The GUP and quantum Raychaudhuri equation

TL;DR

This paper compares quantum corrections to black hole temperature from different approaches, establishing a connection between deformation parameters and suggesting spacetime has a crystal-like structure.

Contribution

It links the quantum Raychaudhuri equation with generalized uncertainty principle parameters, revealing a scale-dependent relation and a correspondence between phenomenological models.

Findings

Quantum Raychaudhuri equation relates to GUP deformation parameters.

The relation between parameters depends on the black hole scale.

Indicates spacetime may have a crystal-like structure.

Abstract

In this paper, we compare the quantum corrections to the Schwarzschild black hole temperature due to quadratic and linear-quadratic generalized uncertainty principle, with the corrections from the quantum Raychaudhuri equation. The reason for this comparison is to connect the deformation parameters $\beta_0$ and $ \alpha_0$ with $\eta$ which is the parameter that characterizes the quantum Raychaudhuri equation. The derived relation between the parameters appears to depend on the relative scale of the system (black hole), which could be read as a beta function equation for the quadratic deformation parameter $\beta_0$. This study shows a correspondence between the two phenomenological approaches and indicates that quantum Raychaudhuri equation implies the existence of a crystal-like structure of spacetime.

Equations

$$\left[\hat{X},\hat{P}\right]=i\hslash \left(1+{\beta }_0\frac{{\hat{P}}^2}{m_{\mathrm{p}}^2}\right),$$

$$[\hat{X},\hat{P}]=i\hslash \left(1-2\alpha \hat{P}+4{\alpha }^2{\hat{P}}^2\right)$$

$$\dot{\theta }=\frac{1}{3}{\theta }^2-{\sigma }^2+{\Omega }^2-R_{\mu \nu }{\xi }^{\mu }{\xi }^{\nu }-\frac{{ϵ}_1{\hslash }^2}{m^2}{h}^{\mu \nu }{R}_{;\mu \nu }-\frac{{\hslash }^2}{m^2}{h}^{\mu \nu }{\left(\frac{\square \mathcal{R}}{\mathcal{R}}\right)}_{;\mu \nu }\text{leavevmode}.$$

$$T_{QGUP}=\frac{\pi }{{\beta }_0}\left(M-\sqrt{M^2-\frac{{\beta }_0}{{\pi }^2}{m}_p^2}\right),$$

$$T_{LQGUP}=\frac{\left(4\pi \hslash \frac{M}{m_p^2}+\frac{\hslash {\alpha }_0}{m_{\mathrm{p}}}\right)-\sqrt{{\left(4\pi \hslash \frac{M}{m_p^2}+\frac{\hslash {\alpha }_0}{m_{\mathrm{p}}}\right)}^2-4\left(\frac{4\hslash {\alpha }_0^2}{m_{\mathrm{p}}^2}\right)\hslash }}{2\left(\frac{4\hslash {\alpha }_0^2}{m_{\mathrm{p}}^2}\right)}\text{leavevmode}.$$

$$T_{QRE}=\hslash \frac{\sqrt{M^2-4\eta m_p^2}}{2\pi {\left(\sqrt{M^2-4\eta m_p^2}+M\right)}^2}$$

$$d{s}^2=\left(1-\frac{2M}{r}+\frac{4\eta m_{\mathrm{p}}^2}{r^2}\right)d{t}^2-{\left(1-\frac{2M}{r}+\frac{4\eta m_{\mathrm{p}}^2}{r^2}\right)}^{-1}d{r}^2-r^{2}d{\Omega }^2$$

$$V(r)\simeq -\frac{M}{r}\left(1-2\eta m_{\mathrm{p}}\frac{m_{\mathrm{p}}}{Mr}\right)\text{leavevmode}.$$

$$d{s}^2=\left(1-\frac{2M}{r}+\epsilon (r)\right)d{t}^2-{\left(1-\frac{2M}{r}+\epsilon (r)\right)}^{-1}d{r}^2-r^{2}d{\Omega }^2$$

$$1-\frac{2M}{r}+\epsilon (r)=0,$$

$$r_h=M+\sqrt{M^2-4\eta m_{\mathrm{p}}^2}\text{leavevmode}.$$

$$T=$$

$$=$$

$$T_{QRE}=\hslash \frac{M}{2\pi {\left(\sqrt{M^2-4\eta m_{\mathrm{p}}^2}+M\right)}^2}+\frac{\hslash {ϵ}^{\prime }(r_h)}{4\pi }.$$

