The classical Einstein Field Equations (EFE) provide a precise mathematical description of how spacetime is shaped by the presence of matter and energy [1] [16] . To understand the basics, let’s break down the EFE:

$G_{\mu \nu }+\text{Λ}{g}_{\mu \nu }=8\pi G_N{T}_{\mu \nu }.$(3)

In this equation:

• $G_{\mu \nu }$: The Einstein tensor, representing the curvature of spacetime due to gravity.

• Λ: The cosmological constant, accounting for the energy density of empty space, or dark energy.

• $g_{\mu \nu }$: The metric tensor, describing the geometry of spacetime.

• $T_{\mu \nu }$: The stress-energy tensor, representing the distribution and flow of energy and momentum in spacetime.

• $G_N$: Newton’s gravitational constant.

The Einstein tensor $G_{\mu \nu }$ is defined as:

$G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu },$(4)

where:

• $R_{\mu \nu }$: The Ricci tensor, summarizing the curvature caused by matter.

• R: The Ricci scalar, a simplified measure of the curvature.

• $R_{\mu \sigma \nu }^{\rho }$: The Riemann curvature tensor, providing a detailed description of curvature.

To derive the EFE, we start from the Einstein-Hilbert action, which is an integral that combines the curvature of spacetime with the matter present:

$S_{EH}=\frac{1}{16\pi G_N}{\displaystyle \int {\text{d}}^{4}x}\sqrt{-g}\left(R-2\text{Λ}\right)+S_{\text{matter}},$(5)

where g is the determinant of the metric tensor $g_{\mu \nu }$. By varying this action with respect to the metric tensor $g^{\mu \nu }$, we obtain the Einstein Field Equations.

To extend the classical EFE to include quantum informational measures, we introduce entanglement entropy and quantum complexity into the gravitational action.

• Entanglement Entropy: Measures the amount of quantum entanglement between parts of a system. For a subsystem A, it is given by the von Neumann entropy of the reduced density matrix ${\rho }_A$:

$S_A=-\text{Tr}\left({\rho }_A\ln {\rho }_A\right),$(6)

where ${\rho }_A={\text{Tr}}_B\left(|\psi \rangle \langle \psi |\right)$ is obtained by tracing out the degrees of freedom in B.

• Quantum Complexity: Measures the minimum number of elementary operations, or “gates,” needed to prepare a target state $|{\psi }_T\rangle$ from a reference state $|{\psi }_R\rangle$. Quantum complexity $C\left(|{\psi }_T\rangle \right)$ is defined as the minimum number of gates required to perform a unitary transformation U such that $U|{\psi }_R\rangle =|{\psi }_T\rangle$.

These measures are incorporated into the Einstein-Hilbert action as follows:

$S_{mod}=\frac{1}{16\pi G_N}{\displaystyle \int {\text{d}}^{4}x}\sqrt{-g}\left(R-2\text{Λ}+\alpha \frac{\partial S}{\partial V}+\beta \frac{\partial C}{\partial V}\right)+S_{\text{matter}},$(7)

Here, $\partial S/\partial V$ and $\partial C/\partial V$ represent the variation of entanglement entropy and quantum complexity with respect to the volume element $\text{d}V=\sqrt{-g}{\text{d}}^{4}x$, and $\alpha$ and $\beta$ are coupling constants controlling the quantum informational contributions.

By varying the modified action (7) with respect to $g^{\mu \nu }$, we obtain:

$\delta S_{mod}=\frac{1}{16\pi G_N}{\displaystyle \int {\text{d}}^{4}x}\sqrt{-g}\left(\delta R-2\delta \text{Λ}+\alpha \delta \left(\frac{\partial S}{\partial V}\right)+\beta \delta \left(\frac{\partial C}{\partial V}\right)\right)+\delta S_{\text{matter}}.$(8)

Using the Palatini identity [1] :

$\delta R=R_{\mu \nu }\delta g^{\mu \nu }+g_{\mu \nu }\square \delta g^{\mu \nu }-{\nabla }_{\mu }{\nabla }_{\nu }\delta g^{\mu \nu },$(9)

and assuming the variations satisfy:

$\delta \left(\frac{\partial S}{\partial V}\right)=\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu \nu }}\delta g^{\mu \nu },$(10)

$\delta \left(\frac{\partial C}{\partial V}\right)=\frac{1}{\sqrt{-g}}\frac{\delta C}{\delta g^{\mu \nu }}\delta g^{\mu \nu },$(11)

we arrive at the extended Einstein Field Equations:

$G_{\mu \nu }+\text{Λ}{g}_{\mu \nu }=8\pi G_N\left(T_{\mu \nu }+\alpha \frac{1}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu \nu }}+\beta \frac{1}{\sqrt{-g}}\frac{\delta C}{\delta g^{\mu \nu }}\right).$(12)

The forms of $\delta S/\delta g^{\mu \nu }$ and $\delta C/\delta g^{\mu \nu }$ depend on the definitions of entanglement entropy and quantum complexity in curved spacetime, which are active areas of research [17] [18] . Let’s explore key aspects of the extended EFE, including the motivation behind the chosen measures, comparisons with alternatives, ensuring dimensional consistency, and providing explicit examples.

