Particle motion in a space-time of a 3D Einstein gravity with torsion

Abstract

We analyze the motion of both massive and massless particles in a model with space-time of 3D Einstein gravity with torsion. We consider the spinor field and the massless scalar field as the source of torsion respectively. Following the Hamilton–Jacobi formalism, we investigate the effective potential of radial motion for test particles in a homogeneous and isotropic space-time with torsion. We show that there are no stable circular orbits for massive and massless particles in the Einstein gravity with torsion induced by the spinor field, in a space-time with two spatial and one time dimensions. In the case of massive particles, we show that stable orbits exist in 3D Einstein gravity with torsion induced by the scalar field.

Appendices

Appendix A: The application of the 4th order Runge–Kutta method on the Einstein gravity with torsion induced by scalar field and the numerical solutions

Fig. 5

The plots of the metrics components v(r) and w(r) are plotted by Runge–Kutta method with respect to r. We set $\Lambda =10^{-8}$, $\varphi (1)=10$, $\varphi '(1)=-2.065$, $v(1)=10^2$, $v'(1)=\Lambda$, $w(1)=10^{-1}$, $J(1)=10^{-2}$ and $J'(1)=10^{-4}$ [29]

Fig. 6

The plots of the angular momentum J(r) and the Ricci scalar R with the angular momentum $J(r)\ne 0$ are plotted by Runge–Kutta method with respect to r. We set $\Lambda =10^{-8}$, $\varphi (1)=10$, $\varphi '(1)=-2.065$, $v(1)=10^2$, $v'(1)=\Lambda$, $w(1)=10^{-1}$, $J(1)=10^{-2}$ and $J'(1)=10^{-4}$ [29]

See Figs. 5 and 6.

We have used 4th order Runge–Kutta method in Ref. [45] in Sect. 10.5.4.2 to solve the Einstein equations for scalar field.

The differential equations $\varphi ''(r)$ (18), $v'(r)$ (19), $w'(r)$ (20) and $J''(r)$ (21) can be described as

$$\begin{aligned}&\varphi ''(r)=f_2\left( r,\varphi (r) ,\varphi '(r),v(r),w(r),J(r),J'(r)\right) ,{}\nonumber \\&v'(r)=f_3\left( r,\varphi (r) ,\varphi '(r),v(r),w(r),J(r),J'(r)\right) ,{}\nonumber \\&w'(r)=f_4\left( r,\varphi (r) ,\varphi '(r),v(r),w(r),J(r),J'(r)\right) ,{}\nonumber \\&J''(r)=f_5\left( r,\varphi (r),\varphi '(r),v(r),w(r),J(r),J'(r)\right) , \end{aligned}$$

(51)

with the initial condition $\varphi (r_0) =a_0$, $\varphi ' (r_0) =a_1$, $J(r_0) =a_2$, $J'(r_0) =a_3$, $v(r_0) =a_4$ and $w(r_0) =a_5$.

If we take $\varphi '(r)=z$ and $J'(r)=u$ we obtain the following equations

$$\begin{aligned}&z=f_1\left( r,\varphi ,z,v,w,J,u\right) ,{}\nonumber \\&z'=f_2\left( r,\varphi ,z,v,w,J,u\right) ,{}\nonumber \\&v'=f_3\left( r,\varphi ,z,v,w,J,u\right) , {}\nonumber \\&w'=f_4\left( r,\varphi ,z,v,w,J,u\right) ,{}\nonumber \\&u'=f_5\left( r,\varphi ,z,v,w,J,u\right) ,{}\nonumber \\&u=f_6\left( r,\varphi ,z,v,w,J,u\right) , \end{aligned}$$

(52)

where

$$\begin{aligned}&f_1\left( r,\varphi ,z,v,w,J,u\right) =z,{}\nonumber \\&f_2\left( r,\varphi ,z,v,w,J,u\right) =\varphi ''(r),\;\;\;\; Eq. \;(18),{}\nonumber \\&f_3\left( r,\varphi ,z,v,w,J,u\right) =v' (r),\;\;\;\;\; Eq.\; (19),{}\nonumber \\&f_4\left( r,\varphi ,z,v,w,J,u\right) =w'(r),\;\;\;\;\; Eq.\; (20),{}\nonumber \\&f_5\left( r,\varphi ,z,v,w,J,u\right) =J'' (r),\;\;\;\;\; Eq.\; (21),{}\nonumber \\&f_6\left( r,\varphi ,z,v,w,J,u\right) =u. \end{aligned}$$

