Get Relation from Stress-Energy Tensor Def.

In summary, the conversation revolves around the derivation of equation (2) from equations (1) and (3), using the definition of stress-energy tensor for a perfect fluid in special relativity. The conversation also touches upon the use of Lagrangian derivative and the conservation of energy. The speaker is seeking assistance in understanding the connection between equations (1), (3), and (4) in order to derive equation (2).
  • #1
fab13
312
6
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
 
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  • #2
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
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:

1. What is the stress-energy tensor and how is it related to stress and energy?

The stress-energy tensor is a mathematical object used in the field of general relativity to describe the distribution of energy and momentum in a given space-time. It is a 4x4 matrix that contains components representing energy density, momentum density, and stress (pressure and shear stress). This tensor is related to stress and energy through Einstein's field equations, which describe the curvature of space-time in the presence of matter and energy.

2. How can the stress-energy tensor be used to understand the behavior of matter and energy in space-time?

The stress-energy tensor provides a mathematical framework for understanding how matter and energy interact with the fabric of space-time. By analyzing its components, we can determine the energy density, momentum flow, and stress distribution in a given region of space-time. This allows us to make predictions about the behavior of matter and energy, such as the trajectory of a moving object or the curvature of space-time in the presence of a massive object.

3. How do we calculate the stress-energy tensor for a given system?

The stress-energy tensor can be calculated using the energy-momentum tensor, which takes into account the energy and momentum of all particles in a system. This tensor is then modified to include other forms of energy, such as electromagnetic fields or pressure, to create the stress-energy tensor. The calculations can be complex and require advanced mathematical techniques, but they ultimately provide valuable insights into the behavior of matter and energy in space-time.

4. How does the stress-energy tensor relate to the concept of spacetime curvature?

According to Einstein's theory of general relativity, the curvature of space-time is determined by the distribution of mass and energy within it. The stress-energy tensor provides a mathematical representation of this distribution, allowing us to calculate the curvature of space-time in a given region. In other words, the stress-energy tensor is one of the key components that helps us understand how matter and energy can bend and warp the fabric of space-time.

5. How does the stress-energy tensor relate to the theory of general relativity?

The stress-energy tensor is a crucial component of Einstein's theory of general relativity, which describes the behavior of gravity as a result of the curvature of space-time. The stress-energy tensor allows us to quantify the effects of matter and energy on space-time, and ultimately helps us understand the behavior of objects in the presence of massive bodies. In this way, the stress-energy tensor plays a fundamental role in our understanding of the universe according to the theory of general relativity.

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