We work on a 4-manifold equipped with Lorentzian metric $g$ and consider a volume-preserving diffeomorphism which is the unknown quantity of our mathematical model. The diffeomorphism defines a second Lorentzian metric $h$, the pullback of $g$. Motivated by elasticity theory, we introduce a Lagrangian expressed algebraically (without differentiations) via our pair of metrics. Analysis of the resulting nonlinear field equations produces three main results. Firstly, we show that for Ricci-flat manifolds our linearised field equations are Maxwell's equations in the Lorenz gauge with exact current. Secondly, for Minkowski space we construct explicit massless solutions of our nonlinear field equations; these come in two distinct types, right-handed and left-handed. Thirdly, for Minkowski space we construct explicit massive solutions of our nonlinear field equations; these contain a positive parameter which has the geometric meaning of quantum mechanical mass and a real parameter which may be interpreted as electric charge. In constructing explicit solutions of nonlinear field equations we resort to group-theoretic ideas: we identify special 4-dimensional subgroups of the Poincare group and seek diffeomorphisms compatible with their action, in a suitable sense.

中文翻译：

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作为物质场的时空微分同胚

我们研究配备洛伦兹度量 $g$ 的 4 流形，并考虑保持体积的微分同胚，这是我们数学模型的未知量。微分同胚定义了第二个洛伦兹度量 $h$，即 $g$ 的回调。受弹性理论的启发，我们引入了通过我们的一对度量以代数方式（无微分）表示的拉格朗日量。对所得非线性场方程的分析产生三个主要结果。首先，我们证明对于 Ricci-flat 流形，我们的线性场方程是具有精确电流的 Lorenz 规范中的 Maxwell 方程。其次，对于闵可夫斯基空间，我们构造了非线性场方程的显式无质量解；这些有两种不同的类型，右手和左手。第三，对于闵可夫斯基空间，我们构造了非线性场方程的显式大规模解；它们包含一个具有量子力学质量几何意义的正参数和一个可以解释为电荷的实参数。在构造非线性场方程的显式解时，我们求助于群论思想：我们识别 Poincare 群的特殊 4 维子群，并在适当的意义上寻求与其作用兼容的微分同胚。