Causal Structure of Spacetime Singularities and Their Observable Signatures

Causal Structure of Spacetime Singularities and Their Observ able Signatures Bina P atel 1 , ∗ Jahn vi Mistry 2 , † Ayush Bidlan 3 , ‡ and P arth Bam bhaniy a 4 § 1 Dep artment of Mathematic al Scienc es, P. D. Patel Institute of Applied Scienc es, Char otar University of Scienc e and T e chnolo gy (CHARUSA T), Changa, Gujar at 388421, India 2 Dep artment of Physic al Scienc es, P. D. Patel Institute of Applie d Scienc es, Char otar University of Scienc e and T e chnolo gy (CHARUSA T), Changa, Gujar at 388421, India 3 Dep artment of Physics, Sar dar V al labhbhai National Institute of T e chnolo gy, Sur at 395007, Gujar at, India, and 4 Instituto de Astr onomia, Ge of ´ ısic a e Ciˆ encias Atmosf ´ eric as, Universidade de S˜ ao Paulo, IAG, Rua do Mat˜ ao 1225, CEP: 05508-090 S˜ ao Paulo - SP - Br azil (Dated: Marc h 24, 2026) W e analyze the causal structure of horizonless compact ob jects via the light-cone geometry and conformal compactification of the Joshi-Malafarina-Nara y an (JMN-1) and Janis-Newman-Winicour (JNW) spacetimes. P enrose diagrams reveal that JMN-1 undergoes a transition from timelike (0 < M 0 < 2 / 3) to null (2 / 3 < M 0 < 4 / 5) singularities, while JNW remains timelike throughout, in con trast to the spacelik e singularity of the Sch w arzsc hild spacetime. W e sho w that photon spheres exist in Sch w arzsc hild and JNW, but arise in JMN-1 only in the null singularit y phase, establishing a direct link b et w een causal character and null geo desic trapping. W e further demonstrate that radial timelik e geo desics develop turning p oin ts for certain parameter regimes in both JMN-1 and JNW spacetimes, indicating the emergence of effective repulsive b eha vior in the strong field region. These features lead to distinct strong field lensing and shado w signatures, p oten tially testable by very long baseline interferometric observ ations such as those of the Ev en t Horizon T elescop e. K ey wor ds : Blac k Holes, Spacetime Singularities, Gravitational Collapse, Shado ws, VLBI-EHT. I. INTR ODUCTION The gravitational collapse of a spherically symmetric homogeneous dust cloud w as first analyzed b y Opp en- heimer, Sn yder, and Datt (OSD) in which the in terior spacetime is describ ed by a closed F riedmann geometry matc hed to an exterior Sc h w arzsc hild region [ 1 , 2 ]. Within their framework, the collapse pro ceeds monotonically to- w ard the formation of a curv ature singularity . During the ev olution, trapp ed t w o-surfaces form b efore the singular ep och is reac hed. The app earance of trapped surfaces directly implies that the future-directed null expansions b ecome negativ e in the neigh borho o d of the center. As a result, the singularity lies entirely within the trapp ed region. It is, therefore, hidden from causal comm unication with the exterior spacetime region. More concretely , an ev en t horizon forms, which acts as the causal b oundary of this spacetime region. In particular, the spacetime singularity is not an or- dinary lo cus in spacetime, but rather it represents the b oundary of the spacetime manifold [ 3 ]. All non-spacelik e geo desics terminate there with a finite affine parameter λ . The asso ciated Jacobi fields χ collapse, and the spacetime v olume element along such congruences v anishes in the singular limit. The global causal structure of an asymp- totically flat spacetime is conv enien tly describ ed with resp ect to future null infinity I + . The even t horizon is ∗ binapatel.maths@charusat.ac.in † jahnvimistry25@gmail.com ‡ i21ph018@phy .svnit.ac.in § grcollapse@gmail.com ; Corresponding author defined as the b ound ary of the causal past of I + , namely H + = ∂ J − ( I + ). It separates the spacetime region from whic h causal signals can reac h distant observ ers from the region that remains p ermanen tly hidden. In collapse sce- narios leading to black hole formation, the ev en t horizon is a n ull hypersurface that encloses the trapp ed region and shields the high-curv ature domain from external observ a- tion. Another important causal boundary in spacetime is the Cauchy horizon , which is defined as the boundary of the domain of dep endence of a spacelike hypersurface. F or an initial spacelike hypersurface Σ, the domain of dep endence D (Σ) consists of all spacetime p oin ts whose ph ysical evolution is uniquely determined by the initial data sp ecified on Σ. The b oundary of this region, kno wn as the Cauc h y horizon, marks the limit beyond whic h the classical evolution of the spacetime geometry cannot b e uniquely predicted from the initial data. The presence of such a horizon therefore signals a breakdo wn of global determinism in general relativit y (GR) [ 4 , 5 ]. In GR, global predictability is ensured if the space- time is globally h yperb olic, meaning that there exists a Cauch y surface whose domain of dep endence co v ers the entire spacetime manifold. In particular, only a re- stricted class of highly symmetric solutions p ossess this prop ert y globally , including Minko wski spacetime, the Sc h w arzsc hild spacetime in its exterior region, and the F riedmann–Lema ˆ ıtre–Robertson–W alk er (FLR W) cosmo- logical mo dels [ 4 , 5 ]. How ev er, sev eral physically im por- tan t exact solutions of the Einstein equations admit inner Cauc h y horizons. Notable examples include the charged blac k hole solution of Reissner–Nordstr¨ om (RN) and the rotating black hole solutions of Kerr and Kerr–Newman. In their maximal analytic extensions, these spacetimes 2 con tain an inner horizon that acts as a Cauch y horizon separating regions where the deterministic ev olution from initial data fails. Beyond this surface, the classical field equations no longer provide a unique contin uation of the spacetime geometry from the initial data sp ecified on a spacelik e hypersurface. The o ccurrence of Cauch y hori- zons in suc h solutions indicates that global predictability is a strong condition in general relativity [ 6 ] and need not b e satisfied b y all mathematically consistent solutions of the Einstein field equations [ 4 , 5 , 7 ]. This feature is closely related to fundamental questions concerning the v alid- it y of the Cosmic Censorship Conjecture (CCC) and the stabilit y of Cauc h y horizons under realistic p erturbations. The nature of a singularity may b e c haracterized by the causal structure of the h ypersurface on which incom- plete causal geo desics terminate. A spacelike singularit y p ossesses a spacelike b oundary and lies entirely to the future of timelike observers, prev en ting an y causal com- m unication with the external spacetime. In contrast, timelik e or null singularities ha v e b oundaries with corre- sp onding causal c haracter, allowing non-spacelik e curv es to originate from the singular region and propagate to other spacetime p oin ts. In particular, null singularities are generated b y null geo desics and may p ermit outgoing radiation that reaches distan t observ ers with arbitrarily large redshift [ 8 , 9 ]. These distinctions play a cen tral role in determining the causal visibility of singularities and in classifying the p ossible end states of gravitational collapse. The analysis of the OSD collapse motiv ated a broader conjecture prop osed by Penrose, known as the Cosmic Censorship Conjecture (CCC) [ 6 ]. The conjecture as- serts that singularities arising in gravitational collapse are generically hidden within even t horizons. This im- plies that the end state of the collapse is exp ected to b e a black hole rather than a horizonless singularit y . De- spite sustained efforts, no precise and universally accepted mathematical formulation has b een established. A gen- eral proof or dispro of remains absent. The status of CCC remains a ma jor op en problem in classical and quantum gra vit y , with significan t implications for blac k hole ph ysics and relativistic astroph ysics. In this con text, the ph ysical nature of ultra-compact ob jects remains unresolved. A singularity is said to b e nake d if future-directed non- spacelik e geo desics