The Lemaître–Tolman–Bondi Metric with a Central Pointlike Mass

Abstract

We present a comprehensive general relativistic analysis of the Lemaître–Tolman–Bondi (LTB) metric, incorporating a cosmological constant $\mathrm{\Lambda }$ and a central pointlike mass $M_{\mathrm{d}}$ at the geometric origin. Within this framework, $M_{\mathrm{d}}$ is identified as the material source of dark matter in cosmology, yielding a scale-dependent total matter–density parameter ${\mathrm{\Omega }}_{\mathrm{m}}\left(L\right)$ characterized by an $L^{-3}$ decay of its dark component ${\mathrm{\Omega }}_{\mathrm{d}}\left(L\right)$. We demonstrate that the Hubble and $S_8$ tensions are not independent anomalies but interconnected consequences of spacetime inhomogeneity. These discrepancies arise from a combination of physical and methodological factors: the probing of radial gradients at different characteristic scales and the subsequent interpretation of these data through a global FLRW template. This approach, compounded by the practice of isotropic sky averaging, masks the underlying LTB geometry and converts the physical variation of the manifold into the observed cosmological tensions. Our framework provides a self-consistent geometric explanation for current anomalies while preserving the Copernican principle, identifying the crisis in cosmology as arising from the application of homogeneous models to a manifold characterized by radial gradients and scale-dependent dynamics, where the observer and probes reside within the same inhomogeneous regime.

1. Introduction

Modern cosmology stands at a crossroads. While the standard $\mathrm{\Lambda }$CDM model has been remarkably successful, current observations reveal systematic discrepancies that challenge the Friedmann–Lemaître–Robertson–Walker (FLRW) paradigm [1]. The most prominent of these are the Hubble tension, characterized by the mismatch between local distance ladder measurements and the cosmic microwave background (CMB) data, and the $S_8$ tension, which relates to the growth of cosmic structures [2,3,4]. Furthermore, recent analyses indicate that the total matter–density parameter ${\mathrm{\Omega }}_{\mathrm{m}}$ exhibits a systematic evolution with redshift z [5].

These tensions, along with the observation of a significant local underdensity, known as the Keenan–Barger–Cowie (KBC) void [6,7], suggest that the assumption of perfect large-scale homogeneity may be an oversimplification. While the standard $\mathrm{\Lambda }$CDM model typically treats such a void as a rare statistical fluctuation, the Lemaître–Tolman–Bondi (LTB) framework [8] allows for its consistent relativistic integration. By employing a scale-dependent curvature $k\left(L\right)$, the LTB metric can effectively model such a local underdensity as an approximation, providing a physical basis for its impact on the expansion rate while addressing the observational degeneracies inherent in inhomogeneous models [9].

In this paper, we provide the relativistic completion of our previous peer-reviewed work [10]. We demonstrate that the presence of a central pointlike mass $M_{\mathrm{d}}$ acts as the physical source of dark matter in cosmology, defining the scale-dependent total matter–density parameter ${\mathrm{\Omega }}_{\mathrm{m}}\left(L\right)$ through the $L^{-3}$ decay of its dark component ${\mathrm{\Omega }}_{\mathrm{d}}\left(L\right)$. A key insight of this work is that the perceived cosmological tensions emerge from a combination of physical and methodological factors. We show that the physical variation of expansion dynamics between the observer and the probes is inherently masked by the standard practice of isotropic sky averaging and the reliance on global FLRW templates. By accounting for these radial gradients and the relativistic shear, we provide a self-consistent resolution to the current crisis in cosmology while preserving the Copernican principle. This framework identifies the discrepancies as a result of applying homogeneous models to a manifold characterized by scale-dependent dynamics, where the terrestrial observer and cosmological probes reside within the same inhomogeneous regime.

2. Theory

2.1. Choice of Coordinates and General Line Element

We consider a spherically symmetric system in geodesic comoving coordinates $x^{\mu }=(ct,L,\theta ,\varphi )$. The most general ansatz [11,12,13] for the line element is

$$d{s}^2=-{(c dt)}^2+b^2(L,t) d{L}^2+r^2(L,t)\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right).$$

(1)

Derivatives with respect to the timelike coordinate $x^0=ct$ are denoted by a dot or by ${\partial }_0$, and spatial derivatives with respect to the coordinate L are denoted by a prime.

2.2. Four-Velocity and Geodesic Condition

In this metric, the geodesic equation for a comoving matter distribution is exactly satisfied by $u^{\mu }=(c,0,0,0)$. The ansatz describes free fall, which excludes hydrodynamic pressure gradients. Since $u^i=0$, this coordinate system does not allow for any peculiar matter flow, e.g., accretion.

2.3. LTB Condition and Einstein Tensor

Einstein’s field equations with a cosmological constant are

$$G_{\nu }^{\mu }=\kappa T_{ \nu }^{\mu }-\mathrm{\Lambda }{\delta }_{\nu }^{\mu } ,$$

(2)

where $\kappa =8\pi G/c^4$. Here, the cosmological constant $\mathrm{\Lambda }$ is used to represent the dark energy density driving the accelerated expansion of the Universe. In comoving coordinates, the absence of energy flux implies $T_L^0=0$.

2.3.1. Momentum Constraint, LTB Condition and Boundary Conditions

To model an inhomogeneous universe anchored by a central inhomogeneity, we posit the existence of a central pointlike mass $M_{\mathrm{d}}$ located at the geometric origin $L_{\mathrm{d}}=0$. This mass acts as a permanent boundary condition for the manifold’s geometry.

