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By means of a family of counter-examples, it is shown that the Reilly upperbound for the first eigenvalue of the Laplace operator for a compactsubmanifold in Euclidean space does not work for $n$-dimensional compactspacelike submanifolds of Lorentz-Minkowski spacetime $\mathbb{L}^m$, $m\geqn+2$. We develop a new suitable technique, based on an integral formula oncompact spacelike sections of the light cone in $\mathbb{L}^m$. Then, a familyof extrinsic upper bounds for the first eigenvalue of the Laplace operator fora compact spacelike submanifold in $\mathbb{L}^m$ is proved. For each one ofthese inequalities, becoming an equality can be characterized in geometricterms. In particular, the eigenvalue achieves one of these upper bounds if andonly if the submanifold lies minimally in certain hypersphere of a spacelikehyperplane.

On the First eigenvalue of the Laplace operator for Compact Spacelike submanifolds in Lorentz-Minkowski Spacetime $\mathbb{L}^{m}$