November 25, 2025
The general uncertainty principle applied to gravity can be implemented as a set of modified Poisson brackets in the canonical formalism. As such, the theory is not canonical and the resulting equations of motion do not lead to a covariant metric. We construct a Hamiltonian that when applying the usual canonical formalism gives a closed algebra and equations of motion that result in the original metric obtained by using distorted Poisson brackets. The resulting theory is thus rendered canonical and covariant. We then covariantly couple scalar matter and dust to the modified gravity to allow the study of dynamics.
Canonical form of a deformed Poisson bracket spacetime
Douglas M. Gingrich
Department of Physics, University of Alberta, Edmonton, AB T6G 2E1 Canada
TRIUMF, Vancouver, BC V6T 2A3 Canada
e-mail: gingrich@ualberta.ca
2026-06-16
One of the grand challenges in physics is to reconcile quantum mechanics and gravity. A theory of quantum gravity is thought necessary to understand extreme conditions like those inside black holes and at the Big Bang. The most promising, or at least most popular, approaches to date have been string theory and loop quantum gravity, both of which are not entirely satisfactory.
To get a hint of which direction to go, it becomes necessary to explore ideas at a more phenomenological level that modify general relativity to achieve the desired goals that we believe a full quantum theory will solve. One necessity of these models, as a bare minimum, is the resolution of singularities that plague general relativity. The models often also incorporate plausible feature we believe a full quantum theory of gravity could possess.
There are a class of long-standing modified gravity theories which take the approach of modify the Einstein-Hilbert action of general relativity. Theories such as \(f(R)\) gravity [1], scalar-tensor gravity [2], and higher-order curvature invariants [3], to name just three, have had varying success and are well established fields of study. As they start from the Lagrangian formalism they are necessarily covariant.
While these studies are of fundamental importance, one should also consider other types of modified gravity beyond simply adding terms to the action. These ideas, often being generated heuristically, are not necessarily covariant [4] and can lack a connection to underlying theories. But at the same time can be highly effective at singularity resolution and incorporating other desirable features we believe a quantum theory of gravity could allow.
Since the uncertainty principle is foundational to quantum mechanics, one might posit incorporating a modified version of the uncertainty principle [5]–[11] into gravity as a method for capturing some quantum aspects of the theory. One approach is to modify the Poisson bracket in the canonical formalism of gravity to incorporate some general uncertainty principle (GUP) considered as an extension of quantum mechanics [12]–[15].
We consider the approach taken in [16], [17]. After modifying the Poisson brackets between conjugate variables to represent a GUP in the black hole interior, solutions to the equations of motion of the triad are found. From these solutions, the interior metric is constructed. The interior metric is then analytically extended to the full spacetime by switching the timelike and radial spacelike coordinates. The singularity is resolved, and the correct classical and asymptotic limits are obtained. Since the Poisson brackets have been distorted, the theory is not canonical and is not covariant (see Appendix 7). Here we remedy this situation.
To connect the GUP spacetime to the Hamiltonian canonical formalism of gravity, we consider the most general Hamiltonian constraint up to second order in the derivatives and quadratic in the first-order derivatives of the phase-space variables. It must form a closed hypersurface deformation algebra with the diffeomorphism constraint, and thus lead to a dynamical flow covariantly defining a spherical symmetric four-dimensional geometry and hence metric on the manifold [18], [19].
Starting from the GUP inspired spacetime, we apply a method that constructs back to the Hamiltonian, such that its dynamical flow covariantly reproduces the given GUP metric. The Hamiltonian is defined as a linear combination of the standard radial diffeomorphism constraint and a deformed Hamiltonian constraint with respect to general relativity which incorporates the GUP corrections. The formalism is covariant. The solution to the constraint equations in different gauges will give the same geometry, though in different coordinate charts.
The Hamiltonian construction provides a dynamical theory. Thus allowing a consistent and covariant coupling to matter fields. The coupling to matter allows the calculation of its evolution using test fields on the spacetime, as well as the backreaction of the matter fields on the geometry. It also allows a determination of whether this particular black hole model may emerge as the end result of dynamical collapse.
An outline of this paper is as follows. In Sec. 2, we summarize the Hamiltonian formalism of classical gravity thus defining the terms and notation that will subsequently be used. We revisit the Poisson brackets leading to the GUP inspired spacetime in Sec. [sec:gup]. The metric is transformed to correspond to a form that can be converted to the Hamiltonian constraint. In Sec. 3, we write down the Hamiltonian constraint for the GUP inspired spacetime and obtain the equations of motion of the emergent gravity phase-space variables. In Sec. 4, we consider the static and homogeneous gauges and obtain the same line element that resulted from a distorted Poisson algebra. By construction the Hamiltonian leads to a covariant theory. We couple matter to the metric in Sec. 5. The deformed Poisson bracket in the static gauge is considered in Appendix 7. Appendix 8 gives more details about the calculation of the hypersurface deformed algebra. Appendix 9 derives the scalar equation of motion in a spherically symmetric spacetime using the Hamiltonian formalism and shows it to be identical to that derived in the same spacetime using the action (Appendix 10). Dust is also coupled to the modified gravity. The paper finishes with a summary and comments on how the results can, and will, be used in future work to study gravitational perturbations and collapse.
In this section, we review the canonical gravity formalism and set the notation which we will use. We follow [18] and in particular the notation of [20]. The phase-space coordinates consist of four fields \(K_x, E^x, K_\varphi\), and \(E^\varphi\), where each is a function of time \(t\) and a radial coordinate \(x\) in the spherically symmetric manifold. In the classical theory, the momenta \(E^x\) and \(E^\varphi\) are components of the densitized triad1, while the configuration variables \(K_x\) and \(K_\varphi\) are directly related to the extrinsic curvature components \(\mathcal{K}_\varphi K_\varphi\) and \(\mathcal{K}_x = 2K_x\).
The symplectic structure2 is chosen to be canonical, such that for a give \(t\) the only nonvanishing Poisson brackets are
\[\{ K_x(t,x_1), E^x(t,x_2) \} = \delta(x_1-x_2) \quad\text{and}\quad \{ K_\varphi(t,x_1), E^\varphi(t,x_2) \} = \delta(x_1-x_2).\]
The diffeomorphism constraint arises from the requirement that the theory is invariant under spatial coordinate transformations. It ensures that the physical states of the theory are independent of the choice of spatial coordinate on the three-dimensional hypersurface. In the ADM formulation of general relativity, the diffeomorphism constraint is obtained by varying the action with respect to the shift vector (see [21]). The resulting diffeomorphism constraint is
\[\mathcal{H}_x = - K_x (E^x)^\prime + E^\varphi K_\varphi^\prime,\] where primes represent derivatives with respect to \(x\). The Hamiltonian constraint \(\mathcal{H}\) is given in [18] and an equivalent version in [19]. Since we do not directly use either of these Hamiltonians, we do not reproduce them here.