The mixed component $G_L^0$ of the metric (1) enforces, due to $T_L^0=0$ in Einstein’s field Equation (2),

$$G_L^0={\displaystyle \frac{2}{r}}\left({\dot{r}}^{\prime }-{\displaystyle \frac{\dot{b}{r}^{\prime }}{b}}\right)=0 \Rightarrow {\displaystyle \frac{\dot{b}}{b}}={\displaystyle \frac{{\dot{r}}^{\prime }}{r^{\prime }}} .$$

(3)

Integration yields the LTB condition with the curvature function $f\left(L\right)$,

$$b(L,t)={\displaystyle \frac{r^{\prime }(L,t)}{\sqrt{1+f\left(L\right)}}} .$$

(4)

Thus, the LTB metric reads

$$d{s}^2=-{(c dt)}^2+{\displaystyle \frac{{r\prime }^2(L,t)}{1+f\left(L\right)}} d{L}^2+r^2(L,t)\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right).$$

(5)

The curvature function $f\left(L\right)$ is chosen as

$$f\left(L\right)=-k\left(L\right) L^2 ,$$

(6)

to establish agreement with the FLRW metric

$$d{s}^2=-{(c dt)}^2+{\displaystyle \frac{A^2\left(t\right)}{1-K{L}^2}} d{L}^2+A^2\left(t\right)L^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right)$$

(7)

in the asymptotic limit $L\to \infty$. The spatial curvature K of the FLRW background is linked to that of the LTB metric by

$$K=\underset{L\to \infty }{lim}k\left(L\right) .$$

(8)

The local scale factor $a(L,t)$ is introduced via

$$r(L,t)=a(L,t)L ,$$

(9)

which implies the radial derivative

$$r^{\prime }(L,t)=a(L,t)+a^{\prime }(L,t)L ,$$

(10)

where the boundary conditions

$$\underset{L\to \infty }{lim}{a}^{\prime }(L,t)=0, \underset{L\to \infty }{lim}{r}^{\prime }(L,t)=\underset{L\to \infty }{lim}a(L,t)=A\left(t\right)$$

(11)

ensure a smooth transition into the asymptotic region.

The boundary of the manifold, $\widehat{L}$, is determined by the global topology. For flat ($K=0$) or open ($K=-1$) universes, the manifold extends to $L\to \infty$. However, for a closed LTB universe, $\widehat{L}$ represents the antipode where the denominator of the radial LTB metric component vanishes, $1+f\left(\widehat{L}\right)=0$. To prevent a divergence of the metric at this boundary, the numerator must satisfy the regularity condition $r^{\prime }(\widehat{L},t)=0$. Substituting Equation (10) into this requirement leads to a fundamental structural consequence: regularity at the antipode enforces the condition $a^{\prime }(\widehat{L},t)=-a(\widehat{L},t)/\widehat{L}$. In contrast, the FLRW paradigm strictly requires $A^{\prime }\left(t\right)=0$ and thus $a^{\prime }(\widehat{L},t)=0$. These two conditions could only be reconciled if the manifold were to extend to $\widehat{L}\to \infty$, which fundamentally contradicts the topology of a closed universe. Consequently, the FLRW metric is mathematically inconsistent with the necessary boundary condition of a closed LTB manifold.

2.3.2. Einstein Tensor of the LTB Metric

Using the metric (1) and the LTB condition (4), the non-vanishing components of the Einstein tensor for the LTB metric can be formulated as

$$\begin{array}{cc}\hfill G_0^0& ={\displaystyle \frac{2r^{″}}{b^{2}r}}+{\displaystyle \frac{{r\prime }^2}{b^2{r}^2}}-{\displaystyle \frac{2b^{\prime }{r}^{\prime }}{b^{3}r}}-{\displaystyle \frac{1}{r^2}}-{\displaystyle \frac{{\dot{r}}^2}{r^2}}-{\displaystyle \frac{2\dot{b}\dot{r}}{br}}=-{\displaystyle \frac{{(r{\dot{r}}^2-rf)}^{\prime }}{r^2{r}^{\prime }}} ,\hfill \end{array}$$

(12a)

$$\begin{array}{cc}\hfill G_L^L& ={\displaystyle \frac{{r\prime }^2}{b^2{r}^2}}-{\displaystyle \frac{2\ddot{r}}{r}}-{\displaystyle \frac{{\dot{r}}^2}{r^2}}-{\displaystyle \frac{1}{r^2}}=-{\displaystyle \frac{2r\ddot{r}+{\dot{r}}^2-f}{r^2}} ,\hfill \end{array}$$

(12b)

$$\begin{array}{cc}\hfill G_{\theta }^{\theta }=G_{\varphi }^{\varphi }& ={\displaystyle \frac{r^{″}}{b^{2}r}}-{\displaystyle \frac{b^{\prime }{r}^{\prime }}{b^{3}r}}-{\displaystyle \frac{\ddot{r}}{r}}-{\displaystyle \frac{\ddot{b}}{b}}-{\displaystyle \frac{\dot{b}\dot{r}}{br}}=-{\displaystyle \frac{{(2r\ddot{r}+{\dot{r}}^2-f)}^{\prime }}{2r{r}^{\prime }}} .\hfill \end{array}$$

(12c)

2.4. Homogenization and Conservation Laws

2.4.1. Justification of a Homogeneous Ideal Fluid

At first glance, the unequal spatial components of the Einstein tensor $G_L^L\ne G_{\theta }^{\theta }=G_{\varphi }^{\varphi }$ would imply, in Einstein’s field Equation (2), the general form of the energy–momentum tensor for an anisotropic fluid, $T_{ \nu }^{\mu }=\mathrm{diag} (-\varrho c^2,P_r,P_t,P_t)$, where one might assume that the mass density $\varrho (L,t)$ and the pressures $P_r(L,t)$ (radial) and $P_t(L,t)$ (tangential) would generally depend on space and time.

