**4. The Ricci Tensor**

Let us express the Ricci tensor:

$$R\_{jk} = \Gamma^i\_{ri}\Gamma^r\_{jk} - \Gamma^i\_{rk}\Gamma^r\_{ji} + \Gamma^i\_{jk,i} - \Gamma^i\_{ji,k'} \tag{19}$$

which is associated with the local curvature of spacetime, in terms of the symmetric and antisymmetric parts of the affinity:

$$\begin{split} \mathsf{R}\_{\mathsf{jk}} &= \quad (\overline{\Gamma}^{i}\_{\vec{n}} + \widehat{\Gamma}^{i}\_{\vec{n}\vec{l}})(\overline{\Gamma}^{r}\_{\vec{jk}} + \widehat{\Gamma}^{r}\_{\vec{jk}}) - (\overline{\Gamma}^{i}\_{\vec{n}k} + \widehat{\Gamma}^{i}\_{\vec{n}l})(\overline{\Gamma}^{r}\_{\vec{jl}} + \widehat{\Gamma}^{r}\_{\vec{jl}}) + (\overline{\Gamma}^{i}\_{\vec{jk}} + \widehat{\Gamma}^{i}\_{\vec{jk}})\_{,i} - (\overline{\Gamma}^{i}\_{\vec{j}i} + \widehat{\Gamma}^{i}\_{\vec{j}i})\_{k\prime} \\ &= \quad \overline{\Gamma}^{i}\_{\vec{n}\vec{l}} (\overline{\Gamma}^{r}\_{\vec{jk}} + \widehat{\Gamma}^{r}\_{\vec{jk}}) - (\overline{\Gamma}^{i}\_{\vec{n}k} + \widehat{\Gamma}^{i}\_{\vec{n}k})(\overline{\Gamma}^{r}\_{\vec{jl}} + \widehat{\Gamma}^{r}\_{\vec{jl}}) + (\overline{\Gamma}^{i}\_{\vec{jk}} + \widehat{\Gamma}^{i}\_{\vec{jk}})\_{,i} - \overline{\Gamma}^{i}\_{\vec{ji}k\prime} \end{split} \tag{20}$$

where we used the fact that Γ *i ri* = 0, as follows from (17). Therefore, now, we have

$$R\_{jk} = R^0\_{jk} - \hat{\Gamma}^i\_{rk}\hat{\Gamma}^r\_{ji} + \hat{\Gamma}^i\_{kr}\overline{\Gamma}^r\_{ji} + \overline{\Gamma}^i\_{rk}\hat{\Gamma}^r\_{ij} - \overline{\Gamma}^i\_{ri}\hat{\Gamma}^r\_{kj} - \hat{\Gamma}^i\_{kj,i\prime} \tag{21}$$

where

$$R^0\_{jk} = \overline{\Gamma}^i\_{ri}\overline{\Gamma}^r\_{jk} - \overline{\Gamma}^i\_{rk}\overline{\Gamma}^r\_{ji} + \overline{\Gamma}^i\_{jk,i} - \overline{\Gamma}^i\_{ji,k} \tag{22}$$

is the usual Ricci curvature tensor associated just with the metric. We now consider the symmetric part of *Rjk* as it is central to our equations:

$$\overline{\mathcal{R}}\_{jk} \equiv \frac{1}{2} (\mathcal{R}\_{jk} + \mathcal{R}\_{kj}) = \mathcal{R}\_{jk}^{0} - \widehat{\Gamma}\_{rk}^{i} \widehat{\Gamma}\_{ji}^{r} = \mathcal{R}\_{jk}^{0} - \mathcal{g}^{si} e\_{\mathrm{srk}\ell} \mathcal{U}^{\ell} \mathcal{g}^{tr} e\_{\mathrm{tjih}} \mathcal{U}^{\ln}.\tag{23}$$

Given an arbitrary point, we can always find a new coordinate system such that the metric is orthogonal at that point. In this new coordinate system at this one point

$$\begin{array}{rcl}\overline{R}\_{11} &=& R^{0}\_{11} - g^{i\overline{i}} e\_{ir1\ell} \mathcal{U}^{\ell} g^{rr} e\_{r1ih} \mathcal{U}^{h},\\\overline{R}\_{12} &=& R^{0}\_{11} - g^{i\overline{i}} e\_{ir1\ell} \mathcal{U}^{\ell} g^{rr} e\_{r2ih} \mathcal{U}^{h},\end{array} \tag{24}$$

where a sum over *i* and *r* is implied. For *eir*1<sup>ℓ</sup> *er*1*ih* to be nonzero, it is necessary that *r* 6= *i* and *ir*ℓ must be a permutation of *rih* (and a permutation of 234), implying ℓ = *h*. Therefore, we obtain

$$\overline{R}\_{11} = R\_{11}^0 - 2g^{22}g^{33}(\mathcal{U}^4)^2 - 2g^{44}g^{22}(\mathcal{U}^3)^2 - 2g^{33}g^{44}(\mathcal{U}^2)^2. \tag{25}$$