Since the phase-space variables are fields, the Poisson brackets are not strictly an algebra but a distribution. To avoid using distributions, one usually smears the fields or functionals of them. Since the Hamiltonian constrains are a function of the fields, we can integrate over them times an arbitrary function, including the spatial determinant since we are on a curved manifold. One thus sees the utility of densitizing the triad. We define the usual smeared forms for the Hamiltonian for arbitrary function \(s\):
\[H_x[s] = \int s \mathcal{H}_x dx \quad\text{and}\quad H[s] = \int s \mathcal{H} dx.\] The hypersurface deformation brackets for spherical symmetry are
\[\begin{align} \{ H_x[s_1], H_x[s_2]\} &= H_x[s_1 s_2^\prime - s_1^\prime s_2],\tag{1}\\ \{ H_x[s_1], H[s_2]\} &= H[s_1 s_2^\prime],\tag{2}\\ \{ H[s_1], H[s_2]\} &= H_x[q^{xx}(s_1 s_2^\prime - s_1^\prime s_2)],\tag{3} \end{align}\] where \(q^{xx}\) is a structure function. Note that we have dropped the tilde which often appears on the top of the structure function to distinguish it from the classical function.
The time evolution of the phase-space variables is given by
\[\begin{align} \dot{E}^x &= \{E^x, H[N] + N_x[N^x] \},\tag{4}\\ \dot{K}_x &= \{K_x, H[N] + N_x[N^x] \},\tag{5}\\ \dot{E}^\varphi &= \{E^\varphi, H[N] + N_x[N^x] \},\tag{6}\\ \dot{K}_\varphi &= \{K_\varphi, H[N] + N_x[N^x] \},\tag{7} \end{align}\] where dots represent time derivatives. The canonical conjugate momenta of the smearing functions \(N\) and \(N^x\) vanish and thus they are nondynamical.
In addition, when evaluated on solutions at any given value of \(t\) (on shell), the diffeomorphism and Hamiltonian are constrained to vanish:
\[\mathcal{H}_x = 0 \quad\text{and}\quad \mathcal{H} = 0. \label{eq:constraints}\tag{8}\] Both \(\mathcal{H}_x\) and \(\mathcal{H}\) are first-class constraints since the Poisson brackets between them close. They thus generate gauge transformations on the phase space.
The dynamical flow on the phase space determines a spherically symmetric geometry on a four-dimensional manifold \(\mathcal{M}\). The topology of this manifold is \(\mathcal{M}^2\times \mathcal{S}^2\), where \(\mathcal{M}^2\) is a two-dimensional manifold and \(\mathcal{S}^2\) a two-sphere. Together, the solution to the equations of motion 4 7 and constraints 8 covariantly define the spherically symmetric metric on \(\mathcal{M}\):
\[ds^2 = -N^2 dt^2 + q_{xx} (dx + N^x dt)^2 + q_{\vartheta\vartheta} d\Omega^2, \label{eq:metric}\tag{9}\] where \(q_{ab}\) is the three-dimensional metric induced on the spatial hypersurfaces foliating the manifold, \(q_{xx} = 1/q^{xx}\) and \(q_{\vartheta\vartheta}\) is a scalar function on the manifold \(\mathcal{M}^2\), \(N\) and \(N^x\) are the lapse function and shift vector field associated to the observer frame with coordinate \(t\), and \(d\Omega^2\) is the metric of the two-sphere on \(\mathcal{S}^2\). The coordinates \((t,x)\) correspond to any generic coordinates on \(\mathcal{M}^2\).
Due to the symmetry reduction of the scalar Hamiltonian to spherical symmetry, the hypersurface deformed bracket does not lead to a \(q^{\vartheta\vartheta}\) component. Thus the scalar \(q_{\vartheta\vartheta}\) does not appear in the phase space and its solution is not determined by the system. Thus one may choose \(q_{\vartheta\vartheta}\) to be any function of \(E^x\), since \(E^x\) is the only phase-space variable that is a spacetime scalar.
When solving the equations of motion 4 7 for a given gauge, and thus picking a lapse \(N\) and shift \(N^x\), the form of the metric 9 in a certain coordinate system is obtained. The gauge freedom on the phase space corresponds to diffeomorphism invariance on the manifold [18]. The algebra is referred to as the hypersurface deformation algebra. The gauge transformations generated by \(\mathcal{H}_x\) on the phase space correspond to deformations on the hypersurface of constant \(t\) on the manifold. For the Hamiltonian constraint \(\mathcal{H}\), gauge transformations generate deformations normal to the hypersurfaces of constant \(t\).
In this section, we write down the GUP Hamiltonian and derive the equations of motion on the phase space. We assume the diffeomorphism constraint remains unmodified by GUP corrections to the classical theory. The general Hamiltonian constraint can be written as [22]
\[\begin{align} \mathcal{H}_{(h_1,h_2,h_2)} &= \left[ -\frac{E^\varphi}{2h_2} \frac{d(h_1 h_2)}{d\bar{r}} - \frac{K_\varphi^2 E^\varphi}{2h_2} \frac{d(h_2 h_3)}{d\bar{r}} + \frac{h_2^2}{2\sqrt{E^x}} \left( \frac{(E^x)^{\prime\prime}}{E^\varphi} - \frac{(E^x)^\prime (E^\varphi)^\prime}{(E^\varphi)^2} \right)\right.\nonumber\\ &\quad \left.\left. - \left( \frac{2h_2}{\sqrt{E^x}} - 3\frac{dh_2}{d\bar{r}} \right) \frac{h_2}{E^x} \frac{[(E^x)^\prime]^2}{8E^\varphi} -2\sqrt{E^x} h_3 K_x K_\varphi \right] \right|_{\bar{r}=\sqrt{E^x}}. \label{eq:hamiltonian} \end{align}\tag{10}\] This expression is derived by starting with a general Hamiltonian with six free functions of \(E^x\). For simplicity, \(\bar{r} = \sqrt{E^x}\) is chosen. Given the form [eq:deform] the free functions are solved in terms of the shape functions and \(\bar{r}\) to give 10 .
The dynamical flow on the phase space given by the Hamiltonian
\[H = \int (N\mathcal{H}_{(h_1,h_2,h_3)} + N^x \mathcal{H}_x) dx\] covariantly defines the deformed Schwarzschild geometry [eq:deform] for any choice of the shape functions. The solution of the Hamiltonian constraints in any gauge will give the same geometry since the construction is covariant. The Hamiltonian is unique up to canonical transformations that do not include derivative terms and leave invariant the diffeomorphism constraint. The expression 10 can be considered as a family of Hamiltonian constraints. The GUP Hamiltonian constraint is one member of the family.
