# Physics 616

• Prof. Andrew W. Steiner
(or Andrew or "Dr. Steiner")
• Office hour: 103 South College, Thursday 11am
• Email: awsteiner@utk.edu
• Homework: Electronically as .pdf
• You may work with each other on the homework, but you must write the solution in your own words

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## Gravitational Deflection of Light

• Remember that $d\phi/d\lambda = L/r^2$ and $$\left(\frac{dr}{d \lambda}\right)^2 = E^2 - \left(1-\frac{2 M}{r} \right) \frac{L^2}{r^2}$$ thus defining $b\equiv L/E$ we have $$\frac{d\phi}{dr} = \pm \frac{1}{r^2} \left[ \frac{1}{b^2} - \frac{1}{r^2} \left(1-\frac{2 M}{r}\right)\right]^{-1/2}$$
• The impact parameter $b$ is the minimum value of $r$ in the Newtonian theory
• Defining $u=1/r$ as before $$\frac{d \phi}{ d u} = \left( b^{-2} - u^2 + 2 M u^3\right)^{-1/2}$$
• Assume $M u \ll 1$ and define $y \equiv u(1-M u)$, then $$\frac{d \phi}{dy} \approx \frac{1 + 2 M y} {\left(b^{-2}-y^2\right)^{1/2}}$$

## Gravitational Deflection of Light II

• This gives $$\phi = \phi_0 + \frac{2 M}{b} + \mathrm{arcsin}(by) - 2 M \left(b^{-2}-y^2\right)^{1/2}$$
• The total deflection is $4 M/b$

## Black Holes and Effacement

• Effacement: ("process of eliminating something", in this case, eliminating complexity) black holes evolve towards a state which can be fully characterizied by $M$, $J$, $Q$
• Also may have other properties, depending on quantum theories of gravity
• Q likely small
• Total area of all horizons cannot decrease in time "area theorem" (violated by Hawking radiation)
• Curvature becomes infinite at the center ("the singularity")
• Area proportional to entropy

## Group Work

• Using the Kerr metric, solve for the radial locations where $g_{rr} \rightarrow \infty$ and $g_{tt} = 0$
• The proper metric for a rotating neutron star is a generalization of the metric. How many metric functions would you guess need to be computed?
• Examine Komatsu, Eriguchi, and Hachisu MNRAS 237 (1989) 355. How many metric functions do they compute?
• What method do they use to solve EFEs?