论文标题
Noether和Hilbert（公制）能量量张量通常不是等效的
Noether and Hilbert (metric) energy-momentum tensors are not, in general, equivalent
论文作者
论文摘要
文献中存在多种用于物理理论的能量弹药张量的方法。最常见的方法是将Noether的第一个定理与4参数Poincaré翻译一起使用，或者在弯曲的时空中写下动作并相对于度量张量进行变化，然后返回Minkowski SpaceTime。这些分别称为Noether和Hilbert（度量/弯曲空间/变异）能量量张量。在电动力学和其他简单模型中，Noether和Hilbert方法产生相同的结果。由于这一事实，通常断言这些方法通常对于所考虑的任何理论都是等效的，并且这使物理学家可以自由地使用任何一种方法来得出能量弹药张量，具体取决于手头的问题。 $ \ dots $ for spin-2，是检查更为复杂的模型的理想候选者，文献中存在许多能量弹药张量，这些量没有量表不变，因此尚不清楚一个人希望从Noether和Hilbert的方法获得哪种表达，而与E.G.的情况不同。电动力学。 It has been shown, however, that the linearized Gauss-Bonnet gravity model (second order derivatives, second rank tensor potential) has an energy-momentum tensor that is unique, gauge invariant, symmetric, conserved, and trace-free when derived from Noether's first theorem (all the same properties of the physical energy-momentum tensor of electrodynamics).这使其成为检查Noether和Hilbert方法是否重合更复杂模型的理想候选人。通过直接计算，在这里证明了使用该模型作为反例子，即noether和Hilbert Energy-Momentum张量通常不是等效的。
Multiple methods for deriving the energy-momentum tensor for a physical theory exist in the literature. The most common methods are to use Noether's first theorem with the 4-parameter Poincaré translation, or to write the action in a curved spacetime and perform variation with respect to the metric tensor, then return to a Minkowski spacetime. These are referred to as the Noether and Hilbert (metric/ curved space/ variational) energy-momentum tensors, respectively. In electrodynamics and other simple models, the Noether and Hilbert methods yield the same result. Due to this fact, it is often asserted that these methods are generally equivalent for any theory considered, and that this gives physicists a freedom in using either method to derive an energy-momentum tensor depending on the problem at hand. $\dots$ For spin-2, the ideal candidate to check this equivalence for a more complicated model, there exist many energy-momentum tensors in the literature, none of which are gauge invariant, so it is not clear which expression one hopes to obtain from the Noether and Hilbert approaches unlike in the case of e.g. electrodynamics. It has been shown, however, that the linearized Gauss-Bonnet gravity model (second order derivatives, second rank tensor potential) has an energy-momentum tensor that is unique, gauge invariant, symmetric, conserved, and trace-free when derived from Noether's first theorem (all the same properties of the physical energy-momentum tensor of electrodynamics). This makes it the ideal candidate to check if the Noether and Hilbert methods coincide for a more complicated model. It is proven here using this model as a counterexample, by direct calculation, that the Noether and Hilbert energy-momentum tensors are not, in general, equivalent.
