3.1. Torsion

We begin with the most pressing issue: how to represent torsion in a classical spacetime. Since the displacement of vectors along one another is given by the derivative operator, to derive a theory of gravity with possibly non-vanishing torsion in the classical context, we must adjust the conditions on our derivative operator. We define a generic (i.e., not specific to the classical context) derivative operator following Malament’s (Reference Malament2012) conventions but drop the last requirement $\left( {DO6} \right)$ —that the action of two derivative operators on a scalar field commute—to allow torsion. This leaves:

Definition. $\nabla $ is a (covariant) derivative operator on $M$ if it satisfies the following conditions:

• (D01) $\nabla $ commutes with addition on tensor fields.

• (D02) $\nabla $ satisfies the Leibniz rule with respect to tensor multiplication.

• (D03) $\nabla $ commutes with index substitution.

• (D04) $\nabla $ commutes with contraction.

• (D05) For all smooth scalar fields $\alpha $ and all smooth vector fields ${\xi ^n}$ , ${\xi ^n}{\nabla _n}\alpha = \xi \left( a \right)$ .

Instead of requiring that $\nabla $ commute in its action on scalar fields $\left( {D06} \right)$ , we set this to be the torsion tensor.

Definition. Let $\nabla $ be a covariant derivative operator on the manifold $M$ . Then, there exists a smooth tensor field ${T^a}_{bc}$ , the torsion tensor, which is defined by

$$2{\nabla _{[a}}{\nabla _{b]}}\alpha = {\nabla _a}{\nabla _b}\alpha - {\nabla _b}{\nabla _a}\alpha = {T^c}_{ab}{\nabla _c}\alpha $$

for all smooth scalar fields $\alpha $ .

Note, from the definition above, that ${T^a}_{bc}$ is anti-symmetric in its lowered indices, $b$ and $c$ :

$${T^a}_{bc} = - {T^a}_{cb}.$$

As in the torsion-free case, the action of any two (possibly torsional) derivative operators, $\nabla $ and $\tilde \nabla $ , can be related by a smooth tensor field ${C^a}_{bc}$ , with the property that for any smooth vector field ${\xi ^a}$ , ${\nabla _a}{\xi ^b} = {\tilde \nabla _a}{\xi ^b} - {C^b}_{an}{\xi ^n}$ (and likewise for other tensor fields). In this case we write $\nabla = \left( {\tilde \nabla, {C^a}_{bc}} \right)$ . Unlike in the torsion-free case, however, this field ${C^a}_{bc}$ need not be symmetric in its lower indices. Instead, we have

$$2{C^a}_{\left[ {bc} \right]} = {T^a}_{bc} - {\tilde{T}^a}_{bc}.$$

3.2. Time and space

We now consider how to represent time and space in a classical spacetime theory with torsion. As in standard NG, we will assume that spacetime has a temporal metric ${t_a}$ and spatial metric ${h^{ab}}$ , both of which will be compatible with the possibly torsional derivative operator $\nabla $ . In standard models of NG, without torsion, it follows from the compatibility of the temporal metric with the (torsion-free) derivative operator that ${t_a}$ is closed, i.e., ${d_a}{t_b} = 0$ , where $d$ is the exterior derivative. This implies that ${t_a}$ is locally exact, i.e., ${t_a} = {\nabla _a}t$ for some smooth time function $t$ . Physically, the availability of a time function means that we can have a well-defined notion of the temporal distance between points. Indeed, if $M$ is simply connected, then we will have a global time function, $t:M \to \mathbb{R}$ . This means our spacetime consists of global simultaneity slices stacked through time, and any two global time functions will differ only in their assignment of the zero-point for the time scale. Compatibility with a torsional derivative operator no longer implies that ${t_a}$ is closed in general. However, we will assume ${t_a}$ is closed and we will only consider derivative operators compatible with closed temporal metrics in what follows.

