March 14: let’s celebrate Einstein’s birthday

Posted 10/03/2023 by Ian Galloway

Mrelativistic= If there is one equation known by the public, it is E=mc2. Created in 1905, the theory of Special Relativity has transformed our understanding of the Universe and the way it works. In that same year, 1905, Einstein explained the photoelectric effect by declaring that energy was quantized, for which he was awarded a Nobel Prize.

Cat in the box
Einstein remained however unhappy about quantum physics for the rest of his life. And he never fully accepted the broad consensus about the nature of quanta, in particular the notion of entangled particles put forward by Schrӧdinger in his famous ‘cat in a box’ thought experiment.

Evolution of Physics
Some books written by great STEM artists have achieved an everlasting impact upon society. One of these is The Evolution of Physics by Albert Einstein and Leopold Infeld, a very easy read for anyone and in my opinion a required text for college Physics students. I am lucky to have a first edition printed in 1938. Just 12 years after Schrӧdinger launched his Wave Mechanics and a mere 3 years after Einstein Podolski and Rosen declared that ‘entanglement’ was impossible. In The Evolution of Physics Einstein writes at the end in the chapter on quanta,

“There is no doubt that quantum physics explained a rich variety of facts…But there is also no doubt that quantum physics must still be based on two concepts: matter and field. It is in this sense a dualistic theory and does not bring our old problem of reducing everything to the field concept even one step nearer realization.”

Today Quantum Field Theory in which particles of matter have been reduced to disturbances of the field, is one of the most successful physical theories of all time. Last year as we know Alain Aspect and others were awarded the Nobel Prize in Physics for their work on entangled particles!

Back to Special Relativity and energy!
Just as most people have heard of E=mc², most people have no knowledge of the two principles on which Special Relativity is based.

  • The first being that it is not possible to determine absolute velocity through space. No reference frame can be absolutely stationary. The concept of motion is that it is strictly relative. I can only determine my motion relative to another object or system.
  • The second is that the speed of light is the same (constant) as measured in all reference frames moving with relative motion one to another. No matter how I measure the speed of light, whether I move towards or away from a light source, the result is always the same.

Special relativity in school
Yet at least one school philosophy course1 taught in France refers to the two principles of special relativity. Advanced placement courses for American universities have optional units on special relativity. I can find very little material on relativity in the curriculum for schools in Europe, this even though the theory is more than 100 years old and very well tried and tested. It is the bedrock of modern science and young people studying science at school have, I believe, a right to have some knowledge of it.

It would be possible to simply provide the experimental fact that the speed of light is constant and invite students to struggle with the cognitive conflict that that presents! There is no need to explain at this stage. There is no need to derive the equations which transform length and time from one reference frame to another, although this is very straightforward. But as a result of time slowing down in frames moving relative to mine and lengths contracting in the direction of motion, mass also undergoes a transformation between frames moving with a relative velocity of v such that

................. mass transformations equation

Mrest is the mass in my frame or rest frame. Mrelativistic is the same object but in a frame moving relative to mine at speed v.

Setting M0 = 1 kg and c2 = 9 *10100 Km2s-2 students can immediately enter this into a graphing calculator and adjust the window to obtain the result below. Using the trace facility, the plotted function shows (bottom right) that at about one tenth light speed masses increase by around 1%. It is clear that as the speed increases further, the mass increases asymptotically to the speed of light.

It can be clearly seen that as an object approaches light speed it simply increases in mass rather than going any faster. Students will be able to apply the binomial theorem to the mass transformation equation. Ignoring terms in v^4/c^4 and higher powers for low relative velocities,

Multiplying both sides by c2 we have
Students will recognise the classical kinetic energy and will realise that each term is an energy term.

The kinetic energy is the mass increase, ∆M times c2. This is the origin of the E = mc2 equation and students may well ask if masses increase when an object is heated or placed in a higher gravitational potential. The answer is yes, but clearly the increase is far too small to be measured on even the most sensitive balance.

  • Question 1. The vast amount of radiation pouring from the Sun results in the Sun losing mass at four million tons per second as a result of nuclear fusion. Students could be supplied with the Sun’s current mass and estimate the percentage mass loss after say a million years.
  • Question 2. The mass of a proton accelerated in the storage ring at CERN is five thousand times greater than its rest mass. If one bunch of protons in the CERN rings contains 100 billion protons (they would be invisible) students could estimate the temperature rise of a 100 kg block of copper if the protons are suddenly brought to rest in it.

1 Rabaudy C., Rolland B. Sophia/ La connaissance Recueil de textes philosophiques. Hatier, Paris. p 331.
Image of Einstein from Wikipedia is in the public domain. Photo E=mc2 courtesy of Unsplash.