标题： 保持零锥的映射，因果张量和代数Rainich理论 显示英文标题

标题： Null cone preserving maps, causal tensors and algebraic Rainich theory

摘要： 在一个洛伦兹流形V上，与n个任意的因果未来方向向量收缩后结果为非负的秩-n张量，被称为具有主导性质。这些张量（不考虑符号）称为因果张量，我们在维度N中确定它们的一般性质。我们证明了将零锥映射到自身的秩-2张量是因果的。已知对于V上的任何张量A，都存在一个对应的“超能量”（s-e）张量T{A}，该张量总是具有主导性质。我们证明了反过来，任何具有主导性质的对称秩-2张量都可以以规范方式表示为N个简单形式的s-e张量之和。我们证明了当N<5时，任何秩-2 s-e张量的平方与度规成比例，并且对于任意N，任何简单形式的s-e张量也是如此。反过来，我们证明了任何其平方与度规成比例的对称秩-2张量，必须是某个简单p-形式的s-e张量（不考虑符号），并且张量T的迹决定了该形式的秩p。这在N和秩p方面推广了经典的代数Rainich条件，这些条件是度规源自某种物理场的必要且充分条件，并且具有几何解释：简单形式的s-e张量集合恰好是保持零锥及其时间定向的张量集合。这也意味着所有对合的洛伦兹变换（LT）都可以表示为简单形式的s-e张量，并且任何秩-2 s-e张量最多是N个共形对合LT的和。非对称的保持零锥的映射被证明具有因果的对称部分，并根据反对称部分的零特征向量进行分类。因此，我们得到了V上所有共形LT和奇异零锥保持映射的完整分类。 显示英文摘要

摘要： A rank-n tensor on a Lorentzian manifold V whose contraction with n arbitrary causal future directed vectors is non-negative is said to have the dominant property. These tensors, up to sign, are called causal tensors, and we determine their general properties in dimension N. We prove that rank-2 tensors which map the null cone on itself are causal. It is known that, to any tensor A on V there is a corresponding ``superenergy'' (s-e) tensor T{A} which always has the dominant property. We prove that, conversely, any symmetric rank-2 tensor with the dominant property can be written in a canonical way as a sum of N s-e tensors of simple forms. We show that the square of any rank-2 s-e tensor is proportional to the metric if N<5, and that this holds for the s-e tensor of any simple form for arbitrary N. Conversely, we prove that any symmetric rank-2 tensor T whose square is proportional to the metric must be, up to sign, the s-e of a simple p-form, and that the trace of T determines the rank p of the form. This generalises, both with respect to N and the rank p, the classical algebraic Rainich conditions, which are necessary and sufficient conditions for a metric to originate in some physical field, and has a geometric interpretation: the set of s-e tensors of simple forms is precisely the set of tensors which preserve the null cone and its time orientation. It also means that all involutory Lorentz transformations (LT) can be represented as s-e tensors of simple forms, and that any rank-2 s-e tensor is the sum of at most N conformally involutory LT. Non-symmetric null cone preserving maps are shown to have a causal symmetric part and are classified according to the null eigenvectors of the skew-symmetric part. We thus obtain a complete classification of all conformal LT and singular null cone preserving maps on V.