F2 The Relativistic Corrections
What renovations does the SRT make on the building of classical physics, in order to eliminate the internal contradictions between mechanics, the relativity principle and the theory of electromagnetism? It is actually few - yet they are very fundamental:
- Time measurements are always relative to a coordinate system and not universal
- Length measurements are always relative to a coordinate system and not universal
- Inertial mass is (likewise) dependent on relative velocity
- Input of energy means also an input of inertial mass
The details to the renovations were precisely developed in the preceding chapters. What however are the consequences for the 4 most important quantities of physics and their associated conservation laws?
- The conservation law of electrical charges goes unchanged
- The conservation law of inertial mass merges with that of energy to give a single conservation law, in which each energy quantity corresponds to a given inertial mass and vice-versa
- The conservation law of momentum goes unchanged, whereby the momentum is to be computed as m(v) • v, thus making the mass dependent on relative velocity
What remains of the three Newtonian laws? Interestingly enough all three remain valid, only the relativistic specification of the momentum must be accounted for. In particular F = dp/dt remains unchanged.
And how is it with the action of forces and the associated force fields? Are there still three? Here, the answer is a “yes, but…”. Since the STR confers Maxwell’s theory with an unqualified validity in all inertial systems, it is no surprise that Coulomb’s law and Lorentz’s law remain valid. There is not the slightest change in the production of the corresponding fields. The 'but' refers to the production of the gravitational field: The instantaneous action at a distance of masses in Newton's gravitation law contradicts the STR result that c is a fundamental speed limit for mass, energy and information transfers. By the way, Newton also found this action at a distance to be somewhat uncanny. At the end of his great work he writes:
“Thus far I have explained the phenomena of the heavens and of our sea by the force of gravity, but I have not yet assigned a cause to gravity. Indeed, this force arises from some cause that penetrates as far as the centers of the sun and planets … and whose action is extended everywhere to immense distances, always decreasing as the squares of the distances. … I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and I do not ‘feign’ hypotheses. … And it is enough that gravity really exists and acts according to the laws that we have set forth and is sufficient to explain all the motions of the heavenly bodies and of our sea.” [03-943]
Einstein started working in 1906 to integrate gravitation into the STR. He found a point of attack in 1907 with the equivalence principle. He still needed years of hard work and the assistance of some mathematician friends, before he could submit at the end of 1915 the equation, which encompasses space, time and gravitation and solves this problem. He later called the equivalence principle “the happiest thought of my life”. More concerning this follows in the next section, which coincidentally has the letter G (like gravitation) assigned to it.
We still want to turn to the new conservation law, which replaces the separate conservation laws for mass and energy. It can be formulated alternatively as the conservation law for the entire inertial mass in a closed system, whereby all amounts of energy ∆Ei with their contribution ∆Ei /c2 to the total mass must be taken into account. Or alternatively seen as a conservation law for total energy, whereby all masses mi with their contribution mi • c2 to the total energy are accounted for. Usually this second representation is preferred. I would like to illustrate the two equivalent possibilities with an example:
We imagine an uncharged capacitor with rest mass m0. What is its contribution to total mass, if it is first charged and then accelerated? During charging the energy ∆E = 0.5 • C • U2 is supplied to it, and therefore its rest mass increases by the amount ∆E / c2. This increased mass must still be divided by the root term, when the capacitor is accelerated. This contributes (m0 + ∆E / c2) / √ to the total mass. The contribution to the total energy is computed as follows: There is the rest energy m0 • c2, and then the energy ∆E = 0.5 • C • U2, supplied by charging of the capacitor at rest, and finally the kinetic energy due to the acceleration. However the acceleration is performed on the already somewhat heavier charged capacitor and therefore we must use (m0 + ∆E / c2) • c2 • (1 / √ - 1) for the kinetic energy. In total we have m0 • c2 + ∆E + (m0 + ∆E / c2) • c2 • (1 / √ - 1) = (m0 + ∆E / c2) • c2 / √, which corresponds exactly to the total mass multiplied by the factor c2 !
It is rather arbitrary, but not wrong, if one still calls this comprehensive conservation law the ‘conservation law of total energy’. The designation ‘conservation law of total mass’ would be just as correct. The two balances differ only by a factor c2 on each side of the equals sign:
∑ Etot,i (before) = ∑ Etot,j (after) or ∑ mv,i (before) = ∑ mv,j (after)
So much for physics from the eagle perspective. The following sections give examples to these (still only) 3 conservation laws. They also show that the world cannot be understood without STR.
Caricature by Sidney Harris