The Arnowitt Deser Misner (ADM) formalism [
1] in general relativity (GR) expresses the Einstein field equations in canonical form, thus permitting a solution of particular field/matter configurations formulated as initial value problems. As a canonical Hamiltonian formulation that splits four-dimensional (4D) spacetime into three-dimensional (3D) space and a selected time direction, ADM provides insight into general features of relativity, but is not always the most convenient of the 3+1 formulations for computation, especially numerical simulation. In this paper we borrow techniques from the 3+1 formalism in order to generalize the Stueckelberg–Horwitz–Piron (SHP) theory of classical electrodynamics [
2,
3,
4,
5,
6,
7] to SHP GR [
8,
9]. The SHP framework is a covariant canonical approach to relativistic classical and quantum mechanics, in which 4D spacetime events are defined with respect to coordinates
and an external evolution parameter
. Events trace out particle worldlines as functions
or
under the monotonic advance of
, producing five
-dependent gauge fields
carrying the interaction between events. (Here and throughout the SHP literature, Greek indices
take the values
, while
run from 0 to 3.) The result is an integrable electrodynamics, instantaneous in the external time
, but recovering Maxwell theory in a
-equilibrium limit. At numerous stages of analysis in SHP, an apparent five-dimensional (5D) symmetry arising from the five variables
must be judiciously broken to 4+1 representations of O(3,1), because the
are coordinates while
is an external parameter. In this paper we apply the lessons of SHP electrodynamics to a 4+1 theory of a local metric
. As we shall see, this approach differs from a 3+2 or (3+1)+1 formalism, in that we do not split 4D spacetime into space and time, maintaining the manifest spacetime covariance of the underlying physical picture in each step. Rather, we construct a purely formal 4+1 ⟶ 5D manifold as a guide to formulating field equations that under the 5D ⟶ 4+1 foliation describe a spacetime metric
evolving with
and preserving the required spacetime symmetries.
1.1. Motivation: The Problem of Time
In summarizing Einstein gravity as “Spacetime tells matter how to move; matter tells spacetime how to curve,” Wheeler [
10] touched on certain general issues in relativity known collectively as the problem of time In nonrelativistic mechanics, space is viewed as the “arena” of physical motion, a manifold with given background metric in some coordinate system, while time is an external parameter introduced to mark the coordinate evolution that characterizes the motion of objects in space. In contrast, time in general relativity retains its traditional Newtonian role as evolution parameter, but also serves as a coordinate, and thus, through the metric, plays a structural role in the spacetime “arena” itself. This dual role is complicated by the principal features of general relativity: the diffeomorphism invariance that eliminates any
a priori distinction between space and time coordinates, and the background independence that regards gravitation as equivalent to motion in the spacetime determined by the local metric.
Because the metric is itself determined by the time parameterized motion of matter, practical approaches to problems in gravitation generally pose the Einstein field equations and the equations of motion for matter as an initial value problem. Beginning with a consistent spacetime geometry at some time, one may solve for the evolution of spacetime and the motions of matter over time. Known as a 3+1 formalism, this approach singles out a time direction, as in standard Hamiltonian formulations of field theory, and so the equations are not manifestly covariant, although general covariance is preserved at each step [
1,
11,
12]. On the one hand, a configuration of matter and spacetime that satisfies the equations of GR represents a 4D block universe, given once and describing all space, past, present, and future. Additionally, on the other hand, we may find such solutions by integrating forward in time from consistent initial conditions at some time. In Wheeler’s words [
13], “A decade and more of work by Dirac, Bergmann, Schild, Pirani, Anderson, Higgs, Arnowitt, Deser, Misner, DeWitt, and others has taught us through many a hard knock that Einstein’s geometrodynamics deals with the dynamics of geometry: of 3-geometry, not 4-geometry.”
Unsurprisingly, the foliation of spacetime into three-geometries of simultaneous points in space further complicates the interpretation of time. Because time is only felt in the evolution from one 3D submanifold to another, the Hamiltonian is constrained to vanish when restricted to any given equal-time three-geometry [
12]. Moreover, there is no preferred criterion for choosing a functional of canonical variables that might be used as an intrinsic time parameter. While one may consider a physical clock that measures the proper time in some reference frame, the proper time depends on a spacetime trajectory that is only known after the equations of motion have been solved. While such a system may be well-posed in classical GR [
11], this is less obvious if the metric is subject to quantum fluctuations.