The curvature of a spacetime is formalized by the Riemann curvature tensor. Intuitively, the Riemann tensor measures the degree to which a vector fails to return to its original value when parallel transported around a closed loop. More formally, it measures the degree to which the second covariant derivatives fail to commute. For a torsion-free spacetime, it is defined as

$${R^a}_{bcd}{\xi ^b} = - 2{\tilde \nabla _{[c}}{\tilde \nabla _{d]}}{\xi ^a}.$$

In spaces with torsion, we adjust the definition of the Riemann tensor to include the contribution from the torsion tensor. ^{Footnote 4} This yields

$${R^a}_{bcd}{\xi ^b} = - 2{\nabla _{[c}}{\nabla _{d]}}{\xi ^a} + {T^n}_{cd}{\nabla _n}{\xi ^a}.$$

There is a valuable formula relating the curvatures of two derivative operators with torsion. If $\nabla = \left( {\tilde \nabla, {C^a}_{bc}} \right)$ , then

$${R^a}_{bcd} = \tilde {R^a}_{bcd} + 2{\tilde \nabla _{[c}}C_{\hskip 3ptd]b}^a + 2C_{[c\left| b \right|}^pC_{\hskip 3ptd]p}^a - \tilde T_{\hskip 4ptcd}^m{C^a}_{mb}.$$

Note that only the torsion of $\tilde \nabla $ appears in this equation.

Let us compare the spatiotemporal geometry of NG and NCT. NG posits that space and time are both flat (i.e., “spacetime is flat”), implying that the Riemann tensor, ${R^a}_{bcd}$ , vanishes entirely. NCT, by contrast, only requires spatial flatness (i.e., “space is flat”). We formalize this condition as ${R^{abcd}} = \bf 0$ , where indices are raised using ${h^{ab}}$ , and interpret it as saying that the parallel transport of spacelike vectors in spacelike directions is, at least locally, path independent. ^{Footnote 5}

To develop a theory most like TPG in the classical context, we will require the curvature of our spacetime to vanish, ${R^a}_{bcd}$ . This is because TPG is set on a flat spacetime background, and we are seeking a classical theory analogous to it.

We have not, thus far, placed any constraints on the torsion tensor. Recall that in NCT, the spatial curvature vanishes. Analogously, we propose that the spatial torsion of our spacetime vanish (i.e., ${T^{abc}} = 0$ ). The vanishing of the spatial torsion will yield a theory like NCT but with torsion, not curvature, encoding gravitational influence.

3.3. Sources and forces

Finally, let us consider how we expect sources to exert (torsional) force in our theory. It will be instructive to consider the treatment of sources and forces in the non-torsional, classical context first. In NG, bodies are subject to gravitational forces and force is mediated by a gravitational potential ( $\phi $ ). The four-velocity, ${\xi ^a}$ , of a particle satisfies

(1) $$ - {\nabla _a}\phi = {\xi ^n}{\nabla _n}{\xi _a},$$

where $\phi $ is a smooth, scalar field and $\nabla $ denotes the flat, torsion-free derivative operator of standard NG. The right-hand side of the equation describes the acceleration that the test point particle undergoes in the presence of the gravitational potential, $\phi $ . The gravitational potential further satisfies Poisson’s equation, relating it to the distribution of matter,

(2) $${\nabla _a}{\nabla ^a}\phi = 4\pi \rho, $$

where $\rho $ is the Newtonian mass density function.

In NCT, like in GR, the curvature of spacetime means that inertial motion is governed by the geodesic principle: in the absence of external (non-gravitational) forces, bodies move along the geodesics of (curved) spacetime. The equation of motion is given as

(3) $${\xi ^n}{\tilde \nabla _n}{\xi ^a} = \bf0,$$

where $\tilde \nabla $ is the curved derivative operator of NCT.

To account for spatiotemporal curvature, NCT adopts a geometrized form of Poisson’s equation, relating the distribution of matter to the curvature of spacetime,

(4) $${R_{ab}} = 4\pi \rho {t_a}{t_b}.$$

As it turns out, models of NG and NCT are systematically related. The Trautman geometrization lemma and degeometrization theorem describe these relations. Let us consider the recovery of models of NG from NCT. This is the direction in which force terms arise and so it will be instructive in formulating torsional forces. To build up to the degeometrization theorem, we will first consider the derivative operators of each theory. One can show that in the non-torsional context, one has the following result.

Proposition 1 (Malament (Reference Malament2012, Proposition 4.1.3)). Let $\left( {M,{t_a},{h^{ab}},\tilde \nabla } \right)$ be a classical spacetime. Let $\nabla = \left( {\tilde \nabla, {C^a}_{bc}} \right)$ be a second derivative operator on $M$ . Then, $\nabla $ is compatible with ${t_a}$ and ${h^{ab}}$ if and only if ${C^a}_{bc}$ is of the form

$${C^a}_{bc} = 2{h^{an}}{t_{(b}}{\kappa _{c)n}},$$

where ${\kappa _{ab}}$ is a smooth anti-symmetric field on $M$ and the parentheses denote symmetrization.

If we permit derivative operators with torsion, a broader class of derivative operators are compatible with the classical metrics. We now have the following generalization of the preceding proposition.

Proposition 2 Let $\left( {M,{t_a},{h^{ab}},\tilde \nabla } \right)$ be a classical spacetime with (possibly) non-vanishing torsion. Let $\nabla = \left( {\tilde \nabla, {C^a}_{bc}} \right)$ be another derivative operator on $M$ also with (possibly) non-vanishing torsion (i.e., $2{C^a}_{\left[ {bc} \right]} = {T^a}_{bc} - {\tilde T^a}_{bc}$ ). Then $\nabla $ is compatible with ${t_a}$ and ${h^{ab}}$ if and only if ${C^a}_{bc}$ is of the form

$${C^a}_{bc} = 2{h^{ar}}{\kappa _{\left[ {r\left| b \right|c} \right]}}.$$

If, in addition, we require ${\tilde T^{abc}} = {T^{abc}} = \bf0$ , then

$${C^a}_{bc} = 2{h^{ar}}\left( {{x_{rb}}{t_c} + {y_{rc}}{t_b}} \right)$$

where ${x_{ab}}$ is an arbitrary smooth tensor field and ${y_{rc}}$ is any smooth anti-symmetric field.

As we can see, in the presence of torsion, there is considerable freedom to define metric-compatible derivative operators. Below, we limit our attention to flat, metric-compatible derivative operators whose torsion has the form ${T^a}_{bc} = 2{F^a}_{[b}{t_{c]}}$ where ${F^a}_b$ is a smooth rank (1,1) tensor field, spacelike in the $a$ index. This is tantamount to stipulating that ${y_{ab}}$ in Proposition 2 vanishes. This restriction clearly satisfies the above-outlined general form and ensures that the spatial torsion vanishes. Furthermore, as will be seen in the following theorem, we can recover the standard, torsion-free connecting field assumed for the degeometrization theorem as a special case of the above.

In describing the difference between the derivative operators, the connecting field is closely related to the force field that arises in the degeometrization of a model of NCT. In the torsion-free context, one typically assumes a connecting field of the form ${C^a}_{bc} = {t_b}{t_c}{\tilde \nabla ^a}\phi $ . ^{Footnote 6} The force term is then just the contracted connecting field: ${C^a}_{rn}{\xi ^r}{\xi ^n} = {\tilde \nabla ^a}\phi $ .

To adapt this to the torsional context, we want to consider the connecting field relating a non-torsional, flat derivative operator to a torsional one. We will capture the impact of the torsion on the trajectories of test bodies using the above-mentioned tensor field, ${F^a}_b$ . In other words, we want ${F^a}_b$ to play the role of a torsional force term. Given a timelike geodesic of $\tilde \nabla $ with unit tangent field ${\xi ^a}$ , the force equation we expect to be satisfied is

(5) $${\xi ^n}{\nabla _n}{\xi ^a} = {\xi ^n}{\tilde \nabla _n}{\xi ^a} - {C^a}_{rn}{\xi ^r}{\xi ^n} = - {F^a}_n{t_c}{\xi ^n}{\xi ^c} = - {F^a}_n{\xi ^n},$$

where $\nabla $ is the derivative operator of our torsional spacetime.

Finally, we want to relate the torsional force term to gravitational sources. In other words, we want to formulate a field equation that is the torsional analog to Poisson’s equation. It will turn out to be

(6) $${\delta _a}^n{\nabla _{[n}}{F^a}_{b]} = 2\pi \rho {t_b}.$$

Again, Poisson’s equation will be recovered as a special case of equation (6), but equation (6) more generally establishes a relation between the first derivative of the force term and the mass distribution along the temporal direction.

3.4. Comparison with other classical theories with torsion

As mentioned in Section 1, a small number of publications have recently emerged in physics surrounding torsional classical spacetime theories. Many in this literature are interested in incorporating torsion in the classical context to address the different notions of time proposed by GR and quantum gravity (QG): though GR does not admit a global notion of simultaneity, in some formulations, QG does. To resolve this difference, some authors have proposed taking the notion of time presented in QG as fundamental and allowing relativistic time to emerge at large distances. Then, motivated by the holographic principle (i.e., that a volume of space can be thought of as encoded in the lower-dimensional boundary of that volume), ^{Footnote 7} this literature considers 5D QG and its 4D reduction. The holography considered is between Hořava–Lifshitz gravity and a new theory of classical gravity: twistless torsional Newton–Cartan (TTNC) theory. ^{Footnote 8}

As noted above, we typically in the classical spacetime context take ${t_a}$ to be closed (i.e., ${d_a}{t_b} = 0$ ), which means it is locally exact and determines a local time function. This is a consequence of its compatibility with any torsion-free derivative operator. We also adopt this assumption in the torsional gravitational theory developed here.

