aa r X i v : . [ phy s i c s . g e n - ph ] M a r On Self Sustained Photonic Globes a) K. Eswaran b) (Dated: 1 April 2019) In this paper we consider a classical treatment of a very dense collection of photons forming a self-sustainedglobe under its own gravitational influence. We call this a “photonic globe” We show that such a densephotonic globe will have a radius closely corresponding to the Schwarzschild radius. Thus lending substanceto 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 tothe radius of the universe and containing radiation of the frequency of the microwave background (160.2GHZ). It so turns out that such a photonic globe has an average density which closely corresponds to theobserved average density of our universe.PACS numbers: 04.20, 42.50, 98.62. 98.80Keywords: photonic globes, Schwarzschild radius, density of universe, black holes, dark matter, dark energy
I. INTRODUCTION: PROBLEM STATEMENT
The possible existence of a self-sustained radiation ex-isting as a spherical or near spherical region was firstconjectured by J.A. Wheeler [1], who gave it the nameGeon. Subsequently many researchers [2-6] have investi-gated this possibility and have studied two types of Geons- gravitational and electromagnetic. All the studies madewere to find out if such structures can exist and be con-sistent with the field equations of general relativity. Theconclusion arrived at was that such structures are essen-tially unstable and at best are not of long duration. Inaddition, Teo [7] has investigated the possibility of pho-tons forming a stable orbit under the influence of a blackhole and has concluded that such stable orbits are possi-ble at a fixed distance which is exactly equal to 1.5 timesthe Schwarzschild radius of the black hole. Such struc-tures were then called ”photonic spheres”, which actuallyis a thin layer of photons (like a thin ballon) at 1.5 timesthe distance of the Schwarzschild radius(SR).We in this paper consider the possibility of photonsexisting under its own gravitational field, and investigateunder what conditions such a collection of photons occu-pying a finite region including the origin,(photonic globe)can exist and be stable.In order to do so we make the following assumptions:(i) The treatment of the problem is classical(ii) The velocity of each photon is the speed of light: c (iii) We assume that each photon is gravitationally at-tracted by another photon, according to Newton’s Grav-itational law and behaves for this purpose as having a“mass” proportional to hν/c ( h and ν being Planck’sconstant, frequency of the photon resp.)(iv) In this brief study it will be assumed that thephotons all have the same frequency, and that the pho- a) Author thanks the management of SNIST b) SNIST, Jawaharlal Univ. of Tech., Yamnampet, Ghatkesar, Hy-derabad 501301, India,Formerly at Dept. of Theoretical Physics, Univ of Madras tons form a self sustained globe of radius R, the numberdensity of the photons σ ν ( r ), will be assumed to be afunction of r alone, r being the radial distance from thecentre 0.It will be shown that by imposing conditions of sta-bility of the system one can show that the radius R ofthe photonic globe corresponds to the Schwarzchilde ra-dius, an expression for the number density σ ν ( r ) is alsoobtained. II. BRIEF DETAILS OF CALCULATION
We herewith assume that a photonic globe, of radius R,consisting solely of photons under its own gravitationalfield and centered about the origin is extant. We define M ( r ) to be the “mass” of an imaginary sphere of radius r , 0 < r ≤ R , then M ( r ) = Z r πr σ ν ( r )( hν/c ) dr (1)Now consider a point P at a distance r from the centre,O, and surrounded by a volume element (in polar coor-dinates), the mass m ∆ = σ ν ( r )( hν/c ) r dr sinθ dθ dφ The gravitational force, F, on this small element m ∆ is given by: F = Gm ∆ M ( r ) r (2)Now imagine the photons at P are moving inward witha velocity c and making an acute angle ψ , with respectto the radial line drawn from P to the origin O. Thenthe tangential velocity, v t , of the photons in this volumeelement will be v t = c sinψ .The centrifugal force (c.f.) on this volume element willbe given by c.f = m ∆ v t /r , but v t = c sin ψ ; substitut-ing the average value of sin ψ over 0 to π as 1 /
2, we seethat v t = c /
2, hence the cenrifugal force on the photonsin the volume element will be : c.f. = m ∆ c r (3)The condition of stability requires that F = c.f , henceby equating (2) and (3), we have: M ( r ) = c G r (4)Substituting for M ( r ), from (1), we have Z r r σ ν ( r ) dr = (cid:18) c πGhν (cid:19) r (5)Since eq. (5) must be true for all r, we can see thatthis is not possible unless the number density, σ ν ( r ), isgiven by the following expression: σ ν ( r ) = (cid:18) c πGhν (cid:19) r (6)It may be noted that though the number density seemsto become infinite as r tends to zero, the number of pho-tons in a very small sphere of radius ǫ will be σ ν ( ǫ ) π ǫ which is finite.Now if we substitute r = R , and noting that M ( R ) = M , the mass of the photonic globe, we have R = 2 GMc (7)It may be noted that the rhs of (7) is nothing but theScwarzchilde radius. III. ON THE PROPERTIES OF THE PHOTONICGLOBE
