Black Holes Introduction: Stable stars which balance pressure due to gravity by quantum pressure(qp) are: 1. White Dwarfs (WD) : QP = electron degeneracy pressure. Mass < 1.4 M_sun This upper limit is called the Chandrashekar limit. 2. Neutron stars (NS): QP = neutron degeneracy pressure. Mass < 2 3 M_sun. Limit is not so well defined. When a Massive star explodes,m > 10 M_sun, explodes (SN type II) and if it leaves behind a star with a mass M > 2 or 3 M_sun in which nuclear energy generation is finished, there is nothing to prevent collapse due to gravity not even neutron degeneracy pressure. This collapse leads to a Black Hole (BH).
A Newtonian Discussion of BH: Escape Velocity: It represents the speed a vertical rocket should have if it is not to return to an object with mass M. Consider a star with mass M and a radius R. A ball on the surface of the star of mass m posses gravitational energy which keeps it pulled to the surface which is : Gravitational energy, ge= G m M R If the ball is thrown straight up by giving it lots of kinetic energy(ke) it will climb up until all its KE is used up doing work against the force of gravity. If KE energy is equal to or greater than ge it will escape to infinity otherwise it will fall back! We define escape velocity by the simple equality: KE = 1 2 m v 2 =G mm escape R or V excape = GM R Escape velocity increases with increasing mass or decreasing radius of the star.
A fundamental postulate of Einstein's theory of relativity is that speed of light, c = 300,000 km/sec is the maximum speed an object can have. So we can ask what is the radius of a star for which escape velocity is the speed of light. v escape =c= GM R which can be solved for R This radius is called the Schwarzchild radius R S and is equal to R S = GM c 2 So from a star of mass M and radius less than Scharzchild radius not even light can escape to reach a distant observer!! Such a star is not visible and has been given the name Black Hole or BH. Any event happening at distances equal to or less than R_s is not observable. So R_s is called the event horizon.
Some numerical values of Scharzchild radius: (1) Consider a star with mass of the sun: M=2 10 30 kg Whata is its event horizon? R S = GM 11 c =[6.6 10 N m 2 /kg 2 ] 2 10 30 kg 2 [9 10 8 m 2 /s 2 ] Schwarzchild radius R S =3 km, For the sun, its radius is ~ 700,000 km! So the sun can shine! If the sun were compressed to 3 km it would be a BH. Its density would be, before it collapsed to a singluarity 1.4 10 20 kg/m 3 What prevents it from collapsing? Nuclear energy. (2) What about a NS? Its radius is 10 km, still larger than 3 km, hence it can stay stable with neutron degeneracy pressure.
Mass vs Circumference for stars in the universe Cirumference = 2 R
Global view of what provides stability and when BH will form: BH regime
According to Einstein's General Theory of relativity Space time near a gravitating object is curved. A two dimensional rubber sheet can depict this space time curvature.
Emission of photons from stars with different radii approaching R_S
Gravitational Red Shift and Time dilation: Gravitational red shift: Light (Electromagnetic wave) is defined by its wavelength and its frequency. Wavelength =, Frequency = f Connection between, f and speed of light, c Speed c = T = f Light is made up of particles called photons. Energy of a photon of frequency f is E photon =h f where h = Planck ' s constant A photon emitted vertically from a star will loose energy due to work done against gravity. So its energy will decrease. Hence its frequency will decrease. And its wavelength will get longer this is called gravitational red shift.
Time dilation due to gravity: Consider two observers, one on the star's surface and one at a very large distance. Gravity causes another peculiar effect: To the distant observer, the clock of the observer on the star's surface will appear to be ticking slower. If the surface of the star is at the Schwrzchild radius then clock will appear infinitely slow to the distant observer as compared to his own. This is called time dilation. So as the star collapses to its Scharzchild radius, to the distant observer it will appear to take an infinitely long time!
Some remarks about BHs