Recession of the spiral nebulæ

Abstract

The very recent and inspiring work of Prof. E. A. Milne on world structure has led us to investigate whether there exists a law connecting the velocity and distance of a particle from an observer which is invariant for the generalised Lorentz transformation. In the usual notation, the only law of the form f(x₁, x₂, x₃, x₄)=0 which is invariant for the infinitesimal Lorentz transformation is known to be x₁² + x² + x₃² + x₄³=0, which gives the propagation of light. Following this, we have investigated whether there exists a law of the form (x₁, x₂, x₃, x₄, u, v, w)=0 which is invariant for the generalised Lorentz transformation; here u=dx₁/dx₄, v=dx₁/dx₄, etc. In its generalised form the transformation is and similarly, v', w' may be obtained. h, a, b are the constants of the transformation. We have found that the following set of equations is the only invariant set of this type, that is, involving both velocities and co-ordinates: The corresponding equations for u', v', x'₁, follow immediately from (1).