We assume that the mass density $\varrho ={\varrho }_{\mathrm{b}}+{\varrho }_{\gamma }$ consists of pressureless baryonic matter ${\varrho }_{\mathrm{b}}$ ($P_{\mathrm{b}}=0$) and radiation ${\varrho }_{\gamma }$ with pressure

$$P_{\gamma }={\displaystyle \frac{1}{3}}{\varrho }_{\gamma }{c}^2.$$

(13)

The latter is naturally isotropic, whereby the energy–momentum tensor simplifies with $P=P_{\gamma }$ to that of an ideal fluid,

$$T_{ \nu }^{\mu }=\mathrm{diag} (-\varrho c^2,P,P,P).$$

(14)

However, an inhomogeneous pressure $P=P(L,t)$ would cause a pressure gradient $P^{\prime }$ that would disturb the free fall in the LTB metric, for which this metric is specifically designed. Therefore, only a homogeneous pressure

$$P=P\left(t\right)$$

(15)

is compatible with the LTB metric.

Although the central pointlike mass $M_{\mathrm{d}}$ implies an inhomogeneous mass distribution at least in its vicinity, cosmological observations confirm that matter is homogeneously distributed on cosmic scales,

$$\varrho =\varrho \left(t\right)={\varrho }_{\mathrm{b}}\left(t\right)+{\varrho }_{\gamma }\left(t\right).$$

(16)

Furthermore, an inhomogeneous radiation density ${\varrho }_{\gamma }(L,t)$ is incompatible with the LTB framework. According to the equation of state (13), any spatial gradient in the radiation density would necessarily generate a pressure gradient. As established in Section 2.2, such gradients would violate the geodesic free-fall condition inherent to the LTB metric, thereby enforcing a homogeneous distribution for the radiation component.

Thus, for this cosmological scenario, the energy–momentum tensor of an ideal fluid is purely time-dependent and therefore homogeneous.

Compatibility Check

By using an ideal fluid, one enforces the equality of the spatial components of the Einstein tensor $G_L^L=G_{\theta }^{\theta }=G_{\varphi }^{\varphi }$ in Einstein’s field Equation (2). In what follows, it is shown that a homogeneous ideal fluid is compatible with the initially seemingly unequal spatial components of the Einstein tensor. To this end, we set them as equal,

$$G_L^L=-{\displaystyle \frac{2r\ddot{r}+{\dot{r}}^2-f}{r^2}}=-{\displaystyle \frac{{(2r\ddot{r}+{\dot{r}}^2-f)}^{\prime }}{2r{r}^{\prime }}}=G_{\theta }^{\theta }=G_{\varphi }^{\varphi } .$$

(17)

By introducing the function

$$F(L,t)=2r\ddot{r}+{\dot{r}}^2-f ,$$

(18)

this equation can be simplified,

$${\displaystyle \frac{F}{r^2}}={\displaystyle \frac{F^{\prime }}{2r{r}^{\prime }}} ,$$

(19)

separated,

$${\displaystyle \frac{F^{\prime }}{F}}={\displaystyle \frac{2r^{\prime }}{r}} ,$$

(20)

and solved,

$$F(L,t)=C\left(t\right)r^2.$$

(21)

Substituting the function F into the spatial components of the Einstein tensor yields the spatial components of the Einstein field Equation (2),

$$G_L^L=G_{\theta }^{\theta }=G_{\varphi }^{\varphi }=-C\left(t\right)=\kappa P\left(t\right)-\mathrm{\Lambda } ,$$

(22)

which proves the compatibility of the spatial components of the Einstein tensor with a homogeneous ideal fluid.

2.4.2. Detailed Derivation of Energy Conservation

The conservation of mass-energy follows for $\nu =0$ from the condition

$${\nabla }_{\mu }{T}_{ \nu }^{\mu }={\displaystyle \frac{1}{\sqrt{-g}}}{\displaystyle \frac{\partial \left(T_{ \nu }^{\mu }\sqrt{-g}\right)}{\partial x^{\mu }}}-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial g_{\mu \lambda }}{\partial x^{\nu }}}{T}^{\mu \lambda }=0$$

(23)

and yields

$$\dot{\varrho }=-(\varrho +{\displaystyle \frac{P}{c^2}})({\displaystyle \frac{\dot{b}}{b}}+{\displaystyle \frac{2\dot{r}}{r}})=-\left(\varrho +{\displaystyle \frac{P}{c^2}}\right){\displaystyle \frac{{\partial }_0\left(b{r}^2\right)}{b{r}^2}} .$$

(24)

The infinitesimal volume element of the LTB metric is found to be

$$dV=\sqrt{g_{LL} g_{\theta \theta } g_{\varphi \varphi }} dL d\theta d\varphi =b{r}^{2}sin\theta dL d\theta d\varphi .$$

