Now we’re ready to explore the significance of Einstein’s famous equation $E=m{c}^2$. FIGURE 36.38 shows two balls of clay approaching each other. They have equal masses and equal kinetic energies, and they slam together in a perfectly inelastic collision to form one large ball of clay at rest. In Newtonian mechanics, we would say that the initial energy 2K is dissipated by being transformed into an equal amount of thermal energy, raising the temperature of the coalesced ball of clay. But Equation 36.42, $E=E_0+K$, doesn’t say anything about thermal energy. The total energy before the collision is $E_{\text{i}}=2m{c}^2+2K$, with the factor of 2 appearing because there are two masses. It seems like the total energy after the collision, when the clay is at rest, should be $2m{c}^2$, but this value doesn’t conserve total energy.

FIGURE 36.38 An inelastic collision between two balls of clay does not seem to conserve the total energy E.

There’s ample experimental evidence that energy is conserved, so there must be a flaw in our reasoning. The statement of energy conservation is

where M is the mass of clay after the collision. But, remarkably, this requires

In other words, mass is not conserved. The mass of clay after the collision is larger than the mass of clay before the collision. Total energy can be conserved only if kinetic energy is transformed into an “equivalent” amount of mass.

The mass increase in a collision between two balls of clay is incredibly small, far beyond any scientist’s ability to detect. So how do we know if such a crazy idea is true?

FIGURE 36.39 shows an experiment that has been done countless times in the last 50 years at particle accelerators around the world. An electron that has been accelerated to $u\approx c$ is aimed at a target material. When a high-energy electron collides with an atom in the target, it can easily knock one of the electrons out of the atom. Thus we would expect to see two electrons leaving the target: the incident electron and the ejected electron. Instead, four particles emerge from the target: three electrons and a positron. A positron, or positive electron, is the antimatter version of an electron, identical to an electron in all respects other than having charge $q=+e$.

FIGURE 36.39 An inelastic collision between electrons can create an electron-positron pair.

In chemical-reaction notation, the collision is

An electron and a positron have been created, apparently out of nothing. Mass $2m_{\text{e}}$ before the collision has become mass $4m_{\text{e}}$ after the collision. (Notice that charge has been conserved in this collision.)

Although the mass has increased, it wasn’t created “out of nothing.” This is an inelastic collision, just like the collision of the balls of clay, because the kinetic energy after the collision is less than before. In fact, if you measured the energies before and after the collision, you would find that the decrease in kinetic energy is exactly equal to the energy equivalent of the two particles that have been created: $\text{Δ}K=-2m_{\text{e}}{c}^2$. The new particles have been created out of energy!

Particles can be created from energy, and particles can return to energy. FIGURE 36.40 shows an electron colliding with a positron, its antimatter partner. When a particle and its antiparticle meet, they annihilate each other. The mass disappears, and the energy equivalent of the mass is transformed into light. In Chapter 38, you’ll learn that light is quantized, meaning that light is emitted and absorbed in discrete chunks of energy called photons. For light with wavelength $\text{λ}$, the energy of a photon is $E_{\text{photon}}=hc/\text{λ}$, where $h=6.63\times {10}^{-34}\text{J}\text{s}$ is called Planck’s constant. Photons carry momentum as well as energy. Conserving both energy and momentum in the annihilation of an electron and a positron requires the emission in opposite directions of two photons of equal energy.

FIGURE 36.40 The annihilation of an electron-positron pair.

If the electron and positron are fairly slow, so that $K\ll m{c}^2$, then $E_{\text{i}}\approx E_0=m{c}^2$. In that case, energy conservation requires

Hence the wavelength of the emitted photons is

This is an extremely short wavelength, even shorter than the wavelengths of x rays. Photons in this wavelength range are called gamma rays. And, indeed, the emission of 0.0024 nm gamma rays is observed in many laboratory experiments in which positrons are able to collide with electrons and thus annihilate. In recent years, with the advent of gamma-ray telescopes on satellites, astronomers have found 0.0024 nm photons coming from many places in the universe, especially galactic centers—evidence that positrons are abundant throughout the universe.

Positron-electron annihilation is also the basis of the medical procedure known as positron-emission tomography, or a PET scan. A patient ingests a very small amount of a radioactive substance that decays by the emission of positrons. This substance is taken up by certain tissues in the body, especially those tissues with a high metabolic rate. As the substance decays, the positrons immediately collide with electrons, annihilate, and create two gamma-ray photons that are emitted back to back. The gamma rays, which easily leave the body, are detected, and their trajectories are traced backward into the body. The overlap of many such trajectories shows quite clearly the tissue in which the positron emission is occurring. The results are usually shown as false-color photographs, with redder areas indicating regions of higher positron emission.