Can the BTZ Black Hole Solutions Be Valid with Different Cosmological Constants?

[erasrot]

Hi everybody. I am well aware that there is only one black hole in 2+1, i.e., the BTZ one. I also know that for vanishing and positive cosmological constants we get solutions with a conical singularity. My question is more about the interpretation of these last results. Assume that in the BTZ solution you just change the value of the cosmological constant from negative to zero or positive. Is that valid? How are the geodesics to be interpreted in these spacetimes? Is just like in any geodesically incomplete background?

 

[bcrowell]

Assume that in the BTZ solution you just change the value of the cosmological constant from negative to zero or positive. Is that valid? How are the geodesics to be interpreted in these spacetimes? Is just like in any geodesically incomplete background?

It wouldn't be a solution to the vacuum field equations. The geodesics would have the same interpretation as in the BTZ solution, because the metric would be the same.

 

[erasrot]

Thanks for your answer. Do you mean that they are not vaccuum solutions because, to be a solution, the conical singularity at r=0 has to be treated as a point particle?

 

[Meselwulf]

Assume that in the BTZ solution you just change the value of the cosmological constant from negative to zero or positive. Is that valid? How are the geodesics to be interpreted in these spacetimes? Is just like in any geodesically incomplete background?

You can change it, but only one of the cosmological values most accurately predict the shape of out universe. If you read up on the Flatness Problem, you will find it. I believe it is +1. My memory could be failing me here, while since I read it.

 

[bcrowell]

Thanks for your answer. Do you mean that they are not vaccuum solutions because, to be a solution, the conical singularity at r=0 has to be treated as a point particle?

No. I mean that they aren't vacuum solutions because they wouldn't satisfy the Einstein field equations with T=0 and the value of \Lambda that actually applied to them.