originate from the singular region and reac h future n ull infinity I + [ 5 , 7 , 10 ]. In such cases, the singularity is not enclosed by an even t horizon and ma y , in principle, influence the external spacetime. Ex- tensiv e inv estigations of gravitational collapse hav e sho wn that the emergence of such horizonless singularities is not excluded within classical gravit y and can arise from regular initial data under physically reasonable conditions [ 4 , 5 , 10 – 16 ]. Moreo v er, the violation of CCC has also b een discussed extensiv ely in quantum gravit y framew orks (see [ 17 – 20 ] and references therein). The o ccurrence of horizonless spacetime singularities raises a fundamental question concerning the causal visibility and physical na- ture of these (ultra) compact ob jects. In particular, the mathematical classification of singularities according to their causal structure, i.e., spacelike, timelike, or null, ma y b e associated with distinct physical and p oten tially observ able signatures. In v estigating this relation b et ween causal structure and observ able prop erties is therefore essen tial for determining whether astrophysical observ a- tions can distinguish blac k holes from horizonless compact ob jects containing differen t types of central singularities. Recen t adv ances in horizon-scale imaging through v ery long baseline interferometry hav e brought these ques- tions into the domain of observ ational inv estigation [ 21 ]. Observ ations by the Ev en t Horizon T elescop e (EHT) of sup ermassiv e compact ob jects, most notably Sgr A* and M87*, hav e rev ealed brigh t ring-like emission structures that are commonly interpreted as the shadows of black holes [ 22 – 24 ]. In particular, the EHT observ ations of Sgr A* provide strong constraints on the geometry of the surrounding spacetime and are broadly consisten t with the Kerr solution predicted by general relativit y [ 25 ]. Nev ertheless, current observ ations do not exclude all possible deviations from the Kerr black hole paradigm. Certain alternative compact ob ject geometries can repro- duce observ ational features similar to those exp ected from Kerr or Sch w arzsc hild spacetimes. These considerations motiv ate con tin ued theoretical inv estigations of compact ob ject solutions, both within and b ey ond general rela- tivit y , esp ecially in anticipation of future high-precision observ ations that may provide significant tests of the underlying spacetime geometry [ 24 ]. Recen t studies indicate that horizonless geometries, including certain classes of compact ob jects, can also pro- duce shado w structures in high resolution images [ 8 , 24 , 26 – 34 ]. W e use the term “shadow” to denote the apparen t dark region in the observer’s sky , irresp ectiv e of the pres- ence of an even t horizon. This observ ational degeneracy highligh ts the need for deep er theoretical in v estigations of alternative compact ob ject mo dels, focusing not only on image morphology but also on their causal structure, geo desic dynamics, pulsar timing dela y , and energy ex- tractions and tidal forces processes in the strong gravit y regime [ 35 – 43 ]. Among the known horizonless compact ob ject solutions, the JMN-1 [ 10 ] and the JNW spacetimes [ 44 , 45 ] provide particularly imp ortan t prob es of strong field gravit y . Both arise as exact solutions of Einstein’s equations with physically motiv ated matter sources and admit parameter ranges in which no even t horizon forms [ 26 , 27 ]. Nevertheless, their geometrical and causal struc- tures differ significantly . The JMN-1 solution represen ts the end state of gravitational collapse of anisotropic mat- ter and can p ossess either null or timelike singularities dep ending on the compactness parameter M 0 [ 46 ]. In con- trast, the JNW geometry describes a static configuration sourced by a scalar field and contains a timelike singu- larit y throughout its ph ysically allow ed parameter range. These spacetimes therefore pro vide a natural framework for in v estigating how observ able prop erties of compact ob jects dep end on the causal nature of the underlying singularit y . 3 P article dynamics in the vicinity of compact ob jects pro- vides a complementary probe of strong gravitational fields [ 47 – 50 ]. In extreme gravitational environmen ts, particle collisions can ac hiev e center of mass energies far exceeding those attainable in terrestrial accelerators [ 51 , 52 ]. Such high-energy pro cesses are b eliev ed to play a significant role in a v ariety of astrophysical phenomena, including relativistic jets, gamma-ray bursts, microquasars, activ e galactic n uclei, and high-energy cosmic ra ys [ 53 , 54 ]. The efficiency of these acceleration mechanisms, as well as their p oten tial observ ational signatures, dep ends sensitively on the underlying spacetime geometry . In particular, the presence of turning p oin ts, photon spheres, and regions of stable or unstable geo desic motion can strongly influence particle tra jectories and collision outcomes [ 51 , 55 – 57 ]. In v estigating particle dynamics in horizonless spacetimes, therefore, provides an additional av enue for assessing the ph ysical viability of alternativ e compact ob ject mo dels. In this w ork, w e presen t a systematic in v estigation of the JMN-1 and JNW spacetimes, fo cusing on three in terconnected asp ects: ( i ) the causal structure of light- cones, ( ii ) the conditions for shadow formation, and ( iii ) particle acceleration in strong gra vit y . First, we analyze the global causal structure through the construction of ligh t-cones and conformal compactification, allowing us to determine the causal nature of the singularities presen t in these geometries. Second, we examine null geo desics to iden tify the parameter regimes that give rise to shadow- lik e observ ational features. Third, we study timelike geo desics and particle collisions to ev aluate the capabilit y of these spacetimes to act as natural particle accelerators. Through this combined analysis, we aim to clarify which observ able prop erties of compact ob jects are gov erned b y the presence of horizons, which are con trolled by photon spheres, and whic h arise from deeper aspects of spacetime geometry . The analysis presented here indicates that shado w formation, the causal nature and strength of singular- ities, and particle acceleration or turning point consti- tute largely indep enden t characteristics of compact ob ject spacetimes and need not coincide within a single class of solutions. Therefore, observ ational signatures commonly asso ciated with blac k holes may also arise in horizonless geometries. This highlights the imp ortance of employing m ultiple observ ational prob es when interpreting strong gra vit y phenomena. Distinguishing among these p ossibil- ities hence requires a syn thesis of theoretical mo deling and high precision observ ations. The present study con- tributes to this ob jective by providing a unified framew ork for analyzing strong gravit y signatures in b oth blac k hole and horizonless compact ob ject geometries. This pap er is organized as follo ws. In Sec. I I , we analyze the causal structure of the JMN-1 spacetime through the construction of null cones and P enrose diagrams. In Sec. I I I , w e p erform a similar causal analysis for the JNW spacetime using n ull cones and conformal diagrams. In Sec. IV , we inv estigate particle acceleration in the vicinit y of the JMN-1 singularity . In Sec. V , we study particle acceleration along timelike geo desics in the JNW spacetime. In Sec. VI , we examine the strength of the singularities present in the JMN-1 and JNW geometries. In Sec. VI I , w e discuss the observ ational implications of these results in the con text of horizon scale imaging with the EHT. Finally , in Sec. VI II , we summarize our results and present our conclusions. Throughout this pap er, w e adopt geometrize units in which G = c = 1. I I. CA USAL STR UCTURE OF THE JMN-1 SP ACETIME In the JMN-1 spacetime, the stress–energy tensor rep- resen ts an anisotropic fluid with zero radial and non-zero tangen tial pressures. Nev