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I am currently taking a course on general relativity with the book "General Relativity: An introduction for physicists" by M.P. Hobson. I am a having a hard time understanding the derivation presented for the Einstein equations. The book states that Einstein proposed the following relation: $$ K_{\mu \nu} = \kappa T_{\mu \nu}$$ Here the tensor ##K_{\mu \nu}## is related to the curvature of spacetime. My book states that this tensor needs to fulfill two conditions. It should contain no terms higher than linear in the second order derivatives of the metric tensor, and the tensor should be symmetric. The book then shows a general form:$$ K_{\mu \nu} = aR_{\mu \nu} + bRg_{\mu \nu} + \lambda g_{\mu \nu} $$ and claims that the last constant ##\lambda## is immediately zero because of the first condition.

I have two questions. I understand that the tensor has to be symmetric, but I am not really sure what they mean with the first condition. I know the tensor is related to the laplacian of the metric tensor, so is that the reason that it should be linear in the second derivatives? Or am I reading this wrong?

As a followup question, why does this condition demand ##\lambda## to equal zero? Why is the last term not linear in the second derivatives of the metric?

Thanks!