Gravitational Collapse in Energy-momentum squared gravity: Nature of singularities

In this paper, we explore a collapsing scenario in the background of energy-momentum squared gravity (EMSG). EMSG claims to have terms that originate from the quantum gravity effects mimicking loop quantum gravity. As a result, the framework admits a bounce at a finite time thus avoiding a singularity. So the question that naturally arises: Is there any realistic chance of the formation of a black hole or the quantum gravity effects are strong enough to totally avoid such a pathology? Motivated by this we are interested in studying a gravitational collapse mechanism in the background of EMSG and investigate the fate of such a process. We model the spacetime of a massive star by the Vaidya metric and derive the field equations in EMSG. Then using the equations we go on to study a gravitational collapse mechanism, on two specific models of EMSG with different forms of curvature-matter coupling. The prime objective is to probe the nature of singularity (if formed) as the end state of the collapse. We see that none of the models generically admit the formation of black holes as the end state of collapse, but on the contrary, they support the formation of naked singularities. This can be attributed to the quantum fluctuations of the gravitational interactions at the fundamental level.

1 Introduction The incompatibility of General Relativity (GR) at cosmological scales came to light with the discovery of the late cosmic acceleration [1,2].This triggered extensive theoretical research to develop a framework that can comprehensively explain the phenomenon.Over the past two decades researchers have proposed two independent mechanism that can satisfactorily explain the cosmic acceleration, namely the theory of dark energy [3] and the theory of modified gravity [4][5][6].The first one proposes the presence of an exotic fluid with negative pressure that can drive the acceleration.The second concept is based on the modification of the Einstein-Hilbert (EH) action of GR, where the the gravity Lagrangian (the Ricci scalar R) is replaced by some other terms.In this work we will concentrate on this second avenue of research that has gained enormous success and popularity is the past decades.
When we talk about modification of the Einstein gravity, the most straightforward modification of the gravity Lagrangian will be to consider an analytical function of the Ricci scalar f (R), which will help explore the non-linear effects of the curvature (may be the most fundamental driving force behind the acceleration).Such theories are termed as f (R) gravity theories in literature [7][8][9].Once such a modification was found to be consistent with the theoretical framework of GR and gave promising results, it was inevitable that other factors were included in the functional forms to extend the theory.Such a crucial extension was proposed by Harko & Lobo in ref. [10], where they considered an even more general functional form in the gravitational Lagrangian by coupling the matter Lagrangian L m with R. Such theories are known as f (R, L m ) theories and significant advances in this area may be found in refs.[11][12][13].Following similar concept Harko et al. [14] proposed the f (R, T ) theory where they considered the trace of the energy-momentum tensor T in place of the matter Lagrangian in the f (R, L m ) theory.Here due to the coupling effects between matter and geometry sectors, the covariant divergence of the energy-momentum tensor (EMT) is non-zero.This indicates non-convergence of EMT leading to extra force, that results in non-geodesic nature of motion of the particles.This interesting feature of the theory attracted a lot of research on it leading to substantial developments [15][16][17][18][19][20][21].In a subsequent modification f (R, T 2 ) theory was proposed in [22], where T 2 = T µν T µν and T µν is the EMT.A specific form of f (R, T 2 ) = R + ηT 2 , was studied in [23], which was termed as energy-momentum squared gravity (EMSG).The study revealed that the field equations of EMSG contained terms similar to those arising from the quantum gravity effects of loop quantum gravity (LQG) and brane gravity.The theory allowed for a minimum length and a maximum energy density in the early universe.In a Friedmann universe it allows for a cosmic bounce alleviating the singularity problem.Subsequent developments in EMSG can be found in literature.Cosmological models in EMSG was studied in [24,25].A dynamical system analysis in the background of EMSG was performed in [26].Other important developments in EMSG may be found in [27][28][29][30][31][32][33].