ertheless, the matter distribu- tion satisfies all energy conditions, confirming its physical consistency . The corresp onding interior JMN-1 metric is giv en by the follo wing line element [ 10 ], ds 2 = − (1 − M 0 )  r R b  M 0 1 − M 0 dt 2 + dr 2 1 − M 0 + r 2 d Ω 2 , (1) where d Ω 2 = dθ 2 + sin 2 θ dϕ 2 , and R b denotes the b ound- ary radius at which the interior JMN-1 spacetime is smo othly matched to the exterior Sc h w arzsc hild geom- etry , thereby specifying the radial extent of the matter distribution. W e sho ws the schematic b eha vior of the ligh t-cone and its asso ciated causal structure within a gravitationally collapsing matter cloud leading to the formation of a Sc h w arzsc hild black hole in Fig. ( 1 ). The corresp onding P enrose diagram of Sch w arzsc hild spacetime, shown in Fig. ( 2 ) for the completeness, uses conformal compactifi- cation to map the en tire spacetime into a finite diagram while preserving the causal structure. Notably , the ev en t horizon is lo cated at r = 2 M and the sp ac elike spacetime singularit y at r = 0. The JMN-1 solution represents a spherically symmetric compact ob ject of radius R b , c haracterized by a dimension- less compactness parameter, and comp osed of gravitating matter with negligible non-gravitational interactions. Im- p osing the condition that the sp eed of sound remains subluminal leads to the constraint R b > 2 . 5 M [ 5 ]. The in terior spacetime is smoothly matched at r = R b to the exterior v acuum Sc h w arzsc hild solution for r ≥ R b . An y static and spherically symmetric in terior solution that is smo othly matc hed to a Sch warzsc hild geometry across a timelik e hypersurface at r = R b , m ust satisfy the appropri- ate junction conditions [ 47 ]. In particular, the contin uity of the metric comp onen ts and the extrinsic curv ature across the b oundary implies that the in terior configu- ration is gov erned b y the T olman-Opp enheimer-V olkoff (TO V) equation [ 10 ], whic h characterizes the condition of hydrostatic equilibrium for a self gravitating fluid in general relativity , − dP r dr = ( P r + ρ ) (4 π P r r 3 + M ) r ( r − 2 M ) + 2 r ( P r − P θ ) , (2) 4 FIG. 1: Schematic representation of the evolution of ligh t-cone structures, highligh ting the causal structure in the gravitational collapse of homogeneous dust. FIG. 2: Penrose diagram of Sch w arzsc hild blac k hole. where P r is radial pressure and P θ is tangential pres- sure, and ρ is the energy densit y . Within the JMN-1 in terior spacetime, the matter distribution is intrinsically anisotropic, indicating that the pressure comp onen ts along differen t spatial directions are not identical. In this study , w e adopt the physically relev ant assumption that the ra- dial pressure v anishes, i.e., P r = 0, whic h implies that dP r dr = 0. Incorp orating this condition into Eq. ( 2 ) leads to a simplified equilibrium relation. The resulting TOV equation dep ends exclusively on the energy density ρ and the tangen tial pressure comp onen ts P θ = P ϕ . Hence, the hydrostatic equilibrium within the JMN-1 interior is main tained by the balance b et w een gravitational forces go v erned by the densit y distribution and the supp orting anisotropic tangential stresses. P θ = P ϕ = M 0 ρ 4(1 − M 0 ) , ρ = M 0 R 2 b . (3) The causal structure of the JMN-1 spacetime can b e analyzed through in v estigating the b eha vior of radial null geo desics near the singular region. This, in principle, can b e done b y imp osing the condition ds 2 = 0, together with d Ω 2 = 0, on the interior JMN-1 line element giv en in Eq. ( 1 ) . Under these conditions, the equation gov erning radial null tra jectories in the ( t, r ) plane reduces to the follo wing form, Z dt = ± 1 1 − M 0 Z  r R b  − M 0 2(1 − M 0 ) dr . (4) In the context of the ligh t-cone structure, the p ositiv e sign represents outgoing null geodesics, while the negative sign corresp onds to ingoing n ull geo desics. Integrating b oth sides of Eq. ( 4 ) leads to the following expression, t = ± 2 R b 2 − 3 M 0  r R b  A + C , (5) where, A = 2 − 3 M 0 2(1 − M 0 ) , and C denotes the constant of inte- gration. Here, b oth ingoing and outgoing null geodesics exhibit a non-trivial functional relationship betw een t and r , making the light-cone structure difficult to visualize in the ( t, r ) co ordinate. T o resolv e this difficulty , w e adopt a construction analogous to the Eddington-Finkelstein (EF) co ordinates, whic h pro vides a b etter description of the causal structure of the spacetime. Accordingly , for an ingoing radial null geodesic corresp onding to the negative sign in Eq. ( 5 ), we define a new null co ordinate p as, p = t + r ∗ , (6) where, r ∗ = 2 R b 2 − 3 M 0  r R b  A . (7) Differen tiating with resp ect to co ordinate time, we obtain dt = dp − dr ∗ , (8) and, dr ∗ = 1 1 − M 0  R b r  M 0 2(1 − M 0 ) dr , (9) substituting these expressions in to the JMN-1 metric giv en in Eq. ( 1 ) and simplifying, we obtain the line element ds 2 = − f ( r ) dp 2 + g ( r ) dp dr + r 2 d Ω 2 , (10) where, f ( r ) = (1 − M 0 )  r R b  M 0 1 − M 0 , g ( r ) = 2  r R b  M 0 2(1 − M 0 ) , (11) 5 (a) Light-cone structure for M 0 = 0 . 33, R b = 6 . 06. (b) Penrose diagram for M 0 = 0 . 33, R b = 6 . 06. (c) Light-cone structure for M 0 = 0 . 75, R b = 2 . 67. (d) Penrose diagram for M 0 = 0 . 75, R b = 2 . 67. FIG. 3: This figure represent s the light-cone structures and corresp onding P enrose diagrams of the JMN-1 spacetime, with the singularity b eing timelik e for M 0 < 2 3 and null for M 0 > 2 3 . Here, r = 0 represents the spacetime singularity and the total mass is considered as M = 1. b y setting ds 2 = 0 and d Ω 2 = 0 in Eq. ( 10 ) obtain the radial null geodesics, whic h up on in tegration giv e p = C , (12) p = 4 R b 2 − 3 M 0  r R b  A + C . (13) W e introduce a new timelike co ordinate t ′ define as t ′ = p − r, (14) whic h gives t ′ = C − r , (15) t ′ = − r + 4 R b 2 − 3 M 0  r R b  A + C . (16) Eq. ( 15 ) sho ws that the ingoing null geo desics are regular, while the outgoing n ull geodesics remain well-behav ed in the retarded EF co ordinate. Since there is no ev en t horizon in the JMN-1 spacetime, a Krusk al-type extension 6 is not needed. The adv anced coordinate is given b y Eq. ( 6 ), and retarded coordinate is, q = t − r ∗ . (17) In this construction, the n ull co ordinates ( p, q ) are ob- tained from the original ( t, r ) co ordinates, where p and q resp ectiv ely represen t the outgoing and ingoing radial n ull geo desics, and b oth extend ov er an unbounded do- main. T o ac hiev e spacetime compactification, b ounded co ordinates are in troduced through the transformations, P = tan − 1 ( p ); Q = tan − 1 ( q ) , (18) whic h map infinite ranges of the null coordinates to finite v alues while preserving the causal structure. The com- pactified coordinates ( P , Q ) are then linearly transformed to define new coordinates ( T , S ) as, T = 1 2 ( P + Q ); S = 1 2 ( P − Q ) , (19) resulting in, T = 1 2  tan − 1 ( p ) + tan − 1 ( q )  , (20) S = 1 2  tan − 1 ( p ) − tan − 1 ( q )  . (21) In sp ecific terms, the parameter M 0 defines the com- pactness of the JMN-1 spacetime configurations and regulates the intensit y of the ov erall effects caused by anisotropic pressure. When M 0 < 2 / 3, the spacetime do es not admit a photon sphere. In this parameter regime, outgoing radial null geo desics emerging from the near by region of cen tral singularit y at r → 0 can reac h future n ull infinit y , implying that the singularity is globally naked [ 58 ]. The corresp onding causal structure for a representa- tiv e v alue M 0 < 2 / 3 is shown in Fig. ( 3a ). The Penrose diagram in Fig. ( 3b ) confirms that the singularit y is time- lik e. In the parameter range 2 / 3 ≤ M 0 < 4 / 5, a photon sphere forms and is lo cated outside the matc hing radius R b . The presence of the photon sphere substan tially alters the b eha vior of n ull geo desics. F ar from the singularity , i.e., r ≫ 0, the light-cone remains undistorted; ho w ev er, as null ra ys propagate tow ards the singularity , the in- creasing spacetime curv ature causes