content_image/327457.jpg content_image/327458.jpg content_image/327459.jpg content_image/327464.jpg content_image/327465.jpg content_image/327466.jpg
## 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 

<img src='content_image/327457.jpg'>

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 

<img src='content_image/327458.jpg'>

<img src='content_image/327459.jpg'>

<img src='content_image/327464.jpg'>

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 

<img src='content_image/327465.jpg'>

For the case of ultrarelativistic motion, 1 + v$_{f}$ ≃ 2, and Eq. (D.27) simplifies to 

<img src='content_image/327466.jpg'>

