*8.1. Plane Wave Solutions*

Here, we consider plane wave solutions to the equations in the weak field approximation. It is to be emphasized that since the equations are non-linear, specifically quadratic in **n**, one cannot generally superimpose our plane wave solutions to obtain another solution.

The simplest case is when the fields only depend on, say, *x*1. Then, we deduce that *T* 1*j* is a constant, i.e.,

$$3n\_1^2 + n\_2^2 + n\_3^2 - n\_4^2 = k\_1, \quad n\_1 n\_2 = k\_2, \quad n\_1 n\_3 = k\_3, \quad n\_1 n\_4 = k\_4. \tag{69}$$

where the *k<sup>i</sup>* are constants. Multiplying the first equation by *n* 2 1 , we obtain

$$m\_1^4 = (k\_1 n\_1^2 - k\_2^2 - k\_3^2 + k\_4^2) / 3,\tag{70}$$

which requires the constants *k<sup>i</sup>* to be such that the right-hand side is non-negative. Thus, *n* 2 1 is constant, and the last three equations in (69) imply that *n* 2 2 , *n* 2 3 , and *n* 2 4 are constants as well, unless *n* 2 <sup>1</sup> = 0. Therefore, the only interesting case is when *n* 2 <sup>1</sup> = 0, implying that *k*<sup>2</sup> = *k*<sup>3</sup> = *k*<sup>4</sup> = 0. Then, according to whether −*k*<sup>1</sup> = *n* 2 <sup>4</sup> − *n* 2 is positive, zero, or negative, the solution will be subluminal, superluminal, or luminal. Thus, subject to the constraint that *n* <sup>2</sup> <sup>≥</sup> *<sup>k</sup>*<sup>1</sup> (relevant only when *<sup>k</sup>*<sup>1</sup> <sup>&</sup>gt; 0), *<sup>n</sup>*2(*x*1) and *<sup>n</sup>*3(*x*1) can be chosen arbitrarily and determine *n* 2 <sup>4</sup> = *n* <sup>2</sup> <sup>−</sup> *<sup>k</sup>*1. In particular, if *<sup>k</sup>*<sup>1</sup> <sup>=</sup> 0, one may choose *<sup>n</sup>*2(*x*1) and *<sup>n</sup>*3(*x*1) to be zero outside an interval of values of *x*1. In a frame of reference moving with velocity −*v*<sup>1</sup> in direction *x*1, this will look like a wave pulse traveling a velocity *v*1, as all the field components will be functions of *x*<sup>1</sup> − *v*1*t*. We call them localized transverse torsion waves, transverse because **n** is perpendicular to the wave front. Unlike longitudinal torsion waves, which can only travel at the speed of light, these can have any velocity less than *c*.

Similarly, when the fields only depend on *t* = *x*4, we deduce that *T* 4*j* is a constant, i.e.,

$$m\_4 \mathbf{n} = \mathbf{k'}, \quad 3n\_4^2 - n^2 = k'\_{4\nu} \tag{71}$$

in which **n** = (*n*1, *n*2, *n*3) and *n* <sup>2</sup> <sup>=</sup> **<sup>n</sup>** · **<sup>n</sup>** and where *<sup>k</sup>* ′ 4 and **k** ′ = (*k* ′ 1 , *k* ′ 2 , *k* ′ 3 ) are constants. Multiplying the last formula by *n* 2 4 shows that

$$n\_4^4 = (k\_4' n\_4^2 + \mathbf{k'} \cdot \mathbf{k'})/3\tag{72}$$

is constant, implying *n* 2 4 is constant and, through the first equation in (71), that *n* 2 1 , *n* 2 2 , and *n* 2 3 are constant as well, unless *n* 2 <sup>4</sup> = 0. When *n* 2 <sup>4</sup> = 0, then **k** ′ = 0. The last formula in (71) then forces *n* 2 to be constant. Subject to this constraint, **n** can have an arbitrary dependence on time (with **n**(*t*) being independent of *x*1, *x*2, and *x*3). However, note that the inevitable spatial variation of **n** may eliminate this arbitrariness, as may going beyond the weak field approximation.