the ligh t-cone to tilt in w ard, indicating a change in the local causal structure of the spacetime. Certain outgoing null tra jectories b ecome trapp ed near the photon sphere and are prev en ted from reac hing future n ull infinity . Although the spacetime still con tains a central horizonless singularit y at r → 0 + , the formation of the photon sphere restricts the causal prop- agation of null rays to distant timelike observers. The resulting causal structure for this parameter range is illus- trated in Fig. ( 3c ). The corresp onding Penrose diagram is sho wn in Fig. ( 3d ), where the singularity appears as a nul l (ligh tlik e) b oundary . How ev er, no well defined Penrose diagram exists at M 0 = 2 3 , as the system lies at a critical threshold marking the transition b et w een distinct causal structures, separating regimes with qualitatively differen t n ull geo desic behavior. I II. CAUSAL STRUCTURE OF THE JNW SP ACETIME In this section, w e examine the causal structure of the JNW spacetime. The JNW geometry is a static and spherically symmetric solution of Einstein’s field equations sourced by a minimally coupled massless scalar field with v anishing scalar potential, V (Φ) = 0. This spacetime represen ts an extension of the Sch w arzsc hild solution in the presence of a scalar field [ 44 ]. The corresp onding line elemen t describing the JNW spacetime is given b y [ 45 ], ds 2 = −  1 − b r  n dt 2 + dr 2  1 − b r  n + r 2  1 − b r  1 − n d Ω 2 , (22) where, b = 2 p M 2 + q 2 , n = 2 M b , (23) the corresp onding scalar field Φ can b e written as Φ = q b √ 4 π ln  1 − b r  . (24) In the limiting case q → 0, which corresponds to n = 1, the scalar field v anishes, i.e., Φ → 0. In this limit, the JNW spacetime reduces smo othly to the Sch w arzsc hild geometry . When the scalar charge q is nonzero, the space- time departs from the Sch w arzsc hild solution. The pa- rameter n = 2 M /b decreases monotonically as the scalar c harge increases. As a result, the scalar field mo difies the spacetime geometry and the cen tral curv ature singularit y at r = b , therefore, b ecomes horizonless. Larger v alues of the scalar charge lead to stronger deviations from the Sc h w arzsc hild spacetime [ 47 ]. The energy-momentum tensor of the scalar field satis- fies the all energy condition ev erywhere in the spacetime [ 47 , 49 ]. Ho w ev er, for 0 < n < 1, b oth the scalar field and the curv ature in v ariants div erge at r = b , indicating the presence of a genuine curv ature singularit y . Since no ev en t horizon forms to enclose this surface, the singularity remains globally visible. The limiting case n = 1 corre- sp onds to the Sc h w arzsc hild blac k hole, whereas the range 0 < n < 1 describ es a spacetime containing a spherical cen tral singularity . No w, by setting ds 2 = 0 and d Ω 2 = 0 in Eq. ( 22 ), the radial n ull geo desic satisfies, dt = ±  1 − b r  − n dr , (25) where the p ositiv e and negativ e signs correspond to the outgoing and ingoing radial null geo desics, resp ectiv ely . W e can obtain time co ordinate from Eq. ( 25 ), t = ± Z dr  1 − b r  n . (26) In this con text, b oth ingoing and outgoing null geo desics exhibit a complex relationship with r, making it difficult to 7 visualize ligh t-cone b eha vior. W e emplo y n ull co ordinates that remain constant along radial null geo desics, similar to the EF coordinates. F or the radial ingoing n ull geo desic represen ted by a negativ e sign in Eq. ( 26 ) , we in troduce a new co ordinate denoted as p . p = t − r ∗ , (27) where, r ∗ = Z dr  1 − b r  n . (28) T aking the differential dt = dp − dr ∗ , (29) where dr ∗ =  1 − b r  − n dr . (30) By substituting abov e Eq. ( 30 ) in to the JNW metric giv en in Eq. ( 22 ) and simplifying, the line element becomes ds 2 = −  1 − b r  n dp 2 + 2 dpdr + r 2 d Ω 2 . (31) T o understand the causal structure of this metric, we can examine the radial null geodesics within this new co ordinate system. By substituting ds 2 = 0 and d Ω 2 = 0 in to Eq. ( 31 ) , w e obtain tw o solutions, and integrating these solutions leads to p = C , (32) p = 2 Z  1 − b r  − n dr + C . (33) No w using the null co ordinate p as in Eq. ( 32 ) and ( 33 ) , w e define a new timelik e coordinate t ′ through, t ′ = p − r , from which one obtains, t ′ = C − r , (34) t ′ = − r + 2 Z  1 − b r  − n dr + C . (35) Here, Eq. ( 34 ) sho ws that all ingoing radial photons are con tin uous throughout the spacetime. In the retarded EF co ordinates, outgoing null geodesics remain contin uous. As the JNW spacetime do es not con tain an even t horizon in this case, a Krusk al extension is not required. The corresp onding adv anced EF co ordinates are given as in Eq. ( 27 ), and retarded EF co ordinates are giv en b y q = t − r ∗ . (36) Since the null co ordinates ( p, q ) are unbounded, a com- pactification is p erformed b y introducing the bounded co ordinates, P = tan − 1 ( p ); Q = tan − 1 ( q ) , (37) whic h map the spacetime to a finite domain. The resulting compactified co ordinates ( P , Q ) are then sub jected to a rotation to define a new set of co ordinates ( T , S ), given as T = 1 2 ( P + Q ); S = 1 2 ( P − Q ) , (38) resulting in, T = 1 2  tan − 1 ( p ) + tan − 1 ( q )  . (39) S = 1 2  tan − 1 ( p ) − tan − 1 ( q )  . (40) The causal structure of the JNW spacetime can b e un- dersto od b y analyzing the b eha vior of radial null geo desics describ ed b y Eq. ( 34 ) and ( 35 ) . As a result, the singular surface lo cated at r = b remains horizonless. Near the singular surface r = b , the ligh t-cones b ecome strongly dis- torted but do not close to form a trapp ed region. There- fore, radial n ull geo desics starting arbitrarily close to the singularity are able to propagate outw ard and reac h n ull infinity without any obstruction. As illustrated in Fig. ( 4a ), for n = 0 . 4 as one of the example of 0 < n < 0 . 5 range, null geodesics emerging arbitrarily close to the sin- gular surface and extend smo othly tow ard null infinity . The absence of any trapp ed region or horizon in this case confirms that no even t horizon is formed, and therefore the singularity is globally naked. In the regime 0 . 5 < n < 1, the dynamical influence of the scalar field becomes comparatively w eak er rela- tiv e to the cases with smaller v alues of n . Despite this reduction in strength, the scalar field con tin ues to play a significant role in determining the lo cal causal struc- ture of the spacetime. In particular, near the central singularit y its con tribution to the spacetime curv ature remains sufficiently strong to substantially distort the ligh t-cone structure. Thus, the tra jectories of outgoing n ull geo desics experience a pronounced deflection in this region, reflecting the strong bending of light-cones in- duced b y the scalar field. This b eha vior indicates that, ev en in this parameter range, the scalar field con tin ues to influence the propagation of null rays and the accessibilit y of the singularit y within the spacetime. This b eha vior mo difies the propagation of radial null geodesics, causing them to deviate from their otherwise regular tra jectories. As shown in Fig. ( 4c ) for n = 0 . 8 as one of the example of 0 . 5 < n < 1 range, the null geo desics show b ending as they pass close to the singular surface at r = b while still propagating outw ard tow ard null infinit y . The cor- resp onding Penrose diagram of the JNW spacetime for differen t v alues of n is shown in Fig. ( 4b ) and ( 4d ). The conformal compactification shows that the curv a- ture singularit y at r = b app ears timelike in nature in the P enrose diagram for the parameter range n ∈ (0 , 1). This indicates that the singularity remains causally connected to the external universe, allo wing ingoing causal curves 8 (a) Light-cone structure for n = 0 . 4, b = 5. (b) Penrose diagram for n = 0 . 4, b = 5. (c) Light-cone structure for n = 0 . 8, b = 2 . 5. (d) Penrose diagram for n = 0 . 8, b = 2 . 