A star at the end of its lifetime exhausts all its nuclear sources of energy and undergoes gravitational collapse due to its own gravitational field and releases enormous amount of energy.Gravitational collapse is a crucial astrophysical phenomenon that is responsible for the structure formation in the universe.Various stellar remnants like the white dwarfs, neutron stars, black holes, etc. are the end products of gravitational collapse of stars.The type of end-product that a gravitationally collapsing system will produce depends on the initial mass configuration of the parent stellar system.A small or an average sized star may end its collapsing process in a white dwarf, where further collapse is hindered by the electron degeneracy pressure.But subsequently if the white dwarf accretes enough mass from the surrounding so that it surpasses the Chandrasekhar limit (> 1.39 solar mass), then the collapse resumes and subsequently it reaches the neutron star stage.In this stage the further collapse is hindered by the neutron degeneracy pressure, which is overcome by again accreting mass from the surrounding.Then further collapse begins with a simultaneous supernova event, and this time it ends in a singularity.So it is clear that throughout the whole collapsing sequence, the most important role is played by mass of the star.The collapse of a supermassive star, having a huge initial mass (> 30 solar masses) will seldom be terminated at any of the intermediate stage (white dwarf or neutron star), but will eventually reach a singularity.The astrophysical significance of the gravitational collapse attracts a lot of exploration on the topic.It all started in 1939 with the pioneering work of Oppenheimer and Snyder [34], studying the collapse of a dust cloud modelled by a Schwarzschild exterior and a Friedmann interior.This was followed by Tolman [35] and Bondi [36] who studied the collapse of an inhomogeneous distribution of dust with a spherically symmetric distribution.All these collapsing scenarios resulted in the formation of black holes.Extensive reviews in gravitational collapse may be found in [37,38].In 1969 Roger Penrose proposed his famous Cosmic Censorship Hypothesis (CCH) [39], which stated that any singularity formed out of a gravitational collapse will always be a trapped inside an event horizon (black hole).This meant a complete censorship of the singularity from the outside observers such that no information of the interior is leaked to the outside.This phenomenon of black holes is termed as the information loss paradox [40][41][42][43].But due to the absence of a comprehensive proof of the CCH, there was widespread doubts regarding its validity.Scientists began searching for ways that can disprove the hypothesis.Now in order to accomplish that, it was necessary to show that a collapsing process does not always result in a trapped surface, but also has provisions for the formation of a naked singularity (NS) [44][45][46][47][48][49][50].For such a singularity, no event horizon forms and there can be a continuous exchange of information to and from the singularity.This directly solves the information loss problem.Not only this, with the continuous supply of data right from the heart of the singularity, our quantum gravity problem will gain impetus.As a result a hunt for NS began with the works of Eardly and Smarr [44] and Christodoulou [45] at the end of the twentieth century.Other significant works in NS can be widely found in literature in the refs.[51][52][53].
The AdS/CFT correspondence [54,55] has been used to gain insights into the behavior of strongly interacting quantum systems, including those that might be analogous to certain aspects of black hole physics.In the context of gravitational collapse, researchers have explored the idea that the dynamics of a collapsing star and the resulting black hole formation might be dual to a description in terms of a non-gravitational quantum field theory living on the boundary of an AdS spacetime.This holographic duality allows physicists to study certain gravitational phenomena, such as black hole formation and evaporation, through the lens of a non-gravitational theory.It provides a different perspective on understanding the quantum nature of gravity in extreme conditions, where classical general relativity alone may not be sufficient.Moreover extensive use of the holographic principle can be found in cosmology [56,57].It's important to note that the application of holography to gravitational collapse and cosmology is an active area of research, and the full understanding of these connections is still evolving.
Schwarzschild metric represented the spacetime outside a spherically symmetric matter distribution possessing a constant mass.Such a solution is static in nature and cannot model the spacetime of a realistic star.A solution to this problem was given by Vaidya in his ground breaking paper in 1951 [58], where he proposed a dynamic spacetime outside a realistic star.Vaidya's metric was aptly termed as the shining or radiating Schwarzschild metric.Here the constant mass parameter in the Schwarzschild metric was replaced by a time dependent dynamic mass parameter, which was the source for the outgoing stellar radiations.Various studied in Vaidya spacetime can be found in the refs.[59][60][61][62][63][64][65][66][67][68].
In this work we intend to study a stellar collapse modelled by Vaidya spacetime in the background of energy-momentum squared gravity.Since EMSG is known to contain some flavours of quantum gravity we are motivated to explore a collapsing scenario in this gravity, that will continue the whole way till the singularity (collapse of a super-massive star).Our basic plan is to study the nature of singularity (BH or NS) that results from the collapse (if at all a singularity forms).This will throw new light on the terms that provide the quantum gravitational effects in EMSG and also the validity of CCH.The fact that GR foretells the occurrence of spacetime singularity at some finite point in the past is one of the theory's most fascinating mysteries.However, because of the anticipated quantum consequences, it turns out that GR itself is no longer valid at the singularity.However, there is currently no accurate description of quantum gravity.When we apply this theory to a homogeneous and isotropic spacetime, we discover that the early Universe has a minimum length scale and a maximum energy density.In other words, there is an early-time bounce and, as a result, the early-time singularity is avoided.This is the main consequence of the correction terms that appear in the theory.This is the reason why these terms are compared to those of the Loop quantum gravity and brane-world gravity.While the underlying physical theory does not incorporate the LQG or brane-world models directly, nor does it reduce to them in a limiting case, it focuses on a type of higher-order matter corrections that modify the Friedmann equations in ways that include phenomenological modifications coming from both Loop quantum gravity and braneworld gravity.It makes sense to assume that this correction term would only matter in high-energy environments, such the early Universe or black holes.Consequently, in the low-curvature regime, there are no deviations from GR.This is the reason why there is a strong motivation of studying the nature of singularities arising from gravitational collapse in the background of this theory.The quantum gravitational effects coming from the correction terms of this theory has crucial physical significance and interpretation in the study of the fate of these singularities.