### *8.2. Solutions with Cylindrical Symmetry, Including Torsion-Rolls*

Consider cylindrical coordinates (*r*, *θ*, *z*, *t*) taking *r* to be the radial distance from the *z*-axis, *θ* to be the angular variable, and *t* to be the time. We seek solutions where **n** = (*n<sup>r</sup>* , *n<sup>θ</sup>* , *nz*) and *n*<sup>4</sup> only depend on *r*, so that

$$\begin{aligned} \nabla \cdot (n\_{\mathbf{t}} \mathbf{n}) &= \frac{1}{r} \frac{d(rn\_{\mathbf{t}} n\_{r})}{dr} \mathbf{\hat{r}},\\ (\mathbf{n} \cdot \nabla) \mathbf{n} &= \left( n\_{r} \frac{dn\_{r}}{dr} - \frac{n\_{r}^{2}}{r} \right) \mathbf{\hat{r}} + \left( n\_{r} \frac{dn\_{\theta}}{dr} + \frac{n\_{r} n\_{\theta}}{r} \right) \boldsymbol{\theta} + \left( n\_{r} \frac{dn\_{z}}{dr} \right) \mathbf{\hat{z}},\\ \mathbf{n} (\nabla \cdot \mathbf{n}) &= -\frac{1}{r} \frac{d(rn\_{r})}{dr} (n\_{r} \mathbf{\hat{r}} + n\_{\theta} \boldsymbol{\theta} + n\_{z} \mathbf{\hat{z}}),\\ \frac{1}{2} \nabla (n^{2} - n\_{4}^{2}) &= -\frac{1}{2} \left[ \frac{d}{dr} (n^{2} - n\_{4}^{2}) \right] \mathbf{\hat{r}},\end{aligned} \tag{73}$$

where we used the standard formulas for the gradient, divergence, and **n** · ∇ in cylindrical coordinates. Then, the conservation laws (57) take the form

$$\begin{aligned} 0 &= \quad \frac{1}{r} \frac{d(rm\_rh\_4)}{dr} ,\\ 0 &= \quad \frac{n\_r^2 - n\_\theta^2}{r} + \frac{1}{2} \frac{d}{dr} \left[2n\_r^2 + n^2 - n\_4^2\right] ,\\ 0 &= \quad \frac{2n\_r n\_\theta}{r} + \frac{d(n\_r n\_\theta)}{dr} ,\\ 0 &= \quad \frac{n\_r n\_z}{r} + \frac{d(n\_r n\_z)}{dr} .\end{aligned} \tag{74}$$

If we consider an interface at a constant radius *r* = *r*0, with outwards unit normal **r**ˆ, then the weak form of the equations *T ij* ,*<sup>j</sup>* = 0 imply the jump conditions on the elements *T ij* that

$$\mathbf{r(t)}$$

must be continuous across the interface, where **T** is given by (49). This implies that the quantities

$$\mathbf{c}\_{4} = \mathfrak{n}\_{4}\mathfrak{n}\_{\prime\prime} \quad \mathbf{c}\_{\theta} = \mathfrak{n}\_{\theta}\mathfrak{n}\_{\prime\prime} \quad \mathbf{c}\_{z} = \mathfrak{n}\_{z}\mathfrak{n}\_{\prime\prime} \quad \mathbf{c}\_{r} = 3\mathfrak{n}\_{r}^{2} + \mathfrak{n}\_{\theta}^{2} + \mathfrak{n}\_{z}^{2} - \mathfrak{n}\_{4}^{2} \tag{75}$$

must all be continuous across the interface *r* = *r*0. Multiplying the last equation by *n* 2 *r* , we see that

$$n\_r^4 = (c\_r n\_r^2 - c\_\theta^2 - c\_z^2 + c\_4^2)/3\tag{76}$$

must be continuous as well, and the first three equations imply that all components of (**n**, *n*4) are continuous across the interface, up to a change of sign, unless *n* 2 *<sup>r</sup>* = 0 at the interface. If *n* 2 *r* is zero at the interface, it follows that *c*<sup>4</sup> = *c<sup>θ</sup>* = *c<sup>z</sup>* = 0 at the interface. Therefore, across *r* = *r*0, any jumps in *n<sup>θ</sup>* (*r*, *t*), *nz*(*r*, *t*), and *n*4(*r*, *t*) that maintain the continuity of *n* <sup>2</sup> <sup>−</sup> *<sup>n</sup>* 2 4 are possible provided *nr*(*r*, *t*) is continuous and *nr*(*r*0, *t*) = 0.