Substitution of the GUP shape functions into 10 gives the GUP Hamiltonian constraint
\[\begin{align} \mathcal{H} &= -\frac{E^\varphi}{2\sqrt{E^x}} p_1^{5/16} - \frac{E^\varphi}{2\sqrt{E^x}} K_\varphi^2 p_0^{-2} p_1^{-21/16} \left[ p_0\left(-5+9p_1\right) - 3p_1 \right]\\ &\quad + \frac{\sqrt{E^x}}{2} \left( \frac{(E^x)^{\prime\prime}}{E^\varphi} - \frac{(E^x)^\prime (E^\varphi)^\prime}{(E^\varphi)^2} \right) p_0^{-1} p_1^{1/4}\\ &\quad + \frac{\sqrt{E^x}[(E^x)^\prime]^2}{8E^x E^\varphi} p_0^{-2} p_1^{-3/16} \left[ -2p_0\left(-3+p_1^{7/16}\right) - 3p_1 \right]\\ &\quad -2 \sqrt{E^x}K_x K_\varphi p_0^{-1} p_1^{-7/8}.\label{eq:ham} \end{align}\tag{11}\] The Hamiltonian constraint has been written so that each of the vacuum general relativity terms are multiply by GUP corrections. In the limit \(Q_b\to 0\) and \(Q_c\to 0\), we recover the Hamiltonian constraint of vacuum spherically symmetric general relativity.
The Hamiltonian cannot be understood as one corresponding to vacuum general relativity plus certain additive corrections. Since the GUP parameters do not appear as additive contribution to the Hamiltonian constraint, they can not be understood as being originated from a minimal coupling to matter fields.
The general structure function is expressed in terms of the six free functions in [22] by computing the hypersurface deformed bracket for the scalar Hamiltonian constraint. Then expressing the free functions again in terms of the shape functions, and ultimately the GUP auxiliary functions, the structure function is
\[q^{xx} = \frac{h_2^2 h_3}{(E^\varphi)^2} = \frac{E^x}{(E^\varphi)^2} p_0^{-2} p_1^{-5/8}.\label{eq:sf}\tag{12}\]
The hypersurface deformation brackets 1 3 should be satisfied by construction. The diffiomorphism bracket 1 is identically satisfied since the classical diffiomorphism constraint has not changed. As a check on the Hamiltonian constraint 11 and structure function 12 , we have explicitly calculated the brackets 2 3 and shown the algebra to close. (See Appendix 8 for some details.)
Using the dynamic-flow equations 4 7 , the GUP equations of motion are
\[\begin{align} \dot{E}^x &= 2 N \sqrt{E^x} K_\varphi p_0^{-1} p_1^{-7/8} + N^x (E^x)^\prime,\tag{13}\\ \dot{E}^\varphi &= \frac{N}{\sqrt{E^x}} \left[ K_\varphi E^\varphi p_0^{-2} p_1^{-21/16} \left[ p_0(-5+9p_1) -3p_1 \right] + 2K_x E^x p_0^{-1} p_1^{-7/8}\right] + (N^x E^\varphi)^\prime,\tag{14}\\ \dot{K}_\varphi &= - \frac{N}{2\sqrt{E^x}} \left[ p_1^{5/16} + K_\varphi^2p_0^{-2} p_1^{-21/16} \left[ p_0(-5+9p_1)-3p_1 \right]\right]\nonumber\\ &\quad -\frac{N}{2\sqrt{E^x}} \left( \frac{(E^x)^\prime}{2E^\varphi} \right)^2p_0^{-2} p_1^{-3/16} \left[ -2p_0\left(-3+p_1^{7/16}\right) - 3p_1 \right] + \left(N\sqrt{E^x}p_0^{-1} p_1^{1/4}\right)^\prime \frac{(E^x)^\prime}{2(E^\varphi)^2}\nonumber\\ &\quad + N^x K_\varphi^\prime.\tag{15} \end{align}\] These equations are written similarly to a typical form of writing the classical equations to allow one to easily view the multiplicative GUP corrections. The equation for \(\dot{K}_x\) has not been written as it is in excess of 50 terms and unlikely to be insightful. Setting \(p_1 = p_2 \to 1\), gives the classical results.
In this section, we determine the phase-space variables, lapse, shift, and thus line element, in different gauges. The construction of the metric in different charts has already been obtained by coordinate transformations [16], [17], but the aim here is not to see the different forms of the metrics themselves but to validate the covariance. Does the theory starting from a covariant GUP Hamiltonian and using the canonical formalism obtain the same results as a non-covariant formalism using GUP distorted Poisson brackets?
Since the constrain equations and equations of motion can sometimes be difficult to solve, we can be guided by the coefficients in the general line element. The spherically symmetric metric in the ADM formalism [23] can be expanded as
\[\begin{align} ds^2 &= -N^2 dt^2 + \frac{1}{q^{xx}} \left(dx + N^x dt\right)^2 + E^x d\Omega^2\\ &= -\left[N^2 - \frac{(N^x)^2}{q^{xx}} \right] dt^2 + \frac{1}{q^{xx}} dx^2 + 2 \frac{N^x}{q^{xx}} dx dt + E^x d\Omega^2. \end{align}\] Written in this form, the line element allows us to identify the correspondence to the GUP metric and thus aid in obtaining the line element for a particular gauge.
To follow [22], we must take \(x = \sqrt{E^x}\). Applying this partial gauge, the diffeomorphism constraint gives
\[K_x = \frac{E^\varphi}{2\bar{r}} K_\varphi^\prime. \label{eq:sdiff}\tag{16}\] Conservation of this gauge condition requires \(\dot{E}^x = 0\), allowing the equation of motion 13 to be written as
\[N^x = - N K_\varphi p_0^{-1} p_1^{-7/8}. \label{eq:seom}\tag{17}\]
In the Schwarzschild gauge, \(N^x = 0\) which giving \(K_\varphi = 0\) by 17 and thus \(K_x = 0\) by 16 .