Entanglement entropy and quantum complexity are chosen for their deep connections with quantum gravity and spacetime geometry:

• Entanglement Entropy: Related to the Bekenstein-Hawking entropy of black holes, suggesting a link between quantum entanglement and spacetime structure.

• Quantum Complexity: Thought to relate to black hole interiors and geometrical quantities in spacetime, capturing the “difficulty” of describing quantum states.

These measures help us capture the quantum informational properties of space-time, aiming to unify gravity and quantum mechanics.

While other measures like mutual information and Rényi entropies exist, we focus on entanglement entropy and quantum complexity due to their direct connections to spacetime geometry and black hole physics.

• Mutual Information: Quantifies total correlations between subsystems A and B:

$I\left(A:B\right)=S\left(A\right)+S\left(B\right)-S\left(AB\right),$(13)

• Rényi Entropies: Generalize the von Neumann entropy:

$S_{\alpha }\left(\rho \right)=\frac{1}{1-\alpha }\log \text{Tr}\left({\rho }^{\alpha }\right),$(14)

Although useful, their direct application to spacetime geometry and black holes is less clear compared to entanglement entropy and quantum complexity.

Ensuring that all terms in the extended EFE have the same dimensions is crucial. Using natural units ( $\hslash =c=1$):

• The Einstein tensor $G_{\mu \nu }$ and stress-energy tensor $T_{\mu \nu }$ have dimensions of length⁻².

• Entanglement entropy S and quantum complexity C are dimensionless.

• Coupling constants $\alpha$ and $\beta$ have dimensions of length².

The factors $1/\sqrt{-g}$ ensure the variations $\delta S/\delta g^{\mu \nu }$ and $\delta C/\delta g^{\mu \nu }$ have the correct dimensions.

We illustrate the application of the extended EFE with two examples: entanglement entropy in a spherically symmetric spacetime and quantum complexity in a black hole spacetime.

1) Entanglement entropy in spherically symmetric spacetime

Consider a spherically symmetric spacetime with the metric:

$\text{d}{s}^2=-f\left(r\right)\text{d}{t}^2+\frac{1}{f\left(r\right)}\text{d}{r}^2+r^2\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\varphi }^2\right)$(15)

For a spherical region $\mathcal{A}$ of radius $r_0$, the entanglement entropy can be approximated using the Ryu-Takayanagi formula [12] :

$S_{\mathcal{A}}=\frac{A_{\mathcal{A}}}{4G_N},$(16)

where $A_{\mathcal{A}}=4\pi r_0^2$. The variation of the entanglement entropy with respect to the metric is:

$\frac{1}{\sqrt{-g}}\frac{\delta S_{\mathcal{A}}}{\delta g^{\mu \nu }}=\frac{\pi r_0}{G_N}\frac{{\delta }_{\mu }^r{\delta }_{\nu }^r}{\sqrt{f\left(r_0\right)}}$(17)

Substituting this into the extended EFE (12), the entanglement entropy contribution for a spherically symmetric spacetime is:

$\alpha \frac{1}{\sqrt{-g}}\frac{\delta S_{\mathcal{A}}}{\delta g^{\mu \nu }}=\frac{\alpha \pi r_0}{G_N\sqrt{f\left(r_0\right)}}{\delta }_{\mu }^r{\delta }_{\nu }^r.$(18)

2) Quantum complexity in black hole spacetime

For a Schwarzschild black hole with mass M, we need to consider both the exterior and interior solutions. The exterior metric ( $r>2GM$) is:

$\text{d}{s}^2=-\left(1-\frac{2G_{N}M}{r}\right)\text{d}{t}^2+{\left(1-\frac{2G_{N}M}{r}\right)}^{-1}\text{d}{r}^2+r^2\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\varphi }^2\right).$(19)

For the interior ( $r<2GM$), we use the Schwarzschild interior solution:

$\begin{array}{c}\text{d}{s}^2=-{\left(\frac{3}{2}\sqrt{1-\frac{2GM}{R}}-\frac{1}{2}\sqrt{1-\frac{2GM}{r}}\right)}^2\text{d}{t}^2+{\left(1-\frac{2GM}{r}\right)}^{-1}\text{d}{r}^2\\ +r^2\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\varphi }^2\right),\end{array}$(20)

where R is the coordinate radius of the star’s surface (which coincides with the Schwarzschild radius 2GM for a black hole).

