Perfect fluids and the stress energy tensor

In summary, the conversation discusses the unclear assumptions in textbooks regarding perfect fluids and the issue of different pressures in different directions. The focus is on the stress energy tensor and its off diagonal components when a rotation is performed around the z axis. The question of which rotation matrix to use is also raised.
  • #1
black_hole
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Homework Statement



"Texbooks that describe perfect fluids are often a little unclear about what is being assumed. It may not be immediately obvious why can't the pressures be different in different directions? Let's examine this. Suppose Tαβ = diag(ρ,(1+ε)P,P,P) . Show that if one performs a rotation around the z axis by an angle phi that the stress energy tensor picks up off diagonal components of order εP."

What rotation matrix should I be using, it's coordinate dependent isn't it?

Homework Equations


The Attempt at a Solution

 
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  • #2
black_hole said:

Homework Statement



"Texbooks that describe perfect fluids are often a little unclear about what is being assumed. it may not be immediately obvious why can't the pressures be different in different directions? Let's examine this. Suppose Tαβ = diag(ρ,(1+ε)P,P,P) . Show that if one performs a rotation around the z axis by an angle phi that the stress energy tensor picks up off diagonal components of order εP."

What rotation matrix should I be using, it's coordinate dependent isn't it?

Homework Equations





The Attempt at a Solution


Since they say rotation about the z axis by an angle phi, I think you can assume the spatial coordinates are cartesian. Just work with those.
 

1. What is a perfect fluid?

A perfect fluid is a hypothetical fluid that has no viscosity or internal friction, and therefore experiences no energy losses due to fluid flow. It is often used as a simplified model for real fluids, such as water or air.

2. What is the stress-energy tensor?

The stress-energy tensor is a mathematical object used in general relativity to describe the energy and momentum distribution of matter and energy in spacetime. It contains 10 components that represent different aspects of the stress and energy in a given region of spacetime.

3. How is the stress-energy tensor related to perfect fluids?

The stress-energy tensor of a perfect fluid is diagonal and has a specific form that depends on the fluid's energy density and pressure. This form allows it to be easily incorporated into the equations of general relativity, making it a useful tool for studying the behavior of perfect fluid systems.

4. What are some examples of perfect fluids?

Some examples of perfect fluids include ideal gases, superfluids, and the early universe in its inflationary phase. While real fluids do have some degree of viscosity and energy losses, these systems can often be well-approximated by the perfect fluid model.

5. How are perfect fluids and the stress-energy tensor used in cosmology?

In cosmology, perfect fluids and the stress-energy tensor are used to model the behavior and evolution of the universe. By studying the stress-energy tensor of different types of matter and energy, we can gain insights into the large-scale structure and dynamics of the universe, as well as its expansion rate and future fate.

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