User:JRSpriggs/sandbox

Re-arrangement
If we form the covariant derivative of the vector (no density) &lambda;&mu;, we get
 * $$ \lambda_{\mu ; \nu} = \lambda_{\mu, \nu} - \lambda_{\sigma} \Gamma^\sigma_{\mu \nu} \, $$.

If we turn this around, we can express the partial derivative in terms of the covariant derivative
 * $$ \lambda_{\mu, \nu} = \lambda_{\mu ; \nu} + \lambda_{\sigma} \Gamma^\sigma_{\mu \nu} \, $$

which will help us to convert more of our equation of motion into an explicitly invariant form. Using our coordinate condition, we get
 * $$ \lambda_{\mu, \nu} g^{\mu \nu} = \lambda_{\mu ; \nu} g^{\mu \nu} + \lambda_{\sigma} g^{\mu \nu} \Gamma^\sigma_{\mu \nu} = \lambda_{\mu ; \nu} g^{\mu \nu} - \tfrac12 \lambda_{\sigma} g^{\sigma \alpha} \Gamma^{\beta}_{\alpha \beta} \, $$

and thus
 * $$ + \tfrac14 g_{(\mu \nu)} g^{\alpha \beta} \lambda_{\alpha, \beta} + \tfrac18 g_{(\mu \nu)} g^{\alpha \beta} \lambda_{\alpha} \Gamma^{\sigma}_{\beta \sigma} = + \tfrac14 g_{(\mu \nu)} g^{\alpha \beta} \lambda_{\alpha ; \beta} $$.

also
 * $$ \lambda_{(\mu, \nu)} = \lambda_{(\mu ; \nu)} + \lambda_{\sigma} \Gamma^\sigma_{(\mu \nu)} = \lambda_{(\mu ; \nu)} + \lambda_{\sigma} \Gamma^\sigma_{\mu \nu} \, $$.

so the correction to the Einstein field equations becomes
 * $$ - \lambda_{(\mu ; \nu)} + \tfrac14 g_{(\mu \nu)} g^{\alpha \beta} \lambda_{\alpha ; \beta} - \lambda_{\sigma} \Gamma^\sigma_{\mu \nu} - \tfrac12 \lambda_{(\mu} \Gamma^{\sigma}_{\nu) \sigma} \, $$

where the first two terms are the stress-energy tensor for an invariant symmetric divergence of &lambda;&mu; (with zero trace and thus zero mass, similar to electromagnetism) and the last two terms are a kind of source term for &lambda;&mu; which is required to comply with our coordinate condition.

Simplification
In one of our preferred coordinate systems, the metric tensor is simply the product of the Minkowski metric and a scale factor
 * $$ g_{\alpha \beta} = \eta_{\alpha \beta} \, s $$.

From this we get that the gravitational force field is
 * $$ \Gamma^\alpha_{\beta \gamma} = \tfrac12 (\ln (s))_{, \rho} ( \delta^{\alpha}_{\beta} \delta^{\rho}_{\gamma} + \delta^{\alpha}_{\gamma} \delta^{\rho}_{\beta} - \eta^{\alpha \rho} \eta_{\beta \gamma} ) \, $$.

So the non-invariant terms in the correction to the Einstein field equations become
 * $$ - \lambda_{\sigma} \Gamma^\sigma_{\mu \nu} - \tfrac12 \lambda_{(\mu} \Gamma^{\sigma}_{\nu) \sigma} = - \lambda_{\mu} (\ln (s))_{, \nu} - \lambda_{\nu} (\ln (s))_{, \mu} + \tfrac12 \eta_{\mu \nu} \lambda_{\sigma} \eta^{\sigma \rho} (\ln (s))_{, \rho} \, $$.

Establishing general invariance
Assuming that $$(\ln (s))$$ is a scalar field (so its gradient is a covariant vector) and observing that when in one of our preferred reference frames
 * $$ \eta_{\mu \nu} \lambda_{\sigma} \eta^{\sigma \rho} (\ln (s))_{, \rho} = \eta_{\mu \nu} s \lambda_{\sigma} \eta^{\sigma \rho} {1 \over s} (\ln (s))_{, \rho} = g_{\mu \nu} \lambda_{\sigma} g^{\sigma \rho} (\ln (s))_{, \rho} \, $$

we can infer that our entire correction to each of the Einstein field equations is equal to
 * $$ - \tfrac12 ( \lambda_{\mu ; \nu} + \lambda_{\nu ; \mu} ) + \tfrac14 g_{\mu \nu} g^{\alpha \beta} \lambda_{\alpha ; \beta} - \lambda_{\mu} (\ln (s))_{, \nu} - \lambda_{\nu} (\ln (s))_{, \mu} + \tfrac12  g_{\mu \nu} \lambda_{\sigma} g^{\sigma \rho} (\ln (s))_{, \rho} \, $$

which is an invariant since it is built entirely from tensors. Thus we have rendered our theory into a generally invariant form so that we can use spherical coordinates or cylindrical coordinates or whatever coordinate system we may need.

However, to properly interpret this correction, we must remember that
 * $$ s = \sqrt[4]{-g} \, $$

so that
 * $$ (\ln (s)) = \tfrac14 \ln ( -g_{tt} \cdot g_{rr} \cdot g_{\theta\theta} \cdot g_{\phi\phi} ) $$

when the metric is diagonal $$g_{\alpha\beta} = \begin{pmatrix} g_{tt} & 0 & 0 & 0\\ 0 & g_{rr} & 0 & 0\\ 0 & 0 & g_{\theta\theta} & 0\\ 0 & 0 & 0 & g_{\phi\phi} \end{pmatrix} \,.$$

Black hole
This would be the case, if one is considering a static spherically-symmetric black hole or collapsar in an asymptotically Minkowskian space-time where: See Schwarzschild coordinates.
 * there is no ordinary matter or radiation outside rmin;
 * $$ g_{tt} = - c^2 ( f (r) )^2 $$ for rmin ≤ r < +&infin; and f (r) → 1 as r → +&infin;;
 * $$ g_{rr} = ( h (r) )^2 $$ for rmin ≤ r < +&infin; and h (r) → 1 as r → +&infin;;
 * $$ g_{\theta\theta} = r^2 $$ for 0 ≤ &theta; ≤ &pi;; and
 * $$ g_{\phi\phi} = r^2 \sin^2 \theta $$ for -&pi; ≤ &phi; ≤ &pi;.

Expanding universe
Or, if our expanding universe (see FLRW) is approximated to be spatially homogeneous and isotropic where:
 * $$ g_{tt} = - c^2 $$ for 0 < t < +&infin;;
 * $$ g_{rr} = ( a (t) )^2 \left( \frac{1}{ 1 - r } \right)^2 $$ for 0 ≤ r ≤ &pi;;
 * $$ g_{\theta\theta} = ( a (t) )^2 \sin^2 (r) $$ for 0 ≤ &theta; ≤ &pi;;
 * $$ g_{\phi\phi} = ( a (t) )^2 \sin^2 (r) \sin^2 \theta $$ for -&pi; < &phi; ≤ &pi;.