Einstein’s energy and space isotropy

Abstract

AbstractIn this note, we derive an extension of the conventional Einstein variation of mass formula with a specific expression arising from a Lorentz invariant equation for the energy rate $$\hbox {d}e/\hbox {d}p$$ d e / d p where $$e = mc^2$$ e = m c 2 is the particle energy, $$p = mu$$ p = m u the particle momentum and u the velocity. This is the simplest one-parameter Lorentz-invariant extension of the Einstein mass–energy relation. Implicit in the new expression is space–time anisotropy such that the particle has different rest masses in the positive and negative x directions. While numerous experiments have been undertaken aimed at testing such hypothesis, and all indicate the veracity of the assumption that space is isotropic, nevertheless since it is generally believed that black-holes exist at the centres of galaxies, space must be intrinsically anisotropic in some sense. Finally, we note a very curious connection with both the conventional Einstein energy–mass expression $$e = e_0/ (1 - (u/c)^2)^{1/2}$$ e = e 0 / ( 1 - ( u / c ) 2 ) 1 / 2 and the new expression derived here with certain singular integral equations usually associated with aero-foil problems, fluid mechanics and punch problems in elasticity, and that this connection is not some vague intangible relationship, but involves an exact correspondence.