Two events have coordinates in frame : event at m and event at m

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Special relativity: spacetime and kinematics · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch Tensors in flat spacetime: index notation and covariant field equations

Problem

Two events have coordinates in frame SS: event AA at (ct,x,y,z)=(0,0,0,0)(ct,x,y,z) = (0,0,0,0) m and event BB at (ct,x,y,z)=(5.0×108, 3.0×108, 0, 0)(ct,x,y,z) = (5.0\times10^8,\ 3.0\times10^8,\ 0,\ 0) m, so Δxμ=(5.0×108, 3.0×108, 0, 0)\Delta x^\mu = (5.0\times10^8,\ 3.0\times10^8,\ 0,\ 0) m. Frame S′S' moves at v=0.6cv = 0.6c along xx relative to SS (γ=1.25\gamma = 1.25, β=0.6\beta = 0.6). ((a)) Compute the invariant interval Δs2=ημνΔxμΔxν\Delta s^2 = \eta_{\mu\nu}\Delta x^\mu\Delta x^\nu in SS. ((b)) Apply the tensor transformation law for the type (1,0)(1,0) object Δxμ\Delta x^\mu (the Lorentz transformation) to find Δx′μ\Delta x'^\mu in S′S', then recompute Δs′2\Delta s'^2 directly in S′S' and confirm it matches part (a). ((c)) Explain, using the fact that ημν\eta_{\mu\nu} is itself a tensor (established by the quotient rule), why this match was guaranteed before doing the arithmetic.

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