(53)

According to the 4th order Runge–Kutta method, we can write the solution as follows:

$$\begin{aligned}&\varphi _{n+1}=\frac{1}{6} h \left( k_{11}+2 k_{12}+2 k_{13}+k_{14}\right) +\varphi _n,{}\nonumber \\&v_{n+1}=\frac{1}{6} h \left( k_{21}+2 k_{22}+2 k_{23}+k_{24}\right) +v_n,{}\nonumber \\&w_{n+1}=\frac{1}{6} h \left( k_{31}+2 k_{32}+2 k_{33}+k_{34}\right) +w_n,{}\nonumber \\&z_{n+1}=\frac{1}{6} h \left( k_{41}+2 k_{42}+2 k_{43}+k_{44}\right) +z_n,{}\nonumber \\&u_{n+1}=\frac{1}{6} h \left( k_{51}+2 k_{52}+2 k_{53}+k_{54}\right) +u_n,{}\nonumber \\&J_{n+1}=\frac{1}{6} h \left( k_{61}+2 k_{62}+2 k_{63}+k_{64}\right) +J_n, \end{aligned}$$

(54)

where

$$\begin{aligned}&k_{1 m}=f_m\left( r_n,\varphi _n,z_n,v_n,w_n,u_n,J_n\right) ,{}\nonumber \\&k_{2 m}=f_m(\frac{h}{2}+r_n,\frac{h k_{11}}{2}+\varphi _n,\frac{h k_{21}}{2}+z_n,\frac{h k_{31}}{2}+v_n,\frac{h k_{41}}{2}+w_n,{}\nonumber \\&\;\;\; \; \;\;\; \;\; \;\; \; \;\;\; \; \;\frac{h k_{51}}{2}+u_n,\frac{h k_{61}}{2}+J_n),{} \nonumber \\&k_{3 m}=f_m(\frac{h}{2}+r_n,\frac{h k_{12}}{2}+\varphi _n,\frac{h k_{22}}{2}+z_n,\frac{h k_{32}}{2}+v_n,\frac{h k_{42}}{2}+w_n,{}\nonumber \\&\;\;\; \; \;\;\; \;\; \;\; \; \;\;\; \; \;\frac{h k_{52}}{2}+u_n,\frac{h k_{62}}{2}+J_n),\nonumber \\&k_{4 m}=f_m(h+r_n,h k_{13}+\varphi _n,h k_{23}+z_n,h k_{33}+v_n,h k_{43}+w_n,{}\nonumber \\&\;\;\; \; \;\;\; \;\; \;\; \; \;\;\; \; \;h k_{53}+u_n,h k_{63}+J_n). \end{aligned}$$

(55)

Our codes were written with Mathematica program.

The metric components v(r), w(r), the angular momentum J(r) and the Ricci scalar R are plotted by Runge–Kutta method with $\Lambda =10^{-8}$, $\varphi (1)=10$, $\varphi '(1)=-2.065$, $v(1)=10^2$, $v'(1)=\Lambda$, $w(1)=10^{-1}$, $J(1)=10^{-2}$ and $J'(1)=10^{-4}$ [29].

Fig. 5 and Fig. 6 show the metric components v(r) and w(r), the angular momentum J(r) and the Ricci scalar R of space-time with torsion.

Appendix B: The Kretschmann scalar for Einstein gravity with torsion induced by spinor field and scalar field

See Figs. 7 and 8.