(25)

From this, one obtains the time-dependent volume within the coordinate L,

$$V(L,t)=4\pi \underset{0}{\overset{L}{\int }}b(l,t)r^2(l,t)dl.$$

(26)

Multiplying Equation (24) by $4\pi c^{2}b{r}^2$ and subsequent integration yields

$$4\pi \dot{\varrho }{c}^2\underset{0}{\overset{L}{\int }}b(l,t)r^2(l,t) dl+4\pi (\varrho c^2+P) {\partial }_0\underset{0}{\overset{L}{\int }}b(l,t)r^2(l,t) dl=0 .$$

(27)

Substituting Equation (26) leads to

$$\dot{\varrho }{c}^{2}V+(\varrho c^2+P)\dot{V}=0 .$$

(28)

This reflects the first law of thermodynamics for an isentropic system,

$$dU=-P dV ,$$

(29)

where $U=\varrho c^{2}V$.

From the radial component and the angular components of Equation (23), no further new insights arise, as these equations are identically satisfied.

2.5. Derivation of the Local Friedmann Equations

2.5.1. The Local First Friedmann Equation (Dynamics of Expansion)

Evaluating the timelike component of the Einstein field Equation (2) using the expression for $G_0^0$ from (12a), we obtain

$$G_0^0=-{\displaystyle \frac{{(r{\dot{r}}^2-rf)}^{\prime }}{r^2{r}^{\prime }}}=-\kappa \varrho c^2-\mathrm{\Lambda } .$$

(30)

A rearrangement yields

$${(r{\dot{r}}^2-rf)}^{\prime }=(\kappa \varrho c^2+\mathrm{\Lambda }) r^2{r}^{\prime } .$$

(31)

The mass function is determined by integrating this expression from the geometric origin $L_{\mathrm{d}}=0$ to the radial coordinate L,

$$M(L,t)=M_{\mathrm{d}}+4\pi \varrho \left(t\right)\underset{0}{\overset{L}{\int }}{r}^2{\displaystyle \frac{\partial r}{\partial l}} dl=M_{\mathrm{d}}+{\displaystyle \frac{4}{3}}\pi \varrho \left(t\right)r^3(L,t) .$$

(32)

The term $M_{\mathrm{d}}$ represents the mass concentrated at the radial coordinate $L_{\mathrm{d}}=0$. It arises mathematically as the lower boundary value of the definite integral, representing a delta distribution of mass that acts as a boundary of the manifold. Integration and subsequent division by $r^3$ yields

$${\displaystyle \frac{{\dot{r}}^2-f}{r^2}}={\displaystyle \frac{2GM}{c^2{r}^3}}+{\displaystyle \frac{\mathrm{\Lambda }}{3}} .$$

(33)

Multiplication by $c^2$, substitution of Equations (6) and (9), and the introduction of the local radial and tangential Hubble parameters,

$$\begin{array}{cc}\hfill \tilde{H}(L,t)& =c {\displaystyle \frac{\dot{b}(L,t)}{b(L,t)}}=c {\displaystyle \frac{{\dot{r}}^{\prime }(L,t)}{r^{\prime }(L,t)}} ,\hfill \end{array}$$

(34a)

$$\begin{array}{cc}\hfill H(L,t)& =c {\displaystyle \frac{\dot{r}(L,t)}{r(L,t)}}=c {\displaystyle \frac{\dot{a}(L,t)}{a(L,t)}} ,\hfill \end{array}$$

(34b)

results in the local first Friedmann equation,

$$H^2(L,t)={\displaystyle \frac{2G{M}_{\mathrm{d}}}{a^3{L}^3}}+{\displaystyle \frac{8\pi G\varrho }{3}}+{\displaystyle \frac{\mathrm{\Lambda }{c}^2}{3}}-{\displaystyle \frac{k\left(L\right)c^2}{a^2}} .$$

(35)

2.5.2. The Local Second Friedmann Equation (Local Acceleration Equation)

Evaluating the radial component of the Einstein field Equation (2) using the expression for $G_L^L$ from (12b), we obtain

$$G_L^L=-{\displaystyle \frac{2r\ddot{r}+{\dot{r}}^2-f}{r^2}}=\kappa P-\mathrm{\Lambda } .$$

(36)

Substituting Equation (33) yields

$$-{\displaystyle \frac{2\ddot{r}}{r}}-{\displaystyle \frac{2GM}{c^2{r}^3}}-{\displaystyle \frac{\mathrm{\Lambda }}{3}}=\kappa P-\mathrm{\Lambda } .$$

(37)

Multiplication by $c^2/2$ and rearranging provides the local acceleration equation,

$$c^2{\displaystyle \frac{\ddot{a}}{a}}=-{\displaystyle \frac{G{M}_{\mathrm{d}}}{a^3{L}^3}}-{\displaystyle \frac{4\pi G}{3}}\left(\varrho +{\displaystyle \frac{3P}{c^2}}\right)+{\displaystyle \frac{\mathrm{\Lambda }{c}^2}{3}} .$$

(38)

2.5.3. The Cosmic Equation of Motion

To represent the local dynamics in a form directly comparable to observed cosmological density parameters, we transform the local first Friedmann equation into a dimensionless equation of motion. We introduce the scaled variable x and the dimensionless time $\tau$ as

$$x(L,\tau )={\displaystyle \frac{a(L,t)}{a_0\left(L\right)}} , \tau \left(L\right)=H_0\left(L\right)t ,$$

