Two dimensional anti de Sitter space without cosmological constant?

[implicitnone]

I have a spacetime which is (I think) AdS_2. The metric is,

I'm trying to find the Einstein tensor, defined as,

and the result is zero! I thought that, for the AdS spacetimes I needed to have a nonzero Einstein tensor which is caused by the cosmological constant.

What is wrong with my calculation? Or can you assure me that my result is correct and this metric is not AdS?

I'm using GRTensorII to do the calculations.

 

[aashay]

hey check your calculations...I tried calculating it manually and I got a non-zero Einstein tensor. I got it as a diagonal matrix.

 

[implicitnone]

hey check your calculations...I tried calculating it manually and I got a non-zero Einstein tensor. I got it as a diagonal matrix.

Thank you for your efforts. I will do so. It will be the first time that grtensor confused me.

 

[aashay]

You are welcome..By the way, what is 'grtensor'?Is it some kind of software?

 

[implicitnone]

You are welcome..By the way, what is 'grtensor'?Is it some kind of software?

GRTensorII is a Maple package for tensor and GR calculations:

http://grtensor.org/

 

[implicitnone]

Very weird but I also tried with two other ways with computer. The internal packages of Maple and EinsteinTensor package of Mathematica library. They both gave zero Enstein tensor again!

 

[bcrowell]

I checked using Maxima and ctensor, and I verified that the Einstein tensor was zero:

load(ctensor);
dim:2;
ct_coords:[t,x];
lg:matrix([-(1+x^2),0],
          [0,1/(1+x^2)]);
cmetric();
einstein(true);

Maxima and ctensor are free and open-source, so anyone who wants to check this and play around with it can do so. E.g., if aashay has calculated the Riemann tensor, he/she could change the code so that it would output that, and we could compare and see if they're the same.

I would assume that spacetime curvature has a very simple characterization in 1+1 dimensions, e.g., there can only be one nonvanishing element in the Riemann tensor.

 

[implicitnone]

Thank you very much bcrowell!

Learning never ends :) It was my first deal with 1+1 gravity and now I checked with some books and saw that the Einstein tensor identically vanishes in 2D gravity because of the definition.

(See for example: "Lower dimensional gravity" by John David Brown or "Diverse topics in theoretical and mathematical physics" by Roman Jackiw. They are both available as Google books.)

 

[bcrowell]

Learning never ends :) It was my first deal with 1+1 gravity and now I checked with some books and saw that the Einstein tensor identically vanishes in 2D gravity because of the definition.

Ah, that's interesting. Kind of counterintuitive, since you *can* have intrinsic curvature in two spatial dimensions.

 

[implicitnone]

Ah, that's interesting. Kind of counterintuitive, since you *can* have intrinsic curvature in two spatial dimensions.

It really is! Knowing that I can still count on GRTensorII feels good :)