Question

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. 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 2RSch . Find an expression for ΔE=Ef−Ei , where Ei ≈ 0 because initially the black holes are far apart and Ef 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 Mc², and you probably recognize that this is related to Einstein's famous E=mc² . The quantity Mc² is the amount of energy that would be released if an entire star of mass M were suddenly converted entirely to energy.

Accepted Answer

Step 1: Begin by understanding the Schwarzschild radius (RSch), which is given by the formula: RSch = 2GMc2, where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light. Step 2: The problem states that the two black holes merge when their separation equals twice their Schwarzschild radius, i.e., 2r = 2RSch. This implies that the separation distance is equal to the sum of their Schwarzschild radii. Step 3: The total energy at the instant of merging (Ef) is the sum of the gravitational potential energy and the rest energy of the system. The gravitational potential energy for two masses M1 and M2 separated by a distance r is given by: U = -GMMr. Here, r is the separation distance, which equals 2RSch. Step 4: The rest energy of the system is the sum of the rest energies of the two black holes, which is approximately $$2Mc^{2}$$, where M is the mass of each black hole. The total energy at the instant of merging is then: E_{f} = $$2Mc^{2} + U$$, where U is the gravitational potential energy. Step 5: The energy radiated away as gravitational waves is the difference between the final energy (Ef) and the initial energy (Ei). Since Ei ≈ 0 (the black holes are initially far apart), the energy radiated is approximately: ΔE = E_{f} = $$2Mc^{2} - $$GMMr. Substitute r = 2RSch to express the result in terms of Mc².

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Key Concepts

Gravitational Waves

Schwarzschild Radius

Energy-Mass Equivalence (E=mc²)