannscalar is independent of the freely selectable position of an observer at ; thus, the principle of cosmology is fullfilled. In the following, we choose . In the exponential function eq. (8), we choose the negative sign. We justify this considering Minkowski spacetime as a limit of curved spacetime for weak fields: there, any observer in his local inertial system ascribes a time dilatation with to any other moved inertial system, even reciprocally. This also applies to differencies of velocities, i.e. to driveless motions in gravitational fields. Geodesic lines in spacetime glide over in Minkowskian woldlines without rupture of the local metric. Due to the negative sign, for , we represent an expanding universe with density and pressure in eqs. (5) and (6) diminishing with time according to . 5. The equations of motion We note the equations of geodesic lines with Christoffel symbols ( : . (9) Here, we look for the equations of a body moving in radial direction ( ) in our time-varying gravitational field: (10) We integrate the first eq. (10) ( ); with an appropriate constant of integration, we get . (11) So, we interprete as the proper time of the moving body, . The two differential equations eq. (10) can be solved iteratively to calculate velocities and positions of a body moving along a geodesic line. By this way, the geodesics have been drawn in Fig. 1. There, the curvature of the surface is held constant. Herewith, the Fig. 2 shows the distance traveled by a recessing body as it is measured in the local coordinates of the pseudosphere. The body starts at r0 = 0 at the time with starting velocity . The plot coincides in good approximation with the straight dashed line which correspondes to a uniform motion of the body. 6. Illustration of the escape motion of bodies In Fig. 1, we present the radial dependence of based on the metric eq. (8) as a snapshot for constant time t. Different to the usual presentation, the time axis is directed to the right side and the distance upwards on the curved surface – the pseudosphere – according to the presentation in graphical timetables. The position of an observer is below at . With increasing , the growing time intervals in the grid correspond to a growing time dilatation, , as seen from the position . The three curves drawn in Fig. 1 are geodesics starting at the time with different velocities , the lower geodesic with ; during their run, the grid is fixed. Due to the locally dependent time dilatation an observer at an arbitrary position will observe a radially symmetric escape movement of all other bodies, even mutually. Seen from outside the curved coordinate grid, this motion appears accelerated in the direction of increasing distance . Fig. 1: Gaussian geodesic grid of the the surface with a constant negative Riemann scalar (pseudosphere). In the physical context of the RW–metric, the vertical geodesic lines are the time axes when positions are fixed and the orthogonal horizontal lines describe the possible positions of bodies at fixed times. Thus, an exponentially expanding universe is illustrated by a radius of the pseudosphere growing exponentially with time. However, in our context, the vertical geodesics show the radial distances of bodies to an observer below at position , the horizontal curved lines represent time axes at the various positions . Three geodesics all starting at the position with different initial velocities are calculated iteratively and drawn in the pseudosphere. Due to the time dilatation growing exponentially with distance , geodesic lines all exhibit an escape drift away from the position at . The lowest starts with . Based on the local coordinates of the pseudosphere, observers move uniformly along their geodesic lines. On the other hand, their motion appears to be accelerated, if considered from outside the surface. 4 7. Interpretation of the stress-energy tensor Whereas usually, energy density and pressure p are given and the metric is derived, here, we have proposed the metric in sec. 2; what can we say about the appropriate stressenergy tensor? Since (see eqs. (5), (6)), it follows that . To interprete this result, we consider a thin spherical mass bubble with radius : Inside the bubble, the gravitational field energy is vanishing, outside, it is negative. If we enlarge the bubble by without changing the mass volume and mass energy of the bubble itself, the field outside of does not change; this is shown, similar to the derivation of the Birkhoff theorem, by writing down the Ricci tensor element for an isotrope, time dependent radially symmetric metric . By increasing the energy density within the spherical mass shell with thickness from negative values to zero, the increase of the gravitational energy of the system, , is completely provided by the pressure work . Using eqs. (10), (11) for the acceleration , we get (12) Here, g is related to the proper time . Due to eq. (12), the field equation eq. (5) can be expressed as . (13) Thus, coincides with the classical formula for the energy density of a gravitational field, , with the local proper time underlying the acceleration in the distance from an observer. Whereas in classical theory, the reference distance , where lies in infinity, here, it is localized at the position of any observer; thus, the sign is reversed. In General Relativity, severe problems arise when gravitational field energy has to be incorporated in the Einstein field equations: there, the energy–stress tensor determinates the curvature of spacetime, but in turn, curvature of spacetime is connected with energy and mass. For the case of weak fields, post-Newtonian approximations are frequently used in which the non-linear, quadratic terms in the Einstein tensor on the left side of the field equations are assumed to be part of the stress-energy tensor on the right side of the field equations. Here, the Einstein tensor elements and which we calculate based on the metric eq. (8) consist exclusively of such terms; this is expected, since in eq. (8), we have not considered the gravitational part of the metric in eq. (2) . 