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On Self Sustained Photonic Globes

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On Self Sustained Photonic Globes On Self Sustained Photonic Globesa) K. Eswaranb) (Dated: 1 April 2019) In this paper we consider a classical treatment of a very dense collection of photons forming a self-sustained globe under its own gravitational influence. We call this a “photonic globe” We show that such a dense photonic globe will have a radius closely corresponding to the Schwarzschild radius. Thus lending substance to the conjuncture that the region within the Schwarzschild radius of a black hole contains only pure radiation. As an application example, we consider the case of a very large photonic globe whose radius corresponds to the radius of the universe and containing radiation of the frequency of the microwave background (160.2 GHZ). It so turns out that such a photonic globe has an average density which closely corresponds to the observed average density of our universe. PACS numbers: 04.20, 42.50, 98.62. 98.80 Keywords: photonic globes, Schwarzschild radius, density of universe, black holes, dark matter, dark energy I. INTRODUCTION: PROBLEM STATEMENT tons form a self sustained globe of radius R, the number density of the photons σν (r), will be assumed to be a The possible existence of a self-sustained radiation ex- function of r alone, r being the radial distance from the isting as a spherical or near spherical region was first centre 0. conjectured by J.A. Wheeler [1], who gave it the name It will be shown that by imposing conditions of sta- Geon. Subsequently many researchers [2-6] have investi- bility of the system one can show that the radius R of gated this possibility and have studied two types of Geons the photonic globe corresponds to the Schwarzchilde ra- - gravitational and electromagnetic. All the studies made dius, an expression for the number density σν (r) is also were to find out if such structures can exist and be con- obtained. sistent with the field equations of general relativity. The conclusion arrived at was that such structures are essen- II. BRIEF DETAILS OF CALCULATION tially unstable and at best are not of long duration. In addition, Teo [7] has investigated the possibility of pho- tons forming a stable orbit under the influence of a black We herewith assume that a photonic globe, of radius R, hole and has concluded that such stable orbits are possi- consisting solely of photons under its own gravitational ble at a fixed distance which is exactly equal to 1.5 times field and centered about the origin is extant. We define the Schwarzschild radius of the black hole. Such struc- M(r) to be the “mass” of an imaginary sphere of radius tures were then called ”photonic spheres”, which actually r, 0 < r ≤ R, then is a thin layer of photons (like a thin ballon) at 1.5 times the distance of the Schwarzschild radius(SR). r 2 2 We in this paper consider the possibility of photons M(r)= 4πr σν (r)(hν/c )dr (1) existing under its own gravitational field, and investigate Z0 under what conditions such a collection of photons occu- Now consider a point P at a distance r from the centre, pying a finite region including the origin,(photonic globe) O, and surrounded by a volume element (in polar coor- can exist and be stable. 2 2 dinates), the mass m∆ = σν (r)(hν/c )r dr sinθ dθ dφ In order to do so we make the following assumptions: The gravitational force, F, on this small element m∆ (i) The treatment of the problem is classical is given by: (ii) The velocity of each photon is the speed of light: c arXiv:1303.3818v1 [physics.gen-ph] 14 Mar 2013 (iii) We assume that each photon is gravitationally at- Gm M(r) tracted by another photon, according to Newton’s Grav- F = ∆ (2) itational law and behaves for this purpose as having a r2 “mass” proportional to hν/c2 ( h and ν being Planck’s Now imagine the photons at P are moving inward with constant, frequency of the photon resp.) a velocity c and making an acute angle ψ, with respect (iv) In this brief study it will be assumed that the to the radial line drawn from P to the origin O. Then photons all have the same frequency, and that the pho- the tangential velocity, vt, of the photons in this volume element will be vt = csinψ. The centrifugal force (c.f.) on this volume element will 2 2 2 2 a)Author thanks the management of SNIST be given by c.f = m∆vt /r , but vt = c sin ψ; substitut- 2 b)SNIST, Jawaharlal Univ. of Tech., Yamnampet, Ghatkesar, Hy- ing the average value of sin ψ over 0 to π as 1/2, we see 2 2 derabad 501301, India, that vt = c /2, hence the cenrifugal force on the photons Formerly at Dept. of Theoretical Physics, Univ of Madras in the volume element will be : 2 point P at an arbitrary distance r in the globe lies on an “event horizon”. The photon number density σν (r), as a m c2 c.f. = ∆ (3) function of r, of such a photonic globe is given by eq.