Randall D. Knight — Newton's Theory of Gravity: Problem 72

September 2015 saw the historic discovery of gravitational waves, almost exactly 100 years after Einstein predicted their existence as a consequence of his theory of general relativity. Gravitational waves are a literal stretching and compressing of the fabric of space. Even the most sensitive instruments—capable of sensing that the path of a 4-km-long laser beam has lengthened by one-thousandth the diameter of a proton—can detect waves created by only the most extreme cosmic events. The first detection was due to the collision of two black holes more than 750 million light years from earth. Although a full description of gravitational waves requires knowledge of Einstein’s general relativity, a surprising amount can be understood with the physics you’ve already learned. (a) Consider two equal masses \( M \) that interact gravitationally and revolve at angular velocity \( \omega \) about their center of mass. Let the distance between them be \( 2r \), so that each is distance \( r \) from the center of mass. Apply Newton’s second law for circular motion to one of the masses to find an expression for \( \omega^2 \) (not \( \omega \)) in terms of \( G, M, \) and \( r \). (b) The total energy \( E \) of the system is the kinetic energy of the two masses, which can be written in terms of \( \omega^2 \), plus their gravitational potential energy. Find an expression for \( E \) in terms of \( G, M, \) and \( r \). This is a bound system, so the total energy should be negative. We’re going to consider two black holes that spiral in until they collide and merge. Because your expression for \( E \) is negative, it will become even more negative as \( r \to 0 \). That is, the system loses energy, and that energy is radiated away as gravitational waves. The details are complex, but we’re interested in only \( E_i \), which we can take to be \( \approx 0 \) because initially the black holes are far apart, and \( E_f \) at the instant they merge. (c) A black hole is a mass so dense that nothing, not even light, can escape from it. In other words, the escape speed exceeds the speed of light, designated \( c \). We define the Schwarzchild radius \( R_{\text{Sch}} \) to be the distance from the center of a black hole to a point where \( v_{\text{escape}} = c \). Find an expression for the Schwarzchild radius of a black hole of mass \( M \). To do this, consider a particle of mass \( m \) that is distance \( R \) from a much larger mass \( M \). What is the minimum speed the particle needs to escape to infinity? Equate this escape speed to \( c \) and solve for \( R_{\text{Sch}} \), the distance from the center of the mass at which the escape speed is \( c \). (d) Two black holes collide and merge when their Schwarzchild radii overlap; that is, they merge when their separation, which we’ve defined as \( 2r \), equals \( 2R_{\text{Sch}} \). Find an expression for \( \Delta E = E_f - E_i \), where \( E_i \approx 0 \) because initially the black holes are far apart and \( E_f \) is their total energy at the instant they merge. This is the energy radiated away as gravitational waves. Your answer will be a fraction of \( Mc^2 \), and you probably recognize that this is related to Einstein’s famous \( E = mc^2 \). The quantity \( Mc^2 \) is the amount of energy that would be released if an entire star of mass \( M \) were suddenly converted entirely to energy. (e) We still need to determine \( M \), and that’s where the experimental data enter. First, combine your expression for \( \omega^2 \) with your expression for \( R_{\text{Sch}} \) to find an equation for the angular velocity \( \omega_{\text{merge}} \) of two revolving black holes at the instant of merger. Then solve this to get an expression for the mass \( M \) in terms of \( G, c, \) and \( \omega_{\text{merge}} \). (f) FIGURE CP13.72 shows the experimental data from LIGO, the Laser Interferometer Gravitational-Wave Observatory in Hanford, Washington. The graph shows the strain, which is the fractional change in length of a 4-km-long laser beam. Even at the peak, the strain is an astoundingly small 1 part in \( 10^{21} \). The blue curve is the measured strain, while the red curve is the theoretical prediction from general relativity—an almost perfect match. The oscillations correspond to the last few orbits of the two black holes that revolve around their center of mass at angular velocity \( \omega \), which is speeding up as they spiral in. Careful measurements of the final revolution before they merge—the peak of the graph—give \( \omega_{\text{merge}} \approx 9000 \text{ rpm} \). Using that value, calculate the black hole mass \( M \), giving your answer as a multiple of \( M_s \), where \( M_s = 1 \text{ solar mass} = 2.0 \times 10^{30} \text{ kg} \). (g) With \( M \) known, calculate \( R_{\text{Sch}} \) (in km) and the speed \( v \) (as a fraction of \( c \)) of the black holes as they merged. You should find that there are many solar masses moving at a significant fraction of the speed of light within an area about the size of a major U.S. city. That is certainly an epic cosmic event. (h) How much energy \( E_{\text{grav wave}} \) was radiated away as gravitational waves? Give your answer as a multiple of \( M_s c^2 \). You should find that several solar masses were converted to energy and radiated away as gravitational waves during the \( \approx 0.2 \text{ s} \) duration of this merger.