The general theory of relativity (GR) is symmetric under smooth coordinate transformations, also known as diffeomorphisms. The general coordinate transformation group has a linear subgroup denoted as the Lorentz group of symmetry, which is also maintained in the weak field approximation to GR. The dominant operator in the weak field equation of GR is thus the d’Alembert (wave) operator, which has a retarded potential solution. Galaxies are huge physical systems with dimensions of many tens of thousands of light years. Thus, any change at the galactic center will be noticed at the rim only tens of thousands of years later. Those retardation effects are neglected in the present day galactic modelling used to calculate rotational velocities of matter in the rims of the galaxy and surrounding gas. The significant differences between the predictions of Newtonian instantaneous action at a distance and observed velocities are usually explained by either assuming dark matter or by modifying the laws of gravity (MOND). In this paper, we will show that, by taking general relativity seriously without neglecting retardation effects, one can explain the radial velocities of galactic matter in the M33 galaxy without postulating dark matter. It should be stressed that the current approach does not require that velocities v are high; in fact, the vast majority of galactic bodies (stars, gas) are substantially subluminal—in other words, the ratio of vc≪1. Typical velocities in galaxies are 100 km/s, which makes this ratio 0.001 or smaller. However, one should consider the fact that every gravitational system, even if it is made of subluminal bodies, has a retardation distance, beyond which the retardation effect cannot be neglected. Every natural system, such as stars and galaxies and even galactic clusters, exchanges mass with its environment, for example, the sun loses mass through solar wind and galaxies accrete gas from the intergalactic medium. This means that all natural gravitational systems have a finite retardation distance. The question is thus quantitative: how large is the retardation distance? For the M33 galaxy, the velocity curve indicates that the retardation effects cannot be neglected beyond a certain distance, which was calculated to be roughly 14,000 light years; similar analysis for other galaxies of different types has shown similar results. We demonstrate, using a detailed model, that this does not require a high velocity of gas or stars in or out of the galaxy and is perfectly consistent with the current observational knowledge of galactic and extra galactic material content and dynamics.

中文翻译：

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洛伦兹对称群、延迟、星系际质量损耗和导致星系旋转曲线的机制

广义相对论 (GR) 在平滑坐标变换下是对称的，也称为微分同胚。一般坐标变换群有一个线性子群，表示为洛伦兹对称群，它也保持在弱场近似中。因此，GR 弱场方程中的主要算子是 d'Alembert（波）算子，它具有延迟的潜在解。星系是具有数万光年尺寸的巨大物理系统。因此，仅在数万年后，银河系中心的任何变化都会在边缘被注意到。在当今用于计算星系边缘和周围气体中物质旋转速度的星系模型中，这些延迟效应被忽略了。牛顿远距离瞬时作用的预测与观察到的速度之间的显着差异通常可以通过假设暗物质或修改万有引力定律 (MOND) 来解释。在本文中，我们将证明，通过认真对待广义相对论而不忽视延迟效应，我们可以在不假设暗物质的情况下解释 M33 星系中星系物质的径向速度。需要强调的是，目前的方法并不要求速度 v 很高；事实上，绝大多数星系体（恒星、气体）基本上是亚光速的——换句话说，vc≪1的比率。星系中的典型速度为 100 公里/秒，这使得该比率为 0.001 或更小。然而，人们应该考虑这样一个事实，即每一个引力系统，即使它是由亚光体构成的，有一个延迟距离，超过这个延迟效果就不能忽略了。每个自然系统，例如恒星和星系，甚至星系团，都会与其环境交换质量，例如，太阳通过太阳风失去质量，星系从星系际介质中吸积气体。这意味着所有自然引力系统都有一个有限的延迟距离。因此，问题是定量的：延迟距离有多大？对于 M33 星系，速度曲线表明，在一定距离之外，延迟效应不可忽视，计算大约为 14,000 光年；对其他不同类型星系的类似分析也显示出类似的结果。我们使用详细的模型演示，