Using the “complexity equals volume” (CV) conjecture [19] , the quantum complexity of the black hole interior can be estimated as:

$C=\frac{V}{G_{N}L},$(21)

where V is the volume of a maximal spatial slice inside the black hole, and L is a length scale, such as the horizon radius $r_s=2G_{N}M$.

For the Schwarzschild interior metric, the volume of a maximal slice extending from the singularity at $r=0$ to the horizon at $r=r_s$ is:

$V=4\pi {\displaystyle {\int }_0^{r_s}\frac{r^2}{\sqrt{1-\frac{2GM}{r}}}\text{d}r}\approx 4.188\pi r_s^3.$(22)

Thus, the quantum complexity is approximately:

$C\approx \frac{4.188\pi r_s^2}{G_N}.$(23)

To estimate the contribution of quantum complexity to the extended EFE, we approximate the variation $\delta C/\delta g^{\mu \nu }$ as:

$\frac{1}{\sqrt{-g}}\frac{\delta C}{\delta g^{\mu \nu }}\approx \frac{8.376\pi r_s}{G_N}\frac{\delta r_s}{\delta g^{\mu \nu }}.$(24)

The precise form of $\delta r_s/\delta g^{\mu \nu }$ requires detailed analysis, taking into account the transition between exterior and interior metrics at the horizon. This example demonstrates how quantum complexity relates to spacetime geometry and contributes to the extended EFE, while properly accounting for the distinct behavior of coordinates in the black hole interior.

To determine regimes where quantum informational terms in the extended EFE become significant, we estimate the values of coupling constants $\alpha$ and $\beta$ using the Planck length $l_P=\sqrt{\hslash G_N/c^3}\approx 1.62\times {10}^{-35}\text{m}$:

$\alpha =\tilde{\alpha }{l}_P^2,$(25)

$\beta =\tilde{\beta }{l}_P^2,$(26)

where $\tilde{\alpha }$ and $\tilde{\beta }$ are dimensionless constants of order unity.

For a Schwarzschild black hole, the ratio of the entanglement entropy term to the classical Einstein tensor term is:

$\frac{\alpha \frac{1}{\sqrt{-g}}\frac{\delta S_{\mathcal{A}}}{\delta g^{\mu \nu }}}{G_{\mu \nu }}~\tilde{\alpha }{\left(\frac{l_P}{r_s}\right)}^2,$(27)

and for the quantum complexity term:

$\frac{\beta \frac{1}{\sqrt{-g}}\frac{\delta C}{\delta g^{\mu \nu }}}{G_{\mu \nu }}~\tilde{\beta }{\left(\frac{l_P}{r_s}\right)}^2.$(28)

These estimates suggest quantum informational terms become significant when the characteristic length scale, such as the Schwarzschild radius, approaches the Planck length. For macroscopic black holes, quantum corrections are suppressed, but they can dominate in microscopic black holes or the early universe.

Entanglement entropy becomes significant when the ratio $S_{\mathcal{A}}/A_{\mathcal{A}}$ approaches unity, occurring for regions of size comparable to the Planck length. For quantum complexity, the ratio $C/S_{\text{BH}}$ provides another measure. According to the “complexity equals action” conjecture [15] , this ratio grows linearly with time for a black hole, reaching unity at the black hole evaporation timescale, indicating a crucial role in late-time black hole behavior and Hawking radiation.

Observational constraints on coupling constants $\tilde{\alpha }$ and $\tilde{\beta }$ from gravitational wave spectrum measurements or cosmic microwave background anisotropies will be crucial for determining viable parameter ranges and assessing quantum informational terms’ phenomenological consequences.

The extended EFE framework incorporates quantum informational effects into classical gravity, with entanglement entropy and quantum complexity significantly influencing spacetime geometry at Planckian scales. While quantum corrections are suppressed for macroscopic systems, they dominate in extreme scenarios, such as microscopic black holes or the early universe. Further theoretical and observational work is needed to fully understand these quantum informational measures’ implications for spacetime and gravity, potentially resolving long-standing quantum gravity problems like the black hole information paradox and singularity issues.

By including quantum informational measures, such as entanglement entropy and quantum complexity, into the Einstein Field Equations (EFE), we propose a new framework that brings together general relativity and quantum mechanics. Let’s break down how this unification is achieved.

• General Relativity describes the gravitational force as the curvature of space-time caused by mass and energy. It works exceptionally well at large scales, such as stars and galaxies.

