A Simple Derivation of Einsteins Mass-Energy Equation

P.S.C. Bruskiewich

Mathematical-physics, University of British Columbia, Vancouver, BC

The Einsteins Mass-Energy Equation 2E mc= can be derived in a number of ways.

One simple derivation was shared with the author by Dr. F. A. Kaempffer and Dr. George

Volkoff during the course of a pleasant conversation in Dr. Kaempffers office in 1984.

1.0 A Mass absorbs Light Energy at Rest

The Mass-Energy Equation 2E mc= , a ubiquitous equation in modern physics, can be

derived in a number of ways. One simple derivation comes from Einstein himself.

Consider a mass M at rest on a frictionless table. Imagine two packets of light energy

being absorbed by the mass in such a fashion that no net momentum is added to the mass

M (refer to Fig. 1: The Mass at Rest Absorbing Two Packets of Light Energy).

Fig. 1: The Mass at Rest Absorbing Two Packets of Light Energy

Each packet delivers momentum p to the mass M, however the two impulses are equal

and opposite. The momentum delivered during a small time interval t comes from a

beam of light of length x c t = .

From Maxwells equations, the momentum delivered by each of the wave packets is

where A is the area of the light beam, P is the light pressure, E is the energy in the light

beam and V is the volume of the light beam (V A x Ac t= = ).

While at rest on the frictionless table, the mass M absorbs a total energy of E. Its net gain

of momentum is zero. This is expected by the Conservation of Momentum. This simple

derivation is interesting in its use of dimensional analysis, a technique not often used in

contemporary times, but was of second nature during the first half of the 20th century.

2.0 A Mass absorbs Light Energy While in Motion

Now let a moving observer approach the table with speed v. By the Special Theory of

Relativity the observer can consider himself at rest with the table moving towards him

with velocity v. It appears now that the mass M is approaching him with velocity v and

momentum Mv.

The observer now sees the two light energy packets approaching the mass M at a slight

angle (refer to Fig. 2: The Mass in Motion Absorbing Two Packets of Light Energy).

( )p F t PA tE EA t A t

V A x

EA t

cA t

E

c

= =

= =

=

=

Fig. 2: The Mass in Motion Absorbing Two Packets of Light Energy

In the observers frame the packets of light appear to approach the mass M at a slope of

v/c. In his frame each packet have momentum with a component in the direction of

motion of

By the conservation of momentum, the total apparent momentum before the absorption

should be equivalent to the total apparent momentum after the absorption of the light,

where an apparent mass m is now added to the initial mass M. From this we find

Acknowledgements:

The author would like to posthumously thank Dr. F.A. Kaempffer and Dr. G.M. Volkoff

for their discussion and encouragement in the writing of this short article. This paper was

originally written in 1984 with their kind assistance. The manuscript remained lost in

the authors papers for several decades, only recently rediscovered and published.

Patrick Bruskiewich 2103

1

2v

E vp

c c

=

( )122

E vMv M m v

c c

+ = +

2

2

Ev mv E mc

c

= =