1.2. Stueckelberg-Horwitz-Piron (SHP) Theory
Stueckelberg–Horwitz–Piron (SHP) theory is a covariant approach to relativistic classical and quantum mechanics developed to address the problem of time as it arises in electrodynamics. In 1937 Fock proposed using proper time as the evolution parameter for a Newton-like force law, succinctly expressing a manifestly covariant formulation of electrodynamics [
14]. But, four years later, Stueckelberg proposed [
2,
3] to interpret antiparticles as particles moving backward in time, and showed that neither the coordinate time
nor the proper time of the motion could serve as evolution parameter for particle/antiparticle pair processes. Because
cannot remain constant during such processes, he introduced an external time
and argued that
can be a
-dependent dynamical quantity, even in flat space. In 1973, Horwitz and Piron [
4] were similarly led to use an external time in formulating a manifestly covariant relativistic mechanics with interactions, in order to overcome
a priori constraints on the 4D phase space that conflict with canonical structure. Thus, writing the eight-dimensional (8D) unconstrained phase space
the O(3,1)-symmetric action for a particle in Maxwell theory
leads to the Lorentz force in the covariant form found by Fock. However, because the potential
is produced by a Maxwell current
depending on the trajectory
that is only given
after the equations of motion have been solved, the system may not be well-posed. To overcome this conflict, Horwitz, Saad, and Arshansky [
5] extended the action (
2) by adding
-dependence to the vector potential, along with a new scalar potential, to obtain the action
where
, and in analogy to
, we write
. Compatibility of SHP electrodynamics with Maxwell theory requires
and we will neglect
where appropriate. If we take the potential to be pure gauge, as
, then the interaction term is just the total
-derivative of
, showing that this theory is the most general U(1) gauge theory on the unconstrained phase space (see also [
15]). Variation with respect to
leads to the Lorentz force [
16] in the form
where the field
is made a dynamical quantity by addition of a kinetic term of the type
to the total action. Because the apparent 5D symmetry of the interaction term
in the action (5) is broken to 4+1 in (
4), SHP electrodynamics differs in significant ways from 5D Maxwell theory. We notice that (7) permits the exchange of mass between particles and fields, and indicates the condition for non-conservation of proper time. It has been shown [
16] that the total mass, energy, and momentum of particles and fields are conserved.
These equations of motion, along with the
-dependent field equations, have been used to calculate [
17] the Bethe–Heitler mechanism for electron-positron production in classical electrodynamics. A positron (an electron with
) propagates backward in coordinate time until entering the bremsstrahlung field produced by another electron scattering off a heavy nucleus. This field leads to
, so the particle gains energy
continuously (and thus
changes sign twice) until emerging as an electron propagating forward in coordinate time with
. At coordinate times prior to the particle’s turn-around (when
) no particles will be observed, but two particles will be observed for subsequent coordinate times, implementing Stueckelberg’s picture of pair creation.
A physical event in SHP is an irreversible occurrence at time with spacetime coordinates . The formalism thereby implements the two aspects of time as distinct physical quantities: the coordinate time describing the locations of events, and the external Stueckelberg time describing the chronological order of event occurrence. This eliminates grandfather paradoxes because for an event at some spacetime point occurs after the event and cannot affect it. Similarly, the 4D block universe occurs at , representing the 4D manifold of general relativity, comprising all of space and coordinate time . A Hamiltonian K generates evolution of occurring at to an infinitesimally close 4D block universe occurring at . The configuration of spacetime, including the past and future of , may thus change infinitesimally from chronological moment to moment in . Thus, it is not unreasonable to expect that will be endowed with a -dependent metric whose dynamics we explore in this paper. On the contrary, a 4D metric given for all would have the character of an absolute background field in this formalism, in violation of the goals of general relativity.
For the kinetic term (
9) we formally raise the five-index of
although we understand the Lagrangian density as
with
simply the choice of sign for the vector-vector term. That is, we bear in mind that in this notation the
index is a formal convenience, indicating O(3,1) scalar quantities, not an element of a 5D tensor, and not a timelike coordinate. In particular,
is constrained to be a constant scalar, identical in all reference frames, and
must not be treated as a dynamical variable. Nevertheless, the contraction on indices
suggests a formal 5D symmetry, possibly O(4,1) or O(3,2) that breaks to O(3,1) in the presence of matter, and for convenience we write
in the form of a 5D flat space metric. Although the higher symmetry is non-physical for matter, it appears in wave equations, much as the wave equations for nonrelativistic acoustics appear to possess a Lorentz symmetry not associated with the physics. In developing an SHP approach to general relativity, we will similarly exploit this notation as a guide to the appropriate extension of GR while respecting the non-dynamical character of
.
Classical and quantum SHP particle mechanics in a spacetime with a
-independent local metric
has been studied extensively by Horwitz [
8,
9] and will not be discussed at length here. Our goal in this paper is to find a consistent prescription for extending general relativity to accommodate a metric
(where
) satisfying
-dependent Einstein equations on a formal 5D manifold whose meaning is explored through particle mechanics and field equations. As in standard approaches to GR, the study of embedded hypersurfaces is central to this program. But, while the 3+1 formalism begins with a 4D block universe
and defines a foliation into embedded spacelike hypersurfaces of equal coordinate time
t, the 4+1 formalism begins with a parameterized family of 4D spacetimes
embedded as hypersurfaces into a 5D pseudo-spacetime. Because the evolution of
is determined by an O(3,1) scalar Hamiltonian
K, with
as an external parameter (Poincaré invariant by definition), there is no conflict with the diffeomorphism invariance of general relativity. This approach will guide us toward the formal structures of a 5D manifold
with coordinates
on which we may perform a 4+1 foliation by choosing
as the unambiguously preferred time direction (See [
18,
19] for discussion of general 5D spacetime with preferred foliation.). We refer to
as a pseudo-spacetime to emphasize that despite the formal manifold structure, in specifying the physics we treat
as a parameter and not a coordinate. Moreover,
represents an admixture of symmetries: 4D spacetime geometry within each
, and canonical dynamics between any pair
,
. We expect no general diffeomorphism invariance for
.