The paper is organized as follows: Section 2 is dedicated in deriving the field equations of Vaidya spacetime in f (R, T 2 ) gravity.In section 3 we explore a collapsing mechanism using the derived field equations.Finally the paper ends with a discussion and conclusion in section 4.
2 Vaidya spacetime in f (R, T 2 ) gravity The action of the model is written as [22,24] where f is a function depending on the square of the energy-momentum tensor T 2 = T µν T µν and the scalar curvature R. Here, κ 2 = 8πG and L m represents the matter Lagrangian.
If we vary the action with respect to the metric we arrive at the following field equations where T is the trace of the energy-momentum tensor.By taking covariant derivatives with respect to the field equation ( 2.2), one finds the following conservation equation As one can see from the above equation that in general, the conservation equation does not hold for this theory.If one chooses f (R, T 2 ) = 2α log(T 2 ), one gets the same result reported in Akarsu et al. [25].
Here we consider perfect fluid and further, we assume L m = p which allows us to rewrite Θ µν defined in eqn.(2.3) as a quantity which does not depend on the function f , as given below [22,24] Θ µν = − ρ 2 + 4pρ + 3p 2 u µ u ν . (2.5) is defined.The Vaidya metric in the advanced time coordinate system is given by [58], where r and using the units G = c = 1.The total energy momentum tensor of the field equation (2.2) is given by the following sum, where µν are the contributions from the Vaidya null radiation and matter respectively defined as, and where ′ ρ ′ and ′ p ′ are the energy density and pressure for matter and ′ σ ′ is the energy density corresponding to Vaidya null radiation.In the co-moving co-ordinates (t, r, θ 1 , θ 2 , ..., θ n ), the two eigen vectors of energy-momentum tensor namely l µ and η µ are linearly independent future pointing null vectors having components and they satisfy the relations Therefore, the non-vanishing components of the total energy-momentum tensor will be as follows Here we consider matter in the form of perfect barotropic fluid given by the equation of state where 'ω' is the barotropic parameter.
The non-vanishing components of the Ricci tensors are given by, where .and ′ represents the derivatives with respect to time coordinate ′ t ′ and radial coordinate ′ r ′ respectively.For this system the Ricci scalar becomes, The trace of the energy momentum tensor is calculated as, The square of the energy momentum tensor will be given by, The relation between density and mass is considered as [69], where n > 0 is the particle number density.

Field equations
From eqn.(2.2), we see that we need to put specific forms of f (R, T 2 ) to get further solutions.Here we will derive the field equations of Vaidya spacetime in EMSG for such specific models depending on the nature of coupling between R and T 2 .We basically consider two general forms, one of which is additive representing minimal coupling and the other is in the product form representing non-minimal coupling.they are discussed below in Model-1 and Model-2 respectively.

Model
Here we report the computed Einstein's field equations of f (R, T 2 ) gravity in the time dependent Vaidya spacetime for the first model.
The (01) component is given by, The ( 22) and ( 33) components are given by, Here we have only reported those components which we have used to find a solution.The other components of the field equations are reported in the appendix section of the paper in order to preserve a better structure of the paper.
Here we report the field equations for the second model.
The (01) component is given by, The ( 22) and ( 33) components are given by, Here also the other components of the field equations are reported in the appendix section of the paper.

Solution of the system
In order to find solution for the field equations we need to consider special toy-models as case studies.Here we can consider various forms of functionalities for f (R) and f (T 2 ) for both types of models.

Model-1
Here we consider the specific functionalities . This model was studied in ref. [26], where a dynamical system analysis was performed.We recover GR for g 2 = 0 and g 1 = β 1 = 1.When g 1 = β 1 = β 2 = 1 then the model reduces to the one used in ref. [24].When g 1 = β 1 = 1 and g 2 = 0, β 2 = 0, then we recover the Energy Momentum Powered Gravity (EMPG) [24].In this model g 1 and g 2 play the role of the coupling parameters between the curvature and the matter sectors.