5. FIG. 4: This figure sho ws the light-cone structures and corresp onding Penrose diagrams of the JNW spacetime, with the singularity being timelik e for 0 < n < 1, where r = b represen ts the spherical singularity . The total mass is considered as M = 1. to terminate at the singular surface while outgoing null geo desics can emerge from regions arbitrarily close to it. In JMN-1 collapse mo del, photon trapping regions can c hange the causal nature of the singularit y . How ev er, no suc h transition o ccurs in the JNW spacetime. Although the causal structure shows some quan titativ e differences for different v alues of n , such as v ariations in the tilting of ligh t-cones and the b eha vior of geodesics near the singular- it y , the qualitative nature of the singularity itself remains the same. In particular, the absence of an ev en t horizon throughout the parameter range n ∈ (0 , 1) ensures that the singularity is globally nak ed. This p ersisten t timelike nature indicates that the non-zero scalar field alters the spacetime curv ature but do es not lead to the formation of a horizon, and therefore the spherical singular surface remains causally connected to the exterior spacetime. IV. TURNING POINTS OF TIMELIKE GEODESICS NEAR THE JMN-1 SINGULARITY The line element describing the JMN-1 spacetime is giv en in Eq.( 1 ). The metric functions are p ositiv e and regular for all r > 0, provided the parameter lies in the range 0 < M 0 < 1. Radial null tra jectories are determined b y the condition ds 2 = 0. Restricting to purely radial 9 motion, this condition reduces to − f ( r ) dt 2 + g dr 2 = 0 , (41) where g = 1 / (1 − M 0 ). Solving this relation yields the co ordinate speed of outgoing radial null ra ys, dr dt = (1 − M 0 )  r R b  α , α ≡ M 0 2(1 − M 0 ) . (42) This explicit relation plays an important role in deter- mining the causal structure of the spacetime. The prefac- tor (1 − M 0 ) is strictly p ositiv e for all admissible v alues 0 < M 0 < 1, while the pow er-la w factor ( r /R b ) α re- mains p ositiv e for every r > 0. Thus, the co ordinate v elo cit y dr /dt is p ositiv e throughout the interior region. This immediately implies that the JMN-1 spacetime do es not contain an even t horizon within the in terior, since there exists no hypersurface at which outgoing radial n ull geo desics satisfy dr /dt = 0. Ho w ev er, the radial dep endence of the factor ( r /R b ) α significan tly affects the orien tation of the lo cal ligh t-cone structure. As r → 0, the p o w er-la w term decreases rapidly , causing the co ordi- nate sp eed of outgoing null ra ys to b ecome progressively smaller. Geometrically , this corresp onds to a strong in- w ard tilting of the light-cones near the central singularit y , indicating that the out w ard propagation of null geo desics b ecomes increasingly suppressed in this region. T o understand this b eha vior in more detail, consider the factor ( r /R b ) α as the radius decreases. F or any fixed compactness parameter M 0 ∈ (0 , 1), the exp onen t α is p ositiv e. Therefore, ( r /R b ) α decreases monotonically as r decreases. As a result, the co ordinate sp eed of out- going light, dr /dt , also decreases tow ard smaller radii. When r b ecomes muc h smaller than R b , the p o w er-la w factor ( r /R b ) α b ecomes v ery small. Consequently , dr /dt b ecomes corresp ondingly small. Physically , this means that outgoing null directions near the cen tral region are almost vertical in a t − r diagram. F uture directed ligh t ra ys then hav e only a very small outw ard radial velocity . The light-cones therefore tilt increasingly inw ard as r de- creases tow ard the singularit y . This b eha vior represen ts a contin uous mo dification of the causal structure rather than the formation of a horizon. Outgoing ra ys still satisfy dr /dt > 0 for all r > 0, but the rate at whic h their radius increases b ecomes arbitrarily small sufficien tly close to the singularity . The strength of this in w ard tilting dep ends on the com- pactness parameter M 0 through the exp onen t α . When M 0 is small, α is also small and the factor ( r /R b ) α remains relativ ely large ov er most of the in terior. Outgoing rays therefore retain a significant outw ard speed until they ap- proac h very close to r = 0. As M 0 increases, the exp onen t α b ecomes larger and the factor ( r /R b ) α decreases more rapidly with decreasing r . The v alue M 0 = 2 / 3 plays a sp ecial role. F or M 0 > 2 / 3, the exponent α exceeds unit y . In this regime, the suppression of dr /dt with decreas- ing r b ecomes v ery strong even at mo derate radii. The ligh t-cones then b egin to tilt inw ard at muc h larger radii compared to smaller v alues of the compactness parameter. This extended inw ard tilting leads to an effective trapping of photons and is closely related to the shadow features asso ciated with the spacetime. T o connect the in terior causal behavior with observ able shado ws, the exterior photon sphere m ust also b e consid- ered. In the exterior Sch w arzsc hild region, the photon sphere is located at r = 3 M . Photons that reac h this radius with insufficient outw ard radial momentum cannot escap e to infinity . Instead, they are either captured or remain near the unstable circular orbit for a long time b efore ev en tually escaping or falling inw ard. When the JMN-1 interior is sufficiently compact, outgoing n ull rays emerging from the interior ma y not p ossess enough out- w ard radial momen tum to cross the photon sphere. Such ra ys therefore do not contribute to the radiation observed at large distances and are effectively remov ed from the observ able flux. F or M 0 > 2 / 3, the suppression of dr /dt inside the JMN-1 region b ecomes v ery strong. Null rays that reac h even mo derate interior radii emerge with extremely small outw ard v elocity . These rays cannot o v ercome the Sc h w arzsc hild photon sphere and are either captured or redirected in w ard. As a result, they do not reac h dis- tan t observers. In this wa y , the shadow arises from the com bined effect of the in terior causal structure and the exterior photon sphere. The in terior geometry strongly reduces the out w ard propagation sp eed of photons, while the photon sphere preven ts the w eakly outgoing rays from escaping to infinity . The motion of massiv e particles and the p ossibilit y of having turning p oin ts are determined by the corre- sp onding effective potential. F or equatorial motion, the conserv ed energy p er unit mass e and angular momen tum p er unit mass L determine the radial equation in the form  dr dτ  2 + V eff ( r ) = E , (43) where, E = e 2 − 1 2 . The effective p oten tial in the JMN-1 in terior is given b y V eff ( r ) = 1 2   (1 − M 0 )  r R b  M 0 1 − M 0  1 + L 2 r 2  − 1   . (44) T urning p oin ts o ccur when ( dr /dτ ) 2 = 0. Consider parti- cles that fall in from rest at infinity . F or such particles, the energy satisfies E = 0. The turning-p oin t condition then gives a relation b et w een the radius r and the angular momen tum L , L ( r ) = ± r   R M 0 1 − M 0 b (1 − M 0 ) r M 0 1 − M 0 − 1   1 / 2 . (45) This expression is real only if the quantit y inside the square brack ets is p ositiv e. Therefore, turning p oin ts 10 exist only at radii where R M 0 1 − M 0 b (1 − M 0 ) r M 0 1 − M 0 > 1 . (46) Since R b and M 0 are fixed parameters of the model, this condition imp oses a non trivial constraint on the allow ed v alues of r and on the compactness parameter M 0 . FIG. 5: Angular momentum L ( r ) required for radial turning p oin ts as a function of the radial co ordinate r for differen t v alues of the parameter M 0 . The color bar indicates the corresp onding v alues of M 0 . A detailed analysis of Eq. ( 45 ) sho ws that the function L ( r ) can possess a minimum. This minimum corresp onds to the critical angular momentum required for a particle to turn its radial motion at some radius inside the matter region. The angular momentum L ( r ) required for radial turning p oin ts in the JMN-1 spacetime for 1 2 < M 0 < 4 5 is shown in Fig. ( 5 ). Such a minim um exists within the in terv al 0 < r ≤ R b only when the compactness parameter lies in the range 1 / 2 ≤ M 0 ≤ 2 / 3. If M 0 < 1 / 2, the radius at which the minim um o ccurs lies outside the b oundary radius R b . In this case, any turning p oin t would o ccur outside the interior region. Therefore, suc h tra jectories cannot pro duce turning p oin ts within the JMN-1 matter region and cannot lead to collisions inside the in terior. If M 0 > 2 / 3, the quan tit y inside the square root in Eq. ( 45 ) nev er becomes p ositiv e for r ≤ R b . As a result, L ( r ) is not real in this region and no turning p oin t exists inside the in terior. Particles with the required angular momentum to turn bac k their motion therefore do not exist. All timelik e geo desics that enter the in terior region plunge monotonically tow ard the central singularit y . This result is consistent with the analysis presented in [ 51 ]. Com bining the null and timelik e analyses leads to the follo wing physical picture. F or compactness in the range 1 / 2 ≤ M 0 ≤ 2 / 3, the interior region supp orts turning p oin t geo desics. A particle arriving from infinity with a suitable angular momentum can decelerate and rev erse its radial motion inside the matter region. This produces an outgoing timelike tra jectory that can intersect with an ingoing particle. Such encoun ters allow high center of mass energy collisions to occur in the strong field region. As M 0 increases within this in terv al, the turning p oin t radius shifts tow ard smaller v alues of r . A t the same time, the spacetime curv ature b ecomes stronger near the in teraction region. Both effects tend to enhance the maximum ac hiev able collision energy . F or M 0 > 2 / 3, the situation c hanges qualitativ ely . The interior causal structure b ecomes strongly tilted inw ard. Outgoing null directions are then strongly suppressed. In this regime, no timelike turning p oin ts exist inside the matter region. P articles that en ter the in terior therefore con tin ue to mo v e inw ard and plunge tow ard the singularit y . Hence, the outgoing tra jectories required for head-on, high-energy collisions cannot form as shown in [ 51 , 55 – 57 ]. In particular, a visible shadow can form for M 0 > 2 / 3. In this regime, the interior geometry strongly suppresses the out w ard propagation of photons. At the same time, the exterior photon sphere captures ra ys with weak out- w ard momentum. Therefore, photons that originate in the interior fail to reac h distant observ ers and con tribute to the shado w appearance. Ho w ev er, the same causal suppression also affects massive particles. The strong in w ard tilting of the light-cones preven ts the formation of outgoing timelik e tra jectories. Therefore, turning p oin t geo desics do not exist inside the matter region when M 0 > 2 / 3. Without such tra jectories, particles cannot rev erse their radial motion within the interior. Thus, the ingoing and outgoing particle paths required for high- energy collisions cannot o ccur. This implies that high cen ter of mass energy collisions are suppressed within the shadow region and hence no sho c k wa v es or pho- tosphere like structure will app ear, that is, within the region b ounded by the photon sphere ( r ≤ 3 M ). This result is consistent with the analysis presen ted in [ 51 , 59 ]. The turning p oin t radii for representativ e v alues of the compactness parameter M 0 are listed in T able I . These v alues corresp ond to particles with the minimum angu- lar momen tum required to rev erse their radial motion inside the JMN-1 interior. The radii are obtained from Eq. (35) by substituting L = L min ( r ) together with the matc hing condition M 0 R b = 2 M . As the compactness pa- rameter increases within the allow ed in terv al, the turning p oin t shifts to smaller radii, indicating that the reversal of particle tra jectories o ccurs deep er in the strong field region. In agreement with the analytical constraints dis- cussed ab o v e, suc h turning p oin ts exist only in the range 1 / 2 ≤ M 0 ≤ 2 / 3. V. TURNING POINTS OF TIMELIKE GEODESICS NEAR THE JNW SINGULARITY The line elemen t of the JNW spacetime is given in Eq. ( 22 ). The dynamics of timelike geo desics are gov erned b y the corresp onding effective potential. This p oten tial 11 T ABLE I: T urning p oin t radii ( r turn ) within the R b in the JMN-1 spacetime for particles with L min . M 0 r turn 0.50 4 . 00 M 0.55 3 . 23 M 0.60 2 . 43 M 0.65 1 . 31 M also determines the conditions under whic h a particle can accelerate and develop a turning p oin t. Because the spacetime is stationary and spherically symmetric, the motion of a test particle admits tw o conserv ed quan tities, the energy p er unit mass e and the angular momentum p er unit mass L . By restricting the motion to the equa- torial plane ( θ = π / 2), the effective p oten tial in the JNW spacetime can b e written as V eff ( r ) = 1 2 "  1 − b r  n L 2 r 2  1 − b r  1 − n + 1 ! − 1 # . (47) The turning p oin t condition then provides a relation b e- t w een the radius r and the angular momen tum L , L ( r ) = ± r "  1 − b r  1 − 2 n −  1 − b r  1 − n # 1 / 2 . (48) FIG. 6: Angular momentum L ( r ) required for radial turning p oin ts as a function of the radial co ordinate r for differen t v alues of the parameter n . The color bar indicates the corresp onding v alues of n . The angular momentum L ( r ) required for radial turning p oin ts in the JNW spacetime for 0 < n < 1 is shown in Fig. 6 . Real turning p oin ts app ear only when n ≥ 0 . 5. F or smaller v alues of n , the expression for L ( r ) b ecomes imaginary (nonphysical) in the region r > b . The case n = 0 . 5 represen ts a critical configuration in whic h the angular momentum remains finite at the singularity . In this situation, L ( r ) attains a minim um at a finite radius. Larger v alues of n shift this L ( r ) minim um to w ard smaller radii and increase its magnitude. Such b eha vior allows particles arriving from infinity to reverse their radial motion closer to the singularity and participate in high- energy collisions. This metho d and result are consistent with the analyses presen ted in [ 55 , 56 ]. In the case of n = 0 . 5, the angular momen tum remains a finite v alue at the singularity r = b . The function L ( r ) reac hes its minim um v alue L min = 3 . 674 at r = 4 . 5 M . Symmetry of the equations under the transformation L → − L implies identical extrema for p ositiv e and neg- ativ e v alues of L . Particles arriving from infinit y with angular momentum in the in terv al − L min < L < L min can therefore penetrate deep in to the strong gravitational region near r ∼ b b efore turning bac k their radial motion. P articles with larger angular momentum are instead scat- tered at larger radii. On the other side, v alues n > 0 . 5 cause the lo cation of the minim um of L ( r ) to mov e pro- gressiv ely closer to the singularity as n increases. The corresp onding magnitude of L min also grows. This b e- ha vior reflects the strengthening of the gravitational field near the singularit y . Particles arriving from infinity with suitable angular momen tum can decelerate, reach a finite turning p oin t radius, and collide head-on with other ra- dially infalling particles. Suc h interactions can generate extremely large cen ter of mass energies in the strong- field region, analogous to the Ba ˜ nados–Silk–W est (BSW) mec hanism [ 55 – 57 ]. These high-energy collisions may giv e rise to sho c k w a v es or an effective photosphere, leading to enhanced electromagnetic emission [ 59 ]. As a result, this may , in principle, lead to observ able signatures in EHT shado w imaging, such as excess luminosit y or bright substructures near the photon sphere, while the o v erall shado w remains largely determined by the underlying photon orbit geometry . T urning point radii for represen tativ e v alues of the pa- rameter n are listed in T able I I . These radii corresp ond to particles p ossessing the minim um angular momentum required to reverse their radial motion in the JNW space- time. The v alues are obtained from Eq. ( 48 ) b y sub- stituting L = L min ( r ). Increasing n shifts the turning p oin t radius gradually in w ard tow ard 4 M , indicating that particle tra jectory turning o ccurs closer to the central region. This result is in agreement with the analytical condition derived ab o ve, according to which suc h turning p oin ts exist only for n ≥ 0 . 