Relativity Lite

Mixmaster Universe | 27 The new triangle has the same shape as the old one. They are called similar triangles, having the same angles between the sides. Then the hypotenuse of this four-dimensional quantity has length mc mc mc mv 2 2 1 2 2 1 2 γ γ ≅ = + , where we have approximated γ = t/τ by the first-order approximation γ 1 from the first part of this chapter. We see that the second term is precisely the kinetic energy we just defined in the last section, so even though we do not at this point know the meaning of the first term, it must be some kind of energy too, or it would not belong next to the KE term. So we have labeled the hypotenuse as an energy E . The horizontal leg of the new triangle has the dimensions (units) of the Newtonian mo- mentum ( m v ) times c , since γ = t/τ is a dimensionless number like 1.25. The new, stretched vertical leg is a relativistic invariant, mc 2 . In the first section of chapter 1, we measured the relative lengths of the sides of the origi- nal triangle to get a relationship between times. We now measure the relative lengths of the sides of the new triangle to get a relationship between energies. If we slow the rocket down so that v is very small, the original triangle is tall and skinny. Since we are stretching each side by the same proportion, keeping the angles the same, the new triangle is also tall and skinny. If we go to the rest frame of the particle, where v = 0 , so that the momentum is zero, r = v t → 0 d = c t H = c τ v t × m c / τ = momentum × c → 0 c τ × m c / τ = m c 2 c t × m c / τ = E Figure 4. A version of figure 3 for when v is very small. we find that the hypotenuse, labeled E , is the same length as the vertical leg, labeled mc 2 . This picture expresses Einstein’s most famous expression, E mc 0 2 = , where the subscript 0 is a reminder that this expression is only valid for v = 0 . This equation (or equivalently, figure 4) expresses the revolutionary concept that objects have energy even when they are at rest, called their rest energy .