(39)

where $a_0\left(L\right)=a(L,t_0)$ and $H_0\left(L\right)=H(L,t_0)$ denote the values of the local scale factor and the local tangential Hubble parameter at the current time $t_0$. Similarly, we denote the current-day local radial Hubble parameter as ${\tilde{H}}_0\left(L\right)=\tilde{H}(L,t_0)$. We further introduce the corresponding reduced Hubble constants

$$h\left(L\right)={\displaystyle \frac{H_0\left(L\right)}{100 \mathrm{km} {\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}}}, \tilde{h}\left(L\right)={\displaystyle \frac{{\tilde{H}}_0\left(L\right)}{100 \mathrm{km} {\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}}.}$$

(40)

The derivative of x with respect to $\tau$ is given by

$${\displaystyle \frac{dx}{d\tau }}={\displaystyle \frac{dx}{dt}}{\displaystyle \frac{dt}{d\tau }}={\displaystyle \frac{c\dot{a}}{a_0}}{H}_0^{-1}={\displaystyle \frac{c\dot{a}}{a}}{\displaystyle \frac{a}{a_0}}{H}_0^{-1}={\displaystyle \frac{H}{H_0}}x.$$

(41)

Substituting the local first Friedmann Equation (35) into the square of Equation (41) yields the cosmic equation of motion,

$${\left({\displaystyle \frac{dx}{d\tau }}\right)}^2={\displaystyle \frac{{\mathrm{\Omega }}_{\gamma }}{x^2}}+{\displaystyle \frac{{\mathrm{\Omega }}_{\mathrm{m}}}{x}}+{\mathrm{\Omega }}_{\mathrm{k}}+{\mathrm{\Omega }}_{\mathrm{\Lambda }}{x}^2 ,$$

(42)

where the local density parameters are defined as

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{\gamma }\left(L\right)& ={\displaystyle \frac{8\pi G{\varrho }_{\gamma }{a}^4}{3H_0^2{a}_0^4}}={\displaystyle \frac{{\varrho }_{\gamma ,0}}{{\varrho }_{\mathrm{c}}}} ,\hfill \end{array}$$

(43a)

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{\mathrm{m}}\left(L\right)& ={\mathrm{\Omega }}_{\mathrm{d}}\left(L\right)+{\mathrm{\Omega }}_{\mathrm{b}}\left(L\right) ,\hfill \end{array}$$

(43b)

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{\mathrm{d}}\left(L\right)& ={\displaystyle \frac{2G{M}_{\mathrm{d}}}{H_0^2{a}_0^3{L}^3}} ,\hfill \end{array}$$

(43c)

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{\mathrm{b}}\left(L\right)& ={\displaystyle \frac{8\pi G{\varrho }_{\mathrm{b}}{a}^3}{3H_0^2{a}_0^3}}={\displaystyle \frac{{\varrho }_{\mathrm{b},0}}{{\varrho }_{\mathrm{c}}}} ,\hfill \end{array}$$

(43d)

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{\mathrm{k}}\left(L\right)& =-{\displaystyle \frac{k{c}^2}{H_0^2{a}_0^2} } ,\hfill \end{array}$$

(43e)

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{\mathrm{\Lambda }}\left(L\right)& ={\displaystyle \frac{\mathrm{\Lambda }{c}^2}{3H_0^2}} .\hfill \end{array}$$

(43f)

In these expressions, ${\varrho }_{\mathrm{b},0}={\varrho }_{\mathrm{b}}\left(t_0\right)$ and ${\varrho }_{\gamma ,0}={\varrho }_{\gamma }\left(t_0\right)$ represent the current baryon and radiation densities, while

$${\varrho }_{\mathrm{c}}\left(L\right)={\displaystyle \frac{3H_0^2}{8\pi G}}$$

(44)

is the local critical density. While the baryonic density ${\varrho }_{\mathrm{b},0}$ remains physically homogeneous, the scale dependency of the total matter–density parameter ${\mathrm{\Omega }}_{\mathrm{m}}\left(L\right)$ is fundamentally anchored by the central pointlike mass $M_{\mathrm{d}}$. It drives the dominant $L^{-3}$ decay of the dark matter density parameter ${\mathrm{\Omega }}_{\mathrm{d}}\left(L\right)$ and simultaneously induces an implicit scale-sensitivity in ${\mathrm{\Omega }}_{\mathrm{b}}\left(L\right)$ through its influence on the local critical density ${\varrho }_{\mathrm{c}}\left(L\right)$.

3. Discussion

3.1. The Physical Nature of the Central Pointlike Mass $M_d$ and the Manifold’s Scale

A defining feature of the local Friedmann equations derived herein is the appearance of the central pointlike mass $M_{\mathrm{d}}$. Building upon our previous work [10], the comprehensive general relativistic treatment presented here identifies $M_{\mathrm{d}}$ fundamentally as the material source of dark matter within the LTB manifold. Within this framework, $M_{\mathrm{d}}$ acts as the singular boundary condition at the geometric origin ($L_{\mathrm{d}}=0$) that influences the manifold’s geometry and determines the scale-dependent dark matter density parameter ${\mathrm{\Omega }}_{\mathrm{d}}\left(L\right)$, which in turn shapes the observed evolution of ${\mathrm{\Omega }}_{\mathrm{m}}\left(L\right)$ across the manifold.

