A Covariant Derivative of Stress Energy Tensor of Scalar Field on Shell

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The discussion revolves around proving a specific formula related to the covariant derivative of the stress-energy tensor of a scalar field on a shell. The original poster shares their work and seeks feedback on potential errors in their approach. Participants are encouraged to provide insights or corrections to assist in the proof. The moderator has elevated the thread's importance level. Clarifications and collaborative problem-solving are central to the conversation.
thatboi
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Hi all,
I am currently trying to prove formula 21 from the attached paper.
My work is as follows:
1.PNG


If anyone can point out where I went wrong I would greatly appreciate it! Thanks.
 

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