^{1}

^{1}

^{3}.

We studied the thermodynamics and spectroscopy of a

Ever since the publication of the seminal papers of Bekenstein and Hawking [

where

where

This study mainly explores the entropy/area spectra of a four-dimensional Lifshitz BH [

The following statements elaborate on the organization of this study. In Section

In this section, we introduce the four-dimensional Lifshitz spacetime and its special case, that is,

The action of the Einstein-Weyl gravity [

where

Now, we focus on the

where the metric function

In the above metric, which is conformal to (A)dS (AdS if

Note that the requirement of

Thus, at spatial infinity, the Ricci and Kretschmann scalars of the

By performing the surface gravity calculation [

Therefore, the Hawking or BH temperature [

The GR unifies space, time, and gravitation and the gravitational force is represented by the curvature of the spacetime. Energy conservation is a sine qua non in GR as well. Because the metric (

Starting with the time-like Killing vector

where

Here,

with

The surface gravity

To have an outward unit vector

On the contrary,

with

Therefore, the nonzero components of

In sequel, the four-vector velocity reads

The nonzero components of

One can verify that

The nonzero components of

After substituting those findings into (

Thus, from (

and, in sequel computing the entropy through the integral formulation (

The above result is fully consistent with the Bekenstein-Hawking entropy. The quasi-local mass

which matches with the quasi-local mass computation of Brown and York [

In this section, we shall study the QNMs and the entropy of a perturbed

Effective potential versus tortoise coordinate graph for various orbital quantum numbers.

where

in which

where

which results in

One may check that the limits of

The effective potential seen in (

which admits these limits:

It is clear that, for any constant

In this section, we shall nudge (perturb) the

We can expand

where

with the parameter

The solutions to the above equation can be expressed in terms of the CH functions of the first and second kinds [

with the parameters

where

With the aid of the limiting forms of the CH functions [

We can alternatively represent (

For QNMs, imposing the boundary conditions that the outgoing waves must vanish at the horizon and no wave at the spatial infinity, that is,

the solution having coefficient

where

Subsequently, using the adiabatic invariant quantity (

we can read the entropy/area spectra of the

Therefore, the minimum spacing of the BH area becomes

Our finding is in agreement with Bekenstein’s conjecture [

In this work, the quantum spectra of the

In addition,

No data were used to support this study.

The authors declare that they have no conflicts of interest.