Furthermore, for *eir*1<sup>ℓ</sup> *er*2*ih* to be nonzero, ℓ must be 2 and *h* must be 1, implying

$$
\overline{R}\_{12} = R\_{12}^0 + 2g^{33}g^{44} \mathcal{U}^3 \mathcal{U}^4. \tag{26}
$$

Of course, similar formulas hold for the other elements of *Rjk*. Hence, at this point, in this coordinate system,

$$\overline{R}\_{\text{jk}} = R^0\_{\text{jk}} + 2\text{g}^{-1} \mathcal{g}\_{\text{j}n} \mathcal{g}\_{km} \mathcal{U}^m \mathcal{U}^n - 2\text{g}^{-1} \mathcal{g}\_{\text{jk}} \mathcal{g}\_{mn} \mathcal{U}^m \mathcal{U}^n,\tag{27}$$

where *g* = *g*11*g*22*g*33*g*<sup>44</sup> is the determinant of the metric tensor, or, introducing a contravariant vector *N<sup>k</sup>* such that *N<sup>k</sup>* = *Uk*/ √ −*g*, we obtain

$$
\overline{R}\_{jk} = R^0\_{jk} + 2\mathcal{g}\_{jk}\mathcal{g}\_{mn}N^mN^n - 2\mathcal{g}\_{jn}\mathcal{g}\_{km}N^mN^n. \tag{28}
$$

This equation being a tensor equation will be true in any coordinate system, as well as at any point since the original point was arbitrarily chosen. Raising indices gives

$$\overline{\mathcal{R}}\_{k}^{j} = (\mathcal{R}^{0})\_{k}^{j} + 2\delta\_{k}^{j} \mathbf{g}\_{mn} \mathcal{N}^{m} \mathcal{N}^{n} - 2\mathbf{g}\_{km} \mathcal{N}^{m} \mathcal{N}^{j}. \tag{29}$$

Finally, contracting indices, we obtain

$$\overline{\mathcal{R}} \equiv \overline{\mathcal{R}}\_{j}^{j} = \mathcal{R}^{0} + 6 \mathbf{g}\_{mn} \mathcal{N}^{m} \mathcal{N}^{n} \, \mathrm{.} \tag{30}$$

where *R* <sup>0</sup> = (*R* 0 ) *j j* . We will call **N** the torsion field.

### **5. The Proposed New Gravitational Equations**

In this section, we investigate how torsion affects the geometry of empty space. For our purposes, it is to be observed that in the Einstein–Cartan–Sciama–Kibble theory, as reviewed in [30,33,34], one has

$$N\_{\circ} = -2\kappa S\_{\circ}.\tag{31}$$

where *S<sup>j</sup>* is the spin axial-vector field, and so, there would be no torsion in empty space. Nevertheless, more general theories of torsion-gravity, in which torsion propagates, do not need to verify such a constraint, and therefore, **N** can still be nonzero even if **S** = 0 identically; see Fabbri [36] and the references therein. Having emphasized that there exist theories in which torsion can still be nonzero, even in a vacuum, we will however not be specifying any particular Lagrangian. Instead, we will work from a very general perspective. The new gravitational field equations are

$$
\overline{R}\_{jk} - \frac{1}{2} \mathcal{g}\_{jk} \overline{R} = \kappa T'\_{jk}.\tag{32}
$$

where the *T* ′ *jk* are the elements of the symmetric stress–energy–momentum tensor **T** ′ and *<sup>κ</sup>* <sup>≈</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup>−43<sup>s</sup> <sup>2</sup>m−1kg−<sup>1</sup> is the gravitational constant. This then has the equivalent form:

$$R^0\_{\vec{j}k} - \frac{1}{2}g\_{\vec{j}k}R^0 - g\_{\vec{j}k}g\_{mn}N^mN^n - 2g\_{jn}g\_{km}N^mN^n = \kappa T'\_{\vec{j}k'} \tag{33}$$

or

$$R^0\_{jk} - \frac{1}{2} \mathcal{g}\_{jk} R^0 = \kappa T\_{jk\prime} \tag{34}$$

with

$$T\_{\rm jk} = T\_{\rm jk}' + \left[ \mathcal{g}\_{\rm jk} \mathcal{g}\_{mn} \mathcal{N}^m \mathcal{N}^n + \mathcal{Z} \mathcal{g}\_{\rm in} \mathcal{g}\_{km} \mathcal{N}^m \mathcal{N}^n \right] / \kappa. \tag{35}$$