Crucially, the mathematical consistency with Einstein’s field equations is maintained because $M_{\mathrm{d}}$ is treated as a boundary of the manifold. In the exterior region ($L>0$), its radial derivative vanishes, ensuring that the local field equations are exactly satisfied by the homogeneous energy–momentum tensor of the fluid background ($\varrho ,P$). To maintain consistency with the observed large-scale isotropy of the cosmic microwave background, the central pointlike mass must be situated far beyond the terrestrial particle horizon.

In this far-field configuration, the terrestrial observer is embedded within the inhomogeneous regime of the manifold, receiving CMB radiation from the surrounding surface of last scattering. Although $M_{\mathrm{d}}$ is distant, its presence remains encoded in the global geometry, acting as the persistent source of the dark matter density parameter ${\mathrm{\Omega }}_{\mathrm{d}}\left(L\right)$. This ensures that the scaling effects are felt consistently by both local and global probes. Furthermore, because the null geodesics of the CMB photons traverse this inhomogeneous volume from a vast distance, the extreme offset to the geometric origin limits the detection of significant directional anisotropies, thereby preserving the observed large-scale isotropy of the Universe.

3.2. Relation to Modified Newtonian Cosmology

The local expansion dynamics exhibit a profound consistency with the results of the modified Newtonian cosmology derived in our previous work [10]. A comparative analysis reveals that the two local Friedmann equations derived herein are formally identical to those obtained in the modified Newtonian framework. This identity confirms that the gravitational influence of the central pointlike mass $M_{\mathrm{d}}$, the homogeneous fluid ($\varrho ,P$), and the cosmological constant $\mathrm{\Lambda }$ is captured consistently in both relativistic and modified Newtonian contexts.

However, the LTB framework provides an essential refinement regarding the physical interpretation and the mathematical structure of the manifold. In the modified Newtonian model, the function $k\left(L\right)$ arises merely as a constant of integration. In contrast, the LTB metric provides the rigorous general relativistic foundation, where $k\left(L\right)$ is identified as the spatial curvature of the inhomogeneous manifold. Furthermore, while the Newtonian model assumes a flat-space background and thus only defines a single Hubble parameter, the LTB metric introduces two distinct expansion rates: the radial Hubble parameter $\tilde{H}(L,t)$ and the tangential Hubble parameter $H(L,t)$. This distinction arises from the metric coefficient $b(L,t)$, representing a purely relativistic effect that is absent in the flat-space approximation.

These structural differences also manifest in the conservation laws. In the Newtonian model, the conservation of mass is defined in a Euclidean volume. In contrast, the LTB framework accounts for the curvature of the manifold through the infinitesimal volume element $dV$ in Equation (25), which depends on both metric coefficients $b(L,t)$ and $r(L,t)$. Consequently, the relativistic continuity Equation (24) incorporates the expansion of the curved spacetime.

A subtle distinction also exists in the treatment of mass and pressure. While the Newtonian approach employs a gravitational mass that explicitly incorporates the pressure to satisfy the modified Poisson equation, the LTB formalism naturally partitions these contributions: the mass function $M(L,t)$ in Equation (32) accounts for the enclosed mass, while the pressure’s gravitational influence emerges inherently through the local acceleration Equation (38). This demonstrates that the LTB metric serves as the rigorous relativistic completion of the heuristic pressure corrections and flat-space limitations of modified Newtonian cosmology, providing a self-consistent geometric basis for the observed scaling effects.

3.3. Dark Matter in Cosmology and the Scale-Dependent ${\mathrm{\Omega }}_m$

The terms involving $M_{\mathrm{d}}$ in the local Friedmann equations provide a compelling geometric explanation for phenomena traditionally attributed to a particle-based dark matter fluid. Our derivation suggests that dark matter in cosmology is a manifestation of the metric gradients and geometric scaling induced by the central pointlike mass $M_{\mathrm{d}}$ within the inhomogeneous LTB framework.

As established in Section 3.1, although $M_{\mathrm{d}}$ is situated far beyond the terrestrial particle horizon to preserve the observed large-scale isotropy of the CMB, its gravitational influence remains encoded in the global scaling behavior of the manifold. A transformative consequence of this model is the recognition that the total matter–density parameter ${\mathrm{\Omega }}_{\mathrm{m}}$ is not a universal global constant but a scale-dependent function. The characteristic $L^{-3}$ dependence, originating from the central pointlike mass $M_{\mathrm{d}}$ at $L_{\mathrm{d}}=0$, defines the dark matter density parameter ${\mathrm{\Omega }}_{\mathrm{d}}\left(L\right)$. This introduces a significant scale sensitivity into the total matter–density parameter ${\mathrm{\Omega }}_{\mathrm{m}}\left(L\right)$ that remains unaccounted for in the conventional $\mathrm{\Lambda }$CDM paradigm.

This theoretical result is consistent with recent empirical findings [5] which suggest that cosmological parameters exhibit systemic evolution when analyzed across different redshift intervals. We argue that the common practice of enforcing isotropic sky averaging likely masks these underlying gradients. Our LTB framework identifies the perceived evolution of ${\mathrm{\Omega }}_{\mathrm{m}}$ as a manifestation of probing the manifold at varying characteristic scales. Crucially, the central pointlike mass $M_{\mathrm{d}}$ is physically decoupled from the background fluid ($\varrho ,P$). While the fluid describes the homogeneous cosmic background, $M_{\mathrm{d}}$ acts as a geometric boundary condition. This ensures that the dark sector’s influence is not the result of a modified equation of state or additional particle species but a manifestation of the manifold’s global geometry. Consequently, the local expansion dynamics are influenced by $M_{\mathrm{d}}$ without necessitating non-standard interactions with baryonic or radiation components, thus preserving the standard thermodynamic evolution of the Universe.