It remains to determine \(E^\varphi\) and \(N\). The Hamiltonian constraint and remaining equation of motion become
\[\begin{align} \mathcal{H} = 0 &= -\frac{E^\varphi}{2x} p_1^{5/16} + \frac{3x}{2E^\varphi} p_0^{-2} p_1^{-3/16} (2p_0-p_1) - \frac{x^2(E^\varphi)^\prime}{(E^\varphi)^2} p_0^{-1} p_1^{1/4}\\ \dot{K}_\varphi &= -\frac{N}{2x} p_1^{5/16} -\frac{Nx}{2(E^\varphi)^2} p_0^{-2} p_1^{-3/16} \left( 2p_0-p_1 \right) + \frac{N^\prime x^2}{(E^\varphi)^2} p_0^{-1} p_1^{1/4}. \end{align}\] The structure function is
\[q^{xx} = \frac{x^2}{(E^\varphi)^2} p_0^{-2} p_1^{-5/8}.\]
The Hamiltonian constraint can be solve for \(E^\varphi\). Given \(E^\varphi\) and the static requirement of \(\dot{K}_\varphi = 0\), we can solve for \(N\). These solutions, along with their derivatives are
\[\begin{align} E^\varphi &= p_0^{-1} p_1^{1/4} p_2^{-1/2} x = \left(1+\frac{Q_b}{r^2}\right)^{-1} \left(1+\frac{Q_bm^2}{r^8}\right)^{1/4} \left[1-\frac{2m}{r}\left(1+\frac{Q_b}{r^2}\right)^{-1/2}\right]^{-1/2},\\ (E^\varphi)^\prime &= -\frac{1}{2}p_0^{-2}p_1^{-3/16} \left[ p_1p_2^{-3/2} - 3(2p_0-p_1)p_2^{-1/2} \right],\\ N &= p_0^{1/2} p_1^{-1/8} p_2^{1/2} = \left(1+\frac{Q_b}{r^2}\right)^{1/2} \left(1+\frac{Q_cm^2}{r^8}\right)^{-1/8} \left[1+\frac{2m}{r}\left(1+\frac{Q_b}{r^2}\right)^{-1/2}\right]^{1/2},\\ N^\prime &= \frac{1}{2} p_0^{-1/2} p_1^{-9/16} \left[p_1p_2^{-1/2} -(2p_0-p_1)p_2^{1/2}\right]x^{-1}. \end{align}\] The structure function becomes
\[q^{xx} = p_1^{-9/8} p_2 = \left( 1 + \frac{Q_cm^2}{r^8} \right)^{-9/8} \left[ 1 - \frac{2m}{r} \left( 1 + \frac{Q_b}{r^2} \right)^{-1/2}\right].\] The line element obtained in the Schwarzschild gauge corresponds the original line element [eq:le2] obtained using deformed Poisson brackets. After writing down the resulting line element, one should transform \(x = \bar{r}\) back to \(r\) to obtained the untransformed line element [eq:le1] in the Schwarzschild coordinate \(r\).
For a \(g_{tt} \neq -1/g_{rr}\) non-symmetric metric, the spatial part of the metric in the Gullstrand-Painlevé gauge is \(-g_{tt} g_{rr}\), rather than flat. With \(E^x = x^2\), the spatial part of the metric is
\[-g_{tt} g_{rr} = \frac{1}{q^{xx}} \quad\Rightarrow\quad E^\varphi = p_0^{-1/2} p_1^{1/8} x = \pm\left(1+\frac{Q_b}{r^2} \right)^{1/2} \left( 1+\frac{Q_bm^2}{r^8} \right)^{1/8} x.\]
The diffeomorphism constraint gives
\[K_x = \frac{K_\varphi^\prime}{2} p_0^{1/2} p_1^{1/8},\label{eq:diffPG}\tag{18}\] which allows us to determine \(K_x\) once \(K_\varphi\) is known.
Conservation of the gauge conditions for \(K_x\) and \(K_\varphi\) gives
\[\begin{align} \dot{E}^x = 0 &= 2x(N K_\varphi p_0^{-1}p_1^{-7/8} + N^x),\\ \dot{E}^\varphi = 0 &= N\left(K_\varphi + 2xK_x p_0^{-1} p_1^{-7/8}\right) + \left(xN^x p_0^{-1/2}p_1^{1/8}\right)^\prime. \end{align}\]
Direct inspection of the line element [17] gives
\[\sqrt{(-g_{tt}g_{rr})(1+g_{tt})} = \frac{N^x}{q^{xx}} \quad\Rightarrow\quad N_x = \sqrt{\frac{1-p_0p_1^{-1/4}p_2}{p_0p_1^{7/8}}}\] and
\[g_{tt} = -N^2 + \frac{(N^x)^2}{q^{xx}} \quad\Rightarrow\quad N = 1.\]
Using \(\dot{E}^x = 0\), gives
\[K_\varphi = -\left( 1 - p_0 p_1^{-1/4} p_2 \right)^{1/2} p_0^{1/2} p_1^{7/16}.\] Thus \(K_x\) can be determined by differentiating \(K_\varphi\) and substituting into 18 . After transform \(x = \bar{r}\) back to \(r\) we obtain the Gullstrand-Painléve line element in [17].
In the homogeneous gauge, the function \(E^x\) on the phase space obeys \((E^x)^\prime = 0\) and \(\dot{E}^x \ne 0\), in general. To complete the gauge fixing we require \((E^\varphi)^\prime = 0\) and \(N^x = 0\). This in turn gives \(K_\varphi^\prime = 0\), \(K_x^\prime = 0\), and \(N^\prime = 0\) on the constraint surface [24].
The diffeomorphism constraint identically vanishes. The Hamiltonian constraint reduces to
\[\mathcal{H} = -\frac{E^\varphi}{2\sqrt{E^x}} p_0^{-2} p_1^{-21/16} \left[ p_0^{2} p_1^{13/8} + K_\varphi^2 \left(p_0(-5+9p_1)-3p_1\right) \right] - 2\sqrt{E^x} K_x K_\varphi p_0^{-1} p_1^{-7/8}.\]
Solving \(\mathcal{H} = 0\) for \(K_x\) gives
\[K_x = -\frac{E^\varphi}{4E^x K_\varphi} p_0^{-1} p_1^{-7/16} \left[ p_0^2 p_1^{13/8} + K_\varphi^2 \left[ p_0(-5+9p_1)-3p_1\right]\right]. \label{eq:static}\tag{19}\]
Recalling that the time coordinate is spacelike and the radial coordinate is timelike in the homogeneous gauge, inspection of the line element gives
\[\begin{align} N &= p_1^{9/16} (-p_2)^{-1/2},\\ E^\varphi &= \sqrt{E^x} p_0^{-1/2} p_1^{-7/16} (-p_2)^{1/2}. \end{align}\] Since \(E^x\), \(p_0\), and \(p_1\) are positive, one could choose to absorb the negative sign into the definition of \(p_2\) to give the usual form
\[-p_2 = \frac{2m}{\sqrt{E^x}}p_0^{-1/2} - 1.\] in which the coefficients are written in the Kantowski-Sachs line element. Since \(\sqrt{E^x}\) should play the role of time, its equation of motion should give
\[\dot{E}^x = 2 \sqrt{E^x}.\] or
\[\begin{align} \dot{E}^x &= 2N\sqrt{E^x} K_\varphi p_0^{-1} p_1^{-7/8}\nonumber\\ &= 2\sqrt{E^x} K_\varphi p_0^{-1} p_1^{-5/16} p_2^{-1/2} \quad\Rightarrow\quad K_\varphi = p_0 p_1^{5/16} p_2^{1/2}. \end{align}\] Substitution into 19 give \(K_x\).