Thus, **T** is the equivalent stress–energy–momentum tensor if we were to reinterpret our equations in the format of Einstein's original gravitational equation. Therefore, if the torsion field **N** is small enough, we recover Einstein's original equations to a good approximation and, hence, those of Newtonian gravity. From here onwards, until the last section, we assume that **T** ′ = 0, i.e., that no ordinary matter is present in the region of spacetime being studied. By multiplying (32) by *g kj* and summing over indices, we see that *R* = 0, and hence, (33) can be rewritten as

$$\overline{\mathcal{R}}\_{\text{jk}} = \mathcal{R}\_{\text{jk}}^0 + 2\mathcal{g}\_{\text{jk}}\mathcal{g}\_{mn}\mathcal{N}^m\mathcal{N}^n - 2\mathcal{g}\_{\text{jn}}\mathcal{g}\_{km}\mathcal{N}^m\mathcal{N}^n = 0,\tag{36}$$

or, raising indices,

$$\overline{R}^{jk} = \{R^0\}^{jk} + 2\mathbf{g}^{jk}\mathbf{g}\_{mn}\mathbf{N}^m\mathbf{N}^n - 2\mathbf{N}^j\mathbf{N}^k = \mathbf{0}.\tag{37}$$

These equations are consistent, for example, with those of Sengupta [39] (see his Equation (27)), which, however, are not the same as they include an extra dimension and incorporate additional fields.

The well-known Bianchi identities between the components of the contracted curvature tensor imply

$$[\{R^0\}^{jk} - \frac{1}{2}g^{jk}R^0]\_{,k} = 0,\tag{38}$$

and as is well known, this implies *T ij* ,*<sup>j</sup>* = 0, reflecting conservation of energy and momentum. Together with (37) and (30), we obtain

$$[\![\mathbf{g}^{jk}\!] \!g\_{mn}\mathbf{N}^{m}\!\!N^{n}+\!\!2N^{j}\!\!N^{k}]\_{,k}=\mathbf{0}.\tag{39}$$

We can view these as the extra four equations needed to determine the four components of **N** in empty space. One slightly unsatisfactory feature of the equations is that **N** is only determined up to a sign change. In other words, given a solution in a spacetime region, another solution can be obtained by reversing the sign of **N** within a subregion. Thus, we do not consider our theory to be complete. At the quantum Planck length scale, it likely needs modification, and the modified theory could prevent abrupt changes in the sign of **N**. Alternatively, one could take the view that there is no torsion, but rather, **N**(**x**) is just a vector field pervading all space. Then, the sign of **N**(**x**) is immaterial, but still, one would expect modifications at the Planck length scale to provide a lower limit to the length scales of "turbulence" in the vector field **N**(**x**).

### **6. The Weak Field Approximation**

Now, consider the weak field approximation where *gαβ* = *g* 0 *αβ* + *<sup>κ</sup>hαβ* and *<sup>N</sup><sup>i</sup>* = √ *κn<sup>i</sup>* where *κ* is a small parameter, and the *g* 0 *aβ* correspond to the Minkowski metric:

$$\mathbf{g}^{0}\_{aa} = \{\mathbf{g}^{0}\}^{aa} = 1, \quad \mathbf{g}^{0}\_{ab} = \{\mathbf{g}^{0}\}^{ab} = 0, \quad \mathbf{g}^{0}\_{a4} = \{\mathbf{g}^{0}\}^{a4} = 0, \quad \mathbf{g}^{0}\_{44} = \{\mathbf{g}^{0}\}^{44} = -1 \tag{40}$$

in which *a*, *b* are indices taking the values 1, 2, or 3 with *a* 6= *b*. There is some freedom in the choice of the *hαβ* due to the coordinate shifts that we can make to first order in *κ*. This freedom can be eliminated by imposing the harmonic gauge that

$$h\_{\vec{k}}^{jk} = \frac{1}{2} \{\mathbf{g}^{0}\}^{jk} h\_{\vec{k}\prime} \tag{41}$$

in which *h* = {*g* 0} *sthst* and *h jk* <sup>=</sup> {*<sup>g</sup>* 0} *js*{*<sup>g</sup>* 0} *kthst*. To first order in *κ* (37) implies

$$0 = \overline{\mathcal{R}}^{jk}/\kappa = -\frac{1}{2} \mathcal{g}^{0}\_{mn} \frac{\partial h^{jk}}{\partial \mathbf{x}\_{m} \partial \mathbf{x}\_{n}} + 2 \{g^{0}\}^{jk} \mathcal{g}^{0}\_{mn} n^{m} n^{n} - 2n^{j} n^{k}.\tag{42}$$

