Schwarzschild解来自Raychaudhuri-Huygens关系
Schwarzschild solution from a Raychaudhuri-Huygens relation
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中文总结 AI 辅助
本文通过Raychaudhuri-Huygens关系，在不使用爱因斯坦方程的情况下，从几何嵌入推导出Schwarzschild解，揭示了引力相空间维度标度维度的非热起源，并可在引力透镜中观测。
中文摘要 AI 辅助
零测地线族编码了时空的因果结构，描述了辐射通道的相空间，其膨胀标量$\theta$满足Raychaudhuri方程。在球对称情形下，面积为$A=4\pi r^2$的波前的膨胀标量满足$\theta = A^\prime/A = 2/r$。对于质量$M$，在紫外-红外一致的耦合$\ell_p^2= G\hbar/c^3$下，这携带一个编码面积$A_E = \lambda \varphi \ell_p^2 = \lambda R_g r \le A$，与其康普顿相位$\varphi = (mc/\hbar)r \propto 1/\hbar$相关。这里，$\lambda = 8\pi $由大距离处的牛顿极限以及基于Rindler时空中等效原理的非局域扩展在关于$M$的视界处的饱和极限$A_E=A$所固定，而无需使用爱因斯坦方程。在球对称情形下，这从$p=A_E/A$的几何嵌入恢复了Schwarzschild度规，该比例在引力透镜中可直接观测。结果表明引力相空间维度为二的标度维度具有非热起源，此前Bekenstein（1973）基于热力学第二定律推导得出，这可能对宇宙学时空中产生影响。
英文摘要
Null-congruences encode the causal structure of spacetime, describing a phase space of radiation channels, whose expansion scalar $θ$ satisfies the Raychaudhuri equation. In spherical symmetry, the expansion scalar of wave fronts of area $A=4πr^2$ satisfies $θ= A^\prime/A = 2/r$. About a mass $M$, this carries an encoding area $A_E = λφ\ell_p^2 = λR_g r \le A$ in a UV-IR consistent coupling $\ell_p^2= G\hbar/c^3$ to its Compton phase $φ= (mc/\hbar)r \propto 1/\hbar$. Here, $λ= 8π$ is fixed by the Newtonian limit at large distances and the saturation limit $A_E=A$ at horizon about $M$ based on the non-local extension of the equivalence principle in Rindler spacetime, without using the Einstein equations. In spherical symmetry, this recovers the Schwarzschild metric from a geometric embedding of $p=A_E/A$, which is directly observable in gravitational lensing. The result shows a non-thermal origin of the scaling dimension of two of gravitational phase space, previously derived based on the second law of thermodynamics by Bekenstein (1973), with possible implications for cosmological spacetime.
