Proving Horizon Structure of Complicated Metric

[latentcorpse]

Hi there,

I have a metric with $$g_{rr}=\frac{1}{r^2-2cr}$$. From this it is clear there exist coordinate singularities at r=0 and r=2c.

I believe that the outer horizon is the event horizon and the inner horizon is a Cauchy horizon. However, I do not know what I need to do in order to prove that this is the case. Can anyone offer any advice?

One option would be to try and draw a Penrose diagram. However, this usually involves lots of horrible coordinate transformations (even for the simplest examples) and my full metric is pretty complicated so if there's an alternative, I'd prefer that!

Thanks.

 

[mark726]

Hi there,

Thank you for your post. It's great to see that you are actively thinking about the implications of your metric. I can offer some advice on how to prove that the outer horizon is the event horizon and the inner horizon is a Cauchy horizon.

Firstly, let's define the event horizon and the Cauchy horizon. The event horizon is a boundary in spacetime beyond which nothing, including light, can escape from the gravitational pull of a black hole. The Cauchy horizon, on the other hand, is a boundary that separates the region of spacetime that can be predicted from the past history of the universe from the region that cannot be predicted.

Now, in order to prove that the outer horizon is the event horizon and the inner horizon is a Cauchy horizon, we need to analyze the behavior of the metric near these points. As you mentioned, the coordinate singularities occur at r=0 and r=2c. However, we need to look at the behavior of the metric in the limit as r approaches these values.

For r=0, we can see that g_{rr} approaches infinity. This indicates that the gravitational pull at this point is infinite, which is a characteristic of the event horizon. This means that nothing, not even light, can escape from this point, making it the event horizon.

For r=2c, we see that g_{rr} approaches zero. This indicates that the gravitational pull at this point is zero, which is a characteristic of the Cauchy horizon. This means that the region beyond this point cannot be predicted from the past history of the universe, making it the Cauchy horizon.

Therefore, by analyzing the behavior of the metric near r=0 and r=2c, we can prove that the outer horizon is the event horizon and the inner horizon is the Cauchy horizon.

In terms of drawing a Penrose diagram, it can be useful in visualizing the spacetime structure, but it is not necessary to prove the above statements. However, if you do decide to draw one, I suggest using Kruskal-Szekeres coordinates, which can simplify the process.

I hope this helps. Good luck with your research!