## E5 Epstein Diagrams for Energy and Momentum

Why not make our lives easy for once? Take the Epstein diagram for mass and momentum of

**E2**and multiply both scales by c

^{2}! Take a close look and realize that we have got an Epstein diagram for energy and momentum: The rest mass m

_{0}becomes m

_{0}• c

^{2}, and therefore the rest energy E

_{0}, the dynamic mass m

_{v}becomes m

_{v}• c

^{2}, and therefore the total energy E

_{tot}, and on the horizontal axis instead of p / c we have p • c:

φ, sin(φ) and cos(φ) keep their previous meaning. We can however check the relations in this new context:

E_{tot} • sin(φ) = m_{v} • c^{2} • v / c = m_{v} • c • v = m_{v} • v • c = p • c

E_{tot} • cos(φ) = m_{v} • c^{2} • √ = (m_{v} • √ ) • c^{2} = m_{0} • c^{2} = E_{0}

Using the Pythagorean theorem we obtain the important relationship between energies and momentum without any effort:

(E_{tot})^{2} = (E_{0})^{2} + (p • c)^{2} xxxxxxxxxx |
( 2 ) |

Also kinetic energy can easily be made visible: One circumscribes a circle around the origin with radius E_{0} and notes that E_{kin} is the difference E_{tot} - E_{0}.

We should again point out that no particle that has a non zero rest mass can ever completely reach the speed of light. One would have to spend an infinite amount of energy on its acceleration! Imagine the angle φ in the above diagram approaching 90º ever more closely and consider how the total energy of the particle increases as it does so!

Reversing our logic we can also conclude that photons do not have a rest mass, since they always have velocity c. From E_{0} = 0 it follows that E_{tot} = p • c from (2). These light particles carry not only energy, but also a well-defined momentum p = E / c within themselves. This momentum of the light particles produces a certain pressure on an illuminated surface. There is a particularly beautiful illustration of the effect: Astronomers have known for a long time that sunlight exerts a radiation pressure on the tail of a comet. The comet tail always flows away from the sun. As a comet moves away from the sun, it does not pull its tail behind it, but instead the tail flies ahead of it! The dust tail, which consists of heavier particles, appears somewhat more lethargic than the gas or ion tail, which consists mainly of water molecules. The following picture beautifully shows the two components of the tail. It is of the comet Hale-Bopp, as seen in March 1997.

One should not think that the total energy of an object

*always*increases as it moves faster. The important thing is whether energy is input to it or not. Thus the total energy and mass of a streetcar which pulls energy from a power line actually does increase as shown in the Epstein diagram above. However, the situation is different for the battery-operated electric vehicle. It takes the energy needed for its acceleration ‘from its own substance’, and thus converts electro-chemical energy into kinetic energy. Neither its total energy nor its mass increases. Try drawing the appropriate Epstein diagram!

Algebra delivers further relationships between the energies, momentum and relative velocity. We present the most important here:

cos(φ) = √ = m

_{0}/ m

_{v}= m

_{0}• c

^{2}/ m

_{v}• c

^{2}= E

_{0}/ E

_{tot}

sin(φ) = v / c = √( 1 - m

_{0}

^{2}/ m

_{v}

^{2}) = √( 1 - E

_{0}

^{2}/ E

_{tot}

^{2}) = √( 1 - 1 / ( 1 + E

_{kin}

^{2}/ E

_{tot}

^{2}) )

m

_{0}

^{2}= m

_{v}

^{2}– p

^{2}/ c

^{2}

E

_{0}

^{2}= E

_{tot}

^{2}– p

_{2}• c

^{2}

p

^{2}= (m

_{v}

^{2}- m

_{0}

^{2})• c

^{2}

Because of its great importance the equation ∆E = ∆m • c

^{2}is always being derived in new ways. Perhaps the most beautiful and simplest derivation is from Einstein himself in 1946 (!) in his very readable book “Out of My Later Years” [22-121ff]. The derivation is not completely accurate (it uses some approximations), however it requires nearly no mathematics and makes few assumptions. It is warmly recommended to the reader.