- #1

George Keeling

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- Is the scale factor a scalar? Assuming so, gets a weird result. I think.

Is the scale factor a scalar?

I think that the answer is no but I want to check because god (or the universe) has been playing tricks on me...

At Sean Carroll's invitation I wanted to check that the tensor$$

K_{\mu\nu}=a^2\left(g_{\mu\nu}+U_\mu U_\nu\right)

$$was a Killing tensor. ##U^\mu=\left(1,0,0,0\right)## is the four velocity of all comoving observers. The FLRW metric in use is given by $$

{ds}^2=-{dt}^2+a^2\left(t\right)\left[\frac{{\rm dr}^2}{1-\kappa r^2}+r^2{d\theta}^2+r^2\sin^2{\theta}{d\phi}^2\right]

$$ For that to be so we need ##\nabla_{(\sigma}K_{\mu\nu)}=0## and the first step is to calculate the components of ##\nabla_\sigma K_{\mu\nu}##. I had the Christoffel symbols to hand and started to compute as follows$$

\nabla_\sigma K_{\mu\nu}=\left(g_{\mu\nu}+U_\mu U_\nu\right)\nabla_\sigma\left(a^2\right)+a^2\nabla_\sigma\left(g_{\mu\nu}+U_\mu U_\nu\right)

$$then ##\nabla_\sigma\left(a^2\right)## became ##2a\partial_\sigma a## which all looked very nice. Unfortunately

The correct result comes from computing ##\nabla_\sigma K_{\mu\nu}=\mathrm{\partial}_\sigma K_{\mu\nu}-\Gamma_{\sigma\mu}^\lambda K_{\lambda\nu}-\Gamma_{\sigma\nu}^\lambda K_{\mu\lambda}## I now know.

* Edit. I did make a mistake. See below.

I think that the answer is no but I want to check because god (or the universe) has been playing tricks on me...

At Sean Carroll's invitation I wanted to check that the tensor$$

K_{\mu\nu}=a^2\left(g_{\mu\nu}+U_\mu U_\nu\right)

$$was a Killing tensor. ##U^\mu=\left(1,0,0,0\right)## is the four velocity of all comoving observers. The FLRW metric in use is given by $$

{ds}^2=-{dt}^2+a^2\left(t\right)\left[\frac{{\rm dr}^2}{1-\kappa r^2}+r^2{d\theta}^2+r^2\sin^2{\theta}{d\phi}^2\right]

$$ For that to be so we need ##\nabla_{(\sigma}K_{\mu\nu)}=0## and the first step is to calculate the components of ##\nabla_\sigma K_{\mu\nu}##. I had the Christoffel symbols to hand and started to compute as follows$$

\nabla_\sigma K_{\mu\nu}=\left(g_{\mu\nu}+U_\mu U_\nu\right)\nabla_\sigma\left(a^2\right)+a^2\nabla_\sigma\left(g_{\mu\nu}+U_\mu U_\nu\right)

$$then ##\nabla_\sigma\left(a^2\right)## became ##2a\partial_\sigma a## which all looked very nice. Unfortunately

**in the ##\nabla_iK_{0j}## components (##i,j=123##).**__it produces exactly the right answer except for a wrong sign__**WTF**. Of course I might have made a mistake*, thus my question: Is the scale factor a scalar? It it's not then ##\ \nabla_\sigma\left(a^2\right)## is nonsense.The correct result comes from computing ##\nabla_\sigma K_{\mu\nu}=\mathrm{\partial}_\sigma K_{\mu\nu}-\Gamma_{\sigma\mu}^\lambda K_{\lambda\nu}-\Gamma_{\sigma\nu}^\lambda K_{\mu\lambda}## I now know.

* Edit. I did make a mistake. See below.

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