In the previous classes we went beyond GR by adding new fields — like scalars or vectors. Here we do something different and, in a sense, more conservative: we add nothing at all. We simply notice that GR quietly switched off a structure that was available from the start, and we switch it back on.

But why would one set null torsion in GR?

Reminders on torsion

As stated already in the trinity class, a smooth manifold can carry two logically independent pieces of structure, and it is worth insisting on how independent they really are:

  1. a metric $g_{\mu\nu}$, which tells you how to measure lengths and angles;
  2. an affine connection $\Gamma^\rho_{\mu\nu}$, which tells you how to parallel-transport a vector from one point to a neighbouring one, and hence how to differentiate.

Nothing in the definition of a manifold forces these to be related. GR imposes two extra conditions by hand:

\[\underbrace{\nabla_\rho g_{\mu\nu} = 0}_{\text{metricity}}, \qquad \underbrace{\Gamma^\rho_{\mu\nu} = \Gamma^\rho_{\nu\mu}}_{\text{no torsion}} .\]

Together these are exactly enough to fix $\Gamma$ uniquely in terms of $g$ as being the Levi-Civita connection (the Christoffel symbols). The point to appreciate is that this is a choice, not a theorem about nature.

As a reminder torsion can be thought as the failure of parallelograms to close and is defined as:

\[\boxed{\;T^\rho{}_{\mu\nu} \;\equiv\; \Gamma^\rho_{\mu\nu} - \Gamma^\rho_{\nu\mu} \;=\; 2\,\Gamma^\rho_{[\mu\nu]}\;}\]

It is a tensor, even though $\Gamma$ is not. This is relevant, because $T^\rho{}_{\mu\nu}$ is a tensor, it cannot be removed by any choice of coordinates. If it is non-zero anywhere, it is non-zero for every observer.

Now, we can identify at least four reasons, of very unequal quality, for why torsion is set to zero within GR:

1. History. Riemannian geometry — Riemann, Ricci, Levi-Civita — was built with $T=0$ from the start, and it was the only geometry available in 1915. Cartan introduced torsion in 1922, seven years after GR. Einstein could not have used what did not exist. (Cartan and Einstein then corresponded about it at length in 1929–1932, without agreeing.)

2. Economy. With $T=Q=0$, the connection is determined by the metric. Allowing torsion means carrying $24$ extra independent components with no obvious source, and GR was already reproducing observations.

3. A version of the equivalence principle. With $T=0$ one can always choose coordinates such that $\Gamma^\rho_{\mu\nu}(p)=0$ at a chosen point — this is the local inertial frame, the mathematical expression of “freely falling observers see no gravity”. With torsion this is impossible, since a tensor cannot be transformed away. But we should be careful here It does not mean torsion violates the WEP for spinless test particles. A structureless test particle obeys the equation of motion derived from its action $S = -m\int\sqrt{-g_{\mu\nu}dx^\mu dx^\nu}$, which contains the metric only, so it follows an ordinary metric geodesic whatever the torsion does. What fails is only the statement “the connection can be flattened at a point”, not “all bodies fall alike”. Torsion couples to spin, not to mass. However two particles with different spins will fall differently and this is a violation of the WEP.

4. The retrospective reason, and the best one. As we will see in the next section, in the minimal theory torsion turns out to be non-propagating and to vanish identically outside matter. Setting $T=0$ was therefore, by accident, almost harmless. Einstein got the right answer for the wrong reason.

Einstein–Cartan theory

We will now add torsion back to general relativity and assess the consequences. This is known as Einstein-Cartan theory. For more on this topic, see Sciama 1964; Kibble 1961 and Hehl, von der Heyde, Kerlick & Nester 1976. Keep the Einstein–Hilbert action exactly as it is,

\[S = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\;R(g,\Gamma) \;+\; S_m[g,\Gamma,\psi],\]

but now treat $g_{\mu\nu}$ and $\Gamma^\rho_{\mu\nu}$ as independent variables and vary with respect to both (the Palatini procedure we already encountered in another lecture). That is the entire modification. No new field, no new parameter, no new scale. Varying with respect to the connection, one obtains

\[T^\rho{}_{\mu\nu} + \delta^\rho_\mu\,T^\sigma{}_{\sigma\nu} - \delta^\rho_\nu\,T^\sigma{}_{\sigma\mu} \;=\; 8\pi G\; s^\rho{}_{\mu\nu},\]
Proof

in prep

where \(s^\rho{}_{\mu\nu} \equiv \delta S_m/\delta K^{\mu\nu}{}_\rho\) is the spin angular momentum density of matter — the intrinsic spin, not orbital.