$$T_{QRE}\simeq \frac{\hslash }{8\pi M}\left[1-{\eta }^2{L}^4-2{\eta }^3{L}^6+\dots \right]$$

$$T_{QGUP}=\frac{\hslash }{8\pi M}\left[1+\frac{1}{4{\pi }^2}{\beta }_0{L}^2+\dots \right],$$

$$T_{LQGUP}=\frac{\hslash }{8\pi M}\left[1-\frac{{\alpha }_0}{2\pi }L+5{\left(\frac{{\alpha }_0}{2\pi }\right)}^2{L}^2+\dots \right]\text{leavevmode}.$$

$${\eta }^2=-\frac{1}{4{\pi }^2}{\beta }_0{L}^{-2}\text{leavevmode}.$$

$${\eta }^2=\frac{{\alpha }_0}{2\pi }{L}^{-3}-5{\left(\frac{{\alpha }_0}{2\pi }\right)}^2{L}^{-2}\text{leavevmode}.$$

Full text

One of the most challenging questions in theoretical physics over the last century is to obtain a fully consistent theory of quantum gravity. Although many attempts to reach such theory have been made, namely loop quantum gravity, string theory and other approaches, it is believed that such programmes are not yet complete and have their own problems. Hence, it appears that it is important to investigate quantum gravity from a phenomenological perspective Hossenfelder:2009nu . In the last few decades, many phenomenological approaches have been developed to investigate quantum gravity at the Plank scale Hossenfelder:2006mi .

One of these phenomenological approaches is the generalisation of Heisenberg uncertainty principle (HUP), which is inconsistent with general relativity, as HUP implies that it is impossible to localise a system without adding a lot of energy to it. However, the energy of this measurement will affect the background space-time according to general relativity, making the smooth structure of the classical space-time impossible. The generalised uncertainty principle (GUP) resolves this inconsistency by changing the HUP such that it has a minimum length or, equivalently, a maximum momentum. The generalisation of HUP has been studied in many contexts GUPearly . The most prominent space deformations of Heisenberg algebra, that imply GUP, consist of either quadratic momentum term VenezGrossMende

[TABLE]

with $\beta_{0}$ to be the dimensionless deforming parameter of GUP, or, both linear and quadratic terms Ali ,

[TABLE]

where $\alpha=\alpha_{0}/m_{\rm p}=\alpha_{0}\ell_{p}/\hbar$ with $\alpha_{0}$ to be the dimensionless GUP parameter and $m_{\rm p}$ to be the Planck mass (we have set $c=1$).

The quadratic deformations were derived from doubly special relativity (DSR), loop quantum gravity, and string theory  VenezGrossMende ; MM ; kempf ; FS ; Adler2 ; SC , while the linear-quadratic deformations were developed later as a generalisation Ali . The GUP has important implications in cosmology and black hole physics alsaleh . Moreover, it has indicated a cyclic non-singular universe  GUPcosmology , black hole remnants and also that the deformed black hole thermodynamics derived from GUP appears to coincide with the one obtained from studying black hole entropy in loop quantum gravity and string theory GUPBH . Furthermore, it has been shown that the GUP produces a logarithmic correction to entropy similar to that derived on statistical grounds Medved:2004yu .

The above-mentioned results derived from the GUP, also appear in other phenomenological approaches (cf. Das:2009hs ). The most recent of which is the quantum Raychaudhuri equation (QRE) Das:2013oda

[TABLE]

The QRE is derived from considering the pilot wave theory of quantum mechanics to assign to each particle moving in the congruence a semi-classical, WKB-type wavefunction  $\psi(x)=\mathcal{R}e^{-i\mathcal{S}x}$. The phenomenological implications of QRE to cosmology Ali:2014qla and black holes Ali:2015tva were very similar to the ones obtained from GUP. On one hand, the deformed black hole temperature can be obtained either from the quadratic GUP and it takes the form Scardiglia

[TABLE]

or, from the linear-quadratic GUP and it becomes alsaleh2

[TABLE]

On the other hand, the deformed black hole temperature can be obtained from studying the quantum-corrected Schwarzschild metric from QRE and it takes the form Ali:2015tva

[TABLE]

with $\eta$ to be the QRE parameter.

It is evident that both deformations predict the existence of a black hole remnant of minimal mass $M_{min}$ and prevent the black hole from a catastrophic evaporation, (see figures 1 and 2).

This could suggest that both approaches describe same physics and one may transform one approach into the other. In order to investigate this would-be correspondence, we will expand the expressions of the deformed black hole temperatures and compare their subleading terms to establish, if this is possible, a relation between GUP and QRE.