The TTNC formalism, by contrast, starts with NC theory but claims that taking ${\partial _\mu }{t_\nu } = 0$ , where $\partial $ is a (torsion-free) coordinate derivative operator, will always result in a torsion-free spacetime. They do not require temporal metrics to be compatible with any torsion-free derivative operator; more generally, they do not require that ${t_a}$ is closed. Consider, for instance, the following claim from Christensen and collaborators: “The boundary geometry becomes Newton–Cartan [i.e., has no torsion] … if and only if ${\tau _{\left( 0 \right)a}}$ is taken to be closed” (Christensen et al., Reference Christensen, Hartong, Obers and Rollier2014, 25). For another illustration of this point, consider the following passage:

They later elaborate that the conditions under which $\tau_\mu$ is closed coincide with torsion vanishing (pp. 10-11).

In order to derive a hypersurface orthogonal temporal metric, such authors appeal to Frobenius’ theorem. This allows them to argue that a spacetime admits a foliation with a time flow orthogonal to the Riemannian spacelike slices if and only if it satisfies the hypersurface orthogonality condition (i.e., ${t_{[a}}{\partial _b}{t_{c]}} = 0$ ). Notably, the “hypersurface orthogonality condition” is a weaker condition than the condition that the temporal metric be closed.

A series of questions emerge from the above discussion: Why does the TTNC literature claim that closed temporal metrics, and metric compatibility more generally, are in tension with torsion? And how does the theory described above incorporate torsion and a closed temporal metric, and thus a notion of absolute time? The answers lie in the form of the connection assumed by the TTNC literature.

Geracie and collaborators define a spacetime derivative operator $\nabla = \left( {\partial, {{\rm{\Gamma }}^a}_{bc}} \right)$ , where they require the form of the connecting field ${{\rm{\Gamma }}^a}_{bc}$ to be

$${\Gamma ^a}_{bc} = {v^a}{\partial _b}{t_c} + {1 \over 2}{h^{an}}\left( {{\partial _b}{{\hat h}_{cn}} + {\partial _c}{{\hat h}_{bn}} - {\partial _n}{{\hat h}_{bc}}} \right),$$

where ${v^a}$ is a unit timelike field, and ${\hat h_{ab}}$ is a spatial projection field determined by ${v^a}$ such that ${h^{an}}{\hat h_{nb}} = {\delta ^a}_b - {v^a}{t_b}$ (Reference Geracie, Thanh Son, Wu and Wu2015, equation (77)). ^{Footnote 10} This definition is motivated by the standard definition of a Levi-Civita derivative operator, and the terms in the parentheses are always symmetric in $b,c$ . It follows that the torsion is given by ${T^a}_{bc} = 2{{\rm{\Gamma }}^a}_{\left[ {bc} \right]} = 2{v^a}{\partial _{[b}}{t_{c]}}$ (see, e.g., Geracie et al., Reference Geracie, Thanh Son, Wu and Wu2015, equation (79)). (Indeed, the name “twistless torsional NCT,” then, comes from the fact that torsion vanishes on spacelike hypersurfaces but not in general.)

And so it is true that if ${t_a}$ is closed, the torsion of this derivative operator would vanish. However, this is only because they have adopted such a strict definition for their connection. Put differently, their connection ensures that the only way to allow torsion is to sacrifice having a closed temporal metric. But there are many other torsional derivative operators that are compatible with a closed temporal metric. Once we allow for a broader class of connections, as is done in the present paper, we recover metric compatibility and a notion of absolute time.

Classical spacetimes with torsion have also been the subject of philosophical analysis. In particular, James Read and Nicholas Teh (Reference Read and Teh2018) have developed a method for “teleparallelizing” in the classical context. A central aim of their project is to show the relation between this classical theory and its relativistic counterpart, TPG. They begin with NCT and teleparallelize to construct a classical spacetime with torsion. Their procedure involves a “mass torsion” term that plays the role of a force. ^{Footnote 11} Contrary to the results derived in the present paper, they claim their teleparallelization method yields standard NG. Their proposal is interesting in its own right. However, we would argue that the spacetime developed in the present paper, in so far as it features spacetime torsion instead of mass torsion, is a stronger analog to a classical TPG.