5. VI. TIPLER’S STRENGTH OF JMN-1 AND JNW SINGULARITIES A spacetime singularit y is c haracterized by the exis- tence of at least one incomplete causal geo desic. Ho w ev er, in the context of collapsing matter, a stronger physical 12 T ABLE I I: T urning p oin t radii ( r turn ) in the JNW spacetime for particles with L min . n r turn 0.5 4 . 50 M 0.6 4 . 43 M 0.7 4 . 30 M 0.8 4 . 18 M 0.9 4 . 08 M 1.0 4 . 00 M requiremen t is often imp osed, namely that any extended ob ject falling into the singularit y is crushed to zer o vol- ume. A singularity satisfying this condition is said to b e gra vitationally strong, in the sense introduced by Tipler [ 60 ]. Let ( N , g ) b e a smo oth spacetime manifold and let Γ( λ ) b e a causal geo desic defined on an interv al [ λ 0 , 0), where λ is an affine parameter, and the endpoint λ = 0 corresp onds to the singularity . Let χ i denote a set of lin- early indep enden t Jacobi vector fields along Γ, orthogonal to the tangen t v ector of the geodesic. The wedge pro duct of these Jacobi fields defines a volume elemen t V = χ 1 ∧ χ 2 ∧ χ 3 . (49) The singularity is said to b e strong in the sense of Tipler if this volume element v anishes in the limit λ → 0. Clarke and Kr´ olak provided a sufficient condition for the o ccur- rence of a Tipler strong curv ature singularity in terms of the growth of the curv ature along causal geo desics [ 61 ]. In particular, at least along one null geo desic with affine parameter λ (suc h that λ → 0 as the singularity is approac hed), the following condition must hold, lim λ → 0 λ 2 R µν K µ K ν > 0 . (50) Here, R µν is the Ricci curv ature tensor, K µ = dx µ /dλ are the tangent vectors to the n ull geo desics, and x µ is the spacetime co ordinate. This condition puts a lo w er b ound on the gro wth of the Ricci curv ature tensor along the n ull geo desic. Within our analysis, we will consider the in terior spacetime of the singularit y to be describ ed by the JMN-1 spacetime geometry with the line element giv en b y Eq. ( 1 ). Using the null geo desic condition, g µν K µ K ν = 0, w e obtain the follo wing relationship b et w een K 0 and K 1 comp onen ts of the null tangen t vector, K 0 = ± 1 (1 − M 0 )  R b r  M 0 2(1 − M 0 ) K 1 . (51) In order to examine the strength of the singularity , we require the Ricci tensor comp onen ts R 00 and R 11 for the JMN-1 metric, which are given b y , R 00 = − M 0 ( M 0 − 2) 4 r 2  r R b  M 0 1 − M 0 , (52) R 11 = (2 − 3 M 0 ) M 0 4 r 2 ( M 0 − 1) 2 . (53) The Tipler strong singularity criterion requires the left hand side of the inequality to approac h a p ositiv e finite v alue as giv en in Eq.( 50 ), substituting the relev ant com- p onen ts, we obtain lim λ → 0 λ 2  M 0 r 2 (1 − M 0 )   dr dλ  2 . (54) T o determine whether this quan tit y tak es a p ositiv e finite v alue near the singular limit λ → 0, w e examine the b eha vior of lim λ → 0  dr dλ  . (55) This is obtained by writing the Lagrangian for the radial n ull geo desics as L null JMN-1 ≡ 1 2 " − (1 − M 0 )  r R b  M 0 1 − M 0  dt dλ  2 + 1 (1 − M 0 )  dr dλ  2 # = 0 . (56) The equation of motion for r is obtained from the Eu- ler–Lagrange equation corresp onding to L null JMN-1 . This giv es, d 2 r dλ 2 + M 0 2 r (1 − M 0 )  dr dλ  2 = 0 . (57) W e consider an ansatz solution to this nonlinear differen- tial equation, with a radial co ordinate r ( λ ) ∼ ˜ α ( λ − β ) γ v alid for all v alues of the parameter M 0 . Here ˜ α , β and γ are p ositiv e constan ts. The first deriv ative then tak es the form, dr dλ ∼ ( ˜ αγ )( λ − β ) γ − 1 . (58) F or simplicity , we set the constant β = 0. Substituting these expressions into the Tipler criterion, we obtain, lim λ → 0 λ 2 R µν K µ K ν = M 0 (1 − M 0 ) > 0 . (59) Since 0 < M 0 < 1, the ab o v e limit is finite and p ositiv e; therefore, the cen tral singularity satisfies the Tipler strong curv ature condition. This result indicates that the JMN-1 spacetime exhibits a strong singularit y for all v alues of the compactness parameter in the allow ed range M 0 ∈ (0 , 1). W e no w consider the JNW spacetime, whose line el- emen t is given by Eq. ( 22 ). The relation b et w een the comp onen ts K 0 and K 1 of the null tangent vector for the JNW metric is giv en by , K 0 = ±  1 − b r  − n K 1 . (60) 13 The Ricci tensor components R 00 and R 11 of JNW metric are given b y , R 00 = 0; R 11 = b 2 (1 − n 2 ) 2 r 2 ( b − r ) 2 . (61) T o determine whether Tipler’s condition given in Eq. ( 50 ) is satisfied near the singularity , we examine the b eha vior of dr dλ along radial null geodesics. F ollo wing the same pro- cedure as in the JMN-1 case, w e construct the Lagrangian describing radial null geo desics in the JNW spacetime, L null JNW ≡ 1 2 " −  1 − b r  n  dt dλ  2 +  1 − b r  − n  dr dλ  2 # = 0 , (62) and the resulting equation of motion for radial n ull geo desics becomes d dλ  dr dλ  = 0 ⇒ r ( λ ) = r 0 + C 1 λ, (63) where r 0 and C 1 are constan ts. Using the ab o ve expression for r ( λ ), we can ev aluate the limit defined in Eq. ( 50 ), whic h gives, lim λ → 0 λ 2 R µν K µ K ν = (1 − n 2 ) > 0 . (64) This represents that the singularity arising in the JNW spacetime satisfies the Tipler strong curv ature condition. In particular, the singularit y remains strong within the allo w ed range 0 < n < 1, implying that an y infalling extended ob ject is crushed to zero volume. Moreov er, the JMN-1 spacetime singularity exhibits similar b eha vior for all v alues of the compactness parameter in the range M 0 ∈ (0 , 1). This demonstrates that the singularities in b oth the JNW and JMN-1 spacetimes are gravitationally strong in the sense of the Tipler criterion. In the next section, we inv estigate the observ ational implications of spacelik e, null, and timelike singularities in the context of EHT-VLBI imaging. VI I. CONNECTION TO EHT IMAGING The remark able progress in horizon-scale imaging, par- ticularly through the observ ations of Sgr A* and M87* b y the EHT, has provided unpreceden ted prob es of the spacetime geometry in the strong gra vit y regime [ 22 , 23 ]. While the current data are broadly consistent with the Kerr black hole solution predicted b y general relativity [ 25 ], they do not yet uniquely rule out all p ossible alterna- tiv es. In particular, certain classes of compact ob jects can pro duce observ ational signatures that closely resem ble those exp ected from Kerr or Sch w arzsc hild spacetimes. These considerations highlight the importance of contin- ued theoretical exploration of compact ob ject solutions b oth within and beyond general relativity , especially in view of future high-precision observ ations that may pro- vide more stringent tests of the underlying spacetime geometry [ 21 , 24 ]. It is shown that b oth JMN-1 and JNW spacetimes can pro duce shado w structures under sp ecific parameter ranges [ 24 , 26 , 27 , 30 , 58 ]. In particular, w e hav e sho wn here that the JMN-1 geometry exhibits suc h features for compactness parameter v alues 2 / 3 < M 0 < 4 / 5, corre- sp onding to the regime in which the singularity is nul l t yp e. Similarly , the JNW spacetime admits shadow for- mation for 0 . 5 < n < 1, where the singularity remains timelike in nature but a photon sphere exists. The JMN-1 and JNW can cast shadow images similar to those of a Sc h w arzsc hild black hole which ha v e sp ac elike singularit y at the cen ter. This suggests that shadow formation is not uniquely determined by the presence of an even t hori- zon or the causal nature of the singularity , but is closely related to the global b eha vior of n ull geo desics and the existence of an upp er b ound of an effectiv e p oten tial [ 31 ]. W e also noticed that recen t work by Bro deric k et al. [ 59 ] argues that many naked singularit y mo dels can b e ex- cluded observ ationally because they possess inner turning p oin ts for timelik e and n ull geo desics. Such turning p oin ts ma y , in principle, lead to the formation of accretion p ow- ered photosphere or sho c ks within the shadow region, whic h would con tradict EHT observ ations of Sgr A* and M87*, where the accretion flo w app ears coherent up to the photon sphere scale. They conclude that only a re- stricted class of naked singularities (classified as type P0j 1 ), including JMN-1 and JNW spacetimes, remain viable b ecause they allegedly lack these turning p oin ts. Ho w ev er, our analysis sho ws that this conclusion must be refined. While JMN-1 solutions indeed lack such turning p oin ts within the shadow pro ducing regime, the JNW spacetime do es admit turning p oin ts when 0 . 