Why can’t we travel faster than the speed of light? Einstein explained this pheonmenon through his famous equation

(1)m=m01(vc)2,

which is the last of his four celebrated works in 1905 (Albert Einstein’s Year of Miracles) titled Does the inertia of a body depend on its energy content?. This paper entails arguably the most famous formula so far: Mass-Energy Equivalence

E=mc2.

From (1), when the velocity of a particle approaches c, the relative mass m will goes to . Therefore any particle with m0>0 can never exceed the speed of light.

For the rest of the post, we use high school physics to proof these famous results!


1. Review from high school physics materials

  • Momentum Conservation Principle (动量守恒定律): For a system (with two objectives) has 0 momentum, it always holds: (2)0=P1P2=m1v1m2v2, multiply both sides by time t, we further have (3)0=m1S1m2S2.
  • Electromagnetic waves: (4)E=Pc
  • Theorem of Momentum: (5)dW=Fdx=dE,F=dPdt.

2. Proof of E=mc2

Einstein’s Thought Experiment. Imagine there is a box with length L, a photon with energy E goes from one side to the other side. Consider the system of photon and the box.

By (4),

(6)P1=Ec

then by (2),

(7)P2=P1=Ec.

Then by the definition of momentum (2)

(8)vbox=P2m2=Em2c.

On the other hand, for photon s1L, then for box we have

(9)s2=vboxt=Em2ct=Em2cLc

Let the mass of photon be m, then by Momentum Conservation Principle,

(10)mLm2ELm2c2=0

Which gives

E=mc2.

This can be generalize to not only photon but all particles!

Quite easy! Right?


Proof of m=m0/1(vc)2

Recall E=mc2, P=mv, hence

(11)EP=c2vE=Pc2v.

Now by (5),

(12)dE=Fdx=dPdtdx=vdP, multiply above by (11), then

EdE=Pc2vvdP=Pc2dP.

Take integral, we obtian

E2=c2P2+const.

When P=0, E=E0=m0c2. Therefore we already derived

(13)E2=c2P2+m02c4

In particular, for photon, by (4) E=Pc, which implies the invariant mass (光子的静止质量) m0=0! This simly says: For any pariticle that has positive invariant mass, it cannot achieve the speed of light!

Finally, use (11), we have P=Evc2, plug in (13) to obtain

E2=E2v2c2+m02c4E=m0c21v2c2.

Again, use E=mc2, we obtain

m=m01(vc)2.

That’s it! Now you know why c is the fastest speed we can get!