Using of the Correction Schwarzschild radius Equation and Its Application of the Black Holes Majida H. Al-kubaisy Dept. of Physic/Collage of Science Al-Mustensiriyh /University Received in:24June 2013,Accepted in :4December 2013 Abstract This study is a try to compare between the traditional Schwarzschild’s radius and the equation of Schwarzschild’s radius including the photon’s wavelength that is suggested by Kanarev for black holes to correct the error in the calculation of the gravitational radius where the wavelengths of the electromagnetic radiation will be in our calculation. By using the different wavelengths; from radio waves to gamma ray for arbitrary black holes (ordinary and supermassive). Key words: black hole, photon, wavelength, Schwarzschild’s radius, mass, general. 121 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Introduction After a star has exhausted its nuclear fuel, it can no longer remain in equilibrium and must ultimately undergo gravitational collapse. The star will end as a white dwarf .If the mass of the collapsing core is less than the famous Chandrashekhar limit of 1.4 M⨀ which represents the solar masses. It will end as a neutron star if the core has a mass greater than the Chandrashekhar limit and less than about 35 times the mass of the sun. It is often believed that a core heavier than about 5 solar masses will end, not as a white dwarf or as a neutron star, but as a black hole [1]. Photon always travels at the speed of light, but they lose energy when travelling out of a gravitational field. The stronger the gravitational field the more energy the photon lose .The extreme case is the black holes where photons from wither radius lose all energy and became invisible. In physical terms, a black hole is a region where gravity is so strong that nothing can escape. In order to make this notion precise, one must have in mind a region of space time to which one can contemplate escaping. Black holes are perhaps the most strange and fascinating objects in the universe [2]. In principle, any object - even a rock - can be made into a black hole, by squeezing it into a tiny enough volume. Under these conditions, the object continues to collapse under its own weight, crushing itself down to zero size. However, according to Einstein's theory, the object's mass and gravity remain behind, in the form of an extreme distortion of the space and time around it. This distortion of space and time is the black hole. The resulting black hole is the darkest black in the universe: No matter how powerful a light you shine on it, no light ever bounces back, because the light is swallowed by the hole. A black hole is a true "hole" in space. Anything that crosses the edge of the hole -called the "horizon" of the hole - is swallowed forever. For this reason, black holes are considered an edge of space, a one-way exit door from our universe; nothing inside a black hole can ever communicate with our universe again, even in principle. A black hole is, by definition a region in space-time in which the gravitational field so strong that is precludes even light from escaping to infinity. Black holes are formed when a body of mass M contracts to size less than the so called gravitational radius Rs= 2GM/C2 (Newton’s gravitational constant) the velocity required to leave the boundary of the black holes and move away to infinity equals to the speed of light [3]. Since signal cannot escape from a black hole, while physical object and radiation can fall into it, the surface boundary the black hole in space-time is called the event horizon, which is a light surface. In Newtonian physics, the escape velocity from a spherical mass M of radius R satisfies 𝑉𝑒𝑠𝑐 = �2𝐺𝑀/ 𝑅, (independent of the mass of the escaping object, by equivalence of inertial and gravitational masses). Vesc exceeds the velocity of light if the R < Rs where The radius Rs is called the Schwarzschild’s radius for the mass M [4],[5]. The New Schwarzschild’s radius It has shown that the equation of the Schwarzschild’s radius for the black holes does not account the photon wavelength [6], so that first we review the ordinary Schwarzschild’s radius. In 1916, Karl Schwarzschild, the German Astronomer offered a formula for the calculation of the gravitational radius Rs of the black holes which is called also Schwarzschild’s radius: 𝑅𝑠 = 2𝐺.