From the above calculation, it so turns out that theradius R of a photonic globe, eq.(7), is nothing but theSchwarzschild radius an expression for which radius wasderived by Schwarzschild in 1916, for a spherically sym-metric body by using equations of general realtivity forregions outside this radius. We have derived the sameexpression for the Schwarzchilde radius by using com-pletely different arguments for regions inside this radiusby considering a collection of photons and using someassumptions detailed above. It is well known that theevent horizon for a black hole occurs at a radius equal tothe Scwarzchilde radius. The physics within this radiusis not well known and can only be guessed at. Also wheneq(4) written as r = 2 GM ( r ) /c , is a valid equation forany radius r centered around the origin, M ( r ) being themass of the imaginary sphere of this radius, we see thatr is the Schwarzschild radius for this sphere, so every point P at an arbitrary distance r in the globe lies on an“event horizon”. The photon number density σ ν ( r ), as afunction of r, of such a photonic globe is given by eq.(6).The above calculation seems to lead to an interestingconjecture: That the region within the Schwarzschild ra-dius of a black hole consists of pure radiation, a stablephotonic globe, sustained within itself by its own “grav-itational” field. IV. APPLICATION REGARDING THE DENSITY OFTHE UNIVERSE
In this section, we will consider a very large photonicglobe which contains photons corresponding to 160.2GHZ, the frequency of the background radiation and as-sume the radius of the globe to be the radius of the uni-verse. If we start from the expression, Eq(6), for thenumber density of photons at frequency ν , and substitute υ ≡ ν B = 160 . ν B is the frequency of thebackground radiation of the universe (which correspondsto a wave length λ = 0 . G = 6 .
673 10 − cgs(cm-gram-second ) units and Planck’s constant h = 6 .
626 10 − erg-sec (cgs units ) , and the value of the velocity of light c = 3 . cms per sec; we see that at this frequency ν B we can write σ ν B ( r ) = 4 . . r (8)which we denote for convenience as σ ν B ( r ) = β/r ,where β = 4 . . .Now to calculate the total number of Photons N R in-side a sphere of radius R, we need to integrate the aboveand obtain N R = Z R πr σ ν B ( r ) dr = 4 π β R (9)The total energy, E total , of radiation is then E total = N R .hυ B ,. The equivalent mass will be M = E total /c .Hence the average “mass density” inside this sphere willbe ρ av = M/ ( π R ). That is ρ av = 3 βhν B R c (10) Now if we take R as the the radius of thevisible universe R=13.5 billion light years, ie. R = 1 .
227 10 cms. Substituting these valuses for R, β , ν B , and c, weget an average mass density for the universe as ρ av =9 .
869 10 − grams/cc. Which is very close to the actualestimated mass density, by the WMAP.[8], as may begathered from the following quotation in the NASA, ar-ticle [8]: “WMAP determined that the universe is flat, fromwhich it follows that the mean energy density in the uni-verse is equal to the critical density (within a 0.5% mar-gin of error). This is equivalent to a mass density of . − g/cm , which is equivalent to only 5.9 protonsper cubic meter.” It is well known that only about 4 percent of the massof the universe consists of Baryonic matter. So, if we, (fora stating approximation), adopt the hypothesis that theuniverse is a photonic globe containing photons of fre-quency of the CMB, then the above calculations give thecorrect average density, and obviously the correct Massfor the universe. However, if one considers the numberdensity of photons of frequency ν B , it turns out to begrossly over estimated, (as can be easily calculated fromthe above equations) from the actual value of 400 photonsper cc (near earth). So here we have a situation wherethe mass and mass-density are correct but the numberof photons estimated are far too much. So where haveall the extra photons gone? It could then be conjecturedthat some unknown physical process has converted allthis extra radiation into dark matter and dark energy,thus keeping the total energy (mass) unchanged. Onlyfurther experimentation and research can resolve such is-sues. V. CONCLUSION
In this paper, we have considered the possibility of theexistence of a stable a selfsustained photonic globe andhave arrived at the following: (i) that such a globe musthave its radius equal to the Schwarzchild radius and (ii)if we consider a photonic globe which contains photonsof frequency equal to 160.2 GHZ and a radius equal tothe radius of the universe then the average mass (energy) density of such a photonic globe is very close to the latestestimate of the average mass density of the universe byNASAs WMAP team[7].