User:JRSpriggs/MOND

As Sabine Hossenfelder has pointed out, MOND appears to be a more successful explanation for the phenomena called dark matter than the Einstein field equations with axions. She has a rather complicated theory called Covariant Extended Gravity. Instead, I would like to try a merely Lorentz-covariant hypothesis which uses just the gravitational force field, Christoffel symbols of the second kind for the Levi-Civita connection $$ \Gamma^\alpha_{\beta \gamma} = \tfrac12 g^{\alpha \epsilon} \left( g_{\epsilon \beta, \gamma} + g_{\gamma \epsilon , \beta} - g_{\beta \gamma , \epsilon} \right)$$, and the Lagrange multiplier &lambda;&mu; for a coordinate condition instead of what she calls the "imposter field".

Our choice for a coordinate condition
Our coordinate condition is inspired by and similar to, but different from, the harmonic coordinate condition. (See also Nordström's theory of gravitation and the Weyl curvature tensor.) It is
 * $$ \left(g^{\mu \rho} \sqrt[4]{-g}\right)_{, \rho} = 0 $$

which was chosen because it is Lorentz-covariant and it ignores the scale of the metric. Consider the covariant derivative
 * $$ 0 = \left(g^{\mu \nu} \sqrt[4]{-g}\right)_{; \rho} = \left(g^{\mu \nu} \sqrt[4]{-g}\right)_{, \rho} + g^{\sigma \nu} \Gamma^{\mu}_{\sigma \rho} \sqrt[4]{-g} + g^{\mu \sigma} \Gamma^{\nu}_{\sigma \rho} \sqrt[4]{-g} - \tfrac12 g^{\mu \nu} \Gamma^{\sigma}_{\sigma \rho} \sqrt[4]{-g} \,.$$

The last term $$ - \tfrac12 g^{\mu \nu} \Gamma^{\sigma}_{\sigma \rho} \sqrt[4]{-g} $$ emerges because $$ \sqrt {-g}$$ is not an invariant scalar, and so its covariant derivative is not the same as its ordinary derivative. Rather, $$ 0 = \sqrt {-g}_{; \rho} = \sqrt {-g}_{, \rho} - \Gamma^{\sigma}_{\sigma \rho} \, \sqrt {-g}$$.

Contracting ν with ρ and applying the coordinate condition, we get:


 * $$ 0 = - \left(g^{\mu \nu} \sqrt[4]{-g}\right)_{, \nu} = + g^{\sigma \nu} \Gamma^{\mu}_{\sigma \nu} \sqrt[4]{-g} + g^{\mu \sigma} \Gamma^{\nu}_{\sigma \nu} \sqrt[4]{-g} - \tfrac12 g^{\mu \nu} \Gamma^{\sigma}_{\sigma \nu} \sqrt[4] {-g} $$

Thus, we get that an alternative way of expressing the coordinate condition is:


 * $$ + g^{\sigma \nu} \Gamma^{\mu}_{\sigma \nu} + \tfrac12 g^{\mu \alpha} \Gamma^{\beta}_{\alpha \beta} = 0 $$.


 * $$ g^{\sigma \nu} \Gamma^{\mu}_{\sigma \nu} = - \tfrac12 g^{\mu \alpha} \Gamma^{\beta}_{\alpha \beta} $$.

Note that this coordinate condition must be satisfied at all times, not just the present, but also the past and the future. Only then, is the theory consistent with special relativity. And only then can one expect it to yield MOND as a possible result.

Lagrangian
The Hilbert action is modified to incorporate our coordinate condition by the addition of the term
 * $$ \left(g^{\mu \nu} \sqrt[4]{-g}\right)_{, \nu} \lambda_\mu \sqrt[4]{-g} $$

where &lambda;&mu; is a Lagrange multiplier. Thus the action becomes


 * $$ S = \int \left[ \frac{1}{2\kappa} R \sqrt{-g} + \mathcal{L}_\mathrm{matter} \sqrt{-g} + \left(g^{\mu \nu} \sqrt[4]{-g}\right)_{, \nu} \lambda_\mu \sqrt[4]{-g} \right] \, \mathrm{d}t \, \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}z $$.

Deriving the equations of motion
The equations of motion will include our coordinate condition (from varying the Lagrange multiplier) and a modified form of the Einstein field equations which will incorporate the result of varying the new terms with respect to the inverse metric tensor. Variation of the new terms gives
 * $$ - \delta g^{\mu \nu} \lambda_{(\mu, \nu)} \sqrt{-g} - \tfrac12 \delta g^{\mu \nu} \lambda_{(\mu} \Gamma^{\sigma}_{\nu) \sigma} \sqrt{-g}

+ \tfrac14 \delta g^{\mu \nu} g_{(\mu \nu)} g^{\alpha \beta} \lambda_{\alpha, \beta} \sqrt{-g} + \tfrac18 \delta g^{\mu \nu} g_{(\mu \nu)} g^{\alpha \beta} \lambda_{\alpha} \Gamma^{\sigma}_{\beta \sigma} \sqrt{-g} $$.

where parentheses indicate symmetrization of the indices. So each of the ten Einstein field equations will have these four terms added to it:
 * $$ - \lambda_{(\mu, \nu)} - \tfrac12 \lambda_{(\mu} \Gamma^{\sigma}_{\nu) \sigma} + \tfrac14 g_{(\mu \nu)} g^{\alpha \beta} \lambda_{\alpha , \beta} + \tfrac18 g_{(\mu \nu)} g^{\alpha \beta} \lambda_{\alpha} \Gamma^{\sigma}_{\beta \sigma} $$.

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;.