𝑀 𝐶2 (1) Where G= 6.67×10-11 N. m2/kg2 is the gravitational constant, M is the star mass and C is the velocity of light. The gravitational radius formula (1) is derived by using the mathematical relation in the law gravitation [7]: 122 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 𝐹𝑔 = 𝐺. 𝑚.𝑀 𝑅2 (2) Here Fg is the gravitational force, m is the photon mass and R is the distance between the center of masses of the bodies which form the gravitation. In order to find the gravitation radius R= Rs of the star , by which its gravitational field retard light it is necessary to find relationship between the gravitational force and the force FF which moves the photon and has the formula [6]: 𝐹𝐹 = 𝑚 𝑐2 𝜆 (3) If we equate equation (2) with equation (3), we will have: 𝐺. 𝑚.𝑀 𝑅𝑠 = 𝑚𝐶2 𝜆 (4) It’s clear that in order to convert the equation of the forces (4) into equation of energies it is necessary to reduce the denominator of the left side part by the Rs and to reduce the denominator of the right side by λ of the photon , Actually it means that it is necessary to equate the gravitational radius to the wavelength of the photon it is possible to do it , Schwarzschild did it and his followers did not notice this error [6]. Thus Kanarev will add the photon wavelength to calculate the gravitational radius of the black hole; it is the result of the mistake that Schwarzschild made when he derived this formula. The final formula derived by Kanarev must add 2π to right side of equations (4) as shown below: 𝑅𝑆 = 1 𝑐 �𝐺.𝑀.𝜆 2𝜋 (5) The density of the black holes according to equation (5) will be: 𝜌 = 3𝑀 4𝜋.𝑅𝑠 3 (6) This the new gravitational radius or new Schwarzschild’s radius which contains the wave length of the photon that corrected the error of the traditional Schwarzschild’s radius formula according to Kanarev work in 2002. The results This work is based on a comparison between classical and new Schwarzschild’s radius for many black holes by using matlap programing to the equations (1) ,(5) and (6). To show the difference between these two radius ( we found some errors in the Kanarev paper’s when he made his calculation for the new Schwarzschild radius of sun at infrared and gamma wavelength, we resolved them as shown in this table below). Equations (1) &(5) are used to find the traditional and the new Schwarzschild radius as shown in table No.(1), for many arbitrary black holes [8] where their masses lie between (4- 20)M⨀. By using wavelength from radio to gamma in equation (5), we found that the new Rs will have less value than the traditional Rs and its value decreases when the wavelength is decreased as it is mentioned in figure (1). It’s clear that the gravitational field in black hole with Rs lie between (10- 19) m (in radio spectrum) will retard the electromagnetic radiation while the smaller wavelength in the spectrum will penetrate easily. Another application for the supermassive black holes [9] for the same equation as shown in table No. (2). The large Rs value will retard the electromagnetic while the small one will penetrate the radiation, here we can see that the black holes will have the larger 123 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 gravitational field for the both traditional Rs and new Rs in the radio , infrared and visible wavelength which is obviously in the figure (2). Conclusion The difference in our calculation of the gravitational field of a black holes between the traditional and new Schwarzschild radius is about 109 times when the wavelength of the electromagnetic radiation taken into account which is close to the value found by Kanarev in 2002 which equal to 108 times, while for the ordinary black holes and for the supermassive black holes 1014 times. This calculation will show us that by using the wavelength in the Schwarzschild radius the black holes will not be black just for the wavelength that penetrates the Rs while the other will shine in new color which refers to the wavelength that retards. Reference 1. Singh T.P. (1999), Gravitational Collapse, Black Holes and Naked Singularities, J. Atrophy’s, Astr., 20, 221–232. 2. Robert M.W. (2001), The Thermodynamics of Black Holes, Published by the Max Planck Institute for Gravitational Physics Albert Einstein Institute, Germany. 3. Frolov p.v., and Nevikov,I.D. (1998), Black Hole Physics (Basic concept and new development) Kluwer Academic Publishers. (Google book). 