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Nothing goes faster than light

Space and time · level 7, lesson 5 of 6

A question to start

A ship moving at \(0.6c\) fires a probe forward at \(0.6c\). Does the probe go at \(1.2c\)?

The idea

Two things stop anything with mass from reaching \(c\).

Speeds do not simply add. If a ship moves at \(u\) and fires something forward at \(v\) (measured on the ship), an outside observer sees

\[w = \frac{u + v}{1 + uv/c^2}.\]

For everyday speeds the bottom is almost exactly 1, and the speeds just add. Near \(c\) the bottom grows, and \(w\) always stays below \(c\).

Energy grows without limit. A body of mass \(m\) moving with Lorentz factor \(\gamma\) has kinetic energy

\[E_k = (\gamma - 1)\,mc^2.\]

As \(v \to c\), \(\gamma\) grows without bound, so the energy needed does too. Reaching \(c\) would take infinite energy.

Light itself has no mass, and always moves at \(c\).

cspeedenergy needed

Worked examples

1. The probe. \(u = v = 0.6c\). \(w = \frac{1.2c}{1 + 0.36} \approx 0.88c\). Not \(1.2c\).

2. Everyday check. \(u = v = 30\) m/s. \(uv/c^2 = 900 \div (9 \times 10^{16}) = 10^{-14}\). The bottom is 1 to 14 decimal places: the speeds just add.

3. Energy for 1 kg. At \(0.8c\), \(\gamma = 1.667\): \(E_k = 0.667 \times 1 \times (3 \times 10^8)^2 \approx 6 \times 10^{16}\) J. At \(0.99c\), \(\gamma \approx 7.09\): about \(5.5 \times 10^{17}\) J, nine times more, for less than a quarter more speed.

Try it

  1. Why can speeds not simply be added near \(c\)?
  2. Add \(0.5c\) and \(0.5c\) using the rule.
  3. What happens to the kinetic energy needed as \(v\) gets close to \(c\)?
  4. Add \(c\) and \(0.5c\) using the rule. What does it show?
Answers
  1. The rule has a bottom line of \(1 + uv/c^2\), which keeps the result below \(c\).
  2. \(\frac{1.0c}{1 + 0.25} = 0.8c\).
  3. It grows without limit.
  4. \(\frac{1.5c}{1 + 0.5} = c\). Light stays at \(c\) whatever the source does.

Common mistake

"If you push long enough, you get past \(c\)." Each push adds less and less speed. The energy goes into ever larger \(\gamma\), not past the limit.

Where it leads