Here the (01) component of the field equations give, Solving the above equation we get for where h 1 (t) is an arbitrary function of time t.The ( 22) or (33) component becomes, Solving the above equation we get for β 1 = β 2 = 1 and for ω = 0 (dust) where h 1 (t) and h 2 (t) are arbitrary functions of time t.It is expected that the solution in eqn.(2.25) will reduce to the solution given in eqn.(2.27) for some suitable initial conditions.So we will use the solution given by eqn.(2.25) for our collapse study because of its generic nature.

Model-2
Here we consider the functional forms Here also we recover GR for g 2 = 0 and g 1 = β 1 = 1.For g 1 = 0, we get the model studied in ref. [26].
Here the (01) component of the field equations give, Solving the above equation for where W [y] is the Lambert W function and h 1 (t) is an arbitrary function of time.The ( 22) or (33) component becomes, Solving the above equation we get for ω = 0 (dust) and where h 1 (t) and h 2 (t) are arbitrary functions of time t.Quite naturally due to its generic nature we will use the solution given by eqn.(2.29) for further analysis.
3 Gravitational collapse: Nature of singularity An object of mass whose internal pressure is not enough to offset the gravitational forces pulling it inward is said to be undergoing gravitational collapse.A black hole's creation is the most wellknown result of gravitational collapse.The formation of a big star marks the start of the process.A star's life is characterized by nuclear fusion, which turns hydrogen into helium and other heavier elements.The star reaches a critical point when its nuclear fuel runs exhausted, at which time the equilibrium between internal pressure pushing outward and gravitational forces pulling inward becomes unstable.When the core of a large star can no longer withstand the force of gravity, it collapses due to the pull of gravity.This collapse can occur quickly, raising the temperature and density.The collapsing core's density greatly rises as it continues to fall.This results in the production of a singularity, or a point with infinite density and curvature, in the context of classical general relativity.Usually, this singularity is concealed within an area known as the event horizon.
An event horizon forms around the singularity if there is not enough angular momentum or charge in the collapsing item to stop it.The event horizon is the point beyond which the tremendous pull of gravity prevents anything from escaping, not even light.This signifies the black hole's creation.
In the presence of density inhomogeneities an event horizon forms and a black hole is created in a homogeneous collapse, in which the collapsing object has a uniform density and spherical symmetry.This process is well-described.The event horizon encloses the singularity.The process of collapse gets more difficult in the presence of asymmetries or density inhomogeneities.It is still feasible for an event horizon to form, but the specifics will depend on the mass and energy distribution.
The Cosmic Censorship Hypothesis raises the question of whether or not a naked singularity (a singularity devoid of an event horizon) can form.According to this theory, which was put forth by Roger Penrose [39], singularities ought to always be concealed within event horizons.If the theory is correct, then any collapsing object would become a black hole with an event horizon, and density inhomogeneities will not be able to resist its formation, thus resulting in a naked singularity.A thorough understanding of general relativity is necessary to comprehend the behavior of gravitational collapse and the creation of horizons in the presence of density inhomogeneities.This is still an area of ongoing theoretical physics research.One important frontier in our attempt to unify general relativity and quantum mechanics is the nature of singularities and horizons, where the effects of quantum gravity may become relevant.
Here we are concerned with the nature of singularity formed as a result of the collapse.So the whole study revolves round the fact that whether or not trapped surfaces are formed surrounding the singularity.Now this is not so straightforward as it seems to be.There are other essential ideas to consider over here.Not only the formation of the horizons, but also the timing of the formation of the horizons play crucial role in determining the nature of the singularity formed.If the horizon forms far before the formation of the singularity, then it is always censored from the external observer, leading to a black hole.But if the formation of the trapped surface is delayed, such that the singularity forms first and then the trapped surface begins to form, then the singularity is free to share information with the external observer for some finite amount of time, thus being naked [37].It is found that inhomogeneities in the initial density profile of the collapsing matter can result in the postponement of the formation of the horizon [37].Moreover the presence of shear in the system can also hinder the formation of the trapped surface for a finite amount of time [37].Since we have seen that EMSG has terms that mimic some properties of loop quantum gravity, there may be some quantum fluctuations at the fundamental level that can create some inhomogeneities in the matter density profile.This can substantially delay the formation of trapped surfaces if not completely hinder its formation, thus making the singularity naked.So it seems that the rate of collapse plays an important role in determining the nature of the singularity formed.A slow collapse will keep the density distribution more or less homogeneous over time, thus resulting in a black hole, whereas a fast collapse will tend to form a naked singularity due to the inhomogeneity in matter density created over time.So we are motivated to explore these effects for EMSG.
