A possible way to capture the effects of quantum gravity in spacetime at a mesoscopic scale, for relatively low energies, is through an energy-dependent metric, such that particles with different energies probe different spacetimes. In this context, a clear connection between a geometrical approach and modifications of the special relativistic kinematics has been shown in the last few years. In this work, we focus on the geometrical interpretation of the relativistic deformed kinematics present in the framework of doubly special relativity, where a relativity principle is present. In this setting, we study the effects of a momentum dependence of the metric for a uniformly accelerated observer. We show how the local Rindler wedge description gets affected by the proposed observer-dependent metric, while the local Rindler causal structure is not, leading to a standard local causal horizon thermodynamic description. For the proposed modified metric, we can reproduce the derivation of Einstein’s equations as the equations of state for the thermal Rindler wedge. The conservation of the Einstein tensor leads to the same privileged momentum basis obtained in other works of some of the present authors, so supporting its relevance.

中文翻译：

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动量相关几何中的时空热力学

一种可能的方法来捕捉时空量子引力在介观上的影响对于相对较低的能量，尺度是通过与能量相关的度量来实现的，这样具有不同能量的粒子会探测不同的时空。在这种情况下，几何方法与狭义相对论运动学的修改之间的明确联系在过去几年中得到了证明。在这项工作中，我们专注于对存在于双狭义相对论框架中的相对论变形运动学的几何解释，其中存在相对性原理。在这种情况下，我们研究了度量的动量依赖性对匀加速观察者的影响。我们展示了局部 Rindler 楔形描述如何受到所提出的观察者相关度量的影响，而局部 Rindler 因果结构则不受此影响，从而得出标准的局部因果视界热力学描述。对于提出的修改度量，我们可以重现爱因斯坦方程的推导作为热 Rindler 楔的状态方程。爱因斯坦张量的守恒导致了在当前一些作者的其他作品中获得的相同的特权动量基础，因此支持了它的相关性。