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Einstein Equation Derivation

May 11, 2013

Introduction to the derivation of Einstein equation

Albert Einstein was an outstanding physicist and one of the famous scientist of 20th century for his theory of relativity. He is remembered for his contribution and work on Special Relativity (1905) and for his work on General Relativity (1915) in addition to his works on the photoelectric effect and Brownian motion.
Albert Einstein

Albert Einstein

Einstein’s equation, E= mc2 , describes the equivalence of mass and energy. It is arguably the most famous equation in physics. Einstein used the postulates of special relativity and other laws of physics to show, E= mc2

Proposition. (Mass-energy equivalence) If a body remains at rest and emits a total energy of E, then the mass of the given body decreases by `E/c^(2)` .

Derivation of Einstein equation

Suppose a motionless box being floated in deep space. A photon gets emitted inside the box and moves from left. As the systems momentum is being conserved, the boxes recoil towards its left due to the emission of a photon. After some time, the photon collides with the opposite side of the box, and transfers whole of the momentum. The total momentum remains conserved. Einstein resolved it by stating that there should have been a ‘mass equivalent’ to energy of a photon.

For momentum of a photon, Maxwell’s expression will be used for momentum of a wave which is electromagnetic, and having a given energy. Let the energy of photon be E and that of the velocity of light be c, the momentum of the photon is:

                                               ` P_(PHOTON)= E/ C`

The box having a mass m, will be recoiled gradually in the opposite direction as of the photon, with speed v. so the box’s momentum would be

                                                    ` P_(BOX)= Mv`

The photon at time, Δt, reaches the later side of the box. Let the distance be Δx. therefore the speed that the box attains is

                                                     ` V= (Delta x)/(Delta t)`

By law of conservation of momentum,

`M (Delta x) / (Delta t) = E/ C`  

If L is the length of the box,

                                                     ` Delta t= L/c`

Substituting these values in the law of conservation of momentum and rearranging:

                                                  `M_(Deltax)= (EL)/c^(2)`

If the photon has mass, m. If the position of the box is x1 and the position of photon is  x2, then the value of centre of mass for whole system:

There should be no change in centre of mass. So,

`(Mx_(1) + mx_(2))/(M+m) = (M(x_(1) – Delta x) + mL)/(M+m)`

The photon initiates from the left of the box, i.e. x2 = 0. So, by rearranging and simplifying the equation,

`mL = M Delta x`

On Substituting:

`mL = (EL)/c^(2)`                                 (1.10)

After Rearranging we get the final equation:

`E= mc^(2)`

Conclusion for the derivation of Einstein equation

As we have derived the Einstein’s equation, we conclude that, if the body remains at rest with zero mass, then it must have zero energy, this results the famous formula, .

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