For any relativistic metric, we know that ds 2 = 0 gives the directions along which the light rays travel.This is evident from the study of light cones in the special theory of relativity.Here we would probe the existence of outgoing light rays (radial null geodesics) from the core of the central singularity.The equation for such geodesics can be found from the metric (2.7) by equating ds 2 = 0 and dΩ 2 2 = dθ 2 + sin 2 θdφ 2 = 0 so as to get This is a differential equation, which possess a singularity at the point r = 0, t = 0.There is a complete breakdown of mathematical and physical structures at the singularity and hence we need to examine the limiting behaviour of the trajectories as one approaches the singularity.In order to conduct such a study, we consider a parameter X = t/r.We will study the limiting behaviour of the function X as one proceeds towards the singularity at r = 0, t = 0 along the trajectory of the radial null geodesic.Suppose the limiting value of X at r = 0, t = 0 is given by X 0 , then L'Hospital's rule yields Using the mass terms that we obtained previously the above limit will provide an algebraic equation in terms of X 0 .We will hunt for the roots of this equation, which actually represent the directions of the tangents to the outgoing geodesics.In principle we should only be interested in the real roots of the equation, since they represent the real exchange of information with the external observers via escaping radiation.In our mathematical set-up, any positive real root of the equation will indicate the gradient of the tangent to an outgoing null geodesic.So it can be concluded that the existence of real positive roots of the obtained algebraic equation corresponds to the creation of a naked singularity as a result of the stellar collapse.
When a single null geodesic get away the singularity, it corresponds to a single wavefront being emanated from the central singularity and illuminating the external observer.In such a case, the singularity would be observable instantaneously to a distant observer, before visibility is obstructed.This will result in a locally naked singularity.Physically speaking, here the formation of the trapped surface is delayed for a little amount of time, when the singularity becomes visible temporarily.But this temporary exposure might not be adequate for a total trade-off of data and information between the singularity and the distant spectator.So for a comprehensive trade-off of information, the singularity needs to be visible for an elongated time interval.This is possible if a bundle of null geodesics emanates from the central singularity, making it globally naked.Our theoretical set-up provides a perfect mechanism to explore this quite easily by examining the count of real positive roots derived from the algebraic equation.The above discussed mathematical formulation for investigating the nature of singularity formed as the final state of a stellar collapse was first proposed by Joshi, Singh and Dwivedi in several of their papers [53,[70][71][72].With the theoretical framework ready, we will now proceed to examine the models separately.
The physical characteristics of the collapsing object, the equation of state of the matter involved, and the particulars of the collapse process all influence the threshold mass at which black hole formation would be prevented.It's crucial to remember that the intricacy of the issue and the range of possibilities that can result in the formation of a black hole make it difficult to provide accurate quantitative estimations.However, a broad rundown of a few important ideas can be provided.For a non-rotating, non-charged star supported by electron degeneracy pressure, the Chandrasekhar limit [73] is a critical mass beyond which electron degeneracy pressure cannot counterbalance gravity, leading to gravitational collapse.This limit is approximately 1.44M ⊙ , where M ⊙ is one solar mass.The pressure of neutron degeneracy opposes the collapse of more massive objects, like neutron stars.The total object mass limit (Tolman-Oppenheimer-Volkof (TOV) limit) [74,75] is the maximum mass that a neutron star can have before collapsing into a black hole.This limit is around 2.16M ⊙ , although the exact number is unclear due to difficulties in the equation of state for neutron-rich matter.A distinct mechanism is involved in the case of big stars.Pair production can result in a drastic drop in pressure for stars with masses around 100 − 250M ⊙ , which can trigger a pair-instability supernova.As a result of the star's total disruption in these situations, black hole creation is prevented.Under some circumstances, black holes can emerge straight ahead of time without the need for an intermediary stage for a supermassive star.This may occur in areas with low metallicity and high density where radiation pressure and nuclear burning do not impede the collapse.Although less clearly defined, the mass threshold for stars that collapse directly to black holes may be in the range of 10 2 − 10 3 M ⊙ .It is crucial to stress that these are only approximations and that the real mass thresholds would vary depending on the particular circumstances and characteristics of the collapsing object.Furthermore, we are constantly improving our understanding of these processes through astrophysical observations.The above quantitative estimates are provided considering standard general relativity as the background theory of gravity.In case of our study, since the correction terms of EMSG possess quantum gravitational effects, they will try to hinder the collapsing procedure thus trying to avoid the ultimate singularity.So the intensity of the collapsing mechanism will be reduced depending on several other factors.Therefore for EMSG there is a possibility that each of the quantitative estimates given above will be raised.For example for a non-rotating, non-charged star the Chandrasekhar Limit may be raised from 1.44M ⊙ .So the new mass limit M n at which the collapse occurs will be M n > 1.44M ⊙ .Similar argument goes for the other cases as well.