3.4. Resolving the Hubble and $S_8$ Tensions

The model developed in this work demonstrates that the most critical discrepancies in modern cosmology—the Hubble and $S_8$ tensions—are not independent anomalies but interconnected consequences of treating an inherently inhomogeneous manifold as a global FLRW background. By integrating the central pointlike mass $M_{\mathrm{d}}$ and the scale-dependent curvature $k\left(L\right)$ of the KBC void into the LTB framework, we provide a unified geometric resolution to both tensions. We show that these discrepancies emerge naturally when scale-dependent radial gradients and relativistic shear are masked by the standard practice of global parameter inference and isotropic sky averaging.

3.4.1. The Hubble Tension

The discrepancy between local measurements of $H_0$ (≈73 $\mathrm{km} {\mathrm{s}}^{-1} {\mathrm{Mpc}}^{-1})$ and those derived from the CMB (≈67 $\mathrm{km} {\mathrm{s}}^{-1} {\mathrm{Mpc}}^{-1})$ is identified as the result of a two-fold effect: a synergistic physical scaling within the LTB geometry and a methodological artifact of standard cosmological analysis. The term $2G{M}_{\mathrm{d}}/\left(a^3{L}^3\right)$ in the local first Friedmann equation (35) ensures that the expansion rate is a radially varying function, naturally enhanced by the $L^{-3}$ influence of the central pointlike mass.

This local increase of the expansion rate is further compounded by two additional factors. First, the negative spatial curvature $k\left(L\right)<0$ of the regional KBC void ($z<0.1$) provides a positive contribution to the expansion rate, as seen from the curvature term $-k\left(L\right) c^2/a^2$ in Equation (35). Second, the LTB framework reveals a relativistic divergence between the radial ($\tilde{H}$) and tangential (H) Hubble parameters. This inherent shear accounts for the local expansion gradients and provides the necessary geometric flexibility to accommodate higher $H_0$ values.

Crucially, the Hubble tension emerges because the local distance ladder and the cosmic microwave background probe the manifold at different radial coordinates. This physical offset yields distinct values for the expansion rate within the inhomogeneous LTB regime. The perceived tension is then generated by interpreting these scale-dependent data through a rigid, global FLRW template. This methodological error is compounded by the common practice of isotropic sky averaging: by enforcing averaging over fixed angular scales, the underlying radial gradients and the relativistic shear—which provides the necessary geometric flexibility to accommodate higher local $H_0$ values—are effectively masked. The Hubble tension is thus not a result of spatial or temporal separation but a consequence of applying a homogeneous model and angular averaging to a manifold where the physical expansion varies between the observer’s location and the scales of the probes.

3.4.2. The $S_8$ Tension

The observation that matter clustering in the late Universe is less pronounced than predicted by Planck data is explained by the local acceleration Equation (38). The term $-G{M}_{\mathrm{d}}/\left(a^3{L}^3\right)$ acts as an additional deceleration component suppressing the growth of the local scale factor $a(L,t)$. Within the LTB framework, this evolution is further governed by the dynamical interplay between the radial ($\tilde{H}$) and tangential (H) expansion rates.

Consequently, the perceived $S_8$ tension is identified as a methodological artifact arising from the assumption of global homogeneity. Standard analyses utilize a constant global total matter–density parameter ${\mathrm{\Omega }}_{\mathrm{m}}$ and a single reduced Hubble constant h to define the clustering scale. However, in our model, ${\mathrm{\Omega }}_{\mathrm{m}}\left(L\right)$ is inherently scale-dependent, reflecting the local geometry influenced by $M_{\mathrm{d}}$ and the scale-dependent critical density ${\varrho }_{\mathrm{c}}\left(L\right)$. Crucially, since the $S_8$ parameter is conventionally defined at the scale of $8 h^{-1}$ Mpc, the fact that the reduced Hubble constants $h\left(L\right)$ and $\tilde{h}\left(L\right)$ are themselves functions of the radial coordinate L directly induces the observed tension. The use of a fixed, global h to calibrate the clustering scale fails to account for the varying expansion dynamics and the inherent relativistic shear within the inhomogeneous manifold, which physically distorts the characteristic volume of the probes.

The tension emerges because the probes used to measure structure growth and the CMB effectively sample the manifold at different radial coordinates. This physical offset, combined with the manifold’s requirement for two distinct, scale-dependent reduced Hubble constants ($h\left(L\right)$ and $\tilde{h}\left(L\right)$) to account for this shear, creates a discrepancy that a rigid FLRW template cannot reconcile. Much like the Hubble tension, this discrepancy is exacerbated by the practice of isotropic sky averaging. By smoothing over the radial gradients of the total matter–density parameter and the expansion shear, the standard analysis enforces a set of global parameters that are physically inconsistent with the inhomogeneous geometry. The $S_8$ tension thus vanishes once the scale dependency of the total matter–density parameter ${\mathrm{\Omega }}_{\mathrm{m}}\left(L\right)$, the functional dependence of the reduced Hubble constants on the coordinate L, and the physical offset between the observer’s location and the characteristic scales of the cosmological probes are properly accounted for, identifying the discrepancy as a result of applying a homogeneous model to a manifold characterized by inherent radial gradients.

