Download
1 / 5
50 likes | 133 Views
One can find resemblance between Poincaré’s theory and Einstein’s one only in the lectures published and edited posthumously. 1) A theory of the Lorentz transformations : This theory was presented only in the 1911 – 1912 lectures. Poincaré wrote the apparent time: (1) t’ = Kt + C .
E N D
One can find resemblance between Poincaré’s theory and Einstein’s one only in the lectures published and edited posthumously. 1) A theory of the Lorentz transformations: This theory was presented only in the 1911 – 1912 lectures. Poincaré wrote the apparent time: (1) t’ = Kt + C. t’ is the apparent time; t is the true time; K and C are constants dependent on ε (the velocity of translation), and C also depends on the position of the point at which the apparent time is evaluated. The velocity of light c = 1, ε = -v.
Pomey, the editor of the 1911 – 1912 lectures: “The reasoning and calculation were given very rapidly by M. Poincaré”. Pomey completed Poincaré’s derivations: (2) t’ = t√(1 – ε²) + εx’. Pomey: “Compare this equation with the Lorentz transformation formulas” (in Poincaré’s notation): (3) x’ = k(x + εt), y’ = y, z’ = z, t’ = k(t + εx), wherek = 1/√(1 – ε²). In (2) x’ is the first Lorentz transformation. Pomey noted that equations (3) lead, by eliminating x, to (2).
2) Constancy of the velocity of light: Poincaré’s 1905 quadratic form followed the Lorentz transformations (1906 – 1907, 1911 – 1912): x'² + y’² + z’² + t’² = x² + y² + z² – t². 1906 – 1907: Poincaré demonstrated using the group properties of the Lorentz transformation that since the true velocity of light (in the ether) was equal to 1, the apparent velocity of light was also equal to 1. The velocity of light is constant in the ether and in all moving systems. Poincaré mathematically proved that the Lorentz transformations form a group, and concluded his entire derivation by saying: “This verifies the principle of relativity”. Poincaré proved the constancy of the velocity of light and then verified the principle of relativity.
3) Longitudinal and transverse masses: After presenting the above derivations in the 1906 – 1907 lectures, Poincaré obtained the longitudinal and transverse masses of the electron. This derivation cannot be found in Poincaré’s 1905 “Dynamics of the Electron”; neither can it be found in Poincaré’s 1908 paper and 1911 – 1912 lectures. But it resembles Einstein’s 1905 theory of the slowly moving electron. Consider the usual law for electrodynamics with constant mass: (7) md²r/dt² = eE, m is the electron's mass, e - its charge, x, y, z - its coordinates (r) and t its time relative to a system at rest. (7) is valid when a magnetic field does not exist.
In the general case: (8) d²r/dt² = aE + b(v/c × B), a and b are coefficients dependent on the velocity for high velocities: (9) a = b = e/m ∙ 1/β. Then: (10) d²x/dt² = a(1 – v²/c²)Ex = e/m ∙ 1/β³ Ex = dEx, d²y/dt² = aEy = e/m ∙ 1/βEy. This gives two coefficients: (11) a = e/m√(1 – v²/c²), d = e/m√(1 – v²/c²)³, from which we arrive at the equations: (12) Ml = m/(√1 – v²/c²)³, Mt = m/(√1 – v²/c²).
