Torsion in the Friedmann Universe

[Dubbelosix]

 

If rotation really should enter the picture, this would produce a torsion field which enters the equation with a negative sign. Rotation would not be unusual if we considered gravity as part of the full Poincare group of spatial symmetries which involves the spin and torsion (Venzo de Sabbata) also see Sivaram. For instance, Torsion enters the Poisson equation

[math]\nabla^2 \phi = 4 \pi G(\rho - \mathbf{k}\sigma^2)[/math]

Where [math]\mathbf{k}[/math] is [math]\frac{G}{c^4}[/math] and [math]\sigma[/math] is the spin density which can be calculated the following way:

[math]\sigma = \frac{J}{V} = \frac{m \omega R^2}{L^3} = \frac{m vR}{L^3}[/math]

so

[math]\mathbf{k}\sigma^2 = \frac{Gm^2v^2R^2}{c^4L^6} = \frac{Gm^2}{c^2L^4}[/math]

With torsion there are contributions to the curvature, a curvature term just looks like

[math]kc^2 = - \frac{2U}{mx^2}[/math]

The torsion term alters the effective density in the following way, using an equation of motion I derived which takes into account the absolute acceleration of a spinning universe

[math](\frac{\ddot{R}}{R})^2 = \frac{8 \pi G}{3}(\rho - k\sigma^2) + a_{eul} + a_{cor} + a_{cent}[/math]

Below is a reference to that derivation.

 

 

 

Edited September 23, 2017 by Dubbelosix

[Mordred]

 Now look into the affect of the above to the corresponding right hand rule with regards to stresses due to torsion. Though that is already accounted for in the references and above via the spin density 

edit or more accurately the stress tensor, in other words make sure you preserve the right hand rule in your model development.

Lol in your case probably needless of that reminder

Edited September 23, 2017 by Mordred

[Mordred]

Here is a decent article on the Raychaudhuri equations to the above

Once again arxiv (phone)

https://www.google.ca/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/gr-qc/0611123&ved=0ahUKEwjqqKOJ-7rWAhWrhFQKHfleBHMQFggiMAE&usg=AFQjCNGRlFqQlIDprhJ94mKefjXR1smlLw

I don't know if you do any mathematica but if so you will find this a bit handy for the Mohr's circle to torsion

https://www.google.ca/url?sa=t&source=web&rct=j&url=http://web.mit.edu/course/3/3.11/www/modules/trans.pdf&ved=0ahUKEwiSkraL87rWAhXhs1QKHZS5CzkQFggfMAE&usg=AFQjCNFI48MwB7AwXYOKzpDAWuPAx4ZPkw

 

there is some excellent sections with regards to hydrodynamics in "Elements of Astrophysics" 

Edited September 23, 2017 by Mordred

[Dubbelosix]

I will read them, thank you.