In flat Minkowski spacetime with ${\eta }_{\mu \nu }=\mathrm{diag}\left(-1,1,1,1\right)$, the free particle action is made maximally U(1) gauge invariant [15,24] by introducing five gauge fields as where we introduce $x^5=c_5\tau$ in analogy to $x^0=ct$ and partition Greek indices such that

This action enjoys the 5D gauge invariance $a_{\alpha }\left(x,\tau \right)⟶a_{\alpha }\left(x,\tau \right)+{\partial }_{\alpha }\mathsf{\Lambda }(x,\tau )$, but because ${\dot{x}}^{\mu }{\dot{x}}_{\mu }$, ${\dot{x}}^{\mu }{a}_{\mu }$, and $a_5$ are O(3,1) scalars, its spacetime symmetry is restricted to 4D. As a guide to posing field equations for a $\tau$-dependent metric in curved spacetime, we may consider equation (4) as a standard 5D action in which we break the symmetry of the matter term by imposing the constraint ${\dot{x}}^5\equiv c_5$ and making the replacement ${\dot{x}}^{\alpha }{\dot{x}}_{\alpha }⟶{\dot{x}}^{\mu }{\dot{x}}_{\mu }$ in the kinetic term, restricting the phase space to $\left(x^{\mu },{\dot{x}}^{\mu }\right)$. The electrodynamics associated with the symmetry-broken action differ in significant ways from standard Maxwell theory in 5D. In particular, the Lorentz force [25] permits mass exchange between particles and fields, while leaving the total mass, energy, and momentum of particles and fields conserved. Compatibility with standard electrodynamic phenomenology places restrictions on $c_5/c\ll 1$ but the strict limit $c_5⟶0$ produces a $\tau$-equilibrium [26] that recovers standard Maxwell theory.

Field dynamics are determined by a kinetic term of the type where $f_{\mu \nu }$ is a second rank tensor, while $f_{5\mu }$ is a vector field strength, because the 5-index signifies an O(3,1) scalar quantity. Raising the 5-index in (7) suggests a 5D flat space metric where $\sigma =\pm 1$. But expanding we may regard $\sigma$ as the choice of sign for the vector-vector interaction, with no inherent significance for the geometry of spacetime.

Following these considerations, we approach the construction of a $\tau$-dependent GR by embedding 4D spacetime $\mathcal{M}$ in a 5D pseudo-spacetime ${\mathcal{M}}_5=\mathcal{M}\times R$ with coordinates $X^{\alpha }=\left(x^{\mu },c_5\tau \right)$ and a metric $g_{\alpha \beta }(x,\tau )$ determined by the standard 5D Einstein field equations on ${\mathcal{M}}_5$. By performing the embedding in a vielbein frame [27], we may specify the metric as (8) and break the 5D spacetime symmetry to O(3,1) at the source, by correcting the flat space metric for the quintrad for the matter terms under the replacement as in electrodynamics. The symmetry broken field equations are transformed to the coordinate frame as using the known vielbein field. The LHS of equation (11) enjoys 5D gauge and spacetime symmetries, while the RHS is O(3,1) covariant. Generalizing the 3+1 formalism in geometrodynamics [28,29,30,31] to 4+1, we take advantage of the natural foliation of ${\mathcal{M}}_5$ into 4D equal-$\tau$ spacetimes homeomorphic to $\mathcal{M}$. Standard techniques in the theory of embedded surfaces enable us to extract an initial value problem in the 4D spacetime sector, describing the $\tau$-evolution of metric ${\gamma }_{\mu \nu }(x,\tau )$ and the extrinsic curvature $K_{\mu \nu }(x,\tau )$ that accounts for aspects of the 5D connection ${\mathsf{\Gamma }}_{\alpha \beta }^{\gamma }$ not contained in the 4D metric.