4. Conclusions

This paper presents a comprehensive general relativistic analysis of the Lemaître–Tolman–Bondi (LTB) metric, incorporating a cosmological constant $\mathrm{\Lambda }$ and a central pointlike mass $M_{\mathrm{d}}$. Our framework demonstrates that the most pressing discrepancies in modern cosmology are not failures of expansion physics, but natural consequences of interpreting an inherently inhomogeneous manifold with homogeneous models.

Our results lead to a transformative interpretation of the cosmic dark sector. We conclude that dark matter in cosmology is fundamentally identified with the central pointlike mass $M_{\mathrm{d}}$ at the geometric origin. This configuration naturally explains the scale dependency of the total matter–density parameter ${\mathrm{\Omega }}_{\mathrm{m}}\left(L\right)$ and the expansion rates $h\left(L\right)$ and $\tilde{h}\left(L\right)$ through the $L^{-3}$ decay of the dark component ${\mathrm{\Omega }}_{\mathrm{d}}\left(L\right)$. To preserve the observed large-scale isotropy of the CMB, $M_{\mathrm{d}}$ must be situated far beyond the terrestrial particle horizon, acting as a global geometric boundary that influences the manifold’s scaling behavior.

The LTB framework successfully resolves both the Hubble and $S_8$ tensions by identifying them as a combination of physical and methodological factors. We show that these tensions emerge as the differential signal of spacetime inhomogeneity, caused by probing radial gradients in $H_0\left(L\right)$, ${\tilde{H}}_0\left(L\right)$, and ${\mathrm{\Omega }}_{\mathrm{m}}\left(L\right)$ at different characteristic scales. We emphasize that the $S_8$ tension, in particular, arises because the conventional clustering scale of $8 h^{-1}$ Mpc is calibrated using a global FLRW template that fails to account for the scale-dependent reduced Hubble constants and the relativistic shear inherent to the LTB manifold. These physical variations are subsequently masked when the data are interpreted through rigid, homogeneous models and the standard practice of isotropic sky averaging, converting the manifold’s geometric information into the observed cosmological anomalies.

Ultimately, our framework provides a self-consistent alternative to the $\mathrm{\Lambda }$CDM paradigm while preserving the Copernican principle. It suggests that the current crisis in cosmology results from applying global models to a manifold characterized by radial gradients and scale-dependent dynamics. This geometric structure also provides a natural basis for observed dipole modulations in the CMB fluctuations [14], although the relative contribution of the central pointlike mass $M_{\mathrm{d}}$ and the effective, spherically averaged curvature profile $k\left(L\right)$ associated with the KBC void remains to be determined in future studies. Consequently, we propose that future analyses, particularly regarding large-scale surveys like DESI or Euclid, should prioritize spatially resolved observations without preemptive angular averaging. This approach would allow for a more rigorous testing of the $L^{-3}$ scaling law and the potential detection of the predicted relativistic shear, providing a self-consistent path forward through the current discrepancies in our understanding of the Universe.

5. Outlook: Einstein–Cartan Theory (ECT)

In Einstein–Cartan theory (ECT), spin-torsion effects introduce repulsive corrections to gravity that have the potential to prevent the formation of singularities [15]. The LTB model presented in this work provides the geometric foundation necessary to describe the transition from such microscopic singularity-avoidance to the observed large-scale expansion. Within this framework, the central pointlike mass $M_{\mathrm{d}}$ at $L_{\mathrm{d}}=0$ represents the macroscopic relic of a primordial inhomogeneity.

A critical aspect for the physical consistency of this transition is the nature of the matter in the early Universe: a macroscopic spin polarization is strictly required for an inflationary phase to have occurred. Such a coherent alignment of spins can be physically induced by primordial magnetic fields, which are generated by the rotation of charged particles in the early Universe. The coupling of the particles’ intrinsic magnetic moments to these self-generated magnetic fields then facilitates the necessary alignment of their spins. If the spins were randomly oriented, forming a macroscopically isotropic spin distribution, the spin-related correction terms would vanish in the acceleration equation when averaged over all orientations [16]. Only a magnetically induced spin polarization can exert the necessary repulsive force to avoid the initial singularity and initiate the observed cosmic dynamics.

From a formal standpoint, such a dense, rotating medium would generate significant pressure gradients. It is important to emphasize that this primordial scenario—characterized by rotation, magnetic fields, and pressure gradients—cannot be formally described by the LTB metric itself, which is strictly designed for geodesic, irrotational motion with homogeneous pressure. However, the LTB metric remains the ideal mathematical framework to describe the long-range relics of such primordial anisotropies in the late-time Universe. In this context, the central pointlike mass $M_{\mathrm{d}}$ is identified as the physical manifestation of the initial singularity-avoidance process—a cosmic central torsion relic originating from a rotating primordial inhomogeneity. Acting as the central source, $M_{\mathrm{d}}$ fundamentally determines the scale dependence of the dark matter density parameter ${\mathrm{\Omega }}_{\mathrm{d}}\left(L\right)$ and shapes the metric gradients across the manifold. The LTB metric thus serves as the bridge between microscopic Einstein–Cartan effects and the macroscopic cosmological tensions observed today.