When comparing the results here with the homogeneous results of [16], [17] there is no direct connection between the phase-space variables or the Hamiltonian constraint. However, the identification \(\tilde{t} = \sqrt{E^x}\) gives a line element identical to the Kantowski-Sachs line element with the same lapse and structure functions. This result demonstrates the covariance of the Hamiltonian constraint presented here 11 and gives an identical line element to that derived by the distorted Poisson bracket motivated by the GUP.
Coupling the GUP Hamiltonian to matter is essential for studying its dynamical behaviour. While expressions for Hamiltonians for various fields are known, it seems here would be a natural place to calculate them in GUP spacetime allowing them to be used in future work.
One typically starts with scalar fields and dust as they are easiest and elucidate basic dynamical properties. To couple to scalar matter, we introduce an additional pair of conjugate variables
\[\{ \phi(t,x_1),P_\phi(t,x_2)\} = \delta(x_1-x_2).\]
The total Hamiltonian becomes
\[H_\text{total} = \int \left[ (\mathcal{H}_x + \mathcal{H}_x^\mathrm{m}) N^x + (\mathcal{H} + \mathcal{H}^\mathrm{m}) N\right] dx,\] where \(\mathcal{H}_x\) and \(\mathcal{H}\) are the gravitational diffeomorphism and Hamiltonian constrains respectively, while \(\mathcal{H}_x^\mathrm{m}\) and \(\mathcal{H}^\mathrm{m}\) correspond to the matter contributions. The scalar diffeomorphism constraint is
\[\mathcal{H}_x^\mathrm{m} = \phi^\prime P_\phi\] and the Hamiltonian constraint is (see Appendix 9)
\[\begin{align} \mathcal{H}^\mathrm{m} &= \frac{1}{2} \left[\;\frac{P_\phi^2}{\sqrt{q}} + \sqrt{q} q^{ab} \partial_a\phi \partial_b\phi + \sqrt{q} V(\phi) \right]\nonumber\\ &= \frac{ p_0^{-1} p_1^{-5/16} \sqrt{E^x}}{2E^\varphi} \left[ \frac{P_\phi^2}{E^x \sin\vartheta} + E^x \sin\vartheta (\phi^\prime)^2 \right] + \frac{p_0p_1^{5/16}\sqrt{E^x} E^\varphi \sin\vartheta}{2} V(\phi). \end{align}\] where \[P_\phi = \frac{\sqrt{q}}{N} \left( \dot{\phi} - N^a\partial_a\phi\right) = \frac{p_0 p_1^{5/16} \sqrt{E^x}E^\varphi \sin\vartheta}{N} \left( \dot{\phi} - N^a\partial_a\phi\right).\]
The Klein-Gordon equation on the curved covariant metric has been shown to have anomaly-free constraint brackets and to respect covariance of both the spacetime and the matter field in spherical symmetry [25]. The gravitational Hamiltonian constraint does not depend on \(\phi\) or \(P_\phi\) as it is the vacuum background, while the scalar Hamiltonian constraint depends only quadratically on \(\phi\) and \(P_\phi\). It can be shown [25] that under these circumstances the back reaction can be neglected, and thus will not be effected by perturbative dynamics.
Likewise for dust, the diffeomorphism constraint is
\[\mathcal{H}_x^\mathrm{m} = T^\prime P_T\] and the Hamiltonian constraint is [26] (see Appendix 9)
\[\mathcal{H}^\mathrm{m} = P_T \sqrt{q^{ab} \partial_a T \partial_b T + 1} = P_T \sqrt{p_0^{-2} p_1^{-5/8} \frac{E^x}{(E^\varphi)^2}(T^\prime)^2 + 1}.\] where \[P_T = \frac{\sqrt{q}M}{N} \left( \dot{T} - N^a\partial_a T\right) = \frac{p_0 p_1^{5/16} \sqrt{E^x}E^\varphi \sin\vartheta M}{N} \left( \dot{T} - N^a\partial_a T\right).\]
Starting from a metric obtained using deformed Poisson brackets, we have derived a Hamiltonian that allows a canonical and covariant formalism to be defined. The Hamiltonian is quadratic in the derivatives and second-order in the first derivatives. Since the formalism used to determine the Hamiltonian involves several functions that need to be determined, the Hamiltonian constraint may not be unique. The dynamical flow covariantly defines a static and spherical symmetric four-dimensional geometry corresponding to the GUP metric.
The lapse, shift, and phase-space variables have been calculated in the Schwarzschild and Gullstrand-Painléve gauges and shown to lead to the correct line elements. The homogeneous gauge has also been determined, and with a proper identification of the time variable reproduces the metric from which the GUP was derived. Thus, three choices of gauge have been shown to give charts that would be obtained from different choices of coordinates.
Simple scalar and dust matter fields have been coupled to the geometry. The procedure used here could help elucidate the dynamical origin of some static geometries. The coupling to a scalar field would allow the study of quasinormal modes or Hawking evaporation, to name just two examples. Coupling to dust allows gravitational collapse to be studied, as well as providing a clock.
Although starting from heuristic arguments, the GUP spacetime, which can give black hole, wormhole, and remnant solutions, has been cast into a consistent canonical formalism that can covariantly give different charts depending on the choice of gauge. The next step is to study the dynamics.
In this appendix, we show that it is not possible to use the same procedure as in [16], [17] to determine GUP corrections using deformed Poisson brackets in the static gauge. The GUP line element was derived using the interior of a static spherically symmetric black hole expressed in Ashtekar-Barbero variables [27] using Schwarzschild coordinates. This is a cosmological based mechanical mini-superspace model with no \(r\) dependence. It is not possible impose a static gauge choice.
To attempt to circumvent this limitation, let’s apply the same deformed Poisson bracket procedure in [16], [17] to a field theory midi-superspace model with \(r\) dependence. We need only consider one of the conjugate phase-space pair of fields \((K_x,E^x)\), say. the canonical Poisson bracket is
\[\left\{ K_x(t,x_1), E^x(t,x_2) \right\}_\mathrm{canonical} = \delta(x_1-x_2).\] The deformed Poisson bracket can be specified by a function \(F\) which in general could be a function of the pair of conjugate phase-space fields and a constant \(\beta\). The deformed Poisson bracket can be written as
\[\left\{ K_x(t,x_1), E^x(t,x_2) \right\}_\mathrm{deformed} = \left\{ K_x(t,x_1), E^x(t,x_2) \right\}_\mathrm{canonical} [ 1 + F(K_x,E^x,\beta)],\] The equation of motion for \(E^x\), say, is
\[\dot{E^x} = \left\{ E^x,H \right\}_\text{deformed} = \left\{ E^x,H \right\}_\text{canonical} [1 + F(K_x,E^x,\beta)].\] In the static gauge,
\[\dot{E^x} = 0 \quad \Rightarrow \quad \left\{ E^x,H \right\}_\text{canonical} = 0.\] The equation of motion is canonical and GUP corrections have no effect in static gauge. A theory based on multiplicative Poisson bracket deformations is thus not canonical nor covariant.