Model
Here we will study a particular toy model based on the model-1 that was discussed in the previous section.The model is given by f (R, T 2 ) = R + αT 2 , where α is the coupling parameter.This model is widely studied in the literature and is cosmologically a successful model.So here we are interested in studying the model in a collapsing scenario.So this model will be retrieved from model-1 for β 1 = β 2 = g 1 = 1 and g 2 = α.Using eqn.(2.25) and eqn.(3.2) for this model we get (on expanding the exponentials and taking only the linear terms), Now we will consider a self-similar collapsing scenario by the choice of arbitrary function h 1 (t) = ξ 1 t −1 , where ξ 1 is an arbitrary constant.Using this choice in eqn.(3.3) we get the following algebraic equation in X 0 Solving the above equation we get the following two roots Here we will consider the positive root as X 01 and the negative root as X 02 .To get a realistic scenario we should have ξ 1 ≤ 0 and ξ 1 ≥ 8. Now depending on the signature of the roots X 01 and Figs. 1 and 2 show the variation of the roots X 01 and X 02 respectively for different values of ξ 1 for Model-1.
X 02 the collapse results in different outcomes.The conditions for getting a locally naked singularity are (X 01 > 0 & X 02 < 0) OR (X 01 < 0 & X 02 > 0).For a globally naked singularity we should have X 01 > 0 & X 02 > 0. Finally for X 01 < 0 & X 02 < 0 the collapse results in a black hole according to our theoretical framework.To understand the trend of the roots they are plotted in Fig. 1 and Fig. 2 respectively against ξ 1 .It clearly seen from the plots that the roots are not realistic in the range 0 < ξ < 8 as discussed before.Hence there are no well-defined trajectories of the roots in that parameter range.From Fig. 1 it is seen that the X 01 < 0 for ξ 1 < 0 and X 01 > 0 for ξ 1 > 8. From Fig. 2 it is evident that X 02 > 0 throughout its domain of definition.
There are two disconnected trajectories (obvious due to the region 0 < ξ 1 < 8 being outside the domain of definition of the root) and they reach asymptotic nature near ξ 1 = 0 and ξ 1 = 8.Now combining the information obtained from both the figures we have ⇒ X 01 > 0, X 02 > 0 for ξ 1 > 8, & X 01 < 0, X 02 > 0 for ξ 1 < 0. So we for ξ 1 > 8, the collapse results in a globally naked singularity while for ξ 1 < 0, a locally naked singularity forms.We further see that formation of a black hole is not an option here.This is probably because the EMSG model has terms which have reminiscence of quantum gravity flavour, which mimic loop quantum gravity [23].So these models have a tendency to bounce at a finite time rather than end in a trapped singularity.
In this case we intend to study a particular toymodel based on model-2 discussed in the previous section.The model is given by f (R, T 2 ) = f 0 RT 2 , where f 0 is the coupling parameter.Now using eqn.(2.22) and (3.2) we get for this model, where h 2 (t) is an arbitrary function of t.Now considering self similar collapsing scenario, we choose h 2 (t) = ξ 2 t 2 , where ξ 2 is an arbitrary constant.Using this choice in eqn.(3.6) we get the following algebraic equation in Solving the above equation we get the roots as, Figs. 3 and 4 show the variation of the roots X 01 and X 02 respectively for different values of ξ 2 for Model-2.
The most striking feature of these roots are the extreme level of constraint that they suffer from in order to take real values.The range in which these root actually take some realistic values are as narrow as 0 < ξ 2 < 0.0078.So we see that this model is highly delicate and requires a considerable amount of fine tuning to derive meaningful result out of it.To gain insights into the trend of the roots, they are plotted in Fig. 3 and Fig. 4 respectively against ξ 2 .From the figures it is evident that in the domain of definition, both the roots remain in the positive level.This shows that we have a globally naked singularity as the outcome of collapse.The quantum gravity effects coming into play restricts the singularity from being trapped inside the horizon.