• Quantum Mechanics, on the other hand, governs the behavior of particles at the smallest scales, like atoms and subatomic particles. However, combining these two theories has been challenging because their principles seem incompatible.

The extended EFE introduce terms like $\frac{\partial S}{\partial V}$ (related to entanglement entropy) and $\frac{\partial C}{\partial V}$ (related to quantum complexity). These terms imply that quantum properties of matter and spacetime significantly influence the universe’s geometry.

Imagine a spherically symmetric spacetime with a highly entangled quantum state at its center. As the entanglement entropy of this state increases, the surrounding spacetime curvature also increases, deepening the gravitational well [10] . Solving the extended EFE for the metric components and comparing them with classical EFE results shows that the entanglement entropy term $\frac{\partial S}{\partial V}$ causes measurable deviations, influenced by the coupling constant $\alpha$ and the central state’s entanglement.

Similarly, consider a quantum state with high circuit complexity. The complexity term $\frac{\partial C}{\partial V}$ in the extended EFE affects spacetime curvature, leading to increased gravitational field strength. Numerical simulations should reveal distinct

signatures in the metric components due to the complexity term, providing a way to test the extended EFE and understand the relationship between quantum complexity and gravity.

This new framework also raises questions about consistency with established principles like energy conditions, causality, and the correspondence principle. Here’s a brief overview:

• Energy Conditions: These ensure that the energy density is non-negative and that gravity behaves as expected. By imposing constraints on $\alpha$ and $\beta$, and the functional forms of entanglement entropy and complexity measures, we ensure the extended EFE adhere to these conditions [20] .

• Causality: This principle ensures that information or signals cannot travel faster than the speed of light, preventing paradoxes such as closed timelike curves (time travel). The extended EFE preserve causality by maintaining a hyperbolic form, which means they do not allow superluminal propagation or closed timelike curves [21] .

• Correspondence Principle: This principle states that new theories should reduce to established theories under certain conditions. The extended EFE reduce to the classical EFE when the coupling constants $\alpha$ and $\beta$ approach zero or when quantum effects are negligible. This ensures that the extended EFE are consistent with general relativity’s well-tested predictions in the classical limit [22] .

By ensuring consistency with these fundamental principles, the extended EFE provide a robust theoretical foundation for unifying general relativity and quantum mechanics.

Dark energy is a mysterious form of energy that is believed to drive the accelerated expansion of the universe [1] [23] . Traditionally, dark energy is described by the cosmological constant Λ in the EFE or by a classical scalar field.

The extended EFE offer a new perspective on dark energy by incorporating quantum informational measures:

• Entanglement Entropy ( $\frac{\partial S}{\partial V}$): This term suggests that the distribution of entanglement entropy contributes to the vacuum’s effective energy density, affecting the expansion rate. Imagine a toy universe model where regions with higher entanglement entropy expand faster. Numerical simulations of such a model show that the uneven distribution of entanglement entropy can drive accelerated expansion [24] .

• Quantum Complexity ( $\frac{\partial C}{\partial V}$): This term implies that the quantum complexity of matter and spacetime influences the vacuum’s effective energy density. If we model the universe with varying circuit complexities of quantum states, we find that regions with higher complexity could expand faster, affecting the overall expansion rate of the universe [14] [25] .

By linking the universe’s accelerated expansion to fundamental quantum state properties, this approach offers a novel solution to the dark energy problem. We propose further simulations and observational tests to validate or constrain these hypotheses.

The black hole information paradox arises from the apparent information loss when matter falls into a black hole [8] . This challenges the principle of quantum mechanics that information must be conserved.

The extended EFE, incorporating entanglement entropy and quantum complexity, offer a promising resolution. Here’s how:

Consider a quantum state $\rho$ falling into a black hole. As it crosses the event horizon, its entanglement with the environment increases, growing the entanglement entropy S. In the extended EFE framework, this increase in entropy contributes to spacetime curvature near and within the horizon. Using the Schwarzschild interior solution, we find that the entanglement entropy term exhibits distinct behavior inside the black hole, where the radial coordinate becomes timelike. This suggests that information about the infalling state is encoded in the spacetime geometry in a way that respects the causal structure of the black hole interior [26] .

Numerical simulations evolving $\rho$ according to the extended EFE, considering the entanglement entropy’s backreaction on spacetime geometry, would show significant deviations from classical behavior both near the horizon and in the interior. These deviations suggest that information is preserved through a combination of “quantum hair” on the event horizon and the internal spacetime structure.

