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In week 2 and 3 we saw that particles that move with speed v have energy and momentum due to their movement. (We are only interested in the energy of movement, the kinetic energy.)

Momentum p=mv and kinetic energy K= \frac12 mv^2

We have also seen that the momentum and the energy are conserved .

Relativistic Energy-Momentum ** do not memorize the equations here. **

Now that we have learned about the special nature of spacetime, how should we define v in these formulas? Normally the speed is

v = distance over time

but which time should we be using? What's more, with the speed of light being always constant, conservation of momentum and energy does not seem to work anymore.

Einstein was facing these tough questions and he found a solution. He found that there was natural definition for momentum and energy such that they would be conserved like before and such that they work perfectly fine with a speed limit. He proposed that the correct formula for momentum (no need to remember that one) is:

p = \gamma m v

where \gamma = \frac{1}{\sqrt{1-v^2/c^2}} which is a factor always greater than 1. As you get closer and closer to the speed of light the factor \gamma grows bigger and so does the momentum!

For the energy, Einstein found the following relation, which should remind you of the Pythagorean theorem:

E^2 = (pc)^2+(mc^2)^2

(this p is \gamma m v now)

• M: This is the most famous equation in physics.
• S: Hmmm, I thought it was simpler than that.
• M: Let's see. If you want the energy of a particle that is not moving, then v=0 and p=0 . Then the formula becomes E=mc^2
• S: Now that's a famous equation!
• M: Indeed. What Einstein found is that even when a particle is not moving it has energy proportional to its mass. This means that mass is just a certain form of energy and one can convert between mass and energy!
• M: But it is really just a special case, the general equation is the one above.

External Resources

Minute Physics has a couple of nice videos regarding special relativity; have a look!

• Pythagorean Theorem for Energy
• Special Relativity