The Schwarzschild radius is the radius of\nthe event horizon of a black hole. Amazingly, we can compute it with Newtonian mechanics,\nwhich is explained below. Consider a big mass M which creates the gravitational\nacceleration a for a small mass m at the distance r\nfrom M, see figure 1 and equation (1) for\ngravitational acceleration a=GM\/r2. For computing v the radial velocity of m in\nthe gravitational field of M we integrate equation (1). <\/p>\n\n\n\n
We compute for the case where m freefalls from\ninfinitely far starting with zero velocity, see (7). With these conditions the radial velocity of\nm at the distance r2<\/sub> f<\/a>rom M is computed in (8), v2 = 2GM\/r2. Reversing (8), r2<\/sub> = 2GM\/v2. The Schwarzschild radius of\nthe event horizon of M is rs<\/sub> such that the Schwarzschild factor equals\ninfinity, see (10).<\/p>\n\n\n\n When v2<\/sub> equals the speed of light c, we apply v2<\/sub>\n= c into (9) and we obtain (11) where r2<\/sub> = rs<\/sub>. So, r2<\/sub>\nequals rs<\/sub> and the Schwarzschild radius is computed with Newtonian\nmechanics. <\/p>\n\n\n\n Although\nthe Schwarzschild radius rs<\/sub> is a relativistic quantity, in the above\nit is derived completely with Newtonian mechanics, which is somewhat weird.\nWhat will be its value if we apply relativistic principle?<\/p>\n\n\n\n In the\nfollowing derivation we will use the formula for relativistic transformation of\nacceleration which is derived in the paper \u00ab Relativistic<\/a> kinematics<\/a> \u00bb linked here: https:\/\/www.academia.edu\/44582027\/Relativistic_kinematics<\/a> <\/p>\n\n\n\n The\nformula is the equation (18) of the paper.<\/p>\n\n\n\n Here,\nthis formula is given by (12) in which the gravitational acceleration of m is a and\nthe acceleration in space is ar<\/sub> and the radial velocity is computed by\nintegration. With the same conditions as (7), the constant of integration k equals 0, see (16). Then, using k = 0 in (15), v is expressed with r in (17).<\/p>\n\n\n\n In the case where\nthe small mass m approaches M, the distance r approaches 0, the radial velocity\nof m approaches the speed of light c, see (18). So,\nv the radial velocity of m does not become bigger than the speed of light c for\nr > 0. The Schwarzschild radius rs<\/sub> is the radius such that v = c.\nSo, rs<\/sub> = r = 0, see (19).<\/p>\n\n\n\n When the\nrelativistic principles are applied correctly, the Schwarzschild radius rs<\/sub>\nequals zero. The gravitational force on m approaches infinity near M. But the\nspeed of m never reaches the speed of light c, which is true for any force\nhowever strong it is and for time of acceleration however long it is. The\nSchwarzschild radius rs<\/sub> must obey relativistic principle and is\nshown to be zero.<\/p>\n\n\n\n That the Schwarzschild radius rs<\/sub> equals\nzero means that the geometrical size of a black hole should be zero and thus, a\nblack hole should not have an interior. I have reached this conclusion and\nexplained it in the paper \u00ab Gravitational time dilation<\/a> and black hole<\/a> \u00bb in which I have\nalso shown that point masses could not\ncoalesce to form a black hole whose volume is zero and that observations\nsupport this conclusion. This paper is linked here:<\/p>\n\n\n\n https:\/\/www.academia.edu\/45434676\/Gravitational_time_dilation_and_black_hole<\/a><\/p>\n\n\n\n The equations and figure are in the paper below<\/p>\n\n\n\n https:\/\/www.academia.edu\/84798805\/Radius_of_a_black_hole_for_relativity_and_Newtonian_mechanics<\/a><\/p>\n\n\n\n For more detailed information, I invite you to read\nthe two cited papers:<\/p>\n\n\n\n \u00ab Gravitational time dilation<\/a> and black hole<\/a> \u00bb https:\/\/www.academia.edu\/45434676\/Gravitational_time_dilation_and_black_hole<\/a><\/p>\n\n\n\n \u00ab Relativistic<\/a> kinematics<\/a> \u00bb https:\/\/www.academia.edu\/44582027\/Relativistic_kinematics<\/a><\/p>\n","protected":false},"excerpt":{"rendered":" For Newtonian mechanics The Schwarzschild radius is the radius of the event horizon of a black hole. Amazingly, we can compute it with Newtonian mechanics, which is explained below. Consider a big mass M which creates the gravitational acceleration a … Continue reading For relativity<\/h1>\n\n\n\n
Conclusion <\/h1>\n\n\n\n