I have a question about the paper:(adsbygoogle = window.adsbygoogle || []).push({});

C. G. . Callan, R. C. Myers and M. J. Perry, “Black Holes In String Theory,” Nucl. Phys. B

311, 673 (1989).

I have attached the relevant section.

I am having trouble using equations (2.1) and (2.4) to derive (2.5) and (2.6). When I do the calculation, I do not get any [itex]\phi'[/itex] terms (i.e. first derivatives --- I do get all the second-derivatives of phi and all other terms).

For example, I don't understand how the (tt) and (θθ) equations can have [itex]\phi'[/itex] terms. Doesn't (2.1a) contain

[itex]\nabla_{a}\nabla_{b}\phi = \partial_a \partial_b \varphi(r) = 0[/itex] unless [itex]a = b = r [/itex]?

Thus shouldn't only the (rr) equation contain a [itex]\phi[/itex]-term... and shouldn't it just be a second derivative?

For another example, since [itex]\nabla \phi[/itex] is first-order in [itex]\lambda[/itex], isn't [itex](\nabla \phi)^2[/itex] second-order in [itex]\lambda[/itex] and thus irrelevant for equation (2.6)? This would lead me to conclude that there should be no first derivative of [itex]\phi[/itex] in (2.6). Where does it come from?

I would greatly appreciate some insight. Tell me where I am going wrong!

Thanks in advance,

Chris

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# Black Holes in String Theory - calculation help needed!

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