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. The ambiguity in selecting one of these two different methods has gained attention in the literature, but the best attempted proofs of equivalence of the two methods require restrictions on the order of derivatives and rank of fields; general equivalence of the Noether and Hilbert methods has not been established. 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.

文献中存在多种推导物理理论的能量动量张量的方法。最常见的方法是将Noether的第一个定理与4参数Poincaré平移配合使用，或者在弯曲的时空中编写动作并相对于度量张量执行变化，然后返回Minkowski时空。这些分别称为Noether和Hilbert（度量/弯曲空间/变分）能量动量张量。在电动力学和其他简单模型中，Noether和Hilbert方法得出的结果相同。由于这个事实，通常会断言这些方法对于所考虑的任何理论通常都是等效的，这使物理学家可以自由地根据所面临的问题使用任何一种方法来导出能量动量张量。选择这两种不同方法中的一种方法的歧义性在文献中得到了关注，但是试图最好地证明这两种方法的等效性要求限制导数的顺序和字段的等级。Noether和Hilbert方法的一般等效性尚未建立。对于自旋2而言，它是检验一个更复杂模型的等价性的理想候选者，文献中存在许多能量动量张量，它们都不是量规不变的，因此尚不清楚人们希望从Noether获得哪种表达希尔伯特的方法不同于例如电动力学的方法。但是，已经证明，线性化的高斯-邦尼引力模型（二阶导数，二阶张量势）具有唯一的，标量不变的，对称的，守恒的能量动量张量。从Noether的第一个定理推导出来，它是无痕的（电动力学的物理能量动量张量的所有相同属性）。这使得它成为检查Noether和Hilbert方法是否符合更复杂模型的理想选择。通过直接计算，在此使用该模型作为反例证明了Noether和Hilbert能量动量张量通常不是等效的。