1. Introduction
A significant obstacle in constructing a quantum theory of gravity centers on a variety of issues known collectively as the problem of time [
1,
2,
3]. Although sometimes presented as a conceptual conflict between general relativity (GR) and quantum theory (see, for example, [
4]), the essential difficulty is already present in the context of classical GR. In relativistic physics, as well as in everyday language, the concept of time play two distinct roles [
5]. On the one hand time, time is one of four coordinates describing a location in spacetime, and in GR, these coordinates are dynamic quantities to be determined by equations of motion. But on the other hand, time is the chronological parameter required for posing and solving these equations. While this system is arguably well-posed in classical GR, its consistency in the presence of quantum fluctuations is a fundamental difficulty facing quantum gravity (see [
1] for a discussion of this question).
The distinction between these two aspects of time was already noted by Stueckelberg [
6,
7] in his foundational work on classical and quantum electrodynamics. In describing antiparticles as particles “going backward in time”, he found that neither the coordinate time
t nor the proper time of the motion
could faithfully parameterize chronological evolution in flat spacetime with
. Instead, he introduced a parameter
, external to the spacetime manifold and independent of the phase space coordinates, whose monotonic increase provides the “arrow of time” that determines chronology. In formulating the Klein–Gordon path integral, Feynman [
8] adopted a similar approach, now known as the Feynman–Stueckelberg interpretation.
In the Stueckelberg framework, a particle worldline is the trajectory of a classical event
or quantum event
generated by a Lorentz scalar Hamiltonian
K in canonical mechanics whose structure is familiar to nonrelativistic physics. Piron and Horwitz [
9] developed Stueckelberg’s work into a relativistic canonical many-body theory [
10,
11,
12,
13,
14] of interacting spacetime events. The resulting Stueckelberg–Horwitz–Piron (SHP) formalism admits
-dependent gauge freedom, leading to an electrodynamic theory in flat spacetime (see [
15,
16,
17] and references contained therein) with five gauge potentials
and
. These fields are designated
with the index convention
Although the free fields enjoy 5D gauge and spacetime symmetries, the description of particles must remain Lorentz-covariant, and so the symmetry of the fields breaks to O(3,1) when coupled with matter. The resulting theory of interacting events is integrable and recovers Maxwell electrodynamics in -equilibrium.
Horwitz extended the SHP framework to curved spacetime [
18,
19] with a background metric on 4D spacetime
, which reduces to
in flat spacetime. The equations of motion are the familiar geodesic equations of GR:
but are parameterized here by the external chronological time
, where
describe the Stueckelberg trajectory for the event
. For simple trajectories, it is often possible to invert
and reparameterize the system using
t, although the resulting description may be significantly more complicated. In the presence of pair creation and annihilation processes, the
trajectory will not be invertible [
17].
To develop this system into a fully dynamical theory framework, we must consider that events
will be associated with an event density
and energy–momentum tensor
that also depend explicitly on
. We must therefore find
-dependent field equations to be solved for a local metric
, reflecting Wheeler’s characterization of geometrodynamics [
20] as “spacetime tells matter how to move; matter tells spacetime how to curve”. In a recent series of papers (see [
21] and references contained therein), we proposed such field equations by constructing a
formalism that builds on the theory of embedded surfaces [
22,
23,
24], as applied in the
Arnowitt–Deser–Misner (ADM) formalism [
25].
In the framework, 4D spacetime is decomposed into a timelike direction, defining a time parameter t, and a 3D spacelike hypersurface whose metric evolves with t for . The well-known Einstein field equations can then be decomposed into an initial value problem for and the extrinsic curvature that expresses geometrical information about the t-direction.
To construct the
formalism, we define a 5D pseudo-spacetime
with coordinates
and a metric
to be discovered. Mimicking the ADM procedure, we take advantage of the natural foliation to decompose
into the
-like direction and the 4D hypersurface
, whose metric
evolves with
for
. The Einstein field equations are extended to 5D in such a way as to ensure that the geometrical structures (the free fields of the theory) retain 5D gauge and spacetime symmetry, while the description of matter possesses O(3,1) spacetime symmetry at most. These field equations can then be decomposed into an initial value problem for
and the extrinsic curvature
that expresses geometrical information about the
-direction. Because
is independent of spacetime in this formalism, the diffeomorphism invariance of general relativity does not raise questions about the evolution of the metric, as might occur under
t-evolution. Moreover, the matter dynamics are determined by a Lorentz scalar Hamiltonian and are similarly unaffected by coordinate transformations. These features should also apply to quantized canonical gravity, sidestepping concerns that quantum fluctuations may affect the evolution equations. For simple spacetime configurations, it may be possible to reparameterize the system by
and eliminate
from the solution for
. As Fock observed when first proposing the parameterization of electrodynamics in proper time [
26], the resulting solutions will generally be significantly more complicated, so that the
formalism is expected to provide more stable numerical solutions for complex energy–momentum configurations.
