The 3+1 formalism in general relativity, including the ADM Hamiltonian formalism and the BSSN formalism in numerical relativity, poses the field equations for the metric as an initial-value problem. It begins by decomposing the ten manifestly covariant Einstein equations in 4D spacetime into space and time sectors: 6 equations describing the -evolution of the space 3D metric and 4 non-evolving constraints. Stueckelberg-Horwitz-Piron (SHP) theory observes that neither clock time nor the proper time of the motion can serve as the evolution parameter in a well-posed relativistic theory and so introduces an external parameter to describe the evolution of classical worldlines and quantum wavefunctions . In this framework, Horwitz has developed classical and quantum Hamiltonian mechanics in a background metric , and we have previously described the -evolution of a 5D metric for linearized Einstein equations.
In this paper, we present a 4+1 formalism exploiting the natural role of the external time as evolution parameter to form a preferred foliation of the 5D pseudo-spacetime. From the 4D spacetime we construct a 5D metric on the 5D manifold , recalling at each step its admixture of geometry (the diffeomorphism invariance of ) and dynamics (the system evolution with the monotonic advance of , a scalar representation of the 4D symmetries). We obtain 10 unconstrained Einstein equations for the -evolution of the 4D metric and 5 constraints to be satisfied by the initial conditions.
A native of New York City, Martin Land received a BA in Physics from Reed College in Portland, Oregon in 1977, an MS in Electrical Engineering from Columbia University in New York in 1979, an MS in Physics from Hebrew University in 1986, and a PhD in Physics from Tel Aviv University in 1996. His dissertation presented a quantum field theory for Stueckelberg-Horwitz-Piron (SHP) theory, and he has continued to elaborate various aspects of the classical and quantum theories. He has also been active in computer hardware engineering and networking since 1975, developing fiberoptic networks at Bell Laboratories and parallel computing platforms at Hebrew University. He joined the faculty of Hadassah College in 1997, teaching computer architecture, networking, and embedded systems. He was a founding member of the International Association for Relativistic Dynamics (IARD) in 1998 and has served as IARD president since 2006.