In this appendix we discuss the calculation of the hypersurface deformation algebra which is rarely discussed, beyond stating that it’s a long but straightforward calculation. The hypersurface deformation brackets 1 3 should be satisfied by construction. The diffiomorphism bracket 1 is identically satisfied since the classical diffiomorphism constraint has not been changed.
Each term in the classical Hamiltonian constraint contains a function of \(\sqrt{E^x}\). The GUP modifications appear as multiplicative factors \(p_0\) and \(p_1\) to various powers, which are also functions of \(\sqrt{E^x}\), and thus just change the function in each term of the classical Hamiltonian constraint. When calculating the mixed hypersurface bracket 2 the differentials of the functions of \(\sqrt{E^x}\) are eliminated by integrating by parts thus leaving the Hamiltonian constraint effectively unaltered. The same can be done with the GUP modified Hamiltonian and the hypersurface bracket 2 is satisfied.
The bracket 3 requires the most work and is less intuitive. The terms in the Hamiltonian constraint can be numbered as \(A_1, A_2, A_3, \ldots A_6\).
\[\mathcal{H} = A_1 + A_2 + A_3 + A_4 + A_5 + A_6,\] where
\[\begin{align} A_1 &= -\frac{E^\varphi}{2\sqrt{E^x}} p_1^{5/16},\\ A_2 &= -\frac{E^\varphi}{2\sqrt{E^x}} K_\varphi^2 p_0^{-2} p_1^{-21/16} \left[ p_0\left(-5+9p_1\right) - 3p_1 \right],\\ A_3 &= \frac{\sqrt{E^x}}{2} \frac{(E^x)^{\prime\prime}}{E^\varphi} p_0^{-1} p_1^{1/4},\\ A_4 &= - \frac{\sqrt{E^x}}{2} \frac{(E^x)^\prime (E^\varphi)^\prime}{(E^\varphi)^2} p_0^{-1} p_1^{1/4},\\ A_5 &= \frac{\sqrt{E^x}[(E^x)^\prime]^2}{8E^x E^\varphi} p_0^{-2} p_1^{-3/16} \left[ -2p_0\left(-3+p_1^{7/16}\right) - 3p_1 \right],\\ A_6 &= -2 \sqrt{E^x}K_x K_\varphi p_0^{-1} p_1^{-7/8}. \end{align}\] The only nonzero brackets are between terms involving \(K\) and the corresponding derivatives of \(E\), that is, \(\{A_2,A_4\}\), \(\{A_6,A_5\}\), \(\{A_6,A_4\}\), and \(\{A_6,A_3\}\). In the classical case, bracket \(\{A_2,A_4\}\) cancels bracket \(\{A_6,A_5\}\). The bracket \(\{A_6,A_4\}\) gives two nonzero terms: the \(K_x\) term is a desired term but the \(K_\varphi\) term is undesirable and must be cancelled. The bracket \(\{A_6,A_3\}\) consists of the product of two total derivatives which can strategically be integrated by parts to give another one of the desired terms and a second term which cancels the previously undesired term. The GUP case is even less intuitive. In this case, \(\{A_2,A_4\}\) and \(\{A_6,A_5\}\) do not cancel, but together generate three terms, none of which are desirable. Similar to the classical case, the bracket \(\{A_6,A_4\}\) gives two nonzero terms: the \(K_x\) term is a desired term but the \(K_\varphi\) term is undesirable and must be cancelled. We now have four undesirable terms. The bracket \(\{A_6,A_3\}\) again gives the product of two total derivatives but this time a strategic integration by parts gives five terms; one desired term and four terms that cancel the previous undesirable terms. The extra terms in the integration by parts come from the differentiation of the \(p_0, p_1\) factors.
In this appendix, we derive the equation of motion for a scalar field using the canonical formalism. The Lagrangian density for a scalar field is
\[\mathcal{L}_\mathrm{m} = -\frac{1}{2} \sqrt{-g} \left[ g^{\mu\nu} \partial \phi_\mu \partial_\nu \phi + V(\phi) \right].\] To evaluate this in curved spacetime, we need the inverse metric. First, the line element in ADM form can be written as
\[ds^2 = -N^2 dt^2 + q_{ab} (dx^a + N^a dt)(dx^b + N^b dt)\] with non-zero metric components
\[g_{00} = -N^2 + q_{ab} N^a N^b, \quad g_{0a} = g_{a0} = q_{ab} N^a, \quad \text{and} \quad g_{ab} = q_{ab}.\] The non-zero inverse metric components are
\[g^{00} = -\frac{1}{N^2}, \quad g^{0a} = g^{a0} = \frac{N^b}{N^2}, \quad \text{and} \quad g^{ab} = q^{ab} - \frac{N^aN^b}{N^2}.\]
We will also need the determinant of the metric \(g\) which can be expressed as
\[\sqrt{-g} = N \sqrt{q}.\]
The Lagrangian density becomes
\[\mathcal{L}_\mathrm{m} = -\frac{1}{2} N\sqrt{q} \left[ -\frac{1}{N^2} (\dot{\phi})^2 + 2\frac{N^a}{N^2} \dot{\phi} \partial_a \phi + \left( q^{ab} -\frac{N^aN^b}{N^2} \right) \partial_a\phi \partial_b\phi + V(\phi) \right].\]
The conjugate momentum of the scalar field is
\[P_\phi = \frac{\delta \mathcal{L}_\mathrm{m}}{\delta\dot{\phi}} = \frac{\sqrt{q}}{N} (\dot{\phi} - N^a \partial_a \phi).\] To eliminate the time derivatives from the Lagrangian, we perform the following manipulations. Inverting the momentum equation gives
\[\dot{\phi} = \frac{N}{\sqrt{q}} P_\phi + N^a \partial_a \phi.\] Squaring gives
\[\frac{\sqrt{q}}{N} (\dot{\phi})^2 = \frac{N}{\sqrt{q}} P_\phi^2 + 2 N^a P_\phi \partial_a \phi + \frac{\sqrt{q}}{N} N^a N^b \partial_a \phi \partial_b \phi.\label{eq:w1}\tag{20}\] To enable the Legendre transformation, we multiply the momentum equation by \(\dot{\phi}\) to get
\[\frac{\sqrt{q}}{N} \left[ (\dot{\phi})^2 - N^a \dot{\phi} \partial_a \phi \right] = P_\phi \dot{\phi}.\label{eq:w2}\tag{21}\] Subtracting one-half of 20 from 21 and substituting the result into the first two terms of the Lagrangian density gives
\[\mathcal{L}_\mathrm{m} = P_\phi \dot{\phi} - N^a P_\phi \partial_a \phi - \frac{N}{2} \left( \frac{P_\phi^2}{\sqrt{q}} + \sqrt{q} q^{ab} \partial_a\phi \partial_b\phi + \sqrt{q}V \right),\] where we can easily identify the diffeomorphism and Hamiltonian constraints.
