Consider a 4-dimensional manifold with metric signature ##- + + +##, and let ##(t,x,y,z)## be coordinates such that the spacetime interval is given by ##ds^2 = -c^2,dt^2 + dx^2 + dy^2 + dz^2##. Define a scalar field ##\Phi(t,x,y,z)## that vanishes everywhere except along two distinct 3-dimensional spatial hypersurfaces: one at ##t = t_0## and one at ##t = t_1##. Suppose we construct a continuous mapping ##F : \mathbb{R}^4 \to \mathbb{R}^4## such that ##F## restricts to the identity on each hypersurface but imposes a phase shift in the metric over an intermediate time interval. Formally, let ##F^\mu(\tau,\sigma,\rho,\chi)## satisfy ##F^0(\tau,\sigma,\rho,\chi) = t_0 + \gamma(\tau)## for some smooth function ##\gamma## with ##\gamma(\tau_0) = 0## and ##\gamma(\tau_f) = t_1 - t_0##. Impose a boundary condition so that ##F## preserves proper lengths on each 3D slice by requiring that the induced metric ##g’{\alpha\beta} = \frac{\partial F^\mu}{\partial \xi^\alpha} \frac{\partial F^\nu}{\partial \xi^\beta} g{\mu\nu}## reduces to ##g_{\alpha\beta}## when ##\tau = \tau_0## or ##\tau = \tau_f##.
We now define a 1-form ##\omega = \Phi,d\eta## for a new coordinate ##\eta##, where ##\Phi## localizes ##\omega## in the time region ##t_0 \le t \le t_1##. This 1-form modifies the geodesic equations by adding a term proportional to ##\omega_\mu u^\mu##, where ##u^\mu## is the 4-velocity. The resulting geodesic equation can be written as
$$
\frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\lambda} \frac{dx^\beta}{d\lambda} = \kappa,\omega_\nu,g^{\mu\nu},
$$
where ##\kappa## is a coupling constant. Because ##\omega## is only active between ##t_0## and ##t_1##, any observer crossing this region evolves along a timelike curve that smoothly lifts them from the hypersurface ##t = t_0## to ##t = t_1##. The constant-time slices ##t = t_0## and ##t = t_1## thus appear as distinct 3-dimensional “copies” of space.