5 ≤ n < 1, precisely in the parameter range where shadows exist. The interpretation in [ 59 ], based on Ref. [ 5 , 56 ], app ears to exclude this p ossibilit y; ho w ev er, a closer examination of Ref. [ 56 ] instead supp orts our result. Therefore, JNW spacetimes in the shadow pro ducing regime ma y still al- lo w inner photosphere structures, and their observ ational viabilit y requires further dedicated mo deling rather than immediate exclusion. VI II. DISCUSSION AND CONCLUSIONS In this work, we carried out a detailed causal and dy- namical inv estigation of the JMN-1 and JNW spacetimes b y analyzing their light-cone structure, particle turning p oin ts, and the corresp onding P enrose diagrams. Our analysis shows that the causal nature of the singularities dep ends sensitively on the underlying parameters of the solutions. In the JMN-1 spacetime, the singularity is n ull in the range 2 / 3 < M 0 < 4 / 5, whereas it b ecomes 1 The P0j-type singularities are defined by a finite angular momen- tum, where timelike geodesics can reac h the singularity . This corresponds to the parameter range in whic h an unstable photon orbit exists [ 59 ]. 14 timelik e for 0 < M 0 < 2 / 3. In contrast, the singularity of the JNW spacetime remains timelike throughout the in terv al 0 < n < 1. Despite these distinct causal characters, b oth geome- tries are capable of pro ducing shadow images that closely resem ble those associated with Sch w arzsc hild blac k holes within appropriate parameter regimes [ 24 , 27 , 28 ]. This result has an imp ortan t conceptual implication. In the Sc h w arzsc hild solution, the cen tral singularit y is spacelike and hidden b ehind an even t horizon, while in the JMN-1 and JNW spacetimes the singularities are horizonless and ma y b e n ull or timelike. The existence of a shadow there- fore do es not dep end on the causal t ype of the singularity , nor on the presence of an ev en t horizon. This conclu- sion is consistent with recent studies demonstrating that shado w formation is not necessarily tied to the existence of a photon sphere, an even t horizon, or even a spacetime singularit y [ 26 , 31 ]. The strength of the singularities w as further examined using Tipler’s criterion. Earlier qualitativ e arguments in the literature hav e suggested that singularities surrounded b y photon spheres may be w eak [ 5 , 62 , 63 ]. The presen t analysis demonstrates that this exp ectation do es not hold for the spacetimes considered here. Both the JNW and JMN-1 geometries p ossess strong curv ature singularities in the sense of Tipler criterion. In the JNW spacetime, the singularity remains strong throughout the entire pa- rameter range 0 < n < 1. A similar result holds for the JMN-1 solution, where the singularit y is strong for all M 0 ∈ (0 , 0 . 8). These findings show that strong curv ature b eha vior is fully compatible with the formation of shadow images and other observ able strong field signatures. As a result, such geometries remain physically relev an t as theoretical mo dels of ultra-compact gravitating ob jects. The existence of radial turning p oin ts for timelike geo desics plays a crucial role in determining the p ossibilit y of high-energy particle interactions in these spacetimes. W e find that, in the JNW spacetime, such turning p oin ts for radially infalling particles exist in the parameter range 0 . 5 ≤ n < 1, while in the JMN-1 spacetime they o ccur for 1 2 ≤ M 0 ≤ 2 3 . In these regimes, particles originating from infinit y with suitable angular momen tum can decelerate and reach finite radial turning p oin ts within the strong- field region, where they may collide with ingoing particles. These interactions can lead to extremely large center of mass energies, analogous to the BSW mechanism [ 55 – 57 ]. Note that such tra jectories require a suitable range of angular momentum, and therefore represen t a kinemati- cally allow ed but not necessarily generic class of particle motion. Therefore, such high-energy collisions may in turn generate shock structures or an effectiv e photosphere, p oten tially enhancing electromagnetic emission [ 59 ]. As a result, observ able signatures may arise in EHT shado w imaging, such as excess luminosity or lo calized bright features near the photon sphere, while the o v erall shadow structure largely determined b y the photon sphere. In contrast, for M 0 > 2 3 in the JMN-1 spacetime, the causal structure undergo es a qualitative change. The strong inw ard tilting of light-cones suppresse s the out- w ard propagation of b oth photons and massive particles. As a result, photons emitted from the interior fail to reac h distant observ ers, contributing to a shado w app ear- ance, while the absence of outgoing timelike tra jectories eliminates radial turning p oin ts within the matter region. Without such turning p oin ts, particles cannot rev erse their motion, and the co existence of ingoing and outgoing tra jectories required for high-energy collisions is no longer p ossible. Therefore, particle acceleration to large center of mass energies is suppressed within the shadow region, and no sho c k formation or photosphere lik e emission is ex- p ected for r ≤ 3 M , as determined by the exterior photon sphere. The presen t w ork is formulated within the framew ork of the test particle appro ximation, neglecting back-reaction and self-force effects, whic h may b ecome imp ortan t at extremely high energies. In realistic astroph ysical settings, v arious dissipativ e pro cesses such as Coulom b interactions, bremsstrahlung emission, plasma effects, and gravitational redshift can significantly alter the electromagnetic signa- tures asso ciated with high-energy particle interactions. These mechanisms ma y mo dify or suppress features such as excess luminosity or lo calized brigh t structures near the photon sphere before the radiation reaches distant observ ers. A detailed quantitativ e treatment of these effects is b ey ond the scope of the presen t work and is left for future inv estigation. Ov erall, our analysis demonstrates that shado w for- mation is primarily dictated by the underlying geodesic structure, rather than by the presence of an ev en t horizon or the causal nature of the singularit y . Both JMN-1 and JNW spacetimes, c haracterized by Tipler-strong singu- larities, provide physically viable strong gravit y configu- rations. Horizonless ultra-compact ob jects can therefore exhibit observ ational signatures typically attributed to blac k holes, while sim ultaneously allowing for extreme particle dynamics in certain parameter regimes. Incor- p orating radiativ e transfer, plasma physics, and realistic observ ational mo deling will b e essen tial in future studies to assess whether such geometries can b e distinguished from black holes using next-generation high resolution imaging and timing observ ations. A CKNO WLEDGMENTS P . Bambhaniy a ac kno wledge supp ort from the S˜ ao P aulo State F unding Agency F APESP (grant 2024/09383- 4). A. Bidlan w ould like to ackno wledge the contribution of the COST Action CA23130 (BridgeQG) communit y . The authors would like to thank A. B. Joshi for pro viding helpful literature related to Penrose diagrams. 15 [1] J. R. Oppenheimer and H. Snyder, Ph ysical Review 56 , 455 (1939) . [2] B. Datt, Zeitsc hrift f ¨ ur Physik 108 , 314 (1938). [3] R. Penrose, Ph ysical Review Letters 14 , 57 (1965) . [4] P . S. Joshi and S. Bhattacharyy a, Journal of Cosmology and Astroparticle Physics 01 , 034 . [5] P . S. Joshi and D. 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