*7.1. Subluminal Regions and the Equivalent Perfect Fluid with Negative Energy That Occupies Them*

Consider a region where *k* = *n* 2 <sup>4</sup> − *n* <sup>2</sup> > 0. We call such a region a subluminal region. Define the 4-velocity **V** with components

$$V\_a = n\_a / \sqrt{k}, \quad V\_4 = n\_4 / \sqrt{k} \tag{50}$$

satisfying *V* 2 <sup>1</sup> + *V* 2 <sup>2</sup> + *V* 2 <sup>3</sup> − *V* 2 <sup>4</sup> = −1. In terms of this velocity, (48) implies

$$T^{aa} = (2V\_a^2 - 1)k\_\prime \quad T^{ab} = 2V\_a V\_b k\_\prime$$

$$T^{44} = (2V\_4^2 + 1)k\_\prime \quad T^{44} = 2V\_a V\_4 k\_\prime \tag{51}$$

By comparison, a perfect fluid moving with 4-velocity **V** has

$$\begin{array}{rcl} T^{aa} &=& (\mu\_0 + p)V\_a^2 + p \quad T^{ab} = (\mu\_0 + p)V\_a V\_b \\ T^{44} &=& (\mu\_0 + p)V\_4^2 - p \quad T^{a4} = (\mu\_0 + p)V\_a V\_{4\nu} \end{array} \tag{52}$$

where *p* = *p* is the pressure and *µ*<sup>0</sup> is the rest density (in the frame with the same velocity as the fluid). Thus, *T* corresponds to a fluid with

$$p = -\mu\_0/\mathfrak{Z}, \quad \mu\_0 = \mathfrak{Z}k.\tag{53}$$

Note that, in this case, it always possible to choose a moving frame of reference with respect to which the fluid is not locally moving, i.e., *n* <sup>2</sup> = 0.

### *7.2. Superluminal Regions and the Equivalent Substance with Negative Mass That Occupies Them*

Consider those regions where *k* = *n* 2 <sup>4</sup> − *n* <sup>2</sup> < 0, which we call superluminal. Then, it is impossible to move to a reference frame such that *n* <sup>2</sup> = 0 at a given point. Rather, we can move to a frame where *n*<sup>4</sup> = 0 at this point. In this frame,

$$\begin{array}{rcl}T^{aa} &=& 2n\_a^2 + n^2 \\ T^{44} &=& -n^2 \end{array} \begin{array}{rcl}T^{ab} = 2n\_a n\_b \\ T^{44} = 0. \end{array} \tag{54}$$

This corresponds to some sort of substance, which, in this frame, has no momentum, a negative mass density −*n* 2 , and a stress:

$$
\sigma = -n^2 \mathbf{I} - 2\mathbf{n} \otimes \mathbf{n} \tag{55}
$$

corresponding to a pressure of *n* <sup>2</sup> and an additional uniaxial compression in the direction **n**.

### *7.3. Luminal Regions*

Finally, consider the regions where *k* = *n* 2 <sup>4</sup> − *n* <sup>2</sup> = 0, which we call luminal. Then,

$$T^{aa} = 2n\_{a\prime}^2 \quad T^{ab} = 2n\_a n\_b \quad T^{44} = 2n\_{4\prime}^2 \quad T^{a4} = -2n\_a n\_4. \tag{56}$$

Clearly, a luminal boundary or luminal region must separate regions that are subluminal or superluminal. In a luminal region, one cannot move to a frame where *n* <sup>2</sup> = 0, nor where *n* 2 <sup>4</sup> = 0, unless both are zero. The momentum density, mass density, and stress are nonzero everywhere, except where the torsion field vanishes.
