Euclidean Action and the Einstein tensor

Abstract

I give a local description of the Euclidean regime $(M,g_{ab},u^a)$ of Lorentzian spacetimes $(M,g_{ab})$ based on timelike geodesics $u^a$ passing through an arbitrary event $p_0\in M$. I show that, to leading order, the Euclidean Einstein-Hilbert action $I_E$ is proportional to the Einstein tensor $G_{ab}{u}^a{u}^b$ of $g_{ab}$. The positivity of $I_E$ follows if $G_{ab}{u}^a{u}^b>0$ holds. I suggest an interpretation of this result in terms of the amplitude $\mathcal{A}[{\Sigma }_0]=\exp [-I_E]$ for a single space-like hypersurface ${\Sigma }_0\in I^{+}(p_0)$ to emerge at a constant geodesic distance ${\lambda }_0$ from $p_0$. Implications for classical and quantum gravity are discussed.

6 pages, 2 figures, added comments and expanded discussion of some implications, matches version accepted in Phys. Rev. D

References in corpus (7)

• Entropy of Null Surfaces and Dynamics of Spacetime
• Emergent Gravity Paradigm: Recent Progress
• Grin of the Cheshire cat: Entropy density of spacetime as a relic from quantum gravity
• Cosmological Constant from the Emergent Gravity Perspective
• The atoms of spacetime and the cosmological constant
• Intrinsic and Extrinsic curvatures in Finsler-esque spaces
• Action and Observer dependence in Euclidean quantum gravity

Cited by in corpus (5)

• Wicked metrics
• Feynman's i-epsilon prescription, almost real spacetimes, and acceptable complex spacetimes
• Probing the Planck scale: The modification of the time evolution operator due to the quantum structure of spacetime
• Information content and minimum-length metric: A drop of light
• Geometric aspects of covariant Wick rotation