New models of general relativistic static thick disks

New families of exact general relativistic thick disks are constructed using the “displace, cut, fill, and reflect” method. A class of functions used to fill the disks is derived imposing conditions on the first and second derivatives to generate physically acceptable disks. The analysis of the function’s curvature further restricts the ranges of the free parameters that allow physically acceptable disks. Then this class of functions together with the Schwarzschild metric is employed to construct thick disks in isotropic, Weyl and Schwarzschild canonical coordinates. In these last coordinates an additional function must be added to one of the metric coefficients to generate exact disks. Disks in isotropic and Weyl coordinates satisfy all energy conditions, but those in Schwarzschild canonical coordinates do not satisfy the dominant energy condition.

Figure 1

(a) The function $\tilde{h}(\tilde{z})$ and (b) its first and (c) second derivatives with $n=1$ for $\tilde{c}=0$, $0.5$, $1$, and $2$. (d) The curvature $\tilde{\kappa }(\tilde{z})$ for the same parameters.

Figure 2

The mass density Eq. (6) for a Newtonian thick disk with parameters $\tilde{m}=1$, $\tilde{b}=2$, $n=1$, and (a) $\tilde{c}=0$, (b) $\tilde{c}=0.5$. Some levels curves of the density are displayed on the right graphs.

Figure 3

The mass density Eq. (6) for a Newtonian thick disk with parameters $\tilde{m}=1$, $\tilde{b}=2$, $n=1$, and (a) $\tilde{c}=1$, (b) $\tilde{c}=2$. Some levels curves of the density are displayed on the right graphs.

Figure 4

(a) The function $\tilde{h}(\tilde{z})$ and its (b) first and (c) second derivatives with $n=2$ for $\tilde{c}=0$, $0.5$, $1$, and $2$. (d) The curvature $\tilde{\kappa }(\tilde{z})$ for the same parameters.

Figure 5

The mass density Eq. (6) for a Newtonian thick disk with parameters $\tilde{m}=1$, $\tilde{b}=2$, $n=2$, and (a) $\tilde{c}=0$, (b) $\tilde{c}=0.5$. Some levels curves of the density are displayed on the right graphs.

Figure 6

The mass density Eq. (6) for a Newtonian thick disk with parameters $\tilde{m}=1$, $\tilde{b}=2$, $n=2$, and (a) $\tilde{c}=1$, (b) $\tilde{c}=2$. Some levels curves of the density are displayed on the right graphs.

Figure 7

(a) Energy density, (b) effective Newtonian density for a thick disk in isotropic coordinates. Parameters: $\tilde{m}=1$, $\tilde{b}=2$, $n=1$, and $\tilde{c}=0$.

Figure 8

(a) Radial and azimuthal pressures, (b) vertical pressure for a thick disk in isotropic coordinates. Parameters: $\tilde{m}=1$, $\tilde{b}=2$, $n=1$, and $\tilde{c}=0$.

Figure 9

(a) Effective Newtonian density, (b) energy density for a thick disk in Weyl coordinates. Parameters: $\tilde{m}=1$, $\tilde{b}=2$, $n=1$, and $\tilde{c}=0$.

Figure 10

(a) Vertical pressures, (b) azimuthal stresses for a thick disk in Weyl coordinates. Parameters: $\tilde{m}=1$, $\tilde{b}=2$, $n=1$, and $\tilde{c}=0$.

Figure 11

(a) Energy density, (b) effective Newtonian density for a thick disk in Schwarzschild coordinates. Parameters: $\tilde{m}=1$, $\tilde{b}=5$, $n=1$, and $\tilde{c}=0$.

Figure 12

(a) The pressure mostly vertical, (b) tension mostly radial for a thick disk in Schwarzschild coordinates. Parameters: $\tilde{m}=1$, $\tilde{b}=5$, and $\tilde{c}=0$.

Figure 13

(a) The azimuthal stresses, (b) the level curve of $|p_{\phi }/ϵ|=1$ and (c) the lines of flow calculated from Eqs. (31a, 31b). Parameters: $\tilde{m}=1$, $\tilde{b}=5$, and $\tilde{c}=0$.