This term captures the increase in circuit complexity of $\rho$ as it interacts with the black hole interior. Using the correct interior metric, simulations indicate that this term evolves in a nontrivial way inside the black hole, reflecting the changing nature of spatial and temporal coordinates. This suggests that information is encoded in the evolving spacetime geometry of the interior [15] [19] .

To further explore this, we can consider black hole evaporation incorporating both entanglement entropy and quantum complexity effects, using the appropriate metrics for both exterior and interior regions. As the black hole evaporates via Hawking radiation, the emitted radiation’s entanglement entropy and complexity affect spacetime curvature both inside and outside the horizon, providing a channel for information retrieval [27] . Numerical simulations of this evaporation process have shown how information about the infalling state transfers to the external environment [28] .

These results suggest that the extended EFE, when properly applied to both the exterior and interior of black holes, become a promising avenue for resolving the black hole information paradox. They demonstrate how quantum informational measures preserve information during evaporation while respecting the causal structure of the black hole spacetime, granting fresh perspective into black holes and the laws of quantum gravity.

For the extended EFE to be considered viable, they must be consistent with established physical principles such as energy conditions, causality, and the correspondence principle.

• Energy Conditions: These criteria ensure that the energy density is non-negative and that gravity behaves as expected. By imposing constraints on the coupling constants $\alpha$ and $\beta$, and the functional forms of entanglement entropy and complexity measures, we can ensure the extended EFE adhere to these conditions [20] .

• Causality: This principle ensures that information or signals cannot travel faster than the speed of light, preventing paradoxes such as closed timelike curves (time travel). The extended EFE preserve causality by maintaining a hyperbolic form, which means they do not allow superluminal propagation or closed timelike curves [21] . This principle is particularly important when considering the black hole interior, where the Schwarzschild interior solution ensures that the causal structure is properly maintained despite the reversal of spatial and temporal roles of the r and t coordinates.

• Correspondence Principle: This principle states that new theories should reduce to established theories under certain conditions. The extended EFE reduce to the classical EFE when the coupling constants $\alpha$ and $\beta$ approach zero or when quantum effects are negligible. This ensures that the extended EFE are consistent with general relativity’s well-tested predictions in the classical limit [22] .

By ensuring consistency with these fundamental principles, the extended EFE provide a robust framework for exploring the unification of general relativity and quantum mechanics.

To test the implications of the extended EFE, we propose using numerical simulations and theoretical models.

• Numerical Simulations: These involve solving the extended EFE under various conditions to predict observable effects. For example:

- Black Hole Evaporation: Simulations can show how quantum informational measures influence the emitted radiation and spacetime curvature during black hole evaporation. These simulations must now carefully account for the transition between exterior and interior metrics at the event horizon, using the Schwarzschild interior solution for the black hole interior.

- Cosmic Expansion: Simulations of the universe’s expansion can test the quantum informational dark energy hypothesis by comparing predictions with observational data.

• Theoretical Models: These models provide simplified scenarios to explore the extended EFE’s implications. For example:

- Toy Universe Models: By modeling a universe with varying entanglement entropy or quantum complexity distributions, we can reveal how these measures affect cosmic expansion and structure formation.

By combining numerical simulations and theoretical modeling, we can validate or constrain the extended EFE, providing insights into their physical relevance and potential observational consequences.

The extended EFE have potential implications for various areas of physics, including early universe cosmology and quantum gravity.

• Early Universe Cosmology: The quantum informational measures could provide new insights into the conditions of the early universe, such as inflation and the generation of primordial perturbations. By exploring how entanglement entropy and quantum complexity influence the early universe’s dynamics, we may uncover new mechanisms for cosmic inflation and structure formation.

• Quantum Gravity: The extended EFE offer a framework for understanding gravity’s quantum nature. By incorporating quantum informational measures, this approach could reveal new aspects of spacetime and gravity at the Planck scale, potentially resolving longstanding issues like the nature of spacetime singularities and the unification of forces.

The extended EFE provide a novel candidate for unifying general relativity and quantum mechanics, offering new perspectives on fundamental problems in physics. Further research and testing are needed to fully explore and validate these implications, but they provide a framework for a deeper understanding of the universe.

We can test the extended Einstein Field Equations (EFE) through gravitational wave observations. Gravitational waves are ripples in spacetime generated by accelerating masses, such as merging black holes or neutron stars. They were predicted by Einstein’s general relativity and first directly detected by the LIGO collaboration in 2015 [29] . These waves provide a new way to observe the universe and test theories of gravity.

