**8. Some Solutions and Perturbative Solutions for the Torsion Field in the Weak Field Approximation**

Let us consider solutions of *T ij* ,*<sup>j</sup>* = 0 in a flat metric given by (40). Using (49), we obtain

$$\begin{aligned} 0 &=& \frac{\partial}{\partial t} [3n\_4^2 - n^2] - 2\nabla \cdot (n\_4 \mathbf{n}), \\ 0 &=& \nabla \cdot (\mathbf{n} \otimes \mathbf{n}) - \frac{\partial (n\_4 \mathbf{n})}{\partial t} + \frac{1}{2} \nabla (n^2 - n\_4^2) \\ &=& (\mathbf{n} \cdot \nabla) \mathbf{n} + \mathbf{n} \nabla \cdot \mathbf{n} - \frac{\partial (n\_4 \mathbf{n})}{\partial t} + \frac{1}{2} \nabla (n^2 - n\_4^2). \end{aligned} \tag{57}$$

where the first equation represents the conservation of energy and the second the balance of forces.

In the subluminal regions, if we look for solutions where **n** = 0 globally and not just at one point, the conservation of energy implies *∂n* 2 4 /*∂t* = 0, while the balance of forces implies ∇(*n* 2 4 ) = 0. Thus, *n* 2 <sup>4</sup> must be a constant in spacetime. On the other hand, if we allow for small values of **n**, with *n* <sup>2</sup> <sup>≪</sup> *<sup>n</sup>* 2 4 , then to first order in the perturbation with **e** = *n*4**n** and *f* = *n* 2 4 , we obtain

$$\frac{\partial \mathbf{e}}{\partial t} = -\frac{1}{2} \nabla f, \quad \mathbf{3} \frac{\partial f}{\partial t} = 2 \nabla \cdot \mathbf{e},\tag{58}$$

giving

$$
\Im \frac{\partial^2 f}{\partial t^2} = -\nabla^2 f. \tag{59}
$$

This has exponentially growing solutions such as

$$f = 1 + \delta e^{\tau} \cos \mathbf{k} \cdot \mathbf{x}, \quad \text{with } k^2 \equiv \mathbf{k} \cdot \mathbf{k} = 3\tau^2,\tag{60}$$

where *δ* is a small parameter. After a finite time, this solution for *f* reaches negative values, but before which, our assumption that *n* <sup>2</sup> <sup>≪</sup> *<sup>n</sup>* 2 4 is violated. Thus, the solution with **n** = 0 is unstable to perturbations.

In the superluminal regions, if we look for solutions where *n*<sup>4</sup> = 0 globally and not just at one point, then the conservation of energy implies that *n* <sup>2</sup> must not vary with time, and the balance of forces implies

$$
\nabla(n^2) + 2\nabla \cdot (\mathbf{n} \otimes \mathbf{n}) = 0. \tag{61}
$$

This provides three equations to be satisfied by the three functions *na*(*x*1, *x*2, *x*3, *t*), *a* = 1, 2, 3. There is a manifold of functions satisfying (61), and we can choose any trajectory **n**(*x*1, *x*2, *x*3, *t*) that lies on this manifold and is such that *n* 2 (*x*1, *x*2, *x*3) = **n**(*x*1, *x*2, *x*3, *t*) · **n**(*x*1, *x*2, *x*3, *t*) is independent of time. Unless **n**(*x*1, *x*2, *x*3, *t*) only depends on *t*, it seems likely that this second condition will generally force **n**(*x*1, *x*2, *x*3, *t*) to be independent of time (up to a sign change in **n**). If we investigate the effect of perturbations, with **n** 2 <sup>4</sup> ≪ *n* 2 and both *n*<sup>1</sup> and *ǫ* depending only on *x*<sup>1</sup> and *t* and say *n*<sup>2</sup> = *n*<sup>3</sup> = 0, we obtain

$$\frac{\partial e\_1}{\partial t} = \frac{3}{2} \frac{\partial d}{\partial \mathbf{x}\_1}, \quad \frac{\partial d}{\partial t} = -2 \frac{\partial e\_1}{\partial \mathbf{x}\_1} \tag{62}$$

where *e*<sup>1</sup> = *n*4*n*<sup>1</sup> and *d* = *n* 2 1 . This gives

$$\frac{\partial^2 d}{\partial t^2} = -3 \frac{\partial^2 d}{\partial x\_1^2} \tag{63}$$

which has exponentially growing solutions such as

$$d = 1 + e^{\tau} \cos \mathbf{k} \cdot \mathbf{x}, \quad \text{with } k^2 = \tau^2/3. \tag{64}$$

