On the Detection Possibility of Extragalactic Objects Redshift Change

Transcription

1 Applied Physics Research; Vol. 7, No. ; 15 ISSN E-ISSN Published by Canadian Center of Science and Education On the Detection Possibility of Extragalactic Objects Redshift Change Jarosla ynecek 1 1 Isetex, Inc., 95 Pampa Drie, Allen, TX 7513, USA Correspondence: Jarosla ynecek, Isetex, Inc., 95 Pampa Drie, Allen, TX 7513, USA. jhynecek@netscape.net Receied: February 11, 14 Accepted: February 6, 15 Online Published: February 8, 15 doi:1.5539/apr.7np58 URL: Abstract This paper describes the detail calculations of expected redshift change in the galaxies or other distant objects light after a certain amount of time between obserations has elapsed. The detection of this phenomenon has been proposed since the ubble s discoery of the galaxies redshift dependence on their distance from Earth and their significant recession elocities. Various astrophysicists hae performed such calculations for seeral cosmological models of the Unierse, but not for the model introduced by the author of this paper. This is now addressed in this publication. Keywords: galaxy redshift change in time, finite size model of the Unierse, Big Bang theory, graitational redshift, Doppler redshift, ubble constant, deformable dark matter, cosmological graitational potential, gamma ray bursts, CMBR temperature 1. Introduction Seeral astrophysicists hae proposed the measurement of change in the galaxies redshift with the elapsed time of obserations. The most comprehensie and early ealuation of this phenomenon was published by Sandage (196, 1 with possibly the earliest ersion suggested by Tolman (193, Seeral recent publications on this subject are by Loeb (1998 and by Lerner, Falomo, and Scarpa (15. oweer, all of these publications are using some ariations of the classical models of the Unierse based either on the main stream Big Bang (BB model or on the Newtonian flat space model. In the preious publication ynecek (1a has introduced a new model of the Unierse that assumes the Unierse being finite in size and filled with a repulsie and deformable Dark (transparent Matter (DM. The DM is repulsie to isible radiating matter but attractie to itself. In this model the galaxies are treated only as small test bodies floating from the bulk of the Unierse to the edge where they explode and generate the well-known immense Gamma Ray Bursts (GRB detected here on Earth. The GRBs that are reflected back to the bulk of the Unierse then contribute to the generation of new matter. This model is in line with the theory proposed by oyle, Burbidge & Narlikar ( of a steady state Unierse with a constant matter creation. The new matter then condenses to stars and eentually to new galaxies endlessly repeating the cycle of creation and destruction. The galaxies explosions residue at the edge of the Unierse generates also the Cosmic Microwae Background Radiation (CMBR with its temperature of.75 K. The repulsie DM density is ery small but its graitational effects dominate the isible matter graitation at large distances. The graitational field of galaxies is thus compensated and screened by the DM after a certain distance. The galaxies thus for the most part moe in this Unierse independently of each other. The one of the significant contributions of this model to the theory of the Unierse, in comparison to other models of the Unierse in particular the BB model, is in the deriation of the relation between the ubble constant and the CMBR temperature (ynecek, 13. These two parameters are typically considered independent of each other. In this model they hae been found dependent and one can be deried from the other. This is only possible in a finite and thermodynamically enclosed model as this one is and not in the open models such as the BB. The details of the model mathematical background and its excellent agreement with the arious obserations hae been published earlier by ynecek (1a, 14. The important equations needed for the deriation of the redshift time dependence are presented in the next section. 58

