GR Cone Singularity Homework: Q1 & Q2 on Setting B(r=0)=0

[binbagsss]

Homework Statement

question attached-
I am stuck on some of the reasoning as to why we set $B(r=0)=0$

please not as described in the attempt I am interested in the concepts and have not posted my solution to A, nor any solutions not relevant to the concepts I am trying to understand

Homework Equations

below

The Attempt at a Solution

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QUESTION 1
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I understand $r=0$ and $r=R$ are going to be the 'critical ' points to look at to loose both integration constants. I am basically stuck on the concepts about the metric descrbing a cone etc.
So I understand that the tip of a cone is a singularity as it is 'geodesically incomplete'. However I am confused with the interpretation of what body\shape one claims that the metric describes, since the coordiantes have no physical meaning as they are just a way of parameterising the manifold.

So, if I can find some coordinates that take the metric to that of a cone, why do I interpret this as significant enough for me to decide that singularity is 'physical' enough for me to use it to impose constraints on $B(r)$ . i.e. there must be multiplie coordinate choices that will allow me to take it to a wide - range of shaped space-times as described by the metric, if these are all regular with no singularities I don't worry, but if there is just one such coordiante transformation that takes me to a body with a singularity I use it as a constraint- so original coordiantes that the metric is in must be such that any possible coordinate transformation from that, yields a metric that has no singularities? Bit confused with this since we said coordiantes are not physically significant.

QUESTION 2
Also, I'm actually stuck as to why this describes a cone. I see the solution comments on the period of the cone,being less than a circle of 2pi, i can see that the new angle defined clearly has a period less than 2pi, however, I have no idea of what you define the period of a cone to be ?!

many thanks

 

[vela]

If you calculate the circumference of the circle $r=r_0$, you'd get a result that's less than $2\pi r_0$.

You could also look at it as if you remove a wedge from a flat plane, since the period of $\theta'$ is less than $2\pi$, and then stitched the two edges together. Do you see that you're going to end up with a cone?

 

[binbagsss]

If you calculate the circumference of the circle $r=r_0$, you'd get a result that's less than $2\pi r_0$.

You could also look at it as if you remove a wedge from a flat plane, since the period of $\theta'$ is less than $2\pi$, and then stitched the two edges together. Do you see that you're going to end up with a cone?

how do you define the period of a cone?
is each cross-section of a cone not a circle?

 

[binbagsss]

how do you define the period of a cone?
is each cross-section of a cone not a circle?

mmm as in the 2-d image i mean by 'cross section' if it's a 3d cross section, i.e. a segment then the diameter of that will decrease as you view it from the circular end of the cone approaching the tip. but would you not have to define a period for each such '2-d cross-section' - i.e. the circle? how else do you define the period?

 

[vela]

It's nothing too mysterious. The angle $\theta$ is the usual polar angle which ranges from 0 to $2\pi$. $\theta'$, on the other hand, only goes from $0$ to $2\pi e^{-B(0)}$ before you return to the same point on the manifold (for constant $r$).

 

[binbagsss]

It's nothing too mysterious. The angle $\theta$ is the usual polar angle which ranges from 0 to $2\pi$. $\theta'$, on the other hand, only goes from $0$ to $2\pi e^{-B(0)}$ before you return to the same point on the manifold (for constant $r$).

Yes exactly , for constant r , and therefore shouldn’t it be a function of r ? Hence my question defining the period for each such ‘ circular cross section ‘ ?