Next, we argue that the deformed QRE metric would imply a quantum correction to Newton’s law, similar to the one obtained from the tree diagrams of graviton exchange in the weak field limit Duff:1974ud ; Dono . It has already been shown that the GUP corrections have a thermal nature  Scardiglia , and that both quadratic and linear-quadratic GUP deformations correspond to the tree-level correction to the Schwarzschild metric Scardiglia ; alsaleh2 ; Ali:2014dfa . To explicitly show the above correspondence between QRE and GUP, we need to relate the two deformation parameters. Thus, we start with the QRE-corrected Schwarzschild metric Ali:2015tva

[TABLE]

which implies the quantum-corrected gravitational potential in the weak field limit (since $V(r)=\frac{1}{2}(g_{tt}-1)$)

[TABLE]

It is easily seen that in the RHS of (7) the second term which is the QRE correction term to the potential in the weak field limit, is scale-dependent since this term contains the relative scale of the system, i.e., $m_{\rm p}/M$. Thus, since the GUP corrections are not scale-dependent, we expect the relation between the QRE parameter $\eta$ and the dimensionless GUP parameter $\alpha_{0}$ to be also scale-dependent. We now rewrite metric (6) as

[TABLE]

where $\varepsilon(r)\ll M/r$. Therefore, the horizon equation takes the form

[TABLE]

and its solution reads

[TABLE]

Furthermore, the black hole temperature is given by the relation Townsend:1997ku

[TABLE]

By taking the limit $r\to r_{h}$, we obtain the expression for the QRE deformed temperature

[TABLE]

Then, considering black holes with $M>>m_{\rm p}$, we expand the above QRE deformed temperature and obtain

[TABLE]

where $L=m_{\rm p}/M$ is the relative scale of the system.

At this point, we expand the quadratic GUP-corrected black hole temperature given by (3), in terms of the relative scale of the system, up to second order,111One can ignore higher order terms since the GUP corrections are most trusted at first and second order Ali:2011fa .

[TABLE]

as well as the linear-quadratic GUP-corrected black hole temperature given by (4), also up to second order,

[TABLE]

Now, we can compare the subleading terms of (13) with the subleading terms of the expansion of the quadratic GUP given by (14) and obtain

[TABLE]

In addition, we can compare the subleading terms of (13) with the subleading terms of (15) and obtain

[TABLE]

At this point a number of comments are in order.

First, as expected from our comments below (7), the QRE parameter, i.e., $\eta$, depends on the relative scale of the system, i.e., $L=m_{\rm p}/M$. Additionally, both relations, namely (16) and (17), reveal a deep connection between GUP and QRE. Furthermore, these relations can be obtained even if higher order terms in the GUP-deformed temperatures were included.

Second, it was recently shown that the dimensionless GUP parameter $\beta_{0}$ cannot be negative since it implies a minimum length of $\Delta x\geq\hbar\sqrt{\beta_{0}}$ which will be imaginary if $\beta_{0}$ is negative Ali:2015zua . However, it was shown in Refs. Jizba:2009qf ; Scardigli:2014qka that one can consider a negative GUP parameter $\beta_{0}$ if the uncertainties are computed on a crystal-like universe and the lattice spacing of the universe is of the order of Planck length, i.e. $\ell_{\rm p}$. Therefore, the negativity of $\beta_{0}$ in (16) can be interpreted as a signal of a lattice structure of space-time. Consequently, this gives the possibility of having a geometric interpretation of the QRE, namely that the Bohmian trajectories of geodesics may be produced by a crystal-like structure of space-time, i.e., a discrete geometry may give Bohmian trajectories.

Third, (16) is a relation between the two parameters, namely QRE and GUP parameters, which can also be viewed as a series expansion in powers of the relative scale $L$ of the system. From this perspective, in the context of statistical mechanics as well as quantum field theory, this relation can be read as the beta function equation for the effective coupling constant $\beta_{0}$ with the energy scale to be the black hole mass, i.e., $\Lambda=M$, and the cut-off scale to be the Planck mass, i.e., $\Lambda_{cutoff}\sim m_{\rm p}$. It should be stressed that in Ref. Faizal:2017dlb the quadratic deformation parameter, i.e., $\beta_{0}$, was found to play the rôle of the coupling constant in an effective field theory.

Acknowledgements

We would like to thank Saurya Das for useful correspondences. This research project was supported by a grant from the “Research Center of the Female Scientific and Medical Colleges”, Deanship of Scientific Research, King Saud University.