Induced Metric Help: Troubleshooting Extrinsic Curvature (12)

[thatboi]

I am having trouble calculating the extrinsic curvature (12) in the following paper: https://arxiv.org/pdf/gr-qc/0310107.pdf
Specifically, I am unsure of what term to plug in for the induced metric h_{ab} in (8). If I am calculating the \sigma term in (12) is h_{ab} all of (4)?
Also I would like assistance in knowing what the {m(a)/a}' term means in (11) because I cannot get (13) either...

 

[Ambitwistor]

Hello,

Thank you for reaching out for help with your calculations. I understand your confusion with the terms in the paper. Let me try to explain them to you.

Firstly, in order to calculate the extrinsic curvature (12), you need to use the induced metric h_{ab} in (8). This metric represents the geometry of a 3-dimensional hypersurface embedded in a 4-dimensional spacetime. In this case, h_{ab} is the metric of the hypersurface defined by the coordinates (x,y,z) in (4). So, you can plug in the values of (4) for h_{ab}.

Moving on to the \sigma term in (12), it is defined as the trace of the extrinsic curvature. In other words, it is the sum of the diagonal elements of the extrinsic curvature tensor. So, once you have calculated the extrinsic curvature using (12), you can simply take the trace of the tensor to get the \sigma term.

Now, for the {m(a)/a}' term in (11), it represents the derivative of the mass function m(a) with respect to the scale factor a. This term is used to calculate (13), which is the time derivative of the extrinsic curvature. In order to understand this term better, I would suggest referring to the definitions and equations in the paper "On the mass function and the extrinsic curvature in general relativity" by B. Dittrich.

I hope this helps clarify your doubts. Please let me know if you need further assistance with your calculations. Best of luck with your research!