Strength of the singularity (Curvature growth near the singularity)
The amount of curvature inflicted on the surrounding spacetime determines the gravitational strength of a singularity.It is the estimate of the destructive capacity of the singularity on any matter that passes by it.Most theories of gravity have been crippled by the presence of singularities in their theoretical set-up.Various theoretical mechanisms of elimination of such singularities have been proposed in literature from time to time, but they are substantially alien in nature and far from being acceptable.We know that singularities are depressions in the fabric of the otherwise continuous and smooth spacetime.In case of a weak singularity the crater is shallow and a continuous extension of spacetime is feasible right through the singularity.Mathematically this idea is analogous to a removable discontinuity, which can rehabilitate the discontinuity of spacetime at a singularity.From the above discussion we see that there is enough motivation in investigating whether a singularity is strong or weak in nature.According to Tipler [76] a curvature singularity will be strong if any object coming in its vicinity or colliding with it is squeezed to zero volume.In Ref. [76] the condition for a strong singularity is given as, where R µν is the Ricci tensor, ψ is a scalar given by the relation ψ = R µν K µ K ν , where K µ = dx µ /dτ represents the tangent to the non spacelike geodesics at the singularity and τ is the affine parameter.In Ref.In ref. [77] it has also been shown that the relation between X 0 and the limiting values of mass is given by, where and ṁ0 is given by the eqn.(3.12).
Studies by Dwivedi and Joshi in Refs.[70,78] revealed that any classical singularity in Vaidya spacetime in the background of Einstein gravity should to be a very strong curvature singularity going by the definition of Tipler.In addition to this, the authors have also shown that the hypothesis [79] that the strong curvature singularities are always trapped by horizons is not always true.It is expected that in the background of the quantum fluctuations of f (R, T 2 ) gravity the strength of the singularity will be considerably weakened if not totally eliminated.The structure of such a naked singularity was explored in Ref. [80] and it was shown that the singularity attains a directional nature as the curvature grows along the geodesics, finally concluding in the singularity.Moreover for a quantum regime it was evident that the singularity is gravitationally weak in nature, which permitted a continuous extension of the spacetime through the singularity [81].Now we will study the strength of the singularities for the respective models.Taking the limit of this as r → 0 we get ξ1e ξ 1 t t 2 e ξ 1 t +2nαω 2 2 .Now taking the limit of this expression as t → 0 we get an indeterminate form, which is not removable by L'Hospital's method.To resolve this issue, we resort to Taylor expansion of the exponential terms of mass given in eqn.(3.15).Expanding and considering the linear terms only we get m(t, r) = . It should be mentioned here that the qualitative features of the collapse mechanism are not hampered due to truncation of the series to linear terms only.This is because we are going to take limits as r → 0 and t → 0 to track the singularity, and all the non-linear terms will behave exactly as the linear term in the limiting scenario for mass.Taking the time derivative for the above Taylor expanded expression for mass we get t +2αnω 2 1+ nr 3 3 2 .Now taking limit as r → 0 we get ξ1 t 2 (1+ ξ 1 t +2αnω 2 ) 2 .We take the limit t → 0 of the obtained expression and see that we have an indeterminate form.But this time the indeterminacy can be removed by using L'Hospital's rule and we get ṁ0 = 0. Using this in eqn.(3.10) we get, Therefore according to the definition of the strength of a singularity by Tipler given in eqn.(3.9), we have a gravitationally weak singularity for this model.This is a direct result of the quantum gravity terms of the theory, that eases the singularity and gives us a less severe pathology to handle.
For this model the mass parameter is given by, Taking time derivative of the above expression we get 2ξ2ωt r 3 6f 0 n +2ξ2ω 2 t 2 .Now taking limit as r → 0 yields Considering ξ 2 > 0, in the above expression we find the t → 0 limit as ṁ0 = √ 2ξ 2 .
For ξ 2 < 0 we do not have real values.So using the result in eqn.(3.10) we get, Now the sign of S completely depends upon whether we take the positive or the negative square root of √ ξ 2 .If we take the positive square root, S > 0, and vice versa.So we see that in this theoretical framework there is provision for both strong and weak singularities depending on the initial conditions.Here we see that the quantum gravitational effects of the model is subdued compared to model-1.This results in both strong and weak singularities depending on the initial conditions.So the nature of coupling between the matter and the geometry sectors are playing crucial roles in determining the nature of singularity formed as a result of the collapse.

Conclusion and discussion
In this work, we have explored a collapsing scenario in the background of f (R, T µν T µν ) gravity.
