Mass, energy and the dilution of radiation

The numbers are, in Guth's phrase, out of sight. One kilogram of mass is equivalent to

For comparison, the world used about 150 petawatt-hours ($1.5 \times 10^{14}$ kWh) in 2008, the latest year Wikipedia had figures for when Guth looked: an average power of about $1.7 \times 10^{10}$ kW. So converting one kilogram per hour completely into energy would supply about 1.5 times the world's power. Guth's illustration: fill a car's tank with matter, convert it completely, and you could power the world for about two days. In practice nothing comes close: a nuclear reactor converts only a fraction of a percent of its uranium's mass into energy.

Guth quotes, without deriving them, the special-relativity results the course needs. Energy and momentum combine into a four-vector

"Four-vector" means it changes from one inertial frame to another exactly as $x^\mu = (ct, \vec x)$ does. Just as $\Delta x^2 + \Delta y^2 + \Delta z^2 - c^2\Delta t^2$ is the same in every frame, so is

The value follows from the simplest frame, the object's rest frame, where $\vec p = 0$ and $E = m_0c^2$. $m_0$ is the rest mass. [On the board an extra factor of $c$ first appeared inside the bracket, $-(m_0c^2)^2$; a student caught it.] In terms of the velocity, with the same $\gamma$ as in time dilation,

At rest, $E_0 = m_0c^2$, which is where we started.

Why redefine energy and momentum at all? Einstein noticed that if you keep Newton's definitions but use Lorentz transformations between frames, energy and momentum conserved in one frame are not conserved in another. To keep conservation as a universal law, both had to be redefined: the new versions agree with Newton's at low speed and are conserved in every frame. Since $p^\mu$ transforms linearly, like $x^\mu$, a change $\Delta p^\mu$ is also a four-vector, and a four-vector that vanishes in one frame vanishes in all.

A hydrogen atom shows $E = mc^2$ at work. Bringing an electron and a proton together from far apart releases 13.6 eV. That energy leaves the system, so the atom has less energy than the separate particles — and therefore less mass:

Guth defines the relativistic mass of any system as its energy divided by $c^2$; for a single particle it is $\gamma m_0$, growing with speed. Many books avoid the idea: it is just $E/c^2$ (in units with $c = 1$, just the energy), students may confuse it with rest mass, and it does not obey $F = ma$. Guth keeps it as a convenient abbreviation because of one property that matters for cosmology: it is what gravitates.

The gravitational field of a single fast-moving star is complicated. You can get it by transforming the Schwarzschild metric to a moving frame, and the result is not isotropic, just as the electric field of a moving charge isn't. But for a gas of particles moving every which way — like the early universe — seen from outside as a box, the directional effects average out, and the box's mass is the sum of the relativistic masses of its particles. Divided by the volume, that is the mass density that produces gravity. In general relativity's own language: the energy density is the source of gravity, and it equals this mass density times $c^2$.

For electromagnetic radiation, then, the mass density is

where $u$ is the energy density of the radiation. Photons have zero rest mass, meaning there is no lower limit to a photon's energy (rest mass sets a particle's lowest possible energy, when it is at rest, and a photon is never at rest). Setting $m_0 = 0$ in the energy formula:

This is already true for classical electromagnetic waves, so it had better be true for the photons they are made of. Because each photon carries momentum, photons bouncing around a box push on its walls: a photon gas has pressure, which the next page calculates.

Put a free photon gas into an expanding universe. Two things happen.

Dilution. Photons are not created or destroyed, so their number density falls as the volume grows, exactly as for matter: $n_\gamma \propto 1/a^3(t)$. ($\gamma$ here is just the usual label for photons of any frequency, from the term "gamma ray".)

Asked about absorption and pair creation, Guth notes that photons can be absorbed and two photons can make an electron–positron pair, though with a rather small cross section. But absorption is a very minor effect in the early universe, where there is little to absorb them, and pair creation and annihilation quickly reach thermal equilibrium, with as many pairs made as destroyed. The universe stays close to thermal equilibrium as long as it cools slowly, with important exceptions that come later.

Redshift. Each photon's frequency falls as the universe expands:

and a photon's energy is $E = h\nu$, with Planck's constant $h = 4.136 \times 10^{-15}$ eV·s. So each photon's energy falls as $1/a$. Together:

(Classical Maxwell theory gives the same answer, with more work.) Matter's density falls as $1/a^3$; radiation's falls one power faster, because of the redshift. Run the film backwards and the ratio of radiation to matter grows as $1/a$. At some early time the two were equal, and before that radiation dominated. When that was is the subject of the next page.

• $E = mc^2$: one kilogram is $9 \times 10^{16}$ J $= 2.5 \times 10^{10}$ kWh, about 1.5 hours of 2008 world power use.
• Energy and momentum form a four-vector $p^\mu = (E/c, \vec p)$ with invariant $|\vec p|^2 - E^2/c^2 = -(m_0c)^2$; $\vec p = \gamma m_0\vec v$, $E = \gamma m_0c^2$.
• Binding energy has mass: hydrogen is lighter than $m_p + m_e$ by $13.6\ \text{eV}/c^2$.
• Relativistic mass $E/c^2$ is what gravitates for a gas; so radiation has mass density $\rho = u/c^2$.
• Photons: $E = c|\vec p|$, $E = h\nu$. Number density $\propto 1/a^3$ and energy $\propto 1/a$ give $u_\gamma \propto 1/a^4$, against $1/a^3$ for matter.