How Einstein’s equations emerge from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>CFT</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:

The of the entanglement entropies between disjoint subsystems enables us to show that the dynamical equation of the entanglement entropy in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:msub> <a:mrow> <a:mi>CFT</a:mi> </a:mrow> <a:mrow> <a:mn>2</a:mn> </a:mrow> </a:msub> </a:mrow> </a:math> is precisely the three-dimensional Einstein equation. We establish a profound relation between the cosmological constant and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:msub> <c:mrow> <c:mi>CFT</c:mi> </c:mrow> <c:mrow> <c:mn>2</c:mn> </c:mrow> </c:msub> </c:mrow> </c:math> entanglement entropy. Thus, entanglement entropies induce internal gravitational geometries in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:msub> <e:mrow> <e:mi>CFT</e:mi> </e:mrow> <e:mrow> <e:mn>2</e:mn> </e:mrow> </e:msub> </e:mrow> </e:math> . Extracting the dual metric from an entanglement entropy becomes a straightforward procedure. Remarkably, we discover that the renormalization group equation is a geometric identity.