The first, third, and last equations in (74) imply

$$rm\_rh\_4 = k\_{4\nu} \quad rm\_rh\_2 = k\_{z\nu} \quad r^2 \eta\_r \eta\_\theta = k\_\theta \tag{77}$$

where *k*4, *kz*, and *k<sup>θ</sup>* are constants. In the case *n<sup>r</sup>* = 0, all are satisfied with *k*<sup>4</sup> = *k<sup>z</sup>* = *k<sup>θ</sup>* = 0. The remaining second equation in (74) becomes

$$\frac{d}{dr}\left[n^2 - n\_4^2\right] = \frac{2n\_\theta^2}{r}.\tag{78}$$

Thus, there is only one constraint among the three functions *n<sup>θ</sup>* (*r*), *nz*(*r*), and *n*4(*r*). We see that *n* <sup>2</sup> <sup>−</sup> *<sup>n</sup>* 2 <sup>4</sup> must monotonically increase with *r*, in a manner controlled by *n* 2 *θ* (*r*), and if it tends to zero at infinity, then *n* <sup>2</sup> <sup>−</sup> *<sup>n</sup>* 2 <sup>4</sup> must be negative for all *r*, corresponding to a subluminal region. If **n** and *n*<sup>4</sup> vanish outside a certain radius, then we call this solution a torsion-roll. Physically, the pressure increases to larger negative values as the radius decreases, and its gradient provides the centripetal force that holds the "fluid" circulating around the *z*-axis with a velocity governed by *n<sup>θ</sup>* . In a moving frame of reference, which is not moving in the *z*-direction, the torsion-roll will appear to be moving.

Of course, if *n* <sup>2</sup> <sup>−</sup> *<sup>n</sup>* 2 4 is constant and positive outside a certain radius (corresponding, for example, to a superluminal region where, say, *n<sup>z</sup>* is constant and *n<sup>θ</sup>* = *n*<sup>4</sup> = 0), then *n* <sup>2</sup> <sup>−</sup> *<sup>n</sup>* 2 4 can remain positive for all r, or can transition from positive to negative values at a particular radius. This example demonstrates that transitions between subluminal and superluminal regions are possible.

Alternatively, if *n<sup>r</sup>* is nonzero, then (77) implies

$$n\_4 = k\_4/(rm\_r), \quad n\_z = k\_z/(rm\_r), \quad n\_\theta = k\_\theta/(r^2n\_r). \tag{79}$$

Substituting these in the second equation in (74) yields

$$\frac{ds}{dr} = \frac{2s(3k\_{\theta}^2 - r^4s^2 - 2kr^2)}{r(3r^4s^2 - k\_{\theta}^2 + kr^2)}, \quad \text{where} \quad s = n\_{r\prime}^2 \quad k = k\_4^2 - k\_z^2. \tag{80}$$

This gives us a flow-field in the (*r*,*s*) phase plane. Note that (80) remains invariant under the transformation

$$r \to \lambda\_1 r, \quad s \to \lambda\_2 s, \quad k\_\theta^2 \to \lambda\_1^4 \lambda\_2^2 k\_{\theta'}^2 \quad k \to \lambda\_1^2 \lambda\_2^2 k. \tag{81}$$

Thus, without loss of generality, we may, by rescaling any solution, take *k<sup>θ</sup>* to be 0 or 1 and *k* to be −1, 0, or 1. If *k* = 0, then there is essentially just one solution: *s*0(*r*) satisfying *s*0(1) = 1 with all other solutions (with *k<sup>θ</sup>* = 1) taking the form *s*(*r*) = *λ* 2 *s*0(*λr*), parametrized by *λ*. The solutions for *s*0(*r*) = *n* 2 *r* (*r*) and *n* 2 *θ* (*r*) = 1/(*r* 4*n* 2 *r* ) are shown in Figure 2 along with the flow field. One can see that the solution does not exist below a critical value of *r*, which looks unsatisfactory. This critical radius is associated with the vanishing of the denominator in (80).

