In four dimensions complexified general relativity (GR) can be non-trivially deformed: there exists an (infinite-parameter) set of modifications all having the same count of degrees of freedom. It is trivial to impose reality conditions that give versions of the deformed theories corresponding to Riemannian and split metric signatures. We revisit the Lorentzian signature case. To make the problem tractable, we restrict our attention to a four-parameter set of deformations that are natural extensions of Ashtekar’s Hamiltonian formalism for GR. The Hamiltonian of the later is a linear combination of EEE and EEB. We consider theories for which the Hamiltonian constraint is a general linear combination of EEE, EEB, EBB and BBB. Our main result is the computation of the evolution equations for the modified theories as geometrodynamics evolution equations for the three-metric. We show that only for GR (and the related theory of self-dual gravity) these equations close in the sense that they can be written in terms of only the metric and its first time derivative. Modified theories are therefore seen to be essentially non-metric in the sense that their dynamics cannot be reduced to geometrodynamics. We then show this to be related to the problem with Lorentzian reality conditions: the conditions of reality of the three-metric and its time derivative are not acceptable because they are not preserved by the dynamics. Put differently, their conservation implies extra reality conditions on higher-order time derivatives, which then leaves no room for degrees of freedom.

在四维中，复杂的广义相对论 (GR) 可以非平凡地变形：存在一组（无限参数）修改，所有修改都具有相同的自由度数。强加现实条件是微不足道的，这些条件给出对应于黎曼和分裂度量签名的变形理论的版本。我们重新审视洛伦兹签名案例。为了使问题易于处理，我们将注意力限制在四参数变形集上，这些变形是 Ashtekar 的 GR 哈密顿形式主义的自然扩展。后者的哈密顿量是EEE和EEB的线性组合。我们考虑哈密顿约束是EEE , EEB , EBB的一般线性组合的理论和BBB. 我们的主要结果是将修正理论的演化方程计算为三度量的几何动力学演化方程。我们表明，仅对于 GR（以及自对偶引力的相关理论），这些方程在它们可以仅根据度量及其一阶时间导数写成的意义上是接近的。因此，修改后的理论被视为本质上是非度量的，因为它们的动力学不能简化为几何动力学。然后我们证明这与洛伦兹现实条件的问题有关：三度量的现实条件及其时间导数是不可接受的，因为它们没有被动力学保存。换句话说，它们的守恒意味着高阶时间导数的额外现实条件，这样就没有自由度的余地。