2 Applied Physics Research Vol. 7, No. ; 15. Mathematical Background Since the long range graitational effects of the isible radiating mater and all of the radiation can be neglected, the space-time metric can be considered static, spherically symmetric, and described by the following differential metric line element (ynecek, 11, 1b: ds ( c g dr g Ω = gtt rr ttd (1 where: dω = dϑ + sin ϑdφ, g tt = exp(φ, g tt g rr = 1, and c is the local intergalactic speed of light. The cosmological Newton graitational potential for the isible matter φ normalized to c is calculated using the well-known equation: ( r 4πκ ϕ ( r = m( ( r ( c ( r where κ is the Newton graitational constant. Due to the deformation of the obsered natural radius r by the DM graity, the physical radius (r must be used in the formula instead of r and this parameter is found from the differential equation that follows from the metric: ϕ = g dr = e dr (3 rr Since any particular galaxy now represents only a small test body in this Unierse, the well-known and many times erified Lagrange formalism can be used to describe the motion of such galaxies. The Lagrangian is therefore as follows: ϕ c ϕ dr ϕ dω (4 L = e e e dτ dτ dτ For a purely radial motion the Lagrangian can be simplified and the first integrals of the corresponding Euler-Lagrange equations easily found using the initial condition at the origin where the recession elocity is zero and where: dτ =. The results are: dτ ϕ = e (5 dr ϕ (6 = c c e dτ Eliminating the non-obserable dτ from these equations then leads to the formula for the recession elocity: dr ϕ = c e ϕ 1 e (7 For the relatiely near objects, where the cosmological graitational potential for the isible matter φ is still small, it holds that: (r ~ r, and m( = m. This simplifies Equation 7 as follows: dr 8 c ϕ = π κ m r = r (8 3 From this result it is then clear that the recession elocity is linearly proportional to the natural coordinate distance r of such nearby objects from the origin and that the ubble constant is related to the DM density m at the origin according to the following Equation: 8 = π κ m (9 3 The recession elocity and the ubble constant are referenced to the DM coordinate system, so the alue of the ubble constant should be corrected and referenced to the Earth's centered coordinate system from where it is actually measured. oweer, the correction is ery small and it will be neglected. Earth and its Milky Way galaxy are located relatiely near the center of the Unierse in comparison to its immense size. In order to proceed further in the model description it is necessary to find the relation for the DM density m( as a function of the physical radius. This is obtained by adapting the well-known approach described, for example, 59

3 Applied Physics Research Vol. 7, No. ; 15 by Zel'doich and Noiko (11 where the DM pressure gradient is expressed as a function of the physical radial distance: dp 4π κ m( = m( (1 After substituting for the DM pressure from the relation: c = P / m, and defining the normalized mass density function: m n ( = m(/m, Equation 1 can be rearranged with the help of the Green's function as: ( m n ( = exp A mn ( ξ ξ ξ / dξ (11 where A is a constant equal to: A = 4πκm /c. There is no known analytic closed form solution for this equation, so it is necessary to use the numerical iteratie approach or find an approximating function. The approximating function approach was selected for the next steps to aoid ery long computing times during iterations. The selected function, howeer, underestimates the true alue of the DM mass density at large, but the error has only a small oerall effect. The first two iterations and the approximating function: m a ( = exp ( h 1 h 35 h 16 h 31 h are shown in a graph in Figure 1..8 ma ( x.6 m1 ( x m ( x Figure 1. The graphs of the first two iterations (dashed and dot-dashed lines and the approximating function describing the dark matter mass density as a function of the physical radius where: x = / h. The introduced parameter defined as: h = c/, is called the ubble distance or the ubble physical radius x Another adantage of using the approximating function is that the DM concentration tail extending past the maximum radial distance can be easily cut off by suitably truncating the power series expansion in the exponent. This feature is adantageous if it is considered that the isible matter debris from explosions of galaxies are accumulating at the edge of the unierse and are forming a loosely bound shell there. Of course, it is possible to add more terms than shown in Equation 1; howeer, this will not be pursued any further in this paper, since the accuracy of the approximation was found reasonable. Once the mass density function is known it is easy to find the normalized graitational potential for the isible matter using the formula in Equation, and for the dark matter using the Green's function formula deried also from the Gauss law as follows: ξ dξ m ( ζ ζ dζ 3.33 = A m ( ξ ( ξ ξ / ξ a a ϕd ( = A d ξ Both potentials are plotted in the graphs shown in Figure. In the next step of the model description it is necessary to find the formula for the Z shift, since this is the parameter that is directly measured by astronomers. The Z shift typically consists of the three components: the star graity (13 6

4 Applied Physics Research Vol. 7, No. ; 15 induced redshift, the cosmological potential induced redshift, and the Doppler redshift resulting from the recession elocity. The star graity induced redshift does not hae to be considered here, since after the star or the galaxy explosions hae occurred when the Supernoa explosions or the GRB data are compiled, most of the principal source of the graitational field has been conerted to radiation and radiated away and only the remnants or the afterglow produce the light that is obsered. The cosmological potential induced redshift does not hae to be considered either, since in this model the galaxies are in a radial free fall and this compensates for the shift..5 1 ϕ( s 1.5 ϕd( s Figure. The graphs of dependencies of normalized graitational potentials for the isible matter (solid line and for the dark matter as functions of the physical radius. The integration constants were adjusted such that the potentials at infinity are zero ( s Ly The only remaining redshift component is thus the Doppler redshift resulting from the radial recession elocity r. The Doppler redshift obsered on Earth, therefore, is: r / cr gtt exp( ϕ Z( = 1 = 1 = 1 (14 / c g exp(ϕ r r where c r indicates the light speed at the galaxy location in reference to Earth. The graph of the Z dependency on the natural coordinate radius r is shown in Figure 3. tt 1 1 Zs ( rs ( Ly Figure 3. The graph of the dependency of redshift on the natural radial distance r. The maximum redshift that can 9 be obsered is: Z mx = The isible matter does not exist at the distances larger than: r mx =.11 1 Ly, since it disintegrates a the Unierse's edge 61