**Figure 2.** Solution for the torsion field with cylindrical symmetry with *n<sup>r</sup>* 6= 0, *k* = 0, and *k<sup>θ</sup>* = 1. (**a**) The flow field when *k* = 0 and *k<sup>θ</sup>* = 1 and the particular solution satisfying *n* 2 *r* = 1 when *r* = 1. (**b**) The same solution for *n* 2 *<sup>r</sup>* on a log–log plot and the accompanying function *n* 2 *<sup>θ</sup>* = 1/(*r* 4*n* 2 *r* ).

To obtain satisfactory solutions that exist for all *r* 6= 0, one may take *k<sup>θ</sup>* = 0 and *k* = 1 to avoid the denominator in (80) vanishing, except at *r* = 0. Then, (80) reduces to

$$\frac{ds}{dr} = -\frac{2s(r^2s^2+2)}{r(3r^2s^2+k)}, \quad \text{where} \quad s = n\_r^2. \tag{82}$$

There is again essentially just one solution: *s*0(*r*) satisfying *s*0(1) = 1 with all other solutions (with *k* = 1) taking the form *s*(*r*) = *λs*0(*λr*), parametrized by *λ*. The solution is graphed in Figure 3. There is a singularity at *r* = 0, and while *n* 2 *r* (*r*) goes rapidly to zero as *<sup>r</sup>* → <sup>∞</sup>, *<sup>n</sup>* 2 4 (*r*) and *n* 2 *z* (*r*) (unless it is zero) diverge to <sup>∞</sup> as *<sup>r</sup>* → <sup>∞</sup>. This solution is satisfactory once one takes into account that the weak field approximation is not valid near the singularity at *<sup>r</sup>* = 0, nor as *<sup>r</sup>* → <sup>∞</sup>, and one should use the full equations (37) there. For this example with *k* = 1 and *k<sup>θ</sup>* = 0, it is interesting that there is a transition from a superluminal region inside to a subluminal region outside according to the sign of

$$n^2 - n\_4^2 = n\_r^2 + \frac{k\_z^2}{r^2 n\_r^2} - \frac{k\_4^2}{r^2 n\_r^2} = n\_r^2 - \frac{1}{r^2 n\_r^2} \tag{83}$$

which is also plotted in Figure 3.

**Figure 3.** Solution for the torsion field with cylindrical symmetry with *n<sup>r</sup>* 6= 0, *k* = 1, and *k<sup>θ</sup>* = 0. (**a**) The graph of *n* 2 *r* = 1 showing its divergence as *r* → 0. (**b**) The plot of *n* <sup>2</sup> <sup>−</sup> *<sup>n</sup>* 2 <sup>4</sup> = *n* 2 *<sup>r</sup>* − 1/(*r* 4*n* 2 *r* ) showing a transition from superluminal to subluminal as *r* increases.

### **9. Extension of the Schwarzschild Solutions With Spherical Symmetry**

Here, we generalize Schwarzschild's solution for a spherically symmetric metric solving Einstein's equations in the absence of matter. The important point is that in appropriate limits, some of the solutions here approach the Schwarzschild solution. Consequently, existing experimental results of black holes do not invalidate our theory, but rather place constraints on the magnitude of the torsion field. This magnitude should be tied to the radius of the universe, and, hence, to the critical acceleration in MOND. Thus, experiments in the near vicinity of a star or black hole would not typically reveal the difference with Schwarzschild's solution. We have not explored the situation regarding rotating black holes.

