Difficult general relativity problem - Weyl's axisymmetric solution

[patrick_gold]

I'm currently doing a course on general relativity, and I'm struggling with the following exercise - I would greatly appreciate the help anyone might offer. Whilst not technically homework (not assessed or in a tutorial set) it is from a collection of difficult questions from a recommended resource. It is as follows:

Homework Statement

Weyl's solution to the field equations in a vacuum has that

$$ ds^{2} = e^{2U}dt^{2} -e^{-2U}(r^{2}d\phi ^{2} + e^{2V}(dr^{2} +dz^2)) $$ $$ where \; \; U=U(r,z),\; V=V(r,z); $$

and the field equations reduce to Laplace's equation in cylindrical coordinates, that being

$$ \nabla^{2}U = U_{,rr} + r^{-1}U_{,r} + U_{,zz}=0, $$

as given U we can find V by integration:

$$ V_{,r}=r((U_{,r})^{2}-(U_{,z})^2)\;, \; \; \; V_{,z}=2rU_{,r}U_{,z} \;. $$

This suggests a way of generating solutions to the Einstein Fields equations: Laplace's
equation is linear, so just add arbitrary (axially symmetric) solutions of it to get U
and then find V .


(a) Why is the Weyl metric static?


(b) Explain why adding a constant to either (or both) U and V is geometrically
(and hence physically) irrelevant.


(c) If $$U\equiv 0 $$ we clearly have flat space. Show that if $$U = V$$ we also have flat space. [Hint: perhaps we might change coordinates (something about hyperbolic trig functions and rapidity?)] This suggests generating GR solutions from Newtonian ones will not be as easy as it seems.


(d)The simplest non-trivial solution to Laplace's equation is the (spherically symmetric) potential of a point particle: $$U=-\frac{m}{\sqrt{r^{2}+z^{2}}}$$. Show this is a solution to $$\nabla^{2}U=0$$, find the corresponding V and explain why the metric is asymptotically flat.


(e) The solution is part (d) is known as the Curzon-Chazy solution. Prove it is not Schwarzschild and so not spherically symmetric. (Thus once again demonstrating that the Newtonian solution and the Einsteinian solutions are not simply related.)


*Nota bene c=1 here.

Homework Equations

See question statement

The Attempt at a Solution

I've struggled a bit, there's some obvious substitution work in here verifying things are solutions but I'm not very confident with the rest of the material.

 

[NateD]

(a) The Weyl metric is static as the metric does not depend on time.(b) Adding a constant to either U or V does not change the metric in terms of its geometry, and thus any physical effects of the metric are unchanged.(c) I'm a bit lost here. (d) $$\nabla^{2}U = U_{,rr} + r^{-1}U_{,r} + U_{,zz}= -\frac{m}{(r^{2}+z^{2})^{\frac{3}{2}}}(-2r^{2}+2z^{2}) + \frac{m}{r(r^{2}+z^{2})^{\frac{1}{2}}} + \frac{m}{(r^{2}+z^{2})^{\frac{3}{2}}}(-2z^{2}+2r^{2}) =0.$$V can be calculated from the given expression for V_{,r} and V_{,z}. The metric is asymptotically flat as it approaches a flat metric as values for r and z become large.(e) I'm also a bit lost here.