| Summary: | In this paper, the generalized helical hypersurfaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">x</mi><mo>=</mo><mi mathvariant="bold">x</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula> with a time-like axis in Minkowski spacetime <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">E</mi><mrow><mn>1</mn></mrow><mn>4</mn></msubsup></semantics></math></inline-formula> are considered. The first and the second fundamental form matrices, the Gauss map, and the shape operator matrix of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">x</mi></semantics></math></inline-formula> are calculated. Moreover, the curvatures of the generalized helical hypersurface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">x</mi></semantics></math></inline-formula> are obtained by using the Cayley–Hamilton theorem. The umbilical conditions for the curvatures of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">x</mi></semantics></math></inline-formula> are given. Finally, the Laplace–Beltrami operator of the generalized helical hypersurface with a time-like axis is presented in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="double-struck">E</mi><mrow><mn>1</mn></mrow><mn>4</mn></msubsup></semantics></math></inline-formula>.
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