As shown by Schwarzschild, the metric in "polar" coordinates spherically symmetric about the origin must be of the form

$$ds^2 = a\,dr^2 + r^2(d\theta^2 + \sin^2\theta \, d\phi^2) - b\,dt^2,\tag{84}$$

in which *a* and *b* are functions of *r* and *t*. Here, we look for solutions where they are functions of *r* only. Setting *x*<sup>1</sup> = *r*, *x*<sup>2</sup> = *θ*, *x*<sup>3</sup> = *φ*, *x*<sup>4</sup> = *t* allows us to use (84) to identify the coefficients:

$$g\_{11} = a, \quad g\_{22} = r^2, \quad g\_{33} = r^2 \sin^2 \theta, \quad g\_{44} = -b. \tag{85}$$

From (36), we obtain the ten equations:

$$\begin{aligned} 0 &=& \overline{R}\_{11} = \frac{a'}{ar} + \frac{a'b'}{4ab} + \frac{(b')^2}{2b^2} - \frac{b'}{2b} + 2a[r^2(N^2)^2 + r^2\sin^2\theta(N^3)^2 - b(N^4)^2], \\ 0 &=& \overline{R}\_{22} = 1 - \frac{1}{a} + \frac{ra'}{2a^2} - \frac{rb'}{2ab} + 2r^2[a(N^1)^2 + r^2\sin^2\theta(N^3)^2 - b(N^4)^2], \\ 0 &=& \overline{R}\_{33} = [1 - \frac{1}{a} + \frac{rd'}{2a^2} - \frac{rb'}{2ab}]\sin^2\theta + 2r^2\sin^2\theta[a(N^1)^2 + r^2(N^2)^2 - b(N^4)], \\ 0 &=& \overline{R}\_{44} = \left(\frac{b'}{ar} + \frac{b''}{2a} - \frac{(b')^2}{4ab} - \frac{d'b'}{4a^2}\right) - 2b[a(N^1)^2 + r^2(N^2)^2 + r^2\sin^2\theta(N^3)^2], \\ 0 &=& \overline{R}\_{mn} = -2g\_{mn}g\_{m}N^mN^n \quad \text{for all } m, n \quad \text{with } m \neq n, \quad \text{no sum on } m, n \end{aligned} \tag{86}$$

where the terms not involving **N** can be identified with the standard formulas for the elements *R* 0 *ij* that are zero when *i* 6= *j*. Here, differentiation with respect to *x*<sup>1</sup> = *r* is denoted by the prime, with the double prime denoting the second derivative. The second and third equations and the last equation force *N*<sup>2</sup> = *N*<sup>3</sup> = 0, which is not surprising considering the symmetry of the problem. Two possibilities remain: either *N*<sup>1</sup> = 0 or *N*<sup>4</sup> = 0. The first case corresponds to a subluminal solution and the second to a superluminal solution.

Let us consider first the case where *N*<sup>1</sup> = *N*<sup>2</sup> = *N*<sup>3</sup> = 0. Multiplying the second last equation in (86) by *a*/*b* and adding it to the first gives

$$\frac{a'}{a} + \frac{b'}{b} - 2q = 0 \quad \text{where } q = rab(N^4)^2 \ge 0. \tag{87}$$

The second equation in (86) implies

$$\frac{a'}{a} - \frac{b'}{b} + 2(a-1)/r - 4q = 0.\tag{88}$$

Adding and subtracting these equations gives

$$\begin{array}{rcl} a'/a &=& \frac{1}{r} - \frac{a}{r} + 3q, \\ b'/b &=& \frac{a}{r} - \frac{1}{r} - q. \end{array} \tag{89}$$

Multiplying the last by *br*, differentiating it, and using the result to eliminate *b* ′′ from the first equation in (86) yield

$$
\sigma' = 2q^2 + \frac{q}{r}.\tag{90}
$$

This has the solution

$$q = \frac{a^2r}{1 - a^2r^2},\tag{91}$$

where *α* is a constant. Furthermore, by replacing *q* with *rab*(*N*<sup>4</sup> ) 2 , one obtains

*q*

$$\begin{split} 2q^2 + \frac{q}{r} &= q' \quad = \
a b(N^4)^2 + (ra'/a)ab(N^4)^2 + (rb'/b)ab(N^4)^2 + rab \frac{d(N^4)^2}{dr},\\ &= \frac{q}{r} [1 + (1 - a + 3qr) + (a - 1 - qr)] + q \frac{d(N^4)^2}{dr} = \frac{q}{r} + 2q^2 + q \frac{d(N^4)^2}{dr}. \end{split} \tag{92}$$

This implies that (*N*<sup>4</sup> ) 2 is a constant that we call *β* 2 , giving

$$\frac{a}{r} = \frac{q}{br^2(N^4)^2} = \frac{a^2}{br\beta^2(1-a^2r^2)}.\tag{93}$$