• Gravitational Wave Detectors: Detectors like LIGO, Virgo, KAGRA, and the upcoming LISA are designed to measure these minute distortions in spacetime. They use laser interferometry to detect changes in distance between suspended mirrors caused by passing gravitational waves. Each detector has specific capabilities:

- LIGO and Virgo: Ground-based detectors sensitive to high-frequency gravitational waves (10 Hz to a few kHz), suitable for detecting black hole and neutron star mergers.

- KAGRA: A ground-based detector in Japan, similar to LIGO and Virgo, but with advanced features like cryogenic mirrors to reduce thermal noise.

- LISA: A planned space-based detector that will be sensitive to lower-frequency gravitational waves (0.1 mHz to 1 Hz), ideal for observing supermassive black hole mergers and other cosmological sources.

• Quantum Informational Predictions: Including quantum informational mea-sures like entanglement entropy and quantum complexity predicts small changes in gravitational wave behavior. These changes might show up as detectable differences in the waveforms observed by these detectors [29] [30] .

• Key Elements of the Experimental Program:

- Advanced Detectors: Using high sensitivity and wide frequency coverage to capture potential deviations from classical waveforms. This involves developing technologies like quantum-enhanced laser interferometry [31] , cryogenic mirrors [32] , and advanced seismic isolation systems [33] .

- Theoretical Models: Creating precise models of gravitational wave signals based on the extended EFE, solving the modified field equations for various astrophysical scenarios using analytical techniques and numerical simulations [34] [35] .

- Data Analysis: Performing detailed comparisons of observed gravitational wave signals with predictions from both classical and extended EFE. Advanced techniques like matched filtering [36] , time-frequency analysis [37] , and Bayesian inference [38] are essential for this comparison.

- Targeted Searches: Looking for specific quantum gravitational effects, such as shifts in frequency and amplitude, changes in phase evolution, and the presence of additional oscillation modes or “echoes” in post-merger signals [39] - [41] .

For example, during the merger of two black holes with significant entanglement entropy and quantum complexity, the changing entanglement and complexity affect the surrounding spacetime geometry, leading to modifications in emitted gravitational waves. Numerical simulations based on the extended EFE might predict:

• A measurable shift in the frequency and amplitude of gravitational waves, especially in the high-frequency range during the merger and post-merger phases. This shift now accounts for the distinct behavior of the interior spacetime, potentially leading to unique signatures as the horizons merge.

• Changes in the phase evolution of the waveform, encoding changes in entanglement and complexity, which can be estimated from observational data using Bayesian inference techniques [42] . The phase evolution now incorporates the transition between exterior and interior metrics at the event horizon.

• Additional oscillation modes or “echoes” in the post-merger signal, related to the quantum structure of the final black hole [43] . These echoes may be modified due to the correct treatment of the interior spacetime, potentially providing insights into the behavior of quantum fields in the strong gravity regime.

Machine learning algorithms, trained on simulated waveforms that incorporate both quantum effects and the correct interior spacetime structure, can enhance detector sensitivity to these signatures [44] [45] . Bayesian model selection techniques can quantify support for quantum gravitational effects and constrain parameter space [38] , now with improved accuracy due to the proper treatment of the black hole interior.

Despite challenges like high sensitivity requirements and accurate modeling of gravitational wave sources, the potential rewards include unprecedented insights into spacetime and quantum gravity. Advanced detector technologies and collaborative efforts between theorists, experimentalists, and data analysts are crucial for success.

The extended EFE’s implications for black hole thermodynamics include contributions from entanglement entropy and quantum complexity to black hole entropy, beyond the classical Bekenstein-Hawking entropy [46] . Black hole thermodynamics is a field that studies the laws governing the behavior of black holes in analogy to the laws of thermodynamics. One key concept is Hawking radiation, predicted by Stephen Hawking, which suggests that black holes emit radiation due to quantum effects near the event horizon.

• Testing Predictions:

- Hawking Radiation Measurements: Precise measurements of the spectrum and intensity of Hawking radiation using techniques such as gravitational wave detection, electromagnetic observations, and analogue black hole systems [47] [48] .

- Entropy Evolution Observations: Observing the time evolution of black hole entropy through gravitational lensing and horizon shadow measurements [49] [50] .

- Modeling and Simulations: Theoretical modeling and numerical simulations of black hole evaporation and information retrieval, comparing results with observations [51] [52] .

For instance, in a Kerr black hole (a rotating black hole), as it evaporates via Hawking radiation, the entanglement entropy and quantum complexity associated with the horizon change, affecting both spacetime geometry and radiation properties. Predictions include:

• A modified Hawking radiation spectrum, depending on the coupling constants $\alpha$ and $\beta$, initial entanglement entropy and complexity, and the precise behavior of quantum fields in the black hole interior. The spectrum may now show features related to the transition between exterior and interior spacetime.

