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PhilDSP
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After suitable study I'd have to say that it is a misnomer to state that de Broglie's theory is not relativistic. But it does express relativity in such a unique way that it becomes a replacement for Newtonian mechanics within any chosen single inertial frame.
The most basic expression is covered on page 40 of "An Introduction To The Study Of Wave Mechanics". For the Lorentz group:
x = \frac{x_0 + vt_0}{\sqrt{1 - \beta^2}} \ \ \ \ \ t = \frac{t_0 + \frac{\beta}{c}x_0}{\sqrt{1 - \beta^2}} \ \ \ \ \ and \ \ \ \ \ x_0 = \frac{x - vt}{\sqrt{1 - \beta^2}} \ \ \ \ \ t_0 = \frac{t - \frac{\beta}{c}x}{\sqrt{1 - \beta^2}}
An EM or energy wave for system with reference to x_0, y_0, z_0, t_0 depends on time only through a factor cos 2\pi f_0(t_0 - \tau_0)
Switching to system x, y, z, t with \tau_0 = 0 the factor becomes cos 2\pi f_0(\frac{t - \frac{\beta}{c}x}{\sqrt{1 - \beta^2}})
Since f = \frac{f_0}{\sqrt{1 - \beta^2}} and v_{phase} = \frac{c^2}{v} = \frac{c}{\beta}
\beta = \frac{c}{v_{phase}} so that the phase factor is cos 2\pi f(t - \frac{x}{v_{phase}})
What de Broglie has done is to account for all relativistic effects on waves, and subsequently particles, within that single chosen inertial reference. There becomes no further need to consider something in a separate inertial frame. The bulk of his thesis is developed in phase space using mathematical ideas of Hamilton and Jacobi, but he carefully relates the variables in the phase space to variables in Euclidean physical space and time. The concept of a secondary inertial frame is not even especially valid then as his theory applies even to accelerating particles.
Thoughts? criticisms? comments?
The most basic expression is covered on page 40 of "An Introduction To The Study Of Wave Mechanics". For the Lorentz group:
x = \frac{x_0 + vt_0}{\sqrt{1 - \beta^2}} \ \ \ \ \ t = \frac{t_0 + \frac{\beta}{c}x_0}{\sqrt{1 - \beta^2}} \ \ \ \ \ and \ \ \ \ \ x_0 = \frac{x - vt}{\sqrt{1 - \beta^2}} \ \ \ \ \ t_0 = \frac{t - \frac{\beta}{c}x}{\sqrt{1 - \beta^2}}
An EM or energy wave for system with reference to x_0, y_0, z_0, t_0 depends on time only through a factor cos 2\pi f_0(t_0 - \tau_0)
Switching to system x, y, z, t with \tau_0 = 0 the factor becomes cos 2\pi f_0(\frac{t - \frac{\beta}{c}x}{\sqrt{1 - \beta^2}})
Since f = \frac{f_0}{\sqrt{1 - \beta^2}} and v_{phase} = \frac{c^2}{v} = \frac{c}{\beta}
\beta = \frac{c}{v_{phase}} so that the phase factor is cos 2\pi f(t - \frac{x}{v_{phase}})
What de Broglie has done is to account for all relativistic effects on waves, and subsequently particles, within that single chosen inertial reference. There becomes no further need to consider something in a separate inertial frame. The bulk of his thesis is developed in phase space using mathematical ideas of Hamilton and Jacobi, but he carefully relates the variables in the phase space to variables in Euclidean physical space and time. The concept of a secondary inertial frame is not even especially valid then as his theory applies even to accelerating particles.
Thoughts? criticisms? comments?
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