Substituting this back in the second equation in (89) gives the linear first-order differential equation:

$$
\frac{db}{dr} + b\left[\frac{1}{r} + \frac{a^2}{1 - a^2r^2}\right] = \frac{a^2}{\beta^2 r (1 - a^2r^2)}.\tag{94}
$$

Multiplying both sides by the integrating factor of *r*/ √ 1 − *α* 2*r* <sup>2</sup> gives

$$\frac{d}{dr}\left[br/\sqrt{1-a^2r^2}\right] = \frac{a^2}{\beta^2(1-a^2r^2)\sqrt{1-a^2r^2}}.\tag{95}$$

Integrating both sides and recalling (93), we obtain

$$\begin{array}{rcl} b &=& \frac{a^2}{\beta^2} - 2m \frac{\sqrt{1 - a^2 r^2}}{r}, \\ a &=& \frac{a^2}{b \beta^2 (1 - a^2 r^2)} \end{array} \tag{96}$$

where *m* is a constant of integration. In particular, with *α* <sup>2</sup> = *β* 2 , this becomes

$$\begin{array}{rcl} b & = & 1 - 2m \frac{\sqrt{1 - a^2 r^2}}{r}, \\ a & = & \frac{1}{b(1 - a^2 r^2)}, \end{array} \tag{97}$$

which in the limit *α* → 0 reduces to the familiar Schwarzschild solution:

$$a = \frac{1}{1 - 2m/r}, \quad b = 1 - 2m/r,\tag{98}$$

which becomes Euclidean at large *r*. Once we allow nonzero *α*, the space is no longer Euclidean at large *r*, but it still has a black hole at the center, with *a* diverging when *r* = 2*m* √ 1 − *α* 2*r* <sup>2</sup> and at *r* = 1/*α* 2 , the latter corresponding to the closed universe studied in the next section.

Now, consider the second possibility that *N*<sup>2</sup> = *N*<sup>3</sup> = *N*<sup>4</sup> = 0. Again, multiplying the second last equation in (86) by *a*/(*b*) and adding it to the first give

$$\frac{a'}{a} + \frac{b'}{b} - 2w = 0 \quad \text{where } w = ra^2(N^1)^2 \ge 0. \tag{99}$$

Furthermore, the second equation in (86) implies

$$\frac{a'}{a} + \frac{b'}{b} + 2(a-1)/r + 4w = 0.\tag{100}$$

Adding and subtracting these equations give

$$\begin{array}{rcl} a'/a &=& \frac{1}{r} - \frac{a}{r} - w, \\ b'/b &=& \frac{a}{r} - \frac{1}{r} + 3w. \end{array} \tag{101}$$

Multiplying the last by *br*, differentiating it, and using the result to eliminate *b* ′′ from the first equation in (86) yield

$$w' = -2w^2 + w(1 - \frac{4}{3}a)/r.\tag{102}$$

The equations (101) and (102) appear to have no simple analytic solution. One may eliminate *a*(*r*) from the two equations that do not involve *b*(*r*) to obtain

$$\frac{w''}{w} = \frac{7(w')^2}{4w^2} - \frac{3w'}{2wr} - \frac{2w}{r} + w^2 - \frac{1}{4r^2} \tag{103}$$

and from a solution *w*(*r*), (102) easily gives *a*(*r*). Alternatively, one may eliminate *w*(*r*) from these equations to obtain

$$\frac{v''}{v} = \frac{\Im(v')^2}{v^2} + \frac{v'}{vr} + (5v' + 2v^2)/3 + v/r.\tag{104}$$

where *v* = *a*/*r*, and given a solution *a*(*r*) = *rv*(*r*), the first equation in (101) yields *w*(*r*). In either case, *b*(*r*) is found by integrating the last equation in (101). Note that if *b*(*r*) is a solution, then so will be *λ* 2 *b*(*r*) for any constant *λ*, i.e., *b*(*r*) is only determined up to a multiplicative constant. This reflects the fact that we are free to rescale the time coordinate, replacing *t* by *t*/*λ* in (84).

