# Einstein Field Equation for a Garfinkle-Horowitz-Strominger theory

When deriving the Einstein field equation from a theory with a dilaton field, we have:

GHS's calculation

However, in my calculation:

My calculation

So I found that GHS has missed $$R$$ and $$(▽φ)^2$$. When setting $$φ->0$$, the conventional Einstein-Maxwell equation can be obtained from my equation but not GHS's.

However, all the articles are using GHS's results. It seems I am wrong. Who can explain this?

• Welcome to PSE. Please write formulas using MathJax instead of posting an image of hand-writing. – Thomas Fritsch Mar 24 at 17:11
• The lhs of GHS equation is Ricci tensor, not Einstein. Therefore the rhs is not $T_{\mu\nu}$ but $T_{\mu\nu}-\frac 12 T g_{\mu\nu}$. – A.V.S. Mar 24 at 18:59