While, after a finite time, our assumption that **n** 2 <sup>4</sup> ≪ *n* <sup>2</sup> becomes violated, the calculation shows that the solution with *n*<sup>4</sup> = 0 is unstable to perturbations.

In luminal regions where *n* <sup>2</sup> <sup>−</sup> *<sup>n</sup>* 2 <sup>4</sup> = 0, we can use this identity to eliminate *n*<sup>4</sup> from (57) and obtain

$$\begin{aligned} 0 &=& \frac{\partial n^2}{\partial t} \pm \nabla \cdot (|\mathbf{n}| \mathbf{n}),\\ 0 &=& \nabla \cdot (\mathbf{n} \otimes \mathbf{n}) \pm \frac{\partial (|\mathbf{n}| \mathbf{n})}{\partial t},\\ &=& (\mathbf{n} \cdot \nabla) \mathbf{n} + \mathbf{n} \nabla \cdot \mathbf{n} \pm \frac{\partial (|\mathbf{n}| \mathbf{n})}{\partial t},\end{aligned} \tag{65}$$

where the plus or minus sign is taken according to whether *n*<sup>4</sup> = ±|**n**|. In the special case where *n*<sup>2</sup> = *n*<sup>3</sup> = 0 (after making a spatial rotation if necessary), we obtain *n*<sup>4</sup> = *n*<sup>1</sup> (or *n*<sup>4</sup> = −*n*1), and (65) reduces to the single equation:

$$\frac{\partial n\_1}{\partial t} = \frac{\partial n\_1}{\partial x\_1} \tag{66}$$

to be satisfied by the function *n*1(*x*1, *x*2, *x*3, *t*), describing a wave propagating at the speed of light in the direction of the *x*1-axis. We call them localized longitudinal torsion waves, longitudinal because **n** is aligned with the direction of propagation. We now look for perturbation solutions with *n*<sup>4</sup> = *n*<sup>1</sup> − *ǫ*, where *ǫ* is a small parameter and both *n*<sup>1</sup> and *ǫ* depend only on *x*<sup>1</sup> and *t* while *n*<sup>2</sup> = *n*<sup>3</sup> = 0. Letting *ǫn*<sup>1</sup> = *η* and *d* = *n* 2 1 , we obtain

$$\frac{\partial(d-\eta)}{\partial t} = -\frac{\partial(d-\eta)}{\partial \mathbf{x}\_1}, \quad \frac{\partial(d-\mathfrak{J}\eta)}{\partial t} = -\frac{\partial(d-\eta)}{\partial \mathbf{x}\_1}.\tag{67}$$

The first wave equation has the solution *η* = *d* + *h*(*x*<sup>1</sup> − *t*), where *h*(*y*) is an arbitrary function, and substituting in the second gives *∂d*/*∂t* = *h* ′ (*x*<sup>1</sup> − *t*), where *h* ′ (*y*) is the derivative of *h*(*y*). We conclude that

$$
\eta = \sigma(\mathbf{x}\_1), \quad d = \sigma(\mathbf{x}\_1) - h(\mathbf{x}\_1 - t), \tag{68}
$$

where *σ*(*x*1) satisfies *h*(*y*) ≤ |*σ*(*y*)| ≪ |*h*(*y*)| for all *y*, to ensure that *d* is non-negative and that the perturbation is small (|*ǫ*(*x*1, *t*)| ≪ |*n*1(*x*1, *t*)| for all (*x*1, *t*), but otherwise is an arbitrary function. Thus, *h*(*y*) can only take negative values, and the perturbation travels at the speed of light in the direction of **n**.

We now present various other solutions of the equations, without investigating their stability.