Rather than dealing with these second-order equations for *w*(*r*) and *v*(*r*), one can numerically solve (101) and (102) directly. Figure 4 shows some typical solutions, excluding unphysical examples where, say, *a*(*r*) or *w*(*r*) remain negative for all *r*

**Figure 4.** Numerical solutions of Equations (101) and (102). (**a**) Graph with *w*(1) = *a*(1) = *b*(1) = 1 showing a "black hole"-type singularity at *r* = 0.5959. (**b**) Same as for (**a**), but on a log–log plot. Note the blow up of *<sup>b</sup>*(*r*) as *<sup>r</sup>* → <sup>∞</sup>. (**c**) Graph with *<sup>w</sup>*(1) = 0.01 and *<sup>a</sup>*(1) = *<sup>b</sup>*(1) = 1. Comparing this with (**b**) and taking note of the different vertical scales, one can see the approach to the usual Schwarzschild solution as *w*(1) → 0. (**d**) Graph with *w*(1) = 0.8, *a*(1) = 0.2, and *b*(1) = 0.3 showing a different type of solution with no critical "black hole" radius, but rather, a singularity at *r* = 0. The solution for *<sup>b</sup>*(*r*) still clearly blows up as *<sup>r</sup>* → <sup>∞</sup>.

### **10. Homogeneous Expanding Universe**

It should be emphasized that the solution given here, which is incompatible with the observations, is for a homogeneous universe devoid of ordinary matter. It does not apply to a universe where spacetime itself has fluctuations that are not due to ordinary matter. Even ignoring ordinary gravitational effects, the perturbation results at the beginning of Section 8 imply such fluctuations occur, and so, one should expect that its cosmological predictions deviate from those presented in this section. This is explained further in the next section.

We take the Friedmann–Lemaître–Robertson–Walker metric in reduced-circumference polar coordinates:

$$ds^2 = a^2 \left[ \frac{dr^2}{1 - kr^2} + r^2(d\theta^2 + \sin^2\theta \, d\phi^2) \right] - dt^2 \tag{105}$$

where *a*, the reduced circumference, can be a function of time, while *r*, *θ*, and *φ* are time independent, and *k* = 1, 0, −1, according to whether the universe is spatially closed, flat, or open with negative curvature. With *x*<sup>1</sup> = *r*, *x*<sup>2</sup> = *θ*, *x*<sup>3</sup> = *φ*, and *x*<sup>4</sup> = *t*, the corresponding metric coefficients are

$$g\_{11} = a^2/(1 - kr^2), \quad g\_{22} = a^2 r^2, \quad g\_{33} = a^2 r^2 \sin^2 \theta, \quad g\_{44} = -1. \tag{106}$$

Assuming *N*<sup>1</sup> = *N*<sup>2</sup> = *N*<sup>3</sup> = 0 and defining

$$P = 2k + (a\sharp + 2d^2),\tag{107}$$

where the dot and double dot denote first and second derivatives with respect to time, the equations become

$$\begin{array}{ll} 0 & = & \overline{R}\_{11} = -2(N^4)^2 a^2 / (1 - kr^2) - P / (1 - kr^2) \\ 0 & = & \overline{R}\_{22} = -2(N^4)^2 a^2 r^2 - Pr^2 \\ 0 & = & \overline{R}\_{33} = -2(N^4)^2 a^2 r^2 \sin^2 \theta - Pr^2 \sin^2 \theta \\ 0 & = & \overline{R}\_{44} = 3 \acute{u} / a \end{array} \tag{108}$$

where the terms not involving *N*<sup>4</sup> can be identified with the standard formulas for *R* 0 *ij*. The last equation in (108) implies *a*˙ is a constant that we define to be *β*. We obtain

$$P = 2k + 2\dot{a}^2 = 2(k + \beta^2), \quad a = \beta t + \gamma,\tag{109}$$

where *γ* is an integration constant that we can choose to be zero by redefining our origin of time appropriately (except in the trivial case of a spatially flat universe independent of time with *N*<sup>4</sup> = 0). From the remaining three equations in (107), which are all equivalent, we obtain

$$(N^4)^2 = -\frac{P}{2a^2} = -\frac{k+\beta^2}{\beta^2 t^2},\tag{110}$$

which implies that *k* = −1 (an open universe, like in anti-de Sitter spacetime) and *β* <sup>2</sup> < 1.