(6). 2 r The above calculation seems to lead to an interesting The condition of stability requires thatF = c.f , hence conjecture: That the region within the Schwarzschild ra- by equating (2) and (3), we have: dius of a black hole consists of pure radiation, a stable photonic globe, sustained within itself by its own “grav- itational” field. c2 M(r)= r (4) 2G IV. APPLICATION REGARDING THE DENSITY OF Substituting for M(r), from (1), we have THE UNIVERSE r 4 In this section, we will consider a very large photonic 2 c r σν (r) dr = r (5) globe which contains photons corresponding to 160.2 Z0 8πGhν GHZ, the frequency of the background radiation and as- Since eq. (5) must be true for all r, we can see that sume the radius of the globe to be the radius of the uni- verse. If we start from the expression, Eq(6), for the this is not possible unless the number density, σν (r), is given by the following expression: number density of photons at frequency ν, and substitute υ ≡ νB = 160.2GHz , where νBis the frequency of the background radiation of the universe (which corresponds c4 1 to a wave length λ = 0.1872 cms), using the value of σν (r) = (6) −8 8πGhν r2 the Gravitational constant G =6.673 10 cgs(cm-gram- second ) units and Planck’s constant h =6.626 10−27erg- It may be noted that though the number density seems sec (cgs units ) , and the value of the velocity of light to become infinite as r tends to zero, the number of pho- c = 3.0 1010 cms per sec; we see that at this frequency 4π ν 3 B tons in a very small sphere of radius ǫ will be σ (ǫ) 3 ǫ ν we can write which is finite. Now if we substitute r = R, and noting that M(R)= 62 1 M, the mass of the photonic globe, we have σνB (r)=4.55 . 10 (8) r2 2 2GM which we denote for convenience as σνB (r) = β/r , R = (7) 62 c2 where β =4.55 . 10 . Now to calculate the total number of Photons NR in- It may be noted that the rhs of (7) is nothing but the side a sphere of radius R, we need to integrate the above Scwarzchilde radius. and obtain III. ON THE PROPERTIES OF THE PHOTONIC R 2 GLOBE NR = 4πr σνB (r) dr Z0 From the above calculation, it so turns out that the radius R of a photonic globe, eq.(7), is nothing but the =4π β R (9) Schwarzschild radius an expression for which radius was The total energy, Etotal, of radiation is then Etotal = derived by Schwarzschild in 1916, for a spherically sym- 2 metric body by using equations of general realtivity for NR.hυB,. The equivalent mass will be M = Etotal/c . Hence the average “mass density” inside this sphere will regions outside this radius. We have derived the same 4π be ρav = M/( R3). That is expression for the Schwarzchilde radius by using com- 3 pletely different arguments for regions inside this radius by considering a collection of photons and using some 3βhνB ρav = (10) assumptions detailed above. It is well known that the R2c2 event horizon for a black hole occurs at a radius equal to the Scwarzchilde radius. The physics within this radius Now if we take R as the the radius of the is not well known and can only be guessed at. Also when visible universe R=13.5 billion light years, ie. eq(4) written as r = 2GM(r)/c2, is a valid equation for R =1.2271028cms. any radius r centered around the origin, M(r) being the Substituting these valuses for R, β , νB, and c, we mass of the imaginary sphere of this radius, we see that get an average mass density for the universe as ρav = r is the Schwarzschild radius for this sphere, so every 9.869 10−30grams/cc. Which is very close to the actual 3 estimated mass density, by the WMAP.[8], as may be density of such a photonic globe is very close to the latest gathered from the following quotation in the NASA, ar- estimate of the average mass density of the universe by ticle [8]: NASAs WMAP team[7].
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