Performing a Legendre transformation gives
\[P_\phi \dot{\phi} - \mathcal{L}_\mathrm{m} = N\mathcal{H}^\mathrm{m} + N^x \mathcal{H}_x^\mathrm{m}\] and the constraints are
\[\begin{align} \mathcal{H}_x^\mathrm{m} &= P_\phi \partial_x \phi,\\ \mathcal{H}^\mathrm{m} &= \frac{1}{2} \left( \frac{P_\phi^2}{\sqrt{q}} + \sqrt{q} q^{ab} \partial_a\phi \partial_b\phi + \sqrt{q} V\right). \end{align}\]
For spherical symmetry, we use
\[\sqrt{q} = \sqrt{q_{xx}} q_{\vartheta\vartheta} \sin\vartheta\] to obtain
\[\dot{\phi} = \frac{NP_\phi}{\sqrt{q}} + N^x\partial_x\phi = \frac{N\sqrt{q^{xx}}P_\phi}{q_{\vartheta\vartheta}\sin\vartheta} + N^x\partial_x\phi. \label{eq:phid}\tag{22}\] and
\[\begin{align} \dot{P}_\phi &= \partial_b\left( N\sqrt{q} q^{ab} \partial_a \phi\right) -\frac{N}{2} \sqrt{q} \frac{\partial V}{\partial \phi} + \partial_x (N^x P_\phi)\nonumber\\ &= \partial_x \left( N\sqrt{q^{xx}} q_{\varphi\varphi} \sin\vartheta \partial_x \phi\right) + N \sqrt{q_{xx}} \partial_\vartheta \left( \sin\vartheta \partial_\vartheta \phi\right) + \frac{ N\sqrt{q_{xx}} \partial_\varphi^2 \phi}{\sin\vartheta}\nonumber\\ &\quad -\frac{N}{2} \sqrt{q_{xx}} q_{\vartheta\vartheta} \sin\vartheta \frac{\partial V}{\partial \phi} + \partial_x (N^x P_\phi)\nonumber\\ &= \sin\vartheta \left[ \partial_x \left( N\sqrt{q^{xx}} q_{\varphi\varphi} \partial_x \phi\right) + N \sqrt{q_{xx}} \left( \frac{\partial_\vartheta \left( \sin\vartheta \partial_\vartheta \phi\right)}{\sin\vartheta} + \frac{\partial_\varphi^2 \phi}{\sin^2\vartheta}\right) \right.\nonumber\\ &\quad \left. -\frac{N}{2} \sqrt{q_{xx}} q_{\vartheta\vartheta} \frac{\partial V}{\partial \phi}\right] + \partial_x (N^x P_\phi)\nonumber\\ &= \sin\vartheta \left[ \partial_x \left( N\sqrt{q^{xx}} q_{\varphi\varphi} \partial_x \phi\right) + N \sqrt{q_{xx}} \Delta^\varphi -\frac{N}{2} \sqrt{q_{xx}} q_{\vartheta\vartheta} \frac{\partial V}{\partial \phi} \right]+ \partial_x (N^x P_\phi), \label{eq:Pd} \end{align}\tag{23}\] where
\[\Delta^\varphi = \frac{1}{\sin\vartheta} \partial_\vartheta (\sin\vartheta \partial_\vartheta )+ \frac{\partial_\varphi^2}{\sin^2\vartheta}.\] is the Laplacian on a two-sphere.
Equations 22 and 23 are two first-order equations. We can write them as a single second-order equation by a partial gauge fixing of \(N^x = 0\):
\[\ddot{\phi} = \frac{N \sqrt{q^{xx}}}{q_{\vartheta\vartheta}} \left[ \partial_x (N \sqrt{q^{xx}} q_{\vartheta\vartheta} \partial_x \phi )+ N \sqrt{q_{xx}} \left( \bar{\Delta} \phi - \frac{q_{\vartheta\vartheta}}{2} \frac{\partial V}{\partial \phi} \right) \right]. \label{eq:2nd}\tag{24}\] This expression is identical to the wave equation for a massive scalar field in curved spacetime as obtain in Appendix 10.
We may further simplify 24 by taking \(q_{\vartheta\vartheta} = x^2\) to obtain
\[\ddot{\phi} = N \sqrt{q^{xx}} \left[ N \sqrt{q^{xx}} \partial_x^2 \phi + \frac{1}{x^2} \partial_x (N \sqrt{q^{xx}} x^2 ) \partial_x \phi + N \sqrt{q_{xx}} \left( \Delta \phi - \frac{1}{2} \frac{\partial V}{\partial \phi} \right) \right],\] where \(\bar{\Delta}^\varphi = \Delta^\varphi /x^2\).
In the classical limit,
\[N = \sqrt{1 - \frac{2M}{x}} \quad\text{and}\quad q^{xx} = 1 - \frac{2M}{x},\] which gives
\[\ddot{\phi} = \left( 1 - \frac{2M}{x} \right) \left[ \left( 1 - \frac{2M}{x} \right) \partial_x^2\phi + \frac{2}{x} \left(1 - \frac{M}{x} \right) \partial_x\phi + \Delta\phi -\frac{\partial V}{\partial\phi} \right].\] This is the well known Klein-Gordon equation in curved Schwarzschild spacetime [21].
In this appendix, we derive the Hamiltonian for dust [26], [28]. The Lagrangian density for dust is
\[\mathcal{L}_\mathrm{m} = -\frac{1}{2} \sqrt{-g} \left[ g^{\mu\nu} M \left( \partial_\mu T \partial_\nu T + 1\right) \right],\] where \(T\) is the dust field. When the dust field is defined in terms of the four-velocity \(U_a = \partial_a T\), \(M\) is the rest mass in the stress-energy tensor.
Using the full ADM form of the metric, the Lagrangian density becomes
\[\mathcal{L}_\mathrm{m} = -\frac{1}{2} M N\sqrt{q} \left[ -\frac{1}{N^2} \left(\dot{T}\right)^2 + 2 \frac{N^a}{N^2} \dot{T} \partial_a T + \left( q^{ab} - \frac{N^aN^b}{N^2} \right) \partial_a T \partial_b T + 1\right].\]
The conjugate momentum of the dust field is
\[P_T = \frac{\delta\mathcal{L}_\mathrm{m}}{\delta \dot{T}} = \frac{\sqrt{q}M}{N} \left( \dot{T} - N^a \partial_a T\right).\]
After some algebra similar to above, the constraints are
\[\begin{align} \mathcal{H}_a^\mathrm{m} &= P_T \partial_x T,\\ \mathcal{H}^\mathrm{m} &= \frac{1}{2} \left[ \frac{P_T^2}{M\sqrt{q}} + M\sqrt{q} \left( q^{ab} \partial_a T \partial_b T + 1 \right) \right]. \end{align}\]
The equation of motion for \(M\) is
\[\frac{\delta\mathcal{L}_\mathrm{m}}{\delta M} = -N \frac{\partial\mathcal{H}}{\partial M} = 0 \quad\Rightarrow\quad M = \frac{P_T}{\sqrt{q}} \left[ q^{ab} \partial_a T \partial_b T + 1 \right]^{-1/2}.\]
Substituting \(M\) back into \(\mathcal{H}^\mathrm{m}\) gives
\[\mathcal{H}^\mathrm{m} = P_T \sqrt{q^{ab} \partial_a T \partial_b T + 1}.\]
In this appendix, we calculate the equation of motion for a scalar field on a curved spacetime background. The wave equation for a massive scalar field is
\[(\Box - m^2)\phi = 0,\] where \(\phi = \phi(r,\vartheta,\varphi,t)\) and \(m\) is the mass of the field.