### **11. Addressing the Dark Matter and Dark Energy Problem**

The result of the previous section giving an expansion rate *a*˙ independent of time agrees with the well-known result that *a*¨ = 0 for a model with *p* = −*µ*0/3. However, this is based on the premise that spacetime is homogeneous. The expansion of the universe appears to be accelerating with measurements indicating *p* = −0.8*µ*<sup>0</sup> [8], and this could be a consequence of our theory, as we now explain.

Dark matter itself is known to be inhomogeneous; see, for example, [45] and the references therein. Spacetime is also inhomogeneous in our model. As the analysis at the beginning of Section 8 shows, if there is a small fluctuation in the torsion vector field in subluminal or superluminal regions of spacetime, then that perturbation will grow. Moreover, ordinary gravitational effects might add to the inhomogeneity: if there is a higher equivalent mass density in two different regions, then there could be gravitational attraction between these regions, leading to accretion. At the same time, "collisions" between accreting regions should tend to disperse the torsion vector field density. Thus, there will be a certain amount of equivalent kinetic energy associated with the torsion field accounting for some additional "dark energy". More importantly, there could be substructures in the torsion field containing differing ratios of "dark energy" to "dark mass". The structures could collide and give rise to different structures. In particular, there might be "negative mass structures", by which we mean structures in the torsion vector field incorporating superluminal regions. Accounting for these effects should reduce the total mass density, providing a higher *p*/*µ*<sup>0</sup> ratio, which may be consistent with the experimental value of −0.8.

It is to be emphasized that both our full equations (37) and their weak field approximations (46), (47), and (57) have no intrinsic length scale. There is a length scale associated with the overall density of the torsion vector field (connected with the mass density of the apparent dark matter and dark energy in our theory), but this is of the order of the radius of the universe. It seems likely that the torsion vector field could be quite turbulent with structures on many length scales, down to some lower cutoff length scale where the current theory breaks down. This cutoff could be the Planck length scale.

To provide quantitative predictions, one needs a better idea of the behavior of the torsion vector field within spacetime, and this will almost certainly require sophisticated numerical simulations to obtain an approximation to the "macroscopic equation of state". Simulations are needed to provide a better understanding of torsion fluid behavior in intergalactic and interstellar regions, as well as around stars, globular clusters, galaxies, and galaxy clusters. These may require the introduction of some parameter that provides a lower length scale to the "turbulence" in the torsion vector field, which ultimately could be taken to be very small. Simulating the dynamics of the torsion vector field over the continuum of length scales may also require a sort of numerical renormalization group approach. While we have not investigated the stability of the torsion waves and torsionrolls, it is not important that they are stable, even in the weak field approximation. The purpose of our exact solutions in the weak field approximation was mainly to illustrate the rich dynamics of the torsion vector field, to give some insight into possible dynamics and to show that one can have transitions between subluminal and superluminal regions, as noted at the end of Section 8.2.

Regarding the question as to whether our model can account for the galactic dark mass problem, an encouraging sign is the apparent cosmological connection between the critical acceleration *<sup>a</sup>*<sup>0</sup> <sup>≈</sup> 1.2 <sup>×</sup> <sup>10</sup>−<sup>10</sup> m/s<sup>2</sup> in MOND, the radius of the universe, and the density of dark matter or energy in the universe, as reviewed in [46]. Thus, the density of dark matter or energy, roughly *̺* <sup>≈</sup> <sup>10</sup>−<sup>27</sup> kg/m<sup>3</sup> , which in our theory is related to the strength of the torsion field **N**, has an associated length scale 1/ p *c* <sup>2</sup>*̺κ* <sup>≈</sup> <sup>6</sup> <sup>×</sup> <sup>10</sup><sup>26</sup> meters (approximately the radius of the universe), which agrees with the length scale *c* <sup>2</sup>/*a*<sup>0</sup> <sup>≈</sup> 7.5 <sup>×</sup> <sup>10</sup><sup>26</sup> meters associated with the critical acceleration *a*<sup>0</sup> in MOND.