5 Applied Physics Research Vol. 7, No. ; 15 The radial distance r, also called in this paper the natural radial distance, which is the obserable parameter, is calculated according to Equation 3 as follows: r( = exp( ϕ (15 To complete the model description the graphs of the galaxies recession elocity and the speed of light as functions of the natural radial distance r are shown in Figure s (1 3 cr(1 s Ly Figure 4. The graphs of the speed of light (dashed line and the galaxies recession elocity in km/sec as functions of the natural radial distance r from the center of the Unierse in light-years rs ( 3. Deriation of the redshift rate change The redshift Z dependency on time can be found by first differentiating Equation 14 with respect to distance and then by differentiating the distance with respect to time. The result of differentiating Equation 14 with respect to distance is as follows: dz dϕ exp( ϕ = (16 (1 exp(ϕ exp(ϕ The differential of the physical distance with respect to natural time t is found from Equation 3 and Equation 7. = c exp( ϕ exp(ϕ (17 By combining Equations 16 and 17 the result for the rate change of the redshift Z is: dz dϕ exp(ϕ = c (18 exp(ϕ To proceed further it is useful to find the expression for the potential of the isible matter in terms of the redshift Z. This is obtained again from Equation 14 with the result: ( Z + 1 exp( ϕ = (19 1+ ( Z + 1 Using this formula in Equation 18 results in a suitable expression for the numerical ealuation as follows: 6

6 Applied Physics Research Vol. 7, No. ; 15 dz n d = c 1+ ( Z + 1 ϕ ( Z + 1 ( It is also possible to find an approximation for the nearby galaxies where the redshift is still small. For this case it holds that the DM density is still reasonably constant and from Equation therefore follows that: ϕ c (1 dϕ = ϕ ( c c By combining these formulas together, using also the logarithm of formula in Equation 19 to substitute for the isible matter potential, the small Z approximation for the redshift rate change is: dz ( Z a ln Z ( + 1 (3 Z Z + 1 Both of these Equations; and 3 were graphically ealuated using conenient Mathcad 15 numerical symbolic calculations and the results are plotted in graphs in Figure ΔZn ( s 1 Yr ΔZa ( s 1 Yr Figure 5. The plots of the dependency of redshift change for the 1 year time period between obserations as a function of the galaxy redshift Z. The dotted line is the Z < 1 approximation Zs ( The ubble constant used in all these calculations was: = 68. km/secmpc (Keel, 7. This alue is in an excellent agreement with the alue deried by ynecek (1a, 13 from the precisely measured CMBR temperature: T =.75±. K. The preiously deried formula relating these two parameters is as follows: T 3 5 exp(ϕ 3 h c = =. 77 K (4 4 k 8 π B κ where k B is the Boltzmann constant, h is the Planck constant, and the isible matter cosmological graitational potential: φ = is calculated, using Equations and 1, at the Unierse s edge. 63

