By John C. Baez and Emory F. Bunny
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Extra resources for The Meaning of Einstein's Equation
Example text
E. in this subcase (unlike the previous one, see Eqs. 25)) one can state a proportionality relation by an overall factor (even if dependent on the metric coefficients) between uiDSR,II (E) and |uDSR,II (E)|∗ . We have therefore shown that the two different procedures of directional separation lead to two different mathematical definitions of maximal causal velocity, an isotropic (w, Eq. 26)) and an anisotropic (u, Eq. 24)) one. The choice between them must be done on a physical basis (see Subsec. 3).
M. d Let us clarify the latter statement. The carriers of a given interaction propagating with the speed u typical of that interaction are expected to be strictly massless only inside the space whose metric is determined by the interaction considered. A priori, nothing forbids that such “deformed photons” may acquire a nonvanishing mass in a deformed Minkowski space related to a different interaction. This might be the case of the massive bosons W + , W − and Z 0 , carriers of the weak interaction, which would therefore be massless in the ˜ (ηweak (E)) related to the weak interaction, but would acquire a space M mass when considered in the standard Minkowski space M of SR (that, as already stressed, is strictly connected to the electromagnetic interaction,e ruling the operation of the measuring devices).
8) Therefore, the generalized expression of the momentum fourvector (in the case of spatial isotropy) is pµ = m0 V µ (E) = (m0 γ˜ (E)u(E), m0 γ˜ (E)v k ). 31) for the interaction considered. e. at energies such that vi ui (E). 10) yields the following NR expression of the energy corresponding to the given interaction: ENR = mu2i (E) = mc2 b20 (E) . g. in the xy plane, at angles θ, θ in frames K, K ) with dispersion relation u = λν = λ ν , where ν, ν are the wave frequencies in K, K . 13) Relativistic Kinematics in a Deformed Minkowski Space 39 • Aberration law: tgθ = sin θ .