Nordström gravity coupled to point particles in ($1+1$) dimensions

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 theory is formally equivalent to scalar field theory in flat spacetime, and we exploit this equivalence to solve the initial value problem for the system for the case of point particle sources. We illustrate our results by obtaining several example solutions to the model.

Figure 1

Diagram of vectors $r^{\mu }$, $w^{\mu }$, and $n^{\mu }$, drawn in the instantaneous rest frame of the particle at the retarded proper time ${\tau }_r(x)$ of event $x^{\mu }$. The curved line is the particle world line.

Figure 2

Stationary particle. (a) $f/f_0$ versus $a{f}_{0}t$ for $B_b/a{f}_0=-1,0,0.5,1,2$. (b, c) $E/a{f}_0$ and $B/a{f}_0$ versus $a{f}_{0}x$ for $B_b/a{f}_0=0.5$ at times $a{f}_{0}t=0,3$.

Figure 3

Moving particle. (a) $f/f_0$ versus $a{f}_{0}t$ for $u_0=1,2,3$. (b, c) $E/a{f}_0$ and $B/a{f}_0$ versus $a{f}_{0}x$ for $u_0=1$ at times $a{f}_{0}t=0,3$.

Figure 4

Two particles. (a) $a{z}^1$ versus $at$. (b) $u$ versus $at$. (c) $f$ versus at.

Figure 5

Two particles. $E/a$ and $B/a$ versus $ax$. (a) $at=0$. (b) $at=1$. (c) $at=4$.