7 Applied Physics Research Vol. 7, No. ; 15 The time interal between the two consecutie obserations was selected to be 1 years. From the graphs it is thus clearly seen that for the small Z, less than unity, the approximation of constant DM density is reasonable, but fails for the larger alues. The interesting result is that there is an optimum redshift rate change for 3 < Z < 4 where the redshift rate is the largest. The obserations should therefore focus on such cosmological objects. The galaxy deceleration together with the slower speed of light at large distances cause the redshift change rate for large Z objects to be lower and eentually drop to zero at the maximum Z shift: Z mx = This finding may seem somewhat counterintuitie, since one would expect the larger rate for the larger redshifts. This result is an unaoidable consequence of the finite size model of the Unierse that is cured by the DM graity. Such an interesting finding will most likely not be the same for the BB model or any other similar Unierse model deried from the BB theory and, therefore, could be used to confirm the eracity of a particular model. For a better clarity another plot of the redshift rate change as a function of the natural radial distance r is shown in Figure 6. From this graph it is clear that the redshift rate drops to zero at the edge of the Unierse een though the redshift itself is the largest there ΔZn ( s 1 Yr r( s Ly Figure 6. The plot of the dependency of redshift change for the 1 year time period between obserations as a function of the natural radial distance r in light-years 4. Discussion It is clear from the graph in Figure 5 that the maximum of the redshift rate that can be obsered is not much larger than1 9 /1 Years. This is an extremely small alue that may be difficult to detect. oweer, with the unprecedented progress in the obserational astronomy and in the atomic clock long term stability, the detection might perhaps be possible. It is, of course, necessary to also subtract the rate changes due to the arious peculiar motions, such as of Earth in the Sun s orbit, the Sun s motion in our galaxy, and our galaxy motion in the Unierse, but this should not present an unsurmountable problem. The reference for subtraction can either be the local galaxy cluster background or the CMBR. Another problem could be the galaxies intrinsic redshift change. But this effect could also be subtracted by obsering the nearby similar galaxies or selected objects. An adantage of the described measurement analysis is that it inoles only one ariable, the redshift, which can be relatiely precisely detected in the light spectrum of distant objects of the Unierse. The successful detection of the redshift rate maximum and the confirmation of the redshift rate dependency on the Z shift as is shown in Figure 5 would represent yet another confirmation of correctness of the finite size model of the Unierse, in addition to the determination of the precise alue of ubble constant from the CBMR temperature, and proe the BB dogma deried from the simplistic extrapolation of obserations to be wrong. 64

8 Applied Physics Research Vol. 7, No. ; Conclusions The article described the detail calculations of the expected redshift change in the galaxies light after a certain amount of time between the obserations has elapsed. This deriation was made for a new model of the Unierse. The maximum alue of the redshift rate was also deried suggesting the best redshifts at which the obserations should be carried out. The possibility of detection of this phenomenon has been proposed since the ubble s discoery of the galaxies redshift dependence on their distance from Earth and their significant recession elocities. The detection of this effect would represent a dramatic confirmation of an alternate model of the Unierse that is static, finite in space and infinite in time. The Unierse without a Big Bang was also recently proposed by Ali and Das (14. References Tolman, R. C. (193. On the estimation of distances in a cured unierse with a non-static line element. Proceedings of the National Academy of Sciences of the United States of America, 16(7, Tolman, R. C. (1934. Relatiity Thermodynamic and Cosmology. Oxford: Oxford Uniersity Press. Sandage, A. (196. The Change of Redshift and Apparent Luminosity of Galaxies due to the Deceleration of Selected Expanding Unierses. The Astrophysical Journal, 136, Loeb, A. (1998. Direct Measurement of Cosmological Parameters from the Cosmic Deceleration of Extragalactic Objects. The Astrophysical Journal, 499, L111-L oyle, F. G. Burbidge, F. G., & Narlikar, J. V. (. A Different Approach to Cosmology. Cambridge, U.K.: Cambridge Uniersity Press. Keel, W. C. (7. The Road to Galaxy Formation (nd ed., pp Springer. Sandage, A. (1. The Tolman surface brightness test for the reality of the expansion. V. Proenance of the test and a new representation of the data for three remote ubble space telescope galaxy clusters. The Astronomical Journal, 139(, Zel doich, Ya. B., & Noiko, I. D. (11. Stars and Relatiity. NY: Doer Publications, Inc., Mineola. ynecek, J. (11. Geometry based critique of general relatiity theory. Physics Essays, 4(, 18. Retrieed from ynecek, J. (1a. Repulsie dark matter model of the unierse. Physics Essays, 5(4, 561. Retrieed from ynecek, J. (1b. The Theory of Static Graitational Field. Applied Physics Research, 4(4, ynecek, J. (13. The Repulsie Dark Matter Model of the Unierse Relates the ubble Constant to the Temperature of the CMBR. Applied Physics Research, 5(, ynecek, J. (14. An Interesting Journey of Discoery of Many Errors in Einstein s General Relatiity Theory. Retrieed from Ali, A. F., & Das, S. (14. Cosmology from quantum potential. Retrieed from Lerner, E., Falomo, R., & Scarpa, R. (15. U V surface brightness of galaxies from the local Unierse to Z ~5. International Journal of Modern Physics D (to be published. Retrieed from Copyrights Copyright for this article is retained by the author(s, with first publication rights granted to the journal. This is an open-access article distributed under the terms and conditions of the Creatie Commons Attribution license ( 65