Get relation from definition of stress-energy tensor

  • #1
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Main Question or Discussion Point

Starting from the following definition of stress-energy tensor for a perfect fluid in special relativity :

$${\displaystyle T^{\mu \nu }=\left(\rho+{\frac {p}{c^{2}}}\right)\,v^{\mu }v^{\nu }-p\,\eta ^{\mu \nu }\,}\quad(1)$$

with ##v^{\nu}=\dfrac{\text{d}x^{\nu}}{\text{d}\tau}## and

##V^{\nu}=\dfrac{\text{d}x^{\nu}}{\text{d}t}## (we have ##v^{\nu}=\gamma\,V^{\nu}##)

So, finally, I have to get the following relation :

$$\dfrac{\partial \vec{V}}{\partial t} + (\vec{V}.\vec{grad})\vec{V} = -\dfrac{1}{\gamma^2(\rho+\dfrac{p}{c^2})} \bigg(\vec{grad}\,p+\dfrac{\vec{V}}{c^2}\dfrac{\partial \rho}{\partial t}\bigg)\quad(2)$$

To get this relation, I must use the conservation of energy : ##\partial_{\mu}T^{\mu\nu}=0\quad(3)##

If someone could help me to find the equation ##(2)## from ##(1)## and ##(3)##, this would be nice to indicate the tricks to apply.

Regards
 

Answers and Replies

  • #2
Orodruin
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Can you show us what you have done so far? Also, I would suggest writing ##\nabla## instead of ##\vec{grad}## (or ##\vec\nabla## if you must).
 
  • #3
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I recognize in the left member of wanted relation ##\quad(2)## the Lagrangian derivative :

$$\dfrac{\text{D}\,\vec{V}}{\text{d}t}=\dfrac{\partial \vec{V}}{\partial t} + (\vec{V}.\vec{\nabla})\vec{V}\quad(4)$$

and I can rewrite ##(1)## with the ##V^{\mu}## components like :

$$T^{\mu\nu}=\left(\rho+\dfrac{p}{c^{2}}\right)\,\gamma^2\,V^{\mu}V^{\nu }-p\,\eta^{\mu\nu}\,\quad(5)$$

But from this point, I don't know how to make the link between ##(4)##, ##(5)##, ##(3)## (the divergence of stress-energy equal to zero), and ##(1)## ...

Any help is welcome
 
Last edited:

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