Applying the Euler-Lagrange equations to the Lagrangian we obtain the 5D geodesic equations for an event $x^{\gamma }\left(\tau \right)$ with Christoffel symbols

We break the 5D symmetry to O(3,1) by asserting as an a priori constraint. The dynamical system is now described by the equations which under an appropriate metric field is consistent with the symmetries of matter, and recovers standard GR when $g_{5\alpha }=0$ and ${\partial }_{\tau }{g}_{\mu \nu }=0$. Defining the canonical momentum the Hamiltonian is which takes the recognizable form if $g^{5\mu }=0$, with $g_{55}(x,\tau )$ playing the role of a $\tau$-dependent potential on 4D spacetime. The canonical equations of motion are and since $p_5\equiv 0$, the Poisson bracket is so for any scalar function $F\left(x,p,\tau \right)$ on phase space generalizing the nonrelativistic result. Therefore, the Hamiltonian is conserved unless K depends explicitly on $\tau$ through $g_{\alpha \beta }\left(x,\tau \right)$. We note that even when K is a constant of the motion, the 4D mass $p^{\mu }{p}_{\mu }/2m$ may vary under $g_{55}$.

As we showed in [2] mass variation can appear in the Newtonian approximation through $\tau$-dependence of the metric. Expanding the geodesic equations as we take ${\partial }_0{g}_{\alpha \beta }=0$ and neglect terms containing ${\dot{x}}^i/c\ll 1$ for $i=1,2,3$, so that inserting the nonzero Christoffel symbols the equations of motion reduce to Comparing with Newtonian gravity, which must obtain exactly when ${\partial }_{\tau }{g}_{\mu \nu }$ and $\nabla g_{55}$ vanish, these become Writing a perturbed source mass parameter $M=M_0+\delta M\left(\tau \right)$ for the source and again neglecting $\dot{r}/c$ we may solve the t equation as so that in spherical coordinates, the radial equation takes the form where we take $\nabla g_{55}=0$ and introduce the conserved angular momentum As required, equations (28) and (29) recover Newtonian gravitation in the absence of the $\tau$-dependent source mass $\delta M$. The Hamiltonian in this coordinate system is with time derivative and as expected, the Hamiltonian for the motion of this test particle is not conserved in the presence of a variable mass gravitational source. This may be interpreted as transfer of mass across spacetime mediated by the metric.

For non-thermodynamic dust (a distribution of geodesically evolving events without mutual interaction), we define $\rho (x,\tau )$ as the number of events per spacetime volume, and write a 5-component event current with mass parameter M with continuity equation where the second equality holds because $j^5=M{c}_5\rho \left(x,\tau \right)$ is an O(3,1) scalar and not the 5-component of a vector with 5D symmetry. Generalizing the 4D energy-momentum tensor to 5D, the mass-energy-momentum tensor is conserved as by virtue of the continuity and geodesic equations. The mass-energy-momentum tensor is thus a suitable O(3,1) covariant source for a 5D field equation.

As indicated in Section 2.1 the derivation of field equations for $g_{\mu \nu }(x,\tau )$ possessing the desired 5D gauge symmetries and 4D spacetime symmetries relies heavily on the theory of embedded surfaces [28,29,30], the 3+1 ADM formalism [31], and the generalization of these techniques to 4+1 [2,3,4,5]. Here we provide a brief overview and refer the reader to the references for details.

We approach the construction of GR with a $\tau$-dependent metric by embedding 4D spacetime $\mathcal{M}$ in a 5D pseudo-spacetime ${\mathcal{M}}_5=\mathcal{M}\times R$ with coordinates $X^{\alpha }=\left(x^{\mu },c_5\tau \right)$. Because the Bianchi identity is independent of dimension [32] and the mass-energy-momentum tensor (35) is conserved, we may combine the Einstein tensor $G^{\alpha \beta }$ and $T_{\alpha \beta }$ to write the field equations for the metric $g_{\alpha \beta }(x,\tau )$ on ${\mathcal{M}}_5$.

