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What is the meaning of expansion, shear and viscosity in context of universe? How can we conclude a result after getting a numerical value of above terms?

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The second question is better than the first here, in a sense. The problem is that "viscosity" is a term for fluids, shear applies to materials and expansion applies to anything with length. So what really is that first question? –  Roy Simpson Mar 21 '11 at 11:14
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@Roy, "shear" applies to fluids as well, without shear you cant measure viscosity. And materials can be liquid, solid or gaseous. –  Georg Mar 21 '11 at 11:55
    
Indeed, so this makes the question extremely general, apparently about all of physics. –  Roy Simpson Mar 21 '11 at 11:57
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The Raychaudhuri equation (en.wikipedia.org/wiki/Raychaudhuri_equation) is often useful in modeling the expanding Universe, when we want to consider the possibility of inhomogeneous and/or anisotropic expansion. Shear and expansion are central there. –  Ted Bunn Mar 21 '11 at 13:12
    
This theorem mentions "vorticity" not "viscosity" though - but it could be the question here. –  Roy Simpson Mar 21 '11 at 16:48
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Our model of the Universe (LambdaCDM) is a hydrodynamical one. That is to say that we treat the universe as a fluid with multiple components like dark matter, ordinary matter and so on. When we say the universe expands we mean the size of a unit volume gets bigger and bigger. Mathematically the determinant of a matrix gives us the volume. What is the matrix that would give the volume of the universe? Well, the $3\times 3$ matrix would have entries like $(p_x, p_y, p_z)$, $\rho(x,y,z)$ describing the pressure in the $x$, $y$ and $z$ directions and density. General relativity is done with time as the fourth dimension, so in the LambdaCDM model the matrix is $4\times 4$ and its determinant gives us the volume. This determinant turns out to be $a(t)^3 r^3$. You see if I fix the radius to say one then the volume still increases because $a(t)$ is a function of time. How do we physically understand this expansion? Well imagine a swarm of particles (dark matter, baryonic matter, even photons (though you'd imagine photons to be coupled to electrons via compton scattering for the fluid concept to apply)), actually imagine a spherical swarm for simplicity, then the volume of this swarm gets larger and larger, this is expansion, or the hubble expansion.

Now shear and viscosity are concepts that we have from ordinary fluid-dynamics (hydrodynamics). Shear is the distortion in the volume, the spherical swarm may change to ellipsoidal, viscosity is the resistance because of internal friction among the fluid components. In cosmology energy transfer can take place between electrons-protons and photons. Photons can't move without striking an electron which may be in the way. Its motion is hindered, this is friction if I may say so, and so our fluid is a viscous fluid.

The spherical swarm may rotate about some axis, this is called vorticity. But note that shear and viscosity only appear in the fluid equations when we consider perturbations to the FLRW metric of our universe. In the background model there is no shear, viscosity or vorticity. There is just expansion, the hubble expansion.

Updated Oct 2, 2013 in response to what is density in the $(x,y,z)$ directions. Density in the $(x,y,z)$ directions is meaning less but you can talk about mass flow in a given direction by considering $\rho(x,y,z)*\vec{v}$ where $\vec{v}$ is the 3-velocity in some direction. Or you may talk about changes in density in a given direction $\vec{v}\cdot\nabla \rho(x,y,z)$. Above the matrix that I am referring to is of course the energy-momentum tensor $T^{\mu\nu}$.

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