As in standard GR, the 5D field equations can be linearized for weak fields, leading to a wave equation for small perturbations to the flat space metric. This equation possesses a Green’s function [
17] that was found in the context of SHP electrodynamics and has been applied to find solutions in GR for sources of various types. For a ‘static particle’—a single event evenly distributed along the
t-axis in its rest frame, with no detailed information about its location in time—we obtain a Schwarzschild-like metric [
21].
However, a free Stueckelberg event in its rest frame is described by
, and we naturally expect that the time coordinate
will advance (or retreat) as
advances monotonically. A free Stueckelberg particle may be modeled as an ensemble of such events [
27] located at some point
in space, but narrowly distributed along the time axis according to some probability distribution
, where
is the nominal time coordinate. In SHP electrodynamics, the 5D wave equation then produces the Coulomb potential in the form
where
has its maximum at
. A test event at some spacetime point
will experience a potential whose support is centered around the chronological time
, the retarded time of the source. An observer located at this static point
x will see the potential rise and fall in strength as the chronological time
approaches and then retreats from the retarded time. But an observer similarly advancing along its worldline with
will see the potential with constant strength determined by
.
In [
21], we showed that this approach is not successful in finding an appropriate metric induced by an event localized on the
t-axis in the
formalism. A source ensemble of the type described above produces an energy–momentum tensor narrowly distributed in
given by
where
is the static energy–momentum tensor found in standard GR. Solving the wave equation in the linearized
formalism thus leads to a Schwarzschild-like metric in which the usual spacetime factor
is modified as
describing a localized gravitational field for a test event with a maximum on the lightcone of the source. In this picture, spacetime is flat and empty, except at the chronological moments
, for which the matter sources and the metric fields they induce have support in the neighborhood of a given spacetime point
. As for the Coulomb potential, an observer advancing along its worldline with
will experience the metric with constant strength determined by
.
Nevertheless, it was also shown in [
21] that the geodesic equations for a test event in any metric of the separable type (
6) lead to unreasonable equations of motion, including a dynamic reversal of the sign of the gravitational acceleration. This problem arises from the structure of the 5D Green’s function, which is itself subject to a number of open questions. As a result, the wave equation in linearized
gravitation does not provide an effective approach to the metric, and we must apply the full 4+1 evolution equations. For weak fields, one may pose the initial value problem by choosing an ansatz for the metric and seeking solutions for the desired metric as a perturbation of the ansatz under the influence of the perturbing source.
In this paper, we propose an ansatz with the desired properties, derive the associated source
from the wave equation, and set up the
metric evolution equations in the weak field approximation. In
Section 2, we briefly review the SHP formalism and
general relativity with invariant evolution.
Section 3 summarizes solutions to the wave equation using the Green’s function, which informs our choice of the ansatz metric. In
Section 4, we present an ansatz metric, examine the resulting geodesic equations, and derive the source using the wave equation. In
Section 5, we set up the
evolution equations by calculating the projected Ricci tensor and extrinsic curvature.
Section 6 presents a discussion of these results.
3. The Metric as Solution to a 5D Wave Equation
We now obtain the metric for a source event of constant velocity using Green’s function (
41) and discuss the difficulties with this approach. For such a source event, expression (
44) for the metric perturbation reduces to
where we denote
as the kinematic and dynamical factors. For a source evenly distributed along the
t-axis in its rest frame, we may write
so that the event density becomes
as is typically written for a “static” particle in standard relativity. Integration of the event density (
57) with Green’s function (
41) leads to
and so taking
the spacetime part of the metric becomes
where
and
. Naturally, this metric is spatially isotropic and
t-independent because the event density is spread evenly along the
t-axis. Transforming to spherical coordinates, (
61) becomes
which, for weak fields, is recognized as the Schwarzschild metric
when expressed in the isotropic coordinates [
30] defined through
This metric is well-known to be Ricci-flat, , a consequence of t-independence.
