User:JRSpriggs/sandbox

Re-arrangement

If we form the covariant derivative of the vector (no density) λ_μ , we get

$\lambda _{\mu ;\nu }=\lambda _{\mu ,\nu }-\lambda _{\sigma }\Gamma _{\mu \nu }^{\sigma }\,$.

If we turn this around, we can express the partial derivative in terms of the covariant derivative

$\lambda _{\mu ,\nu }=\lambda _{\mu ;\nu }+\lambda _{\sigma }\Gamma _{\mu \nu }^{\sigma }\,$

which will help us to convert more of our equation of motion into an explicitly invariant form. Using our coordinate condition, we get

$\lambda _{\mu ,\nu }g^{\mu \nu }=\lambda _{\mu ;\nu }g^{\mu \nu }+\lambda _{\sigma }g^{\mu \nu }\Gamma _{\mu \nu }^{\sigma }=\lambda _{\mu ;\nu }g^{\mu \nu }-{\tfrac {1}{2}}\lambda _{\sigma }g^{\sigma \alpha }\Gamma _{\alpha \beta }^{\beta }\,$

and thus

$+{\tfrac {1}{4}}g_{(\mu \nu )}g^{\alpha \beta }\lambda _{\alpha ,\beta }+{\tfrac {1}{8}}g_{(\mu \nu )}g^{\alpha \beta }\lambda _{\alpha }\Gamma _{\beta \sigma }^{\sigma }=+{\tfrac {1}{4}}g_{(\mu \nu )}g^{\alpha \beta }\lambda _{\alpha ;\beta }$.

also

$\lambda _{(\mu ,\nu )}=\lambda _{(\mu ;\nu )}+\lambda _{\sigma }\Gamma _{(\mu \nu )}^{\sigma }=\lambda _{(\mu ;\nu )}+\lambda _{\sigma }\Gamma _{\mu \nu }^{\sigma }\,$.

so the correction to the Einstein field equations becomes

$-\lambda _{(\mu ;\nu )}+{\tfrac {1}{4}}g_{(\mu \nu )}g^{\alpha \beta }\lambda _{\alpha ;\beta }-\lambda _{\sigma }\Gamma _{\mu \nu }^{\sigma }-{\tfrac {1}{2}}\lambda _{(\mu }\Gamma _{\nu )\sigma }^{\sigma }\,$

where the first two terms are the stress-energy tensor for an invariant symmetric divergence of λ_μ (with zero trace and thus zero mass, similar to electromagnetism) and the last two terms are a kind of source term for λ_μ 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

$\Gamma _{\beta \gamma }^{\alpha }={\tfrac {1}{2}}(\ln(s))_{,\rho }(\delta _{\beta }^{\alpha }\delta _{\gamma }^{\rho }+\delta _{\gamma }^{\alpha }\delta _{\beta }^{\rho }-\eta ^{\alpha \rho }\eta _{\beta \gamma })\,$.

So the non-invariant terms in the correction to the Einstein field equations become

$-\lambda _{\sigma }\Gamma _{\mu \nu }^{\sigma }-{\tfrac {1}{2}}\lambda _{(\mu }\Gamma _{\nu )\sigma }^{\sigma }=-\lambda _{\mu }(\ln(s))_{,\nu }-\lambda _{\nu }(\ln(s))_{,\mu }+{\tfrac {1}{2}}\eta _{\mu \nu }\lambda _{\sigma }\eta ^{\sigma \rho }(\ln(s))_{,\rho }\,$.

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

$-{\tfrac {1}{2}}(\lambda _{\mu ;\nu }+\lambda _{\nu ;\mu })+{\tfrac {1}{4}}g_{\mu \nu }g^{\alpha \beta }\lambda _{\alpha ;\beta }-\lambda _{\mu }(\ln(s))_{,\nu }-\lambda _{\nu }(\ln(s))_{,\mu }+{\tfrac {1}{2}}g_{\mu \nu }\lambda _{\sigma }g^{\sigma \rho }(\ln(s))_{,\rho }\,$

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

$(\ln(s))={\tfrac {1}{4}}\ln(-g_{tt}\cdot g_{rr}\cdot g_{\theta \theta }\cdot g_{\phi \phi })$

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:

• there is no ordinary matter or radiation outside r_min;
• $g_{tt}=-c^{2}(f(r))^{2}$ for r_min ≤ r < +∞ and f (r) → 1 as r → +∞;
• $g_{rr}=(h(r))^{2}$ for r_min ≤ r < +∞ and h (r) → 1 as r → +∞;
• $g_{\theta \theta }=r^{2}$ for 0 ≤ θ ≤ π; and
• $g_{\phi \phi }=r^{2}\sin ^{2}\theta$ for -π ≤ φ ≤ π.

See Schwarzschild coordinates.

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 < +∞;
• $g_{rr}=(a(t))^{2}\left({\frac {1}{1-r}}\right)^{2}$ for 0 ≤ r ≤ π;
• $g_{\theta \theta }=(a(t))^{2}\sin ^{2}(r)$ for 0 ≤ θ ≤ π;
• $g_{\phi \phi }=(a(t))^{2}\sin ^{2}(r)\sin ^{2}\theta$ for -π < φ ≤ π.