Appendix A: Einstein’s Equations

A number of concepts included in the H2 syllabus are closely related to, or “discovered” through Einstein’s special theory of relativity. While the H2 syllabus only requires students to apply these concepts, I guess some students may be interested in their derivation as well.

But we’ll still skip all the relativity stuff, and start from

\displaystyle K=\frac{{m{{c}^{2}}}}{{\sqrt{{1-{{v}^{2}}/{{c}^{2}}}}}}-m{{c}^{2}}\cdots \cdots \cdots (1)

This is the equation for the kinetic energy K of a particle, derived from work-energy theorem (using relativistic force \displaystyle F=\frac{{ma}}{{\sqrt{{1-{{v}^{2}}/{{c}^{2}}}}}}). [1] If you do a binomial expansion, equation (1) will collapse to become the familiar \displaystyle K=\frac{1}{2}m{{v}^{2}} for \displaystyle v<<c. Hopefully this bolsters your trust in the equation.

Anyway, Einstein interpreted the first term in the equation (1) to represent the total energy E of the particle, and the second term to be energy the particle has even when it is at rest (so that K is the energy due to motion only). Re-arranging the equation, we have

\displaystyle K+m{{c}^{2}}=E=\frac{{m{{c}^{2}}}}{{\sqrt{{1-{{v}^{2}}/{{c}^{2}}}}}}\cdots \cdots \cdots (2)

The energy mc2 comes to be known as the rest energy. It is from here that we get all that mass-equivalence stuff.

But this is not all that Einstein milked from this equation.

He wrote equation (2) as

\displaystyle \frac{E}{{m{{c}^{2}}}}=\frac{1}{{\sqrt{{1-{{v}^{2}}/{{c}^{2}}}}}}\cdots \cdots \cdots (3)

Then, from relativistic momentum \displaystyle p=\frac{{mv}}{{\sqrt{{1-{{v}^{2}}/{{c}^{2}}}}}}, he wrote

\displaystyle \frac{p}{{mc}}=\frac{{v/c}}{{\sqrt{{1-{{v}^{2}}/{{c}^{2}}}}}}\cdots \cdots \cdots (4)

Then by doing \displaystyle {{(3)}^{2}}-{{(4)}^{2}}, he got rid of v in the equations and arrived at

\displaystyle {{E}^{2}}={{(pc)}^{2}}+{{(m{{c}^{2}})}^{2}}\cdots \cdots \cdots (5)

For a particle at rest (\displaystyle p=0), the equation becomes

\displaystyle E=m{{c}^{2}}

So \displaystyle E=m{{c}^{2}}, the most famous physics equation, is actually a special case of the more general equation (5).

For the other special case of massless particles (\displaystyle m=0),equation (5) becomes

\displaystyle E=pc

This suggests that a massless particle may have momentum despite not having any mass!

For photons which has energy \displaystyle E=\frac{{hc}}{\lambda }, we obtain

\displaystyle \frac{{hc}}{\lambda }=pc\text{ }\Rightarrow \text{ }p=\frac{h}{\lambda }

This is where we obtain the formula for the momentum of a photon.

All de Broglie did was to assume that \displaystyle p=\frac{h}{\lambda } is equally valid for even massful particles travelling at non c speed, and voilà, de Broglie got his namesake formula (and a Nobel Prize).

\displaystyle \lambda =\frac{h}{p}

[1] You’ll see the Lorentz factor \displaystyle \gamma =\frac{1}{{\sqrt{{1-{{v}^{2}}/{{c}^{2}}}}}} all over the place. Don’t be too intimidated. Just take it as the factor by which stuff gets “distorted” when particles travel at speed close to c.

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