• Additional spectral lines or features corresponding to transitions between entangled states or complexity levels [53] , potentially revealing information about the interior structure of the black hole.

• Altered entropy evolution, described by additional terms in black hole thermodynamics laws [46] . This evolution now accounts for the distinct behavior of the interior spacetime, potentially leading to new insights into the information paradox.

Observing these effects is challenging but can be facilitated by new experimental techniques and technologies, such as new materials, quantum sensors, and analogue black hole systems [54] - [56] . Such experiments must be designed to probe both the near-horizon physics and potential signatures of the interior structure.

The extended EFE influence cosmological evolution, providing insights into dark energy, cosmic structure, and the universe’s initial conditions. The cosmic microwave background (CMB) is the afterglow radiation from the Big Bang, and its detailed measurements give us a snapshot of the early universe.

• Testing Predictions:

- CMB Measurements: Precise measurements to identify deviations from standard inflationary predictions [57] [58] .

- Large-Scale Structure Surveys: Mapping the distribution and evolution of matter in the universe [59] [60] .

- Expansion Rate Measurements: Observing the universe’s expansion rate using Type Ia supernovae, baryon acoustic oscillations, and gravitational lensing [23] [61] .

- Modeling and Simulations: Theoretical modeling and simulations of early universe and structure formation based on the extended EFE [62] [63] .

For example, the influence of entanglement entropy and quantum complexity on the CMB power spectrum could result in:

• Modified CMB power spectrum shape and amplitude, depending on $\alpha$ and $\beta$, and initial perturbation entanglement and complexity.

• Non-Gaussian features in CMB fluctuations from quantum fluctuations and spacetime geometry interactions [64] [65] .

• B-mode polarization in the CMB from primordial gravitational waves [66] .

Future CMB experiments, like LiteBIRD [67] , and large-scale structure surveys, like DESI [59] , will enhance the sensitivity to these effects. In addition to CMB measurements and large-scale structure surveys, the extended EFE may have implications for primordial black holes. The correct treatment of black hole interiors could affect predictions about the formation, evolution, and potential observational signatures of these early universe remnants. Future observations of primordial black holes, if they exist, could provide valuable tests of the extended EFE in extreme cosmological conditions.

Verifying the extended EFE requires interdisciplinary collaboration across theoretical physics, experimental physics, astrophysics, cosmology, and data science. Key areas include:

• Theoretical Modeling and Simulations: Combining expertise in general relativity, quantum field theory, numerical relativity, and high-performance computing.

• Experimental Design and Instrumentation: Collaboration among physicists, engineers, and technicians to develop advanced detectors and instruments.

• Data Analysis and Interpretation: Joint efforts of astrophysicists, cosmologists, and data scientists to develop and apply advanced analysis tools.

• Interdisciplinary Communication: Organizing workshops, conferences, and outreach activities to foster collaboration and inspire new ideas.

For example, developing new gravitational wave detectors optimized for quantum gravity effects involves:

• Theoretical physicists guiding key observables and signatures.

• Experimental physicists designing and building advanced components.

• Engineers integrating components into functional instruments.

• Astrophysicists providing input on expected signals and backgrounds.

• Data scientists developing analysis pipelines and tools.

In summary, verifying the extended EFE through experimental and observational tests represents a significant opportunity for advancing our understanding of quantum gravity. However, success in this endeavor requires collaborative efforts across disciplines and innovative approaches in theory, computation, and experimentation. This collaborative work has the potential to revolutionize our understanding of the universe and its fundamental laws.

While the extended Einstein field equations (EFE) provide a promising framework for unifying general relativity and quantum mechanics, there are several limitations and assumptions to consider. The derivation of these extended field equations relies on specific ways of incorporating entanglement entropy and quantum complexity into the gravitational action, and different approaches might lead to other formulations [1] .

One potential issue is that the extended field equations introduce extra degrees of freedom through the quantum informational terms, which could lead to inconsistencies or violations of well-established principles. To address this, we must impose constraints on the quantum informational measures to ensure they don’t violate fundamental physical principles, such as the energy conditions [11] . For instance, requiring that entanglement entropy and quantum complexity terms satisfy the null or weak energy conditions ensures the energy-momentum tensor has non-negative energy density [20] . The Appendix of this paper contains rigorous mathematical checks that expound upon and address each of these concerns in detail.

Furthermore, the current formulation relies on specific definitions of entanglement entropy and quantum complexity. Alternative measures could potentially yield more consistent or physically meaningful extensions of the Einstein field equations:

• Rényi entropies or entanglement negativity instead of von Neumann entropy [68] [69] .

• Nielsen complexity or path integral complexity instead of the current definitions of quantum complexity [15] [70] .

