Nordström gravity in $(1+1)$ dimensions coupled to matter

We consider a $(1+1)$-dimensional model of general relativity that is based on a geometric theory of gravity due to Nordström. We show that the model can be reinterpreted as a scalar field theory in flat spacetime in which the scalar field couples to the trace of the total energy-momentum tensor, and we use the flat-spacetime interpretation to formulate the initial value problem for the model. We illustrate our results by using computer simulations of the model to obtain several example solutions.

Figure 1

Coordinate transformation $x^{\mu }\to {\overline{x}}^{\mu }$ described by Eq. (15). Shown are lines of constant $x$ and lines of constant $t$ in the $\overline{x}-\overline{t}$ plane for the case of a right-moving Gaussian wave packet $\lambda$.

Figure 2

Collision of two spatially separated mass distributions. Shown are the scalar field $\varphi$, the field derivatives $B={\partial }_t\varphi$ and $E={\partial }_x\varphi$, and the energy-momentum tensor component ${\Theta }_{mi}^{00}$ versus $x$ at times $t=0$, 1.5, 3.2, 10.0.

Figure 3

Propagation of a gauge wave packet through a static mass distribution. Shown are the scalar field $\varphi$, the field derivatives $B={\partial }_t\varphi$ and $E={\partial }_x\varphi$, and the energy-momentum tensor component ${\Theta }_{mi}^{00}$ versus $x$ at times $t=0$, 2.5, 3.0, 4.9.

Figure 4

Propagation of a gauge wave packet through a static mass distribution. Shown are the proper energy density $\rho$ and the curvature scalar $R$ versus proper distance $s$ at times $t=0$, 2.5, 3.0, 4.9.