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## EINSTEIN RADIUS FOR GRAVITATIONAL LENSING IN A CURVED UNIVERSE

I suspect that gravitational lensing data used to support dark matter can be explained by gravitational lensing in a closed static (Einstein) spacetime.  In order to make progress, we begin with getting a formula for (ahe Einstein radius in a curved spacetime.  It is

$\theta_E = \sqrt{ \frac{4GM}{a(z^L)}c^2 \frac{f^{SL}}{f^Sf^L}}$

where the quantities $f^S,f^L,f^{SL}$ replace the distance quantities for the formula in the flat spacetime.  Specifically for curvature $K>0$ one considers the Robertson-Walker metric

$g = -c^2 dt^2 + a^2( d\chi^2 + f^2_K(\chi)[ d\beta_1^2 + \cos^2(\beta_1) d\beta_2^2])$

$f_K(\chi) = \frac{1}{\sqrt{K}} \sin(\sqrt{K} \chi)$

$f^{S} = f_K(\chi^S), f^L = f_K(\chi^L), f^{SL} = f_K(\chi^S-\chi^L)$.

All this is known material that I took from S_Hilbert_Gravitational_Lensing.  The interesting question is whether this can be used to fit the actual data.  The data can be found here: gravitational lensing datagravitational lensing data

The dataset has three columns of interest to us: (i) redshift of the object deflecting the light, (ii) redshift of the source, and (iii) the ‘size’ of the deflection which is essentially the Einstein radius.  We can show that a static universe with a finite produces smaller estimates for the mass of the deflecting object  than a flat space.  The significance of this is that it is possible that the ‘dark matter’ that is inferred from gravitational lensing may be explained by the curvature of the universe.

Here is the way to test this: we compare the inferred mass from the flat space model:

$\theta_E \cdot (10^{11.09} M_{\odot})^{1/2} ( D^L D^S/D^{LS}/Gpc )^{1/2} = \sqrt{M}$

with the corresponding inferred mass formula in curved space:

$\theta_E \cdot (10^{11.09} M_{\odot})^{1/2} ( 1/a f^L f^S/f^{LS}/Gpc )^{1/2} = \sqrt{M}$

Now it is easy enough to produce a reduction of inferred mass that is proportional to the inferred mass from a flat universe by changing the parameter $a$ in the Friedmann-Robertson-Walker metric.  For example, if we set $a=1.25$ and consider the metric

$g = -c^2 dt^2 + a^2 ( d\chi^2 + f_K^2(\chi)[d\beta_1^2 + \cos^2(\beta_1) d\beta_2^2])$

where the curvature is set by $K=1/R^2$ then we can take more or less any radius we want, including $R=28.5 Gpc = 28.5 \times 3 \times 10^{25} m$ which is a generic estimate of the radius of the universe and find that the curved space estimate of the mass is $1/4$ that of the flat space mass with the same Einstein radius.  That is, if dark matter be universally proportional to mass of the object then we can remove any dark matter halo needed to explain the gravitational lensing in a STATIC Robertson-Walker metric.  This is very simple to check on actual dataset.  Read in the dataset for lensing into R and extract the columns for zs,zl, and size ($\theta_E$), then just set $a=1.25$ and compare $\theta_E \cdot (D^L D^S/D^{LS})^{1/2}$ with $\theta_E \cdot (\frac{1}{a} \frac{f^L f^L}{f^{LS}})^{1/2}$.  The ratio will be obviously be determined by $a$.  So this is a trivial way to get rid of the necessity of a halo of dark matter to explain the gravitational lensing.  What is more important is that a static constant (spatial) curvature universe automatically removes DARK ENERGY as well and explains redshift.  This is the right sort of idea because you do NOT need modified gravity at least for the case of lensing.  I suspect that if we get rid of thinking of the ‘cosmological redshift’ as expansion, a revisiting of the galactic rotation curves will probably clarify whether there really is a serious discrepancy with more classical predictions.