In [
21], we attempted to modify the calculation for the “static” event by specifying a trajectory narrowly distributed along the
t-axis at the spatial origin
. We wrote the event density
with support in a neighborhood around
. The kinematic factors are
where
, and the dynamic factors are
The leading term is easily found as
producing a gravitational field with a maximum at
for a given
. Thus, for a source located at
, a test event will feel the strongest gravitational force when moving on the lightcone of the source, accounting for the propagation time of the gravitational field. The similarity to the Coulomb force in (
4) in electrodynamics is clear. Detailed analysis of
shows [
21] that this term vanishes for
, while for
which drops off as
, leaving the contribution from
dominant at a long distance. Neglecting the subdominant term, the metric becomes
which seems reasonable as it modifies the Schwarzschild-like metric (
61) by restricting the support of its influence to
. At some chosen point
in spacetime, the metric is flat at
, rises to a maximum at
, and again returns to the flat metric at
. Thus, the effect of the metric in
GR depends on the trajectories of the source and test events moving through spacetime. In particular, for a test event evolving as
, the factor
and the gravitational field will be determined by the time-independent Schwarzschild metric.
The equations of motion for a nonrelativistic test event are found by expanding (
19) as
and neglecting terms containing
. The nonzero Christoffel symbols are
where we used
and dropped
. The equations of motion split into time and space components
which differ from (
27) because, in this case,
is explicitly
t-dependent. In spherical coordinates, the equations of motion are
where again, the conserved angular momentum is
. For a nonrelativistic test event, we can neglect
and take
. Then
and again, using (
60) for
, we find
for the radial equation. Any normalized distribution with maximum
will satisfy
when the test event is precisely on the source event’s lightcone and so the radial equation will recover Newtonian gravitation. However, for a narrow distribution,
will become large and negative just slightly away from the lightcone, and so the terms in parentheses in (
83) will likely become very large and possibly negative.
For example, if we consider the Gaussian distribution
where
is a time scale representing the width of the event distribution along the
t-axis, then for an event precisely on the lightcone of the source
However, if gravity accelerates the test event away from the lightcone and toward the source, so that
, then
which depends on the ratio
. Since we expect the radial distance
R to be large, while the width of the event distribution
is small by assumption, this term magnifies the gravitational field by a large factor. Worse still, if the initial conditions are such that
, then the gravitational force will become repulsive. To eliminate these problems from this model would require taking
, which puts
. But with this restriction, the metric components (
72) and (73) recover the
-independent metric (
61) and the
t-localization is lost, thus defeating the goal of the model.
We conclude that these problems will be present whenever the metric takes the separable form
, which must apply to any solution derived from the truncated Green function. As seen in (
4), the equal-
leading term in (
41) provides an approximate solution to the 5D wave function adequate for electrodynamics. However, it also provides an exact solution to the 4D wave equation on spacetime, as appears in the linearized form of standard 4D Einstein GR
where
is the 4D Ricci tensor. The 4D wave equation thus fails to capture the full geometrical content of
weak field GR, where the 5D wave equation is obtained from the linearized 5D Ricci tensor (
37). By separating the spacetime terms as
we immediately notice the reappearance of the Christoffel symbol
that is absent from the geodesic Equation (20) because of the constraint
. Since
represents contributions to the 5D curvature from
-evolution (extrinsic curvature), we see that (
86) must fail to take proper account of the
-evolution of the 4D geometry, regardless of the
-dependence of the source. So while
provides approximate solutions that are adequate for electrodynamics, it misrepresents the evolving spacetime geometry we seek to describe. For this reason, it becomes necessary to approach the metric through the 4 + 1 evolution equations described in
Section 2.4.
4. Ansatz Metric
We now switch directions, and instead of specifying the expected source for a gravitational field, we propose an ansatz for the metric that possesses certain expected properties. Using the 5D wave equation in the weak field approximation, we can then derive the source for the ansatz metric, evaluate the Ricci tensor and extrinsic curvature, and set up the
evolution equations for general mass–energy configurations. While the ansatz metric for the derived source satisfies the evolution equations exactly, the procedure leads to alternative metrics found as perturbations under general sources. The construction of general expressions characterizing perturbed metrics is discussed in standard texts on numerical relativity, such as [
22].
We propose an ansatz metric in the form
where
is a constant kinematic term,
is a spherically symmetric function containing the dependence on
t,
, and
. We specify the component structure of
to be
as found from (
67) for a source event evolving on the
t-axis.