8. Information theoretical analysis of eq. (5) Elektromagnetic waves transport information; here, we consider the connection between transported information and time dilatation by means of an thought experiment using toy units: a wave, which is emitted with frequency Hz at a distant position A is detected at position B with frequency 0,5 Hz. An observer at A may assume that the wave transports an information of 2 bit per s (e.g. in two halfwaves); then, B will receive in his second only 1 bit. On the other hand, the information of 1 bit disposes over a time intervall of 0,5 s at position A, but of 1 s in the time scale of position B. This growth of temporally freedom corresponds to the growth of the availible volume e.g. in a diffusion process with growing entropy. Since is doubled when going from A to B, we may connect the content of information (entropy) per second with time dilatation by . (14) If for example we have , the time scale is shortened and bit. We define as the flux density of entropy. Now, we wish to investigate the connection between the entropy eq. (14) and the radial field equation eq. (6). To do so, we first consider a general smooth function which obeys the Fokker-Planck equation with vanishing drift coeffizient : . (15) Here, the dot means the partial derivate with respect to the time and the dashes the derivatives with respect to the distance . If is constant, eq. (15) reduces to the diffusion equation. Eq. (15) can be transformed into a flux equation of entropy : Dividing by , we get or . With eq. (15) can be written as . (16) Here, ´ describes the change of the flux of entropy and is the source term of entropy. -0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0 0,2 0,4 0,6 0,8 1 distance r proper time ds/c Fig. 2. Distance traveled by a body; it is calculated iteratively using eq. (10) based on the local coordinates of the pseudosphere. The body starts at at the time with starting velocity . The units of the distance and the velocity in Fig. 2 are not fitted to experimental data but calculated in order to display the essential features. 5 We compare eq. (6) with eq. (16): for large , the radial field equation (6) can be interpreted as constituent of the FokkerPlanck equation , (17) where is the flux density of information. The source term of entropy, , is represented by the tensor element G00 = G11 and the flow change ´ is given by the Ricci Tensorelement R11. The temporal evolution of entropy (18) is proportional to the Rieman scalar ; it is positve definite, as required by the principle of entropy. In our approach, the acceleration of the escape drift of the universe, (19) is opposite to the attraction produced by distant gravitating masses, described by the factor ; it is connected to the flux of entropy . 9. Thermodynamical aspects We now incorporate attracting gravitation in our model and compare it with a simple thermodynamical description. Due to eq. (2), the gravitation is described by a timescale growing with distance r from the position of an observer at . For weak fields, , (20) where is the potential of the gravitational field . Combing eq. (20) with eq. (8), the entropy, describing the content of transmitted information per time unit is now (21) For weak fields, ; due to eq. (19), we get two opposite accelerations . (22) They express the gravitational compression and the entropic expansion for . In a simple model, we consider the present universe as a spherical system with a constant number of identical point particles, the galaxies, with gravitating mass , radius and constant spatial density. The averaged energy of one particle due to the gravitational interaction is . Since the averaged distance of two particles in the universe is proportional to , we have with an appropriate constant . So, the gravitational acceleration in eq. (22), , is proportional to . In the future, for large values of the coordinate time , ; so, the proper time flows proportional to the coordinate time and the averaged distance of masses moving on geodesics is growing proportional to the proper time (see Fig. 2); so, due to the entropic acceleration, in the future, dissipation dominates gravitation in eq. (22). Comparing our geometrical approach with thermodynamics, we require thermalization of the particles and infer a uniform temperature . The one-particle entropy is given by . (23) Here, is the inner energy of one particle, is the Boltzmann constant and is the free Helmholtz energy. In an ideal gas, describes the number of possible positions which can be randomly ´choosen´ if is the minimal required volume per particle. As is well known, in an ideal gas, dissipation is driven by the growth of entropy without any change of energy. In an real gas, entropy is responsible, that the gas dissipates against the attracting molecular forces thereby cooling down. Thus, there are two forces : (24) a contracting potential force and a dissipating force , which is connected with the change of free Helmholtz energy. In our approach, we replace the ´free choice´ of positions by the ´free choice´ of the disponible time interval, so by . Then, using eq. (19) . (25) The energy afforded to the expansion against the contracting force leads to the cooling of the system. On the other hand, the growth of entropy in irreversible processes is not necessarily accompanied by a change of energy; nevertheless, it can be measured in terms of energy, if an alternative reversible process is considered with initial and final states equal to those in the irreversible process. In our context, the energetic relation eq. (13) gives the energy according to an imagined reversible expansion of the universe, which is driven by the entropic force . It is positive, expressing the energy availible in such a reversible process due to the expansion. 10. Effects leading to redshift of radiation We consider monochromatic light which is emitted by a source moving along a geodesic line. The frequency of light emitted from distance at the moment and detected at at the later time is shifted due to a) The alteration of the time dilatation during the runtime of the light from the source to the detector due to the time-dependence of the metric. b) The time dilation in the distance due to the location dependence of the metric, see Fig. 1. c) The Doppler shift caused by the escape velocity of the source, see Fig. 2. 6 The shift of the wavelength of light emitted in the past from distance and detected at is . (26) Underlying our metric eq. (8), the time development and the distance dependence of are balanced during the flying time of light traveling back from the distance in the direction of an observer at ; so, we are intitled to neglect the effect a) and b). By using local coordinates , we are allowed to look at the propagation of light as in a Minkowski diagram (Fig. 3). Regarding c), the observer notices individual velocities of the bodies rushing away; at his local time , the observer receives monochromatic light which has been emitted at different times from the bodies. With growing flying time of the light and thus growing distance , the escape velocities of the light emitting bodies increase and thus the Doppler shifts (27) of the emitted radiation. A value corresponds to ; This escape velocity is much higher than the relative velocities of individual galaxies, inversely as in the diffusion of molecules in gases. We assume that bodies can drift apart at different initial speeds after the big bang. Based on Fig. (2), neglecting the retarding effect of gravity and relative velocities, we assume that the motion of different bodies is uniform. In linear approximation, we take into account the influence of gravity by assuming that the escape velocities at time are somewhat reduced compared to the uniform motion with . We express this effect by a correction factor . So, in Fig. 3, light is emitted from a body with velocity (28) in distance at time and registrated at time . Since the redshift data cover distances with , we have . (29) The Hubble constant is . Fig. 4 shows the graph of eq. 29 with value . The graph is not fitted to experimental data, but shows the feature, that deviates from the Hubble law of the cosmological redshift. Underlying the RW–metric, this feature is responsible for the assumption of dark energy to exist. Since the Hubble residual for is small, is only slightly less than one. For small distances , we may expand eq. (29) to get . (30) This Hubble approximation is marked in Fig. 4 as dashed line. In our approach, the Doppler shift of the radiation corresponds to the emission time . So, we can detect bodies emitting radiation only at times . Recently, new observations showed massive galaxies with only 500 – 700 Myr after the big bang, when interpreted in the framework of the standard model of cosmology . In our approach, such high values of need not correspond to times . Fig. 3. Sketch of the expanding universe as seen by an observer based on its local coordinate system. Fig. 4. Redshift of the radiation which is emitted from bodies in the distance . The straight line is drawn assuming the Hubble law. References Easson D. A., Frampton P. H., Smoot G. F.: Phys. Lett. B696, 273-277 (2011), arXiv 1002.4278v3 Farnes J.S.: A&A 620, A92 (2018), arXiv 1712.07962v2 Transregio research 33: the dark universe (2018) Hauret C., Magain P. and Birneaux J.: entropy (2017), 19(7) Wheeler, J. A.: A journey into gravity and spacetime, p. xi, (W.H. Freemann, N.Y., 1990) Fließbach T.: Allgemeine Relativitätstheorie, Springer, 6. edition (2012) Risken H.: The Fokker-Planck equation, Springer-Verlag (2013) Will, S. M.: Proc. Nat. Acad. Sci. (US) 108, 5938 (2011), arXiv 1102.5192v1 Betoule, M. et al.: A&A, 568, A22 (2014), arXiv 1401.4064v2 Labbe, I.: nature 616, 266-269 (2023) 0 0,2 0,4 0,6 0,8 1 1,2 1,4 0 5 10 15 z r/Gpc distance light escaping bodies light big bang time