In curved spacetime,
\[\left(\nabla_\mu \nabla^\mu -m^2 \right)\phi = 0.\] Since \(\nabla^\mu\phi = \partial^\mu\phi\) for a scalar,
\[\left(\nabla_\mu \partial^\mu - m^2\right)\phi = 0.\] Using the identity
\[\nabla_\mu \partial^\mu \phi = \frac{1}{\sqrt{-g}} \partial_\mu \left( \sqrt{-g} \partial^\mu\phi \right)\] we obtain
\[\frac{1}{\sqrt{-g}} \partial_\mu (\sqrt{-g}\partial^\mu \phi) - m^2\phi = \frac{1}{\sqrt{-g}} \partial_\mu (g^{\mu\nu} \sqrt{-g} \partial_\nu \phi) - m^2\phi = 0.\]
For a spherically symmetric static diagonal line element, the metric functions can be written as
\[g_{tt}(r), \quad g_{rr}(r), \quad g_{\vartheta\vartheta}(r), \quad\text{and}\quad g_{\varphi\varphi}(r) = g_{\vartheta\vartheta}(r)\sin^2\vartheta.\] The inverse metric components are
\[g^{tt}(r) = \frac{1}{g_{tt}(r)}, \quad g^{rr}(r) = g_{rr}(r), \quad g^{\vartheta\vartheta}(r) = \frac{1}{g_{\vartheta\vartheta}(r)}, \quad\text{and}\quad g^{\varphi\varphi}(r) = \frac{1}{g_{\vartheta\vartheta}(r)\sin^2\vartheta}.\] From now on the dependence on \(r\) will be implicit.
The determinant is
\[\sqrt{-g} = \sqrt{-g_{tt} g_{rr}} g_{\vartheta\vartheta} \sin\vartheta\, .\]
Evaluating each component of the wave equation gives
\[\begin{align} \partial_t (g^{tt}\sqrt{-g}\partial_t\phi) & = & \frac{1}{g_{tt}} \sqrt{-g} \frac{\partial^2\phi}{\partial t^2} \to \frac{1}{g_{tt}} \frac{\partial^2\phi}{\partial t^2},\\ \partial_r (g^{rr}\sqrt{-g}\partial_r\phi) & = & \frac{\partial}{\partial r} \left( \frac{1}{g_{rr}} \sqrt{-g_{tt} g_{rr}} g_{\vartheta\vartheta} \sin\vartheta \frac{\partial\phi}{\partial r} \right) = \sin\vartheta \frac{\partial}{\partial r} \left( \sqrt{\frac{-g_{tt}}{g_{rr}}} g_{\vartheta\vartheta} \frac{\partial \phi}{\partial r} \right)\nonumber\\ & \to & \frac{1}{\sqrt{-g_{tt} g_{rr}}g_{\vartheta\vartheta}} \frac{\partial}{\partial r} \left( \sqrt{ \frac{-g_{tt}}{g_{rr}} } g_{\vartheta\vartheta} \frac{\partial \phi}{\partial r} \right),\\ \partial_\vartheta \left( g^{\vartheta\vartheta} \sqrt{-g}\partial_\vartheta \phi \right) & = & \frac{1}{g_{\vartheta\vartheta}} \frac{\partial}{\partial\vartheta} \left( \sqrt{-g_{tt} g_{rr}} g_{\vartheta\vartheta} \sin\vartheta \frac{\partial\phi}{\partial\vartheta} \right) = \sqrt{-g_{tt} g_{rr}} \frac{\partial}{\partial \vartheta} \left( \sin\vartheta \frac{\partial\phi}{\partial\vartheta} \right)\nonumber\\ & \to & \frac{1}{g_{\vartheta\vartheta}\sin\vartheta} \frac{\partial}{\partial\vartheta} \left( \sin\vartheta \frac{\partial \phi}{\partial \vartheta} \right),\\ \partial_\varphi (g^{\varphi\varphi}\sqrt{-g}\partial_\varphi\phi) & = & \frac{1}{g_{\vartheta\vartheta}\sin^2\vartheta}\sqrt{-g} \frac{\partial^2\phi}{\partial \varphi^2} \to \frac{1}{g_{\vartheta\vartheta}\sin^2\vartheta} \frac{\partial^2 \phi}{\partial \varphi^2},\\ & & -\sqrt{-g} m^2 \phi \to -m^2\phi. \end{align}\] The arrow indicates that the expression has been divided by \(\sqrt{-g}\).
The wave equation becomes
\[\frac{1}{g_{tt}} \frac{\partial^2 \phi}{\partial t^2} + \frac{1}{ \sqrt{ -g_{tt} g_{rr} } g_{\vartheta\vartheta} } \frac{\partial}{\partial r} \left( \sqrt{\frac{-g_{tt}}{g_{rr}}} g_{\vartheta\vartheta} \frac{\partial \phi}{\partial r} \right) + \frac{1}{g_{\vartheta\vartheta}} \Delta^\varphi \phi - m^2 \phi = 0,\] where \[\Delta^\varphi = \frac{1}{\sin\vartheta} \frac{\partial}{\partial\vartheta} \left( \sin\vartheta \frac{\partial \phi}{\partial\vartheta} \right) + \frac{1}{\sin^2\vartheta} \frac{\partial^2 \phi}{\partial\vartheta^2}\] is the Laplacian on a two-sphere. This form of the scalar wave equation corresponds to motion in a general spherically symmetric static spacetime.
I thank Saeed Rastgoo for helpful discussions. We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). Nous remercions le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG) de son soutien.
A triad is the space part of a tetrad. They are orthogonal vector fields that can be used to construct the spatial part of the matrix. Triads are densitize by multiplying the triad by the square root of the determinant of the spatial metric.↩︎
A canonical symplectric structure provides a geometrical form for Hamiltonian mechanics and phase space. The symplectic structure and Poisson brackets are canonically connected, thus linking the geometrical structure to the algebraic structure.↩︎