Get relation from definition of stress-energy tensor

• I
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

Orodruin
Staff Emeritus
Homework Helper
Gold Member
2021 Award
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).

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: