Hamiltonian Form of Gravity around a Singularity

To set the notation and variables, let us briefly recall the standard Hamiltonian representation of Hilbert-Palatini gravity. In terms of the tetrad $e_{\mu}^{I}$ and spin-connection fields $\omega_{\mu}^{~{}IJ}$, the Lagrangian density reads:

Here $\kappa$ is the gravitational coupling and $R_{\mu\nu}^{~{}~{}~{}IJ}(\omega)=\partial_{[\mu}\omega_{\nu]}^{~{}IJ}+\omega_{[\mu}^{~{}IK}\omega_{\nu]K}^{~{}~{}~{}J}$ is the $SO(3,1)$ field strength.

The standard redefinition of the tetrad fields in terms of a scalar Lapse $N$ and the shift $N^{a}$ is given by Peldan (1994):

The spatial metric is defined as: $q_{ab}~{}:=~{}V_{a}^{I}V_{bI}$, and the tetrad determinant as: $e:=det(e^{I}_{\mu})=N\sqrt{q}=\sqrt{-g}$.

The resulting Hamiltonian density is given by a sum of constraints:

where the constraints have the following expressions:

In the above the momenta conjugate to the canonical coordinates $\omega_{a}^{~{}IJ}$ are given by:

Next, we introduce a set of new variables as Barros e Sa (2001); Date et al. (2009); Kaul and Sengupta (2012):

In these variables, the symplectric form reads:

Note that $q=\det E^{a}_{i}:=E$. In the above, the L.H.S. contains eighteen canonical pairs, whereas the R.H.S. has only twelve. Thus, the six redundant components ($N_{kl}$) of $\omega_{a}^{~{}ij}$ correspond to vanishing momenta:

A consistent time evolution of these leads to additional secondary constraints $\phi^{kl}\approx 0$:

Next, we fix the boost freedom ($G_{0i}=0$) by choosing time-gauge $\chi_{i}=0$. This in turn fixes the conjugate momenta as: $\zeta_{i}=\partial_{a}E^{a}_{i}$. The constraints $\phi^{kl}$ defined above then determine $N^{kl}$ (and hence $\omega_{a}^{~{}ij}(E)$) completely:

The resulting constraints are displayed below:

where in the last line $\omega_{a}^{~{}ij}(\zeta^{k},N^{lm})\equiv\omega_{a}^{~{}ij}(E)$ are defined completely in terms of the densitized triad.

Let us now introduce the following scaling of the spatial triads:

where the index $a$ in the scaling parameter $\delta_{(a)}$ (the round brackets indicate that these indices are not summed when repeated) runs over the three space coordinates. To emphasize, the scaling parameters along the spatial directions are allowed to be anisotropic and inhomogeneous in general. Using the requirement that the symplectic form (or, Poisson brackets) be preserved under the scaling (9), it acts on the phase space variables as:

where $(\delta=\prod_{a}\delta_{(a)})$. We define the limit to a vanishing triad determinant ($q\rightarrow 0$) as: $\frac{\delta}{\delta_{(a)}}\rightarrow 0$ for any $a$. This implies that as the singularity is approached, the determinant $\sqrt{q}=\delta\sqrt{q^{\prime}}$ vanishes faster than the rate of contraction or expansion of any of the three spatial directions.

The dependent variables $\zeta_{i}(E)$ and $N^{kl}(E)$ read:

where we have assumed the derivatives of the scaling parameters to satisfy the following properties:

($a,b,c$ above may or may not be equal).

Next, let us first rewrite the constraints in the rescaled variables and then take the limit. Note that the rotation constraints remain unaffected. Using (11), the diffeomorphism constraint (upto a rotation) is found to transform as:

Similarly, it is straightforward to check that the last two terms in the Hamiltonian constraint in eq.(II.1) are subleading, leading to a genuine truncation. Noting that the natural partial derivative on the rescaled variables is $\partial^{\prime}_{a}\equiv\frac{\delta}{\delta_{(a)}}\partial_{a}$, as fixed by the transformation of the connection, the new set of constraints finally read:

where the lapse and shift get rescaled as: $N=\delta.N^{\prime},~{}N^{a}=\frac{\delta}{\delta_{(a)}}.N^{{}^{\prime}a}$. The Hamiltonian constraint is purely algebraic (upto an overall density factor) in the canonical variables. In other words, the non-polynomiality characteristic of the Hilbert-Palatini Hamiltonian constraint owing to the spatial scalar curvature term is absent here.

In this context, let us note that a Hamiltonian constraint of the same form as above may also be obtained in the first order form of ‘electric’ Carroll gravity associated with the $c\rightarrow 0$ limit Sengupta (2023). However, the latter limit corresponds to a constant scaling parameter acting on the variables. Hence, the $q\rightarrow 0$ limit is more general, and the $c\rightarrow 0$ limit might be interpreted as a special case with $\delta_{(a)}=const.$, as apparent from the scaling laws explicitly presented in ref.Sengupta (2023) ¹¹1We observe that a generalization of the ‘magnetic’ Carroll limit Bergshoeff et al. (2017); Henneaux and Salgado-Rebolledo (2021); Sengupta (2023) may be defined within the limiting formalism here, where the coordinates scale differently: $Q_{a}^{i}=\delta_{(a)}Q_{a}^{{}^{\prime}i}$. In this case, the Poisson brackets are not preserved. However, it is neither relevant nor our purpose here to discuss this limit any further..

As an example of an inhomogeneous and anisotropic universe, let us consider the following spacetime Kasner (1921); Belinsky et al. (1970); *bkl1:

Such a solution can exist in presence of matter in Einstein gravity. Assuming that the effect of matter could be neglected close to the spacelike curvature singularity at $t=0$, we may use the Hamiltonian theory constructed in the previous section to analyze how the singularity is approached in this case.

The nontrivial components of the momenta are given by (suppressing the primes which are implicit in the variables and constraints):

The time evolution of the canonical pair with respect to the Hamiltonian constraint (II.2) leads to (in the gauge $\omega_{t}^{~{}ij}=0$):

From the first equation above, the nontrivial components of the coordinates are obtained as:

The second evolution equation in (III.1) for the coordinates, along with the Hamiltonian constraint, imply:

Assuming $\alpha\neq 0,~{}\beta\neq 0,~{}\gamma\neq 0$, these are solved as: $\alpha+\beta+\gamma=1=\alpha^{2}+\beta^{2}+\gamma^{2}$. Next, the spatial diffeomorphism constraints lead to:

These restrict the Kasner exponents as: $\alpha=\alpha(y,z),~{}\beta=\beta(z,x),~{}\gamma=\gamma(z,x)$, as the curvature singularity is approached. The rotation constraints are trivially satisfied.

To understand the behaviour of the derivatives ofthe scaling parameters, let us note that the approach to the singularity here could be parametrized as $\epsilon\rightarrow 0$ where $t=\epsilon t^{\prime}$ ($t^{\prime}$ being finite). Inserting this in the metric solution just obtained above, we find:

It is now straightforward to verify that the space derivatives satisfy all the conditions encoded in (11).

Here we consider the essential model underlying inflationary cosmology, namely, gravity coupled to a scalar field:

The Hamiltonian theory in the time-gauge is now defined by the following set of constraints:

While the canonical variables for gravity scale as in (10), the matter phase space pair $(\phi,~{}\pi:=\frac{\sqrt{q}}{N}(\dot{\phi}-N^{a}\partial_{a}\phi))$ do not transform. Rescaling the variables, taking the degenerate limit $\frac{\delta}{\delta_{(a)}}\rightarrow 0$ and then redefining the derivative and Lagrange multipliers exactly as earlier, we find once again that only the Hamitonian constraint gets modified (we suppress the primes as earlier):

For an FLRW spacetime with a spatial curvature $K$ is given by:

the non-vanishing components of the canonical pairs are the following:

Using the above expressions, the Hamiltonian constraint and the time evolution of $Q_{a}^{i}$ lead to the following equations of motion:

For a comoving ideal fluid characterized by a density $\rho$ and pressure $P$, this implies a stiff equation of state: $\rho=P=\frac{\pi^{2}}{2E}$ (implying $a\sim t^{\frac{1}{3}}$). The spatial diffeomorphism constraints simplify as:

Finally, the scalar equations of motion are given by:

The solution to the above is: $\phi\sim lnt$.

To emphasize, the $q\rightarrow 0$ limit rids the equations of motion of the spatial derivatives, the spatial curvature and the scalar potential. Hence, for any arbitrary scalar potential, it predicts the existence of a stiff phase right after the birth of the Universe (singularity).

As our next example, we consider a spherically symmetric geometry where the radial coordinate is timelike:

Proceeding as in the previous example, we obtain the only nontrivial components of the triad and connection as (in the gauge $\omega_{T}^{~{}ij}=0$):

The Hamiltonian constraint reads:

whose solution is: $e^{\lambda}=\frac{C(R)}{T}$, $C(R)$ being an arbitrary function. All the remaining constraints are trivial, except the radial component of the diffomorphism:

Finally, from the evolution equations for $Q_{a}^{i}$, we obtain:

Comparing (24) and (25), we find the solution: $e^{\mu}=\frac{T}{2M}$, where $M=const.$. The solution for the metric (III.3) finally reads:

where the function $C(R)$ has been absorbed (upto a const.) into a redefinition of the radial coordinate.

Note that the above is precisely the Schwarzschild metric in the limit $r\rightarrow 0$, upon the identification $T\equiv r$, $R\equiv t$ to the usual Schwarzschild coordinates and $M$ to the black hole mass. Thus, from the limiting theory, we find that the spacetime around the singularity could only be the interior Schwarzschild geometry when the radial coordinate is timelike.