Read this equation carefully, because everything follows from its form. It contains no derivatives of the torsion. It is algebraic. Therefore:

This is a very unusual situation among the theories in this course: a modification of gravity that costs nothing observationally, because it is invisible by construction rather than by screening.

Eliminating torsion algebraically leaves an ordinary Einstein equation with an extra piece:

\[G_{\mu\nu} = 8\pi G\left(T_{\mu\nu} + T^{\rm spin}_{\mu\nu}\right),\qquad T^{\rm spin}_{\mu\nu} \sim -\,8\pi G\,s^2 .\]

The spin correction is a four-fermion contact interaction (the Hehl–Datta term), schematically $\sim G\,(\bar\psi\gamma^5\gamma^\mu\psi)^2$, and — this is the crucial sign — it enters with a negative effective energy density. Gravity acquires a repulsive component at very high fermion density.

Now, when does any of this matter? Let us estimate it. Spin density $s\sim \hbar n/2$ for $n$ fermions per unit volume. The spin term competes with the ordinary one when

\[\underbrace{G\,s^2}_{\text{spin}} \;\sim\; \underbrace{m\,n}_{\text{mass}} \qquad\Longrightarrow\qquad n \;\sim\; \frac{m}{G\hbar^2} \;=\; \frac{1}{\overline{\lambda}_{\rm C}\,\ell_{\rm Pl}^2},\]

with \(\overline{\lambda}_{\rm C}=\hbar/mc\) the Compton wavelength and $\ell_{\rm Pl}$ the Planck length. For nucleons this gives the Cartan density

\[\boxed{\;\rho_C \;\sim\; 10^{54}\ \mathrm{g/cm^3}\;}\]

To put this, in context, a neutron star core has a density of $10^{15}$ g.cm\({^-3}\) while the Planck density is of $10^{93}$ g.cm\({^-3}\). So torsion is utterly negligible for stars — including neutron stars, by nearly forty orders of magnitude. But notice the second comparison, which is the interesting one: $\rho_C$ is roughly $10^{-40}$ of the Planck density. Since $\rho\sim T^4$, that means it is reached at a temperature

\[T_C \sim 10^{-10}\,T_{\rm Pl} \sim 10^{9}\ \mathrm{GeV},\]

ten orders of magnitude below the Planck scale. Whatever torsion does in the early universe, it does before quantum gravity is required. That is what makes Einstein–Cartan cosmology worth taking seriously rather than dismissing as speculation. However, such phenomena could be relevant in the very early Universe, for example as alternative to inflation (Popławski (2010)).

Now, how about the geodesic equation with torsion? With torsion, “straightest” and “shortest” come apart:

Which one do particles follow? The question sounds deep but the answer is prosaic: you do not get to choose — the matter action decides. A structureless particle has $S=-m\int d\tau$, which knows only about $g_{\mu\nu}$, so it follows the metric geodesic. Autoparallels are a different curve that nothing forces particles onto.

(To be discussed: for the totally antisymmetric torsion that a Dirac field actually generates, the two coincide anyway — the symmetric part of the contorsion vanishes identically.) Spinning bodies are another matter: their spin precesses relative to the metric-parallel-transported frame. This is the only way torsion could ever be detected directly.

Because torsion couples to spin, the tests are spin-polarised experiments, not orbital ones:

These bounds constrain a hypothetical background torsion. They say almost nothing about Einstein–Cartan theory itself, which predicts zero torsion outside matter and is therefore, in its minimal form, essentially untestable with present technology.

If you want torsion to do something observable, you must give it dynamics — that is, put derivatives of torsion into the action. This means quadratic invariants:

\[\mathcal{L} = a_0 R + \sum_i a_i\, T^2 + \sum_j b_j\, R^2 ,\]

the general Poincaré gauge theory (PGT), with about ten free parameters. Now torsion propagates and carries massive modes of spin-parity $0^\pm, 1^\pm, 2^\pm$.

The price is severe: generic parameter choices give ghosts or tachyons. Only a small number of combinations are healthy (Sezgin & van Nieuwenhuizen 1980; see Obukhov 2018 for the modern overview). This is the same Ostrogradsky-flavoured problem that motivated the Horndeski construction, reappearing in the connection sector.

Gauge theory of gravity

in prep

Further reading