We are interested in a metric that recovers the
-dependence of Newtonian gravitation but whose support is restricted to a neighborhood of
for a given test particle. We consider a source event in its rest frame evolving along its time axis as
, and a test event similarly evolving along its time axis at a spatial distance
r. This suggests a possible functional form of the type
where we denote
for convenience. Evaluating the connection for this metric, we study the trajectory of a test event determined by the geodesic Equation (
19) with the initial conditions
for which
, and
takes on its maximum value. If the test event deviates from the trajectory
, then the strength of the metric will diminish. Because
has a maximum at
with respect to
t but not with respect to
r, this functional form does not suffer from the difficulties associated with the geodesic Equation (
83), as discussed in
Section 3.
To write the geodesic equations, we use the derivatives
leading to the connection in the explicit form
so that splitting the equations of motion (
76) into time and space components, we arrive at
In the neighborhood of the initial conditions, the equations of motion reduce to
so that neglecting
in the nonrelativistic regime the time equation reduces to
and the space equation becomes
which recovers Newtonian gravitation if we take
.
Now, considering a relativistic test event with the initial condition
, and again keeping only lower order terms in
, the equations of motion become
Since
by assumption, while for non-tachyonic particles
, we see here that the event trajectory may differ from the post-Newtonian relativistic form seen in (
80).
In linearized GR, we may attribute the ansatz metric to a source by writing the field Equation (
16) in the linearized form (
40)
with spacetime components
for any spacetime functional
. Combining the derivatives of (
92)
and
we are led to
and if we again take
along with
, then
where only the energy component
is non-vanishing. The structure of this source is easiest to see for
, in which case
where
has units of length
−3, as expected for a particle density in space. Although
appears to vanish at
, we recall that under this condition,
so that
becomes independent of
t and
. In this case, the wave Equation (
111) from which we derive the source reduces to
which describes a point source evenly spread along the
t-axis. In contrast, at a small spatial distance
, we have
, which describes a narrow particle density along the
t-axis centered at
. This localization at a coordinate time
t determined by the chronological time
expresses the desired
-evolution. As expected, the source (
117) describes a matter distribution evolving with
, leading to the metric (
91), which similarly evolves with
, and the geodesic Equations (
102) and (
103), whose coefficients evolve with
.
5. 4 + 1 Evolution Equations
In
numerical relativity, we obtain the metric induced by a given source
by writing the evolution Equations (
50) and (
51) for a metric
expressed as a perturbation of a known metric
. In
relativity, it is common to use the York–Lichnerowicz [
31,
32] conformal decomposition
, for which the evolution equations become an initial value problem for the factor
, with initial conditions for the projected Ricci tensor
and the extrinsic curvature
derived from the unperturbed metric.
In weak field approximation, the
evolution equations are simplified [
29] by discarding terms of the order
. Combining Equations (
49) and (
91), we express the 5D metric as
from which
the Lie derivative of the metric reduces to
and the metric evolution Equation (
50) becomes
confirming that
is of the order
. As a result, we may neglect the Lie derivative in the second evolution Equation (
51)
as well as quadratic terms in
. Equation (
51) for the extrinsic curvature now reduces to
and the constraints become
with source terms
As an aside, we note that the 4 + 1 initial value problem for weak fields can be simply related to the 5D wave equation found from the Ricci tensor [
29]. To see this, we write the five components of Equations (
16) and (
39) as
from which it follows that any weak field solution with the component structure (
91) will satisfy the constraints. Before imposing the Lorenz gauge for the spacetime terms, we may split (
37) as
in which the first term is the projected Ricci tensor
and the second term contains the form of the extrinsic curvature
in the weak field approximation. Using the terms in parentheses for
, this expression can be rearranged as
which recovers the first evolution Equation (
122). Similarly, combining (
130) with (
16) and absorbing the negligible term
into
, we obtain
which recovers (
124). We thus see that the first-order evolution equations can be found by introducing the auxiliary variable
(almost, but not quite conjugate to
) and rearranging terms in the linearized field equations.
The initial value problem requires the Ricci tensor and extrinsic curvature computed from the unperturbed metric as initial conditions. Splitting the Lorenz gauge condition into spacetime and five parts
where
is the 5D trace, the projected Ricci tensor may be written as
providing a relatively simple means to calculate
. As an aside, we note that combining (
132) and (
134) to eliminate
, we recover the wave Equation (
40)
much as the elimination of momentum in favor of velocity in the Hamiltonian equations leads to second-order differential equations in particle mechanics.
