Relativity Lite

28 | Relativity Lite Another way to say this is that mass and energy are interchangeable. In fact, when we work with particles, we usually express their masses in a unit called an electron volt (eV)—­ the kinetic energy gained by an electron by “falling” through a one-volt charge difference—­ divided by the speed of light squared, rather than the kilogrammass unit that is usually used for lumps. Most particles have masses greater than 1 MeV/c 2 , where M stands for a million. The expression E mc 0 2 = is precisely the source of energy that fuels our Sun, providing the solar energy that warms our bodies and allows our food to grow. A proton, the nucleus of a hydrogen atom, crashes into and sticks to a particle called a deuteron, which has a pro- ton and a neutron bound together. The proton’s mass is 938.7 MeV/c 2 , and the deuteron’s mass is 1,876.0 MeV/c 2 . The resulting triton’s mass is 2,809.2 MeV/c 2 , which is less than the sum of the masses of the proton and deuteron, 2,814.7 MeV/c 2 . This is OK if the mass difference, 5.5 MeV/c 2 , is given off as some other form of energy—in this case, a particle of light that is massless and has 5.5 MeV of energy. Another application of this principle is in particle accelerators that bang two protons together at high speed, converting some of their motional (or kinetic) energy into massive particles, such as the Higgs boson that was in the news in 2014. The rest energy, like proper time, is a relativistic invariant whose value everyone can measure and agree upon. The general expression for the total energy of a particle is given by the expression E = γ E 0 , just as t = γ τ . * We can use this, for example, in find- ing the time-dilation factor for the muons created in the Earth’s atmosphere. The muons have an average total energy of 2,000 MeV, † and the mass of a muon is 105 MeV/ c 2 . Then γ = = = × = = E E E mc MeV MeV c c MeV MeV 0 2 2 2 2000 105 2000 105 19 / , which is where we got the value we used in the first chapter. The relativistic relation between a particle’s momentum and energy may be found by taking the ratio of the horizontal leg of figure 3 divided by c to the hypotenuse: p E m v m c v c = = γ γ 2 2 . Note that for a particle of light (a photon) whose velocity is v = c , this expression gives a well-defined value of the momentum of a massless particle: p = E/c . You may have read about the spaceship LightSail 2 that demonstrated in 2019 that the sunlight reflecting off its thin Mylar sail transferred its momentum to the spacecraft. One might, thus, use a laser to send a spaceship to Alpha Centauri. * An alternate but equivalent form is obtainable from figure 4 by using the Pythagorean theorem: E m v c m c p c m c 2 2 2 2 2 2 4 2 2 2 4 = + = + γ . † National Council on Radiation Protection and Measurements, Report No. 94, Exposure of the Population in the United States and Canada fromNatural Background Radiation (NCRP, Bethesda, MD, 1987), p. 12.