pdf-file - Institut für Theoretische Physik
pdf-file - Institut für Theoretische Physik
pdf-file - Institut für Theoretische Physik
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
∂νΠ µν = 0 (7)<br />
and is symmetric. (The greek indices go from 0 to 3, the latin ones from 1 to<br />
3.)<br />
The Lorentz transformation will now yield these thermodynamic expressions<br />
for an arbitrary inertial frame. Denoting the rest frame quantities with the<br />
superscript 0 , the Lorentz transformation<br />
Π µν =Λ µ<br />
α<br />
<br />
Π αβ 0<br />
Λ ν<br />
β , (8)<br />
for v = v ex employs with the matrix<br />
where<br />
Λ ν<br />
⎛<br />
⎞<br />
⎜ γ γβ 00⎟<br />
⎜<br />
⎟<br />
⎜<br />
⎟<br />
⎜ γβ γ 00⎟<br />
µ = ⎜<br />
⎟<br />
⎜<br />
⎟ ,<br />
⎜ 0 0 1 0 ⎟<br />
⎝<br />
⎠<br />
(9)<br />
0 0 0 1<br />
<br />
β = v/c , γ =1/ 1 − β2 . (10)<br />
Up to order v2<br />
c 2 ,wehave<br />
ɛ tot =(1+v 2 /c 2 )ɛ tot,0<br />
+2 v [1 + v 2 /c 2 ] · g tot,0<br />
+vi Π 0 ij vj/c 2 , (11)<br />
cg tot<br />
i = Qi/c = vi/c (1 + v 2 /c 2 ) ɛ tot,0<br />
+[1 + v 2 /(2 c 2 )] cg tot,0<br />
i<br />
+3 vi cg tot,0<br />
l vl/(2 c 2 )<br />
+[1 + v 2 /(2 c 2 )] vl Π 0 li/c<br />
+vi vl (Π 0 lk/c) vk/(2 c 2 ), (12)<br />
Πik = vi vk ɛ tot,0 /c 2 +[1+v 2 /(2 c 2 )] vi g tot,0<br />
k<br />
+[1 + v 2 /(2 c 2 )] vk g tot,0<br />
i<br />
+ vi vk g tot,0<br />
l<br />
vl/(vc 2 )<br />
+Π 0 ik +Π 0 il vl vk/(2 c 2 )<br />
+Π 0 lk vl vi/(2 c 2 ) . (13)<br />
As ε ≪ ρc2 , we shall only include terms of linear order in v,<br />
except for terms<br />
c<br />
connected to the rest mass, ρc2 , where quadratic order will be included. (In the<br />
5