This paper provides derivation of some basic identities for complex four-component vectors defined in a complex four-dimensional spacetime frame specified by an imaginary temporal axis. The resulting four-vector ident...This paper provides derivation of some basic identities for complex four-component vectors defined in a complex four-dimensional spacetime frame specified by an imaginary temporal axis. The resulting four-vector identities take exactly the same forms of the standard vector identities established in the familiar three-dimensional space, thereby confirming the consistency of the definition of the complex four-vectors and their mathematical operations in the general complex spacetime frame. Contravariant and covariant forms have been defined, providing appropriate definitions of complex tensors, which point to the possibility of reformulating differential geometry within a spacetime frame.展开更多
Analysis of a four-dimensional displacement vector on the fabric of space-time in the special or general case into two Four-dimensional vectors, according to specific conditions leads to the splitting of the total fab...Analysis of a four-dimensional displacement vector on the fabric of space-time in the special or general case into two Four-dimensional vectors, according to specific conditions leads to the splitting of the total fabric of space-time into a positive subspace-time that represents the space of causality and a negative subspace-time which represents a space without causality, thus, in the special case, we have new transformations for the coordinates of space and time modified from Lorentz transformations specific to each subspace, where the contraction of length disappears and the speed of light is no longer a universal constant. In the general case, we have new types of matric tensor, one for positive subspace-time and the other for negative subspace-time. We also find that the speed of the photon decreases in positive subspace-time until it reaches zero and increases in negative subspace-time until it reaches the speed of light when the photon reaches the Schwarzschild radius.展开更多
文摘This paper provides derivation of some basic identities for complex four-component vectors defined in a complex four-dimensional spacetime frame specified by an imaginary temporal axis. The resulting four-vector identities take exactly the same forms of the standard vector identities established in the familiar three-dimensional space, thereby confirming the consistency of the definition of the complex four-vectors and their mathematical operations in the general complex spacetime frame. Contravariant and covariant forms have been defined, providing appropriate definitions of complex tensors, which point to the possibility of reformulating differential geometry within a spacetime frame.
文摘Analysis of a four-dimensional displacement vector on the fabric of space-time in the special or general case into two Four-dimensional vectors, according to specific conditions leads to the splitting of the total fabric of space-time into a positive subspace-time that represents the space of causality and a negative subspace-time which represents a space without causality, thus, in the special case, we have new transformations for the coordinates of space and time modified from Lorentz transformations specific to each subspace, where the contraction of length disappears and the speed of light is no longer a universal constant. In the general case, we have new types of matric tensor, one for positive subspace-time and the other for negative subspace-time. We also find that the speed of the photon decreases in positive subspace-time until it reaches zero and increases in negative subspace-time until it reaches the speed of light when the photon reaches the Schwarzschild radius.
