206 Appendix D: Formation Time and Brick Estimates D.2.2 Static Nonconstant Medium We note in passing the analytically interesting case of a time-independent medium of nonconstant density, µ = µ ( ⃗ x ). Then dp p = − µ ( x ) dt = − µ 1 v dx. (D.23) v (1 + γ $^{2}$v$^{2}$) γmdv γmv = − µdx. (D.24) ∫ v 2 1 − v $_{2}$dv = 1 2 ln 1 + v 1 − v − v, (D.25) Since p = γmv we have that dp = mvdγ + γmdv = (1 + γ $^{2}$v$^{2}$) dv . Then Eq. (D.23) becomes Moreover so we have that where the integral on the RHS is really a line integral through the medium from the initial starting point ⃗ x$_{i}$ to the final point of interest, ⃗ x$_{f}$ . Therefore 1 2 ln 1 + v 1 − v$^{∣}$ ∣ ∣ v$_{f}$ v = v$_{i}$ = − ∫ x$_{f}$ x$_{i}$ µdx, (D.26) 1 + v$_{f}$ 1 − v$_{f}$ = 1 + v$_{i}$ 1 − v$_{i}$ e − 2 ∫ $^{µdx}$. (D.27) v$_{f}$ = 1 − (1 − v$_{i}$ ) e − 2 ∫ $^{µdx}$. (D.28) For the case of ultrarelativistic motion, 1 + v$_{f}$ ≃ 2, and Eq. (D.27) simplifies to