## Wednesday, January 30, 2013

### gravity in 2d and 3d

why can't gravity rotate galaxy images in gravitational lensing, but introduce physical rotation of galaxies?

bonus question: where's the global minimum of $x^x$, $x>0$?

1. bonus question: differentiation of

x^x = \exp(x\ln(x))

and setting it to zero yields

\frac{\mathrm{d}}{\mathrm{d}x}x^x = x^x(\ln(x)+1) = 0

the result $x=\exp(-1)$ for the position of the minimum.

2. following a mail CQW received from one of our readers I wanted to add: the question was aiming at weak field, Newtonian gravity.

3. well, this question is a very misleading one and CQW tries its best to explain this. first of all, gravitational lensing with classical Newtonian potentials can in fact rotate images, that's because the lensing equation (giving the deflection angle as a function of distance) is implicit, and is solved by a perturbation series. at lowest order, one carries out the integration for collecting up the individual deflections along a straight line (the Born-approximation), and in this limit lensing is not rotating images. but higher order corrections to the geodesic do in fact rotate images: the physical reason is that the light ray does not interact locally with the potential, but non-locally, therefore the lensing Jacobian made up from the derivative of the deflection angle with position, is not necessarily symmetric (because the derivative don't interchange anymore).

4. for the second part CQW would like to point out that it's in fact weird for a scalar potential to rotate objects, after all: at what stage does the object decide to rotate clockwise or counterclockwise? assuming that the rotation of galaxies is linked to the vorticity of the large-scale structure then necessarily requires motion for generating vorticity. similarly to what's been written on lensing, the motion of CDM particles is vorticity free up to second order in perturbation theory and only provides vorticity starting from third order.

5. if one assumes, however, that the rotation of galaxies is linked to angular momentum, there's a different mechanism at work called tidal shearing. the key thing to understand is that one can have angular momentum even if particles follow straight lines, as long as the point relative to which one calculates the angular momentum does not lie on the particle trajectory. relative to a galaxy's centre of mass it is possible to integrate up the angular momentum contributions from all particles, giving an estimate of the total angular momentum, even if all particles are moving perfectly laminar.