Furthermore, to first order in *κ*, (39) implies

$$[\{g^0\}^{jk}g^0\_{mn}n^mn^n+2n^jn^k]\_{,k}=0.\tag{43}$$

Not all 10 equations in (42) are independent, as a consequence of the Bianchi identities (38). To see this directly, multiply (42) by *g* 0 *hj* and contract indices to give

$$0 = \overline{\mathcal{R}}/\kappa = -\frac{1}{2}\mathcal{g}\_{mn}^{0}\frac{\partial h}{\partial \mathfrak{x}\_{m}\partial \mathfrak{x}\_{n}} + 6\mathcal{g}\_{mn}^{0}n^{m}n^{n} \tag{44}$$

which is also implied by taking the first-order approximation to (30). Thus, we have

$$0 = \left(\overline{\mathcal{R}}^{\vec{k}} - \frac{1}{2}g^{\vec{\mu}}\overline{\mathcal{R}}\right)/\mathfrak{x} = -\frac{\partial}{\partial \mathfrak{x}\_m \partial \mathfrak{x}\_n}(h^{\vec{\mu}} - \frac{1}{2}\{g^0\}^{\vec{\mu}}h) - [\{g^0\}^{\vec{\mu}}g^0\_{mn}n^mn^n + 2n^jn^k].\tag{45}$$

With (41), we recover (43). In summary, we should first use the four equations (43) to determine the *n i* (**x**), *i* = 1, 2, 3, 4. Then, we should use the 16 Equations (41) and (42), of which only 10 are independent, to determine the 10 functions *hij*(**x**). Writing out Equation (42) explicitly, we obtain

$$\begin{aligned} \nabla^2 h^{ab} - \frac{\partial^2}{\partial t^2} h^{ab} &= 4[\delta\_{ab}(n^2 - n\_4^2) - n\_a n\_b], \\ \nabla^2 h^{a4} - \frac{\partial^2}{\partial t^2} h^{a4} &= 4n\_a n\_{4\prime} \\ \nabla^2 h^{44} - \frac{\partial^2}{\partial t^2} h^{44} &= -4n^2. \end{aligned} \tag{46}$$

where the indices *a* and *b* take values from 1 to 3, *n* <sup>2</sup> = *n* 2 <sup>1</sup> + *n* 2 <sup>2</sup> + *n* 2 3 , and *n<sup>i</sup>* = *g* 0 *ijn j* . As we used the harmonic gauge, there is the additional restriction that the *h jk* satisfy (41), i.e., that

$$\begin{array}{ccccccccc}h\_{,1}^{a1} + h\_{,2}^{a2} + h\_{,3}^{a3} + h\_{,4}^{a4} &=& \frac{1}{2}(h^{11} + h^{22} + h^{33} - h^{44})\_{,a} & a = 1,2,3, \\ h\_{,1}^{41} + h\_{,2}^{42} + h\_{,3}^{43} + h\_{,4}^{44} &=& -\frac{1}{2}(h^{11} + h^{22} + h^{33} - h^{44})\_{,4}. \end{array} \tag{47}$$

The identities (43) imply *T ij* ,*<sup>j</sup>* = 0 with, to zeroth order in *κ*,

$$T^{aa} = \begin{aligned} 2n\_a^2 + n^2 - n\_{4\prime}^2 & \quad T^{ab} = 2n\_a n\_{b\prime} \\ 2T^{44} &= \ 3n\_4^2 - n^2 \end{aligned} \quad T^{ab} = -2n\_a n\_{4\prime} \tag{48}$$

Equivalently, the matrix **T** with elements *T ij* takes the block form:

$$\mathbf{T} = \begin{pmatrix} 2\mathbf{n} \otimes \mathbf{n} + (n^2 - n\_4^2)\mathbf{I} & -2n\_4\mathbf{n} \\ -2n\_4\mathbf{n}^T & 3n\_4^2 - n^2 \end{pmatrix} \tag{49}$$

where **n** *T* is the row vector, which is the transpose of **n**, defined as **n** = (*n*1, *n*2, *n*3).

### **7. Subluminal, Luminal, and Superluminal Regions of Spacetime**

In this section, we do not make the weak field approximation, but we consider any point *P* in spacetime and choose the Minkowski metric (40) at that point.
