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 $\Gamma _{\beta \gamma }^{\alpha }={\tfrac {1}{2}}g^{\alpha \epsilon }\left(g_{\epsilon \beta ,\gamma }+g_{\gamma \epsilon ,\beta }-g_{\beta \gamma ,\epsilon }\right)$, and the Lagrange multiplier λ_μ for a coordinate condition instead of what she calls the "imposter field".

Lagrangian

$S=\int (\,\kappa _{G}R{\sqrt {-g}}+\kappa _{D}L_{D}+L_{O}\,)\,\mathrm {d} t\,\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z$

where: L_O is the Lagrangian in classical physics of ordinary matter and radiation, excluding gravity and dark matter. (Here, L_O includes quantum effects. That is, this is classical in the sense of excluding gravity, not quantum.) κ_G and κ_D are proportionality constants to match the units of gravity and dark matter to those of classical physics.

If dark matter is described as a merely Lorentz-covariant field, then L_D might take the form

$L_{D}=({g_{\alpha }}_{\beta }-s{\eta _{\alpha }}_{\beta }){M^{\alpha }}^{\beta }$

or

$L_{D}=(g^{\mu \nu }s)_{,\nu }s\lambda _{\mu }$

where s is a scale factor, the classical Newtonian gravitational field. M and λ are Lagrange multipliers. (See also Nordström's theory of gravitation and the Weyl curvature tensor.)

Really special relativity

Let us now assume that there is a non-empty set of preferred coordinate systems in which

${g_{\alpha }}_{\beta }=s{\eta _{\alpha }}_{\beta }$

holds everywhere and always where η_αβ is the Minkowski metric (not a tensor, but just a mathematical constant).

Moving towards general invariance

We need to transform to an arbitrary coordinate system rather than confining ourselves to the preferred coordinates. This is necessary because: we may not initially know which are the preferred coordinates; we may want to test whether there are in fact such preferred coordinates; and the symmetries of the systems we want to study may require the use of different coordinate systems. The main difficulty with such a change is caused by the need to calculate the covariant derivatives with respect to the new coordinates. That, in turn, requires knowledge of

$\Gamma _{\beta \gamma }^{\alpha }={\tfrac {1}{2}}g^{\alpha \epsilon }\left(g_{\epsilon \beta ,\gamma }+g_{\gamma \epsilon ,\beta }-g_{\beta \gamma ,\epsilon }\right)$

for arbitrary coordinates. To obtain that, we can first calculate it for our preferred coordinates and then transform it to arbitrary coordinates.

$\Gamma _{\beta \gamma }^{\alpha }=$

${\tfrac {1}{2}}\left(g^{\alpha \epsilon }g_{\epsilon \beta ,\gamma }+g^{\alpha \epsilon }g_{\gamma \epsilon ,\beta }-g^{\alpha \epsilon }g_{\beta \gamma ,\epsilon }\right)$

${\tfrac {1}{2}}\left(g^{\alpha \epsilon }\eta _{\epsilon \beta }s_{,\gamma }+g^{\alpha \epsilon }\eta _{\gamma \epsilon }s_{,\beta }-g^{\alpha \epsilon }\eta _{\beta \gamma }s_{,\epsilon }\right)$

${\tfrac {1}{2}}\left({\tfrac {1}{s}}{\delta _{\beta }^{\alpha }}s_{,\gamma }+{\tfrac {1}{s}}{\delta ^{\alpha }}_{\gamma }s_{,\beta }-{\tfrac {1}{s}}\eta ^{\alpha \epsilon }\eta _{\beta \gamma }s_{,\epsilon }\right)$

${\tfrac {1}{2}}\left({\tfrac {1}{s}}{\delta _{\beta }^{\alpha }}s_{,\gamma }+{\tfrac {1}{s}}{\delta ^{\alpha }}_{\gamma }s_{,\beta }-{\tfrac {1}{s}}g^{\alpha \epsilon }g_{\beta \gamma }s_{,\epsilon }\right)$

${\tfrac {1}{2}}{\tfrac {1}{s}}s_{,\epsilon }\left({\delta ^{\alpha }}_{\beta }{\delta _{\gamma }}^{\epsilon }+{\delta ^{\alpha }}_{\gamma }{\delta ^{\epsilon }}_{\beta }-g^{\alpha \epsilon }g_{\beta \gamma }\right)$

${\tfrac {1}{2}}\ln(s)_{,\epsilon }\left({\delta ^{\alpha }}_{\beta }{\delta _{\gamma }}^{\epsilon }+{\delta ^{\alpha }}_{\gamma }{\delta ^{\epsilon }}_{\beta }-g^{\alpha \epsilon }g_{\beta \gamma }\right)$

where δ^ε_α is the Kronecker delta. Notice that the last line is meaningful for arbitrary coordinates.

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.

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

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

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 (in such a preferred coordinate system) 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.