Fig. 7

The Kretschmann scalar K for a space-time of a 3D Einstein gravity with torsion induced by the spinor is plotted with respect to r. We set $\alpha = 0.6$, $\Lambda = 10^{-8}$, $C_2= -\Lambda$ and $s_1=1$. We get $r_-=0.790569$ and $r_+=1.58114$. The Kretschmann scalar has a gap between $r_-$ and $r_+$

Fig. 8

The Kretschmann scalar K for a space-time of a 3D Einstein gravity with torsion induced by the scalar field is plotted by 4th order Runge–Kutta method with respect to r, for the values of $\Lambda =10^{-8}$, $\varphi (1)=10$, $\varphi '(1)=-2.065$, $v(1)=10^2$, $v'(1)=\Lambda$, $w(1)=10^{-1}$, $J(1)=10^{-2}$ and $J'(1)=10^{-4}$

The Kretschmann scalar plays an important role to know whether the space-time is regular or not. By regular space-time it means that the space-time must have regular curvature invariants are finite at all space-time points [46].

The Kretschmann scalar, $K=R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }$, of this metric (9) is derived

$$\begin{aligned} K= & {} -\Lambda ^2 (32 \Lambda (2 \Lambda r^2 (\Lambda -m^2)-s_1\sqrt{\Lambda (\Lambda -m^2)} \sqrt{\Delta })\nonumber \\&+C_2 (93 \Lambda ^2+48 m^4-8 \Lambda m^2))/(4 C_2 m^4). \end{aligned}$$

(56)

Note that the roots of $\Delta$ (13) are the horizons $r_-$ and $r_+$ (14). Since $\Delta <0$ in the region $r_-<r<r_+$, K assuming imaginary values, is unphysical. The Kretschmann scalar should have the gap between the horizons $r_-$ and $r_+$.

Substituting $m^2$ (15) into Eq. (56) the plot of the Kretschmann scalar K for a space-time of a 3D Einstein gravity with torsion induced by the spinor field is given in Fig. 7. By using the methods of 4th order Runge–Kutta, we further plot the Kretschmann scalar K for a space-time of a 3D Einstein gravity with torsion induced by the scalar field in Fig. 8.

Appendix C: The 4th order Runge–Kutta method on the Einstein gravity with torsion induced by spinor field

See Figs. 9, 10 and 11.

We have also used 4th order Runge–Kutta method on the spinor case for massive particles $\varepsilon =1$ to check our analytical solutions in Sect. 3.1.1. By using the method of 4th order Runge–Kutta, numerical solutions are given in Fig. 9 and Fig. 10.

The metric components v(r) (10), w(r) (11) and the angular momentum J(r) (12) are plotted by Runge–Kutta method. We set $\alpha = 0.6$, $\Lambda = 10^{-8}$, $C_2= -\Lambda$, $s_1=1$, $w(1)=10^{-1}$ and $J(1)=10^{-2}$.

Fig. 9

The plots of the metrics components v(r) and w(r) are plotted by 4th Runge–Kutta method with respect to r, for the values of $\alpha = 0.6$, $\Lambda = 10^{-8}$, $C_2= -\Lambda$, $s_1=1$, $w(1)=10^{-1}$ and $J(1)=10^{-2}$

Fig. 10

The plot of the angular momentum J(r) is plotted by 4th Runge–Kutta method with respect to r, for the values of $\alpha = 0.6$, $\Lambda = 10^{-8}$, $C_2= -\Lambda$, $s_1=1$, $w(1)=10^{-1}$ and $J(1)=10^{-2}$

Fig. 11

The effective potential V(r) for radial motion in a space-time of a 3D Einstein gravity with torsion induced by the spinor field for massive particles $\varepsilon =1$ is plotted by Runge–Kutta method with respect to r. We set $\alpha = 0.6$, $\Lambda = 10^{-8}$, $C_2= -\Lambda$, $s_1=1$, $w(1)=10^{-1}$, $J(1)=10^{-2}$. The figure shows the effective potential for the four different curves correspond to $L=1, 3, 6$, and 9 from the up bottom

From the plot of w(r) in the right of Fig. 9, we can see that the metric component w(r) has the gap between $r\approx 1820$ and $r\approx 1900$.

The effective potential V(r) for radial motion in a space-time of a 3D Einstein gravity with torsion induced by the spinor for massive particles with the value $\epsilon =1$ is plotted by 4th Runge–Kutta method in Fig. 11.

The behavior of the effective potential V(r) in Fig. 11 and V(r) in the right of Fig. 1 are similar the values of r grater than 15.