In order to break the spacetime symmetry of the field equations to O(3,1), we first transform from a coordinate frame tangent to the manifold ${\mathcal{M}}_5$ to the constant quintrad frame where by convention Latin letters indicate reference to the quintrad. To facilitate foliation of ${\mathcal{M}}_5$ into spacetime hypersurfaces ${\mathsf{\Sigma }}_{\tau }$ of equal-$\tau$, we extend the partition of coordinate indices to the quintrad indices, leading to the combined index convention where the five index with respect to the quintrad frame are denoted $\overline{5}$. The transformation between frames is provided by the vielbein field where the spacetime hypersurface (quatrad) is spanned by the ${\mathbf{e}}_k$ while ${\mathbf{e}}_5$ points in the direction of $\tau$-evolution orthogonal to ${\mathsf{\Sigma }}_{\tau }$. Introducing the ADM parameterization where $N^{\mu }$ generalizes the shift 3-vector, N is the lapse function with respect to $\tau$, and $n={\mathbf{e}}_5$ is the unit normal, the vielbein field becomes leading to the coordinate metric which generalizes the ADM decomposition. Since N and $N^{\mu }$ are arbitrary functions acting as Lagrange multipliers whose choice is comparable to gauge freedom [31], the dynamical content in the vielbein field is contained entirely in the spacetime vierbein field $E_{\mu }^k$.

In the quintrad frame, the Einstein equations take the form and the spacetime symmetry may be broken to O(3,1) by making the replacement (10) as in the matter terms, leading us to where $\widehat{T}={\widehat{\eta }}^{ab}{T}_{ab}={\eta }^{kl}{T}_{kl}$. Using (44), this expression transforms back to the coordinate frame to provide the O(3,1) symmetric field equations with the transformed metric that acts as a projection operator from ${\mathcal{M}}_5$ onto the 4D spacetime hypersurface ${\mathsf{\Sigma }}_{\tau }$ and thus breaks any higher symmetry to O(3,1).

Given the foliation of the pseudo-spacetime into equal-$\tau$ spacetimes, the initial value problem is found by using $P_{\alpha \beta }$ to project the geometrical structures from ${\mathcal{M}}_5$ onto ${\mathsf{\Sigma }}_{\tau }$:

The evolution equations are where ${\mathcal{L}}_{\mathbf{N}}$ is the Lie derivative along $N^{\mu }$. Their solutions must satisfy the Hamiltonian constraint the momentum constraint Expressions (52) and (53) differ slightly from those found from the standard 5D Einstein equations (38), expressing the breaking of spacetime symmetry to O(3,1). The differences are in (52) and in (53).

In some cases, such as a diagonal metric in Cartesian coordinates, it is possible to formulate the evolution equations directly in the quatrad frame [5] in the simplified form with constraints providing an initial value problem for $E_{\mu }^k$ and $K_{kl}$, with the metric obtained from the vierbein field.

As in standard GR, the weak field approximation [4,18] poses the local metric as a small perturbation $h_{\alpha \beta }$ of the flat metric so that In this approximation, the 5D Ricci tensor takes the form where $h={\eta }^{\alpha \beta }{h}_{\alpha \beta }$ and we imposed the Lorenz gauge condition permitted by invariance of the metric under a 5D translation $x^{\alpha }⟶x^{\alpha }+{\mathsf{\Lambda }}^{\alpha }(x,\tau )$. Conveniently exploiting the Ricci tensor in this form, the SHP field equation (49) becomes the wave equation which admits the principal part Green’s function [33] in which the leading term, denoted $G_{\mathrm{Maxwell}}$, has lightlike support at equal-$\tau$ and is dominant at long distance. The second term, denoted $G_{\mathrm{correlation}}$, drops off as $1/{\mathrm{distance}}^2$ and has spacelike support for $\sigma =-1$ or timelike support for $\sigma =+1$.

For a general trajectory, we may consider an event distribution moving in tandem in the neighborhood of a point with 5D coordinates and for the shared velocity we introduce the notation The spacetime event density is leading to the mass-energy-momentum tensor which is seen to be conserved by simply noting that ${\partial }_{\tau }\rho (x,\tau )=-{\xi }^{\mu }{\partial }_{\mu }\rho (x,\tau )$. The generic solution for the metric perturbation is thus where ${\xi }^{\alpha }={\xi }^{\alpha }\left({\tau }^{\prime }\right)$ and $\widehat{\xi }{}^2={\widehat{\eta }}^{\mu \nu }{\xi }_{\mu }{\xi }_{\nu }$.