4. Jacobson, T. (1996), Introductory Lectures on Black Hole Thermodynamics, Institute for Theoretical Physics University of Utrecht. 5. Hooft, G. (2009), Introduction to the theory of black holes, Lectures presented at Utrecht University. 6. Kanarev Ph. M., (2002), The gravitational Radius of Black Holes, Journal of Theoretics Vol.4-1. 7. Ginsburg, V.L. (1985) On Physics and Astrophysics, M, NAUKA. 8. Ziolkowski, J. (2008) Mass of Black Holes in the Universe, Chin. J. Astron. Atrophy’s. Vol.8, Supplement, 273-280. 9. Nicholas, J.; McConnell, Chung-pei Ma, Karl Gebhardt, Shelley, A.; Wright, Jeremy D.; Murphy, Tod, R. Lauer, James, R. Graham and Douglas, O. Richstone, (2011),Two ten- Billion Solar Mass Black Holes at the Centre’s of Giant elliptical Galaxies, arXiv:1112.1078 V1 [astro-ph.CO.] 5 Dec. 10. Hillwing, T. C.;Gies, D. (2006), Bell. AAS, 38,954. 11. Ziolkowski, J. (2005), Evolutionary constraints on the masses of the components of HDE 226868/Cyg X-1 binary system, MNRAS, 358, 851. 12. Hynes, R.L.;Steghs, D.;Casares, J.; Charles, P. A. and O’Brien K., (2003), Dynamical Evidence for a Black Hole in GX 339-4 , AP J , 583, L95. 13. Masetti,N.; Bianchini, A.; Bonibaker, J.; Della Valle, M. and Vio, R. (1996), The superhump phenomenon in GRS 1716-249, A&A, 314, 123 14. Gelino, D.M.;Balman, S.; Kilziloglu, U.;Yılmaz, A.;Kalemci, E. and Tomsick, J. (2006), The Inclination Angle of and Mass of the Black Hole in XTE J1118+480 , Ap. J., 642, 438. 124 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Table No. (1) Schwarzschild radius (Rs& new Rs) for black holes in binary system for given wavelengths Note: All the reference in the table exists in [8]. Table No. (2) Schwarzschild radius (Rs & new Rs) for supermassive black holes in binary system for given wavelengths Note: All reference in the table exists in [9] for the supermassive black holes masses. Black Holes (BH) BH mass (Solar mass) (kg) Traditional Rs (m) Radio Infrared visible x-ray Gamma Ref. New Rs (m) New Rs (m) New Rs (m) New Rs (m) New Rs (m) SS433 4.4±0.8 1.3061e+4 10.6925 1.0195 0.0204 1.0195e-4 1.0195e-5 10 GX 339-4 ≥ 6.0 1.7810e+4 12.4861 1.1905 0.0238 1.1905e-4 1.1905e-5 12 GS1124-684 7±0.6 2.0779e+4 13.4866 1.2859 0.0257 1.2859e-4 1.2859e-5 14 XTE J1118+480 8.5±0.6 2.5231e+4 14.8615 1.4170 0.0283 1.4170e-4 1.4170e-5 14 A 0620-00 11±2 3.2652e+4 16.906 1.6120 0.0322 1.6120e-4 1.6120e-5 12 GRS 1915+105 14±4 4.1558e+4 19.0729 1.8185 0.0364 1.8185e-4 1.8185e-5 13 CygX-1 20±5 5.9368e+4 22.7965 2.1736 0.0435 2.1736e-4 2.1736e-5 11 Black Holes (BH) BH mass (Solar mass) (kg) Traditional Rs (m) Radio Infrared visible x-ray Gamma Ref. New Rs (m) New Rs (m) New Rs (m) New Rs (m) New Rs (m) Milky Way 4.1e+6 1.2170e+10 1.0322e+04 984.1205 19.6824 0.0984 0.0098 8,38,40 Circinus 1.7e+6 5.0463e+09 6.6463e+03 633.6958 12.6739 0.0634 0.0063 4,41 N1194 6.8e+7 2.0185e+11 4.2035e+04 4.0078e+03 80.1569 0.4008 0.0401 34 N1023 14.6e+7 4.3339e+11 6.1593e+04 5.8726e+03 117.4526 0.5873 0.0587 8,46 N224(N31) 1.5e+8 4.4526e+11 6.2431e+04 5.9525e+03 119.0507 0.5953 0.0595 8,44 N524 8.3e+8 2.4638e+12 1.4686e+05 1.4002e+04 280.0434 1.4002 0.1400 45 N1332 1.45e+9 4.3042e+12 1.9410e+05 1.8507e+04 370.1437 1.8507 0.1851 49 N3842 9.7e+9 2.8793e+13 5.0204e+05 4.7868e+04 957.3532 4.7868 0.4787 37 125 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Figure No. (1) The difference for the Rs for many black holes with wavelength Figure No.(2) The difference for the Rs for many supermassive black holes with wavelength 0.000001 0.001 1 1000 1E-14 1E-10 0.000001 0.01 1E-14 1E-10 0.000001 0.01 Wavelength (m) The new R s The new R s Wavelength (m) 126 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 باستخدام المعادلة المصححة لنصف قطر شوارزجایلد وتطبیقاتھا على الثقوب السوداء ماجدة حمدان الكبیسي الجامعة المستنصریة /كلیة العلوم /قسم الفیزیاء 2013كانون االول 4، قبل في :2013حزیران 24استلم في : الخالصة شوارزجایلد التي التقلیدي ومعادلة نصف قطر شوارزجایلدراسة محاولة ألجراء مقارنة بین نصف قطر دھذه ال نصف تتضمن الطول الموجي للفوتون والمقترحة من كنیرف للثقوب السوداء وذلك لتصحیح الخطأ في الحسابات لقیاس االطوال الموجیة لإلشعاع الكھرومغناطیسي ستستعمل في حساباتنا. باستخدام االطوال الموجیة قطر التجاذبي، حیث ( االعتیادیة وفائقة الكتلة). ھا بشكل اعتباطي للثقوب السوداءشعة كاما وتطبیقالمختلفة من الموجات الرادیویة والى ا الكلمات المفتاحیة: الثقب االسود، الفوتون، نصف قطر شوارزجایلد، الكتلة، عام. 127 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014