For an ansatz metric with the component structure (
91), the source can be found directly from (
111) as we achieved in expression (
115). The projected Ricci tensor is found using (
134)
and the extrinsic curvature is
from which
We see that the off-diagonal terms in
and
mutually cancel, leaving the source diagonal as required. Inserting the explicit functional form (
92) for
we obtain
and
Using any suitable functional
to derive an ansatz metric (
91), its unperturbed source (
111), along with initial conditions (
136) and (
137), the weak field evolution Equations (
122) and (
124) are satisfied exactly. Writing the unperturbed source as
, the perturbed source
can be found using standard approximation methods. For example, we may write a source parameterized by
as
where
, and we seek a perturbed metric
up to the desired order in
. For example, writing
where
leads to
to first order. In the linearized theory, we also have the first-order
where
and
are derived from
. And since the evolution equations are linear and solved exactly by
for
, the evolution equations for the perturbed metric reduce to the evolution equations for the perturbation itself.
where we may use (
136) and (
137) to express
and
in terms of the perturbation
. We note that in order to preserve the structure of the evolution equations as an initial value problem, we must preserve
as an independent dynamical quantity and choose the initial values for
and
.
6. Discussion
In this paper, we considered the relationship between the evolving 4D metric
and the evolving matter source
in weak-field
general relativity and discussed an approach to determining the metric for a given source. We demonstrated the difficulties associated with applying the principal part Green’s function (
41) for the 5D wave equation to a source localized and evolving uniformly in its rest frame. We saw that the geodesic equations for a test event moving in the spacetime described by the resulting metric are highly sensitive to initial conditions and become unstable under general choices for
-synchronization with the source. This issue is attributed to the product structure
for general solutions found from the leading term
of (
41), where
is a narrow distribution that localizes the source along the
t-axis at a given
. This product structure was seen to result from the fact that in addition to approximate solutions to the 5D wave equations,
provides exact solutions to the 4D wave function. Consequently, these solutions lead to metric solutions to the 4D Einstein equations, not correctly accounting for the evolution of spacetime under
. Thus, while solving the wave equation for the linearized field equations is normally the simplest way to obtain the metric, we concluded that the shortcomings of the available Green’s function for the 5D wave equation make this method impractical. As for the case of general fields in 4D GR, the most suitable approach to the metric is to solve the ADM-like first-order evolution equations.
In order to apply the 4 + 1 formalism, we must begin with an unperturbed metric ansatz. Thus, we proposed an ansatz with many of the expected properties for the metric and found the associated matter source from the linearized field equations. In general, the source was seen to describe an event density evolving at a constant velocity along the t-axis in its rest frame, within a small volume . At a given point in the rest frame of the source, only has support in a narrow region of chronological time around . Consequently, the induced metric at a given point drops off as , but its support is similarly restricted to this narrow region around . As a test event evolves along some trajectory , it experiences the spacetime geometry as a function of , as well as its coordinate location. The geodesic equations for this metric were seen to recover Newtonian gravitation in the nonrelativistic limit, with possible deviations from standard models at relativistic energies.
We next proceeded to characterize the evolution equations in the weak field limit and demonstrate their relationship with the 5D Ricci tensor. We showed that the first-order initial value problem emerges from the linearized 5D field equations by introducing the extrinsic curvature as an auxiliary variable, much in the way that Hamiltonian mechanics poses first-order equations of motion for position and its conjugate momentum, treated as an independent quantity. The 5D wave equation is easily found by combining expressions for the projected Ricci tensor and , in such a way that off-diagonal terms cancel. Given the linearized evolution equations and the ansatz metric that solves them exactly for a given source, the initial value problem for a perturbed metric is just the linearized evolution equations for the perturbation terms themselves. We showed that with a presumed general structure for the perturbed metric, the evolution equations become a set of first-order (in ) partial differential equations in the parameters of the perturbation.
While the ansatz chosen for study in this paper contains the general features expected in an evolving metric, these methods can be applied to other forms. For example, writing the functional dependence of the metric as
for some short time scale
, we easily find the source to be
In the neighborhood of
, the geodesic equations become
and
which is identical to the equation of motion (
109) found using the functional form for
in (
92). These results suggest that other forms for
may lead to phenomenological behavior of a similar type. As we showed, perturbative modifications to (
149) may be considered by again replacing
, as we performed in (
144). Numerical solutions for perturbed metrics will be presented in a subsequent paper.