Since the correction terms of these models are reminiscent of those coming from the quantum gravity effects in loop quantum gravity or braneworld models, the early radiation dominated universe is not supposed to start from a singularity.Motivated from this we wanted to investigate the effect of the quantum geometry on the collapsing mechanism of a star in the background of this gravity.The basic idea is to probe the possibility of a singularity-free collapse.Even if a singularity forms, we wanted to find out whether there is any possibility of formation of trapped surfaces as the outcome of collapse.We modelled the spacetime of a massive star by the Vaidya metric.The reason behind this choice is that, the time dependent Vaidya metric represents the spacetime of a realistic star.Moreover a collapsing procedure being a time-dependent phenomenon is ideally suited for this metric.The field equations were found and the mass term was calculated from them.Now using this mass term we devised a theoretical framework of the collapsing mechanism.The idea was to probe the existence of outgoing radial null geodesics emanating from the central singularity.This was obtained by considering the line element ds 2 = 0 and dΩ 2 2 = dθ 2 + sin 2 θdφ 2 = 0.In the process we reached a differential equation in t and r.To explore the effects near the singularity, we considered a limiting condition as t → 0 and r → 0. This led us to an algebraic equation in terms of a collapsing parameter X 0 , whose signature actually determined the nature of the singularity formed.A positive value of X 0 indicated the presence of outgoing radial null geodesics and hence showed the absence of trapped surfaces, i.e. black holes.This would indicate that either there is no singularity formed or even if there is a singularity it naked in nature.If X 0 assumed negative values, it showed the absence of any outgoing null geodesics.This indicated the probable formation of event horizon, which meant the existence of a black hole.So the whole mechanism was based on the nature of roots obtained from our algebraic equation.
We applied this collapsing mechanism on two specific models of f (R, T 2 ) gravity with different types of couplings.In the first model, we studied the functional form f (R, T 2 ) = R + αT 2 .It was found that there was no possibility of the formation of a black hole.Either the collapse did not result in a singularity or if it did so it was a very weak naked singularity.Depending on the initial conditions the naked singularity can be local or global.We think that this can be totally attributed to the quantum geometry effects coming from the correction terms.The quantum effects as expected have either totally cured the singularity or even if it is present, it does not have the ability to form trapped surfaces, due to the loss of its extreme curvature effects (strength).In the second model, we probed the functional form f (R, T 2 ) = f 0 RT 2 .It can be clearly seen that the coupling effect between R and T 2 is quite different here in contrast to the first model.From the collapse analysis it was seen that for this model also formation of trapped surfaces is hindered, by the quantum gravity effects.The naked singularity formed for this model was globally naked irrespective of the initial conditions.However in the strength analysis it was observed that the singularity may be weak or strong depending on the initial conditions.This shows that the nature of coupling between matter and geometry plays an important role in determining the outcome of collapse.For minimal coupling the quantum gravity effects are far more dominant compared to the non-minimal coupling.This is the reason why the singularity considerably eases out (weakens) for minimal coupling, while for the non-minimal case there is a possibility for the formation of both strong and weak singularities depending on the initial conditions.
If naked singularities exist, they would not have an event horizon to conceal them from view, unlike black holes, which are distinguished by this feature.Finding visible indicators to differentiate between the two is a difficult endeavor since it requires going beyond what is currently known about physics and observational skills.However some observable indicators could be useful in differentiating between naked singularities and black holes.The Presence or absence of event horizon is the easiest way to distinguish between naked singularities and black holes.There could be differences in attributes between the accretion disks surrounding naked singularities and black holes.For instance, in the case of naked singularities, the structure and properties of the accretion disk may be affected by the lack of an event horizon.A system with a bare singularity may produce different gravitational waves than a system with a black hole.Finding these variations via gravitational wave measurements could yield insightful information.A naked singularity may not have the same gravitational pull on surrounding stuff as a black hole.Tracking the movement of adjacent objects and the surrounding area may show interesting patterns.Unlike black holes, bare singularities may not produce Hawking radiation.A closer look at the radiation characteristics close to the proposed singularity may reveal further information about its existence.Analyzing quantum effects close to the singularity may help distinguish bare singularities from black holes.This would necessitate a greater comprehension of quantum gravity, a notion that physicists are presently unable to grasp.In contrast to events around black holes, events near naked singularities may be more chaotic or display surprising behaviors if they defy the Cosmic Censorship Hypothesis.It is noteworthy that the aforementioned recommendations are theoretical in nature and that there is currently a lack of a solid theoretical foundation for naked singularities.Furthermore, it's possible that our existing observational capabilities won't be enough to find and identify these unusual things.Such severe astrophysical phenomena demand a greater understanding of the interactions between quantum mechanics and general relativity, an area of ongoing theoretical physics research.
From the above discussion, we see that energy momentum squared gravity can be a cure for the singularity problem existing in the literature of cosmology.The asymptotic isolation of the singularity for a distant observer is significantly eased out in this case due to the presence of quantum fluctuations at the fundamental level.Although in the early stages of research, but still it has shown enough promise to be a potential candidate with its quantum geometric correction terms which mimic the loop quantum gravity or braneworlds.From the perspective of the cosmic censorship hypothesis, this work seems to be a substantive development.Hereby we reserve ourselves and do not go on to declare the complete failure of cosmic censorship hypothesis, but this work should be considered as more than a significant counterexample of the hypothesis.