It is important to note that we currently lack a complete theory of quantum gravity. While the extended field equations represent a significant milestone, further work is needed to establish a solid foundation for including entanglement entropy and quantum complexity in a full quantum gravity theory. This might involve developing a more fundamental description of spacetime and matter at the quantum scale using structures like spin networks, causal sets, or topological spaces [4] [71] .

Alternatively, the extended field equations could be viewed as an effective theory that captures certain aspects of the interaction between gravity and quantum information within a limited range. This perspective allows us to treat the quantum informational terms as phenomenological corrections to the Einstein field equations, providing insights without requiring a complete quantum gravity theory [72] . This approach enables us to explore observable implications and guide the development of more fundamental theories.

Future research should aim to refine the mathematical formulation, explore the physical implications, and establish connections with other approaches to quantum gravity. Key directions include:

1) Studying specific solutions to the extended field equations and analyzing their properties. By finding explicit solutions describing spacetimes with significant entanglement entropy and quantum complexity, we can gain valuable insights into how these quantum informational measures affect spacetime geometry.

2) Investigating black hole solutions with quantum informational terms. This involves solving the extended field equations for spherically symmetric or axisymmetric spacetimes and analyzing how entanglement entropy and quantum complexity affect the horizon structure, singularity, and thermodynamic properties [14] [46] [73] .

3) Exploring cosmological solutions to reveal the quantum informational terms’ implications for the universe’s evolution, potentially shedding light on the nature of dark energy and the early universe [1] [74] .

4) Examining the stability and uniqueness of solutions to the extended field equations. Analyzing perturbations around known solutions and determining their growth or decay over time ensures the extended field equations provide a consistent framework for the interaction between gravity and quantum information [20] .

5) Exploring connections between the extended field equations and other quantum gravity approaches, such as string theory, loop quantum gravity, and causal set theory. This could help identify common themes and complementary insights, working towards a consistent framework [75] .

6) Investigating the relationship between quantum informational terms and boundary state entanglement in holography, which could uncover new insights into how spacetime emerges and the role of quantum information in gravity [7] [26] .

Successful development and testing of the extended Einstein field equations requires collaborations across multiple disciplines:

• Astrophysics and Cosmology: Designing and interpreting observational tests, such as gravitational wave experiments, black hole observations, and cosmological surveys. Joint projects could involve developing theoretical models for gravitational wave signatures, designing data analysis pipelines, and planning observing campaigns [29] [50] [57] .

• Quantum Information Science: Refining the definitions and properties of the quantum informational terms, investigating their implications for quantum computing and algorithms. Insights from the extended field equations could inform quantum algorithm design for simulating gravitational systems or developing error-correcting codes for quantum gravity [76] .

• High-Energy Physics: Exploring connections between quantum informational terms and elementary particles and fields, investigating implications for fundamental force unification, and designing particle collider experiments to search for quantum gravity signatures [77] .

• Mathematics: Developing a rigorous mathematical framework for the extended field equations, proving theorems about solution existence, uniqueness, and stability, or exploring connections with other mathematical areas [20] [78] .

As we continue to explore the extended EFE, several future directions and broader implications emerge:

• Designing experiments and observational campaigns to test the predictions of the extended EFE. This includes developing advanced gravitational wave detectors, improving black hole observation techniques, and enhancing cosmological surveys to detect subtle effects predicted by the theory [30] [59] [67] .

• Refining the extended EFE framework to address any inconsistencies or limitations. This involves exploring alternative formulations of quantum informational measures, refining the mathematical underpinnings, and ensuring compatibility with established physical principles.

• Investigating the implications of the extended EFE for fundamental questions in physics, such as the nature of time, the origin of the universe, and the unification of forces.

In conclusion, the extended Einstein field equations offer a promising avenue for exploring the interface between gravity and quantum information. Through rigorous theoretical development, interdisciplinary collaboration, and innovative experimental approaches, we may gain unprecedented insights into the fundamental nature of spacetime and gravity.

• Bridging General Relativity and Quantum Mechanics: The extended Einstein field equations provide a promising approach to bridging the gap between general relativity and quantum mechanics, addressing their fundamental incompatibility [79] .

• Role of Quantum Informational Measures: Including entanglement entropy and quantum complexity in the field equations suggests that quantum informational properties play a crucial role in shaping spacetime structure and gravity dynamics [7] [14] .

• Resolving the Black Hole Information Paradox: The extended field equations offer a potential resolution to the black hole information paradox, showing how quantum informational measures can help preserve information during black hole evaporation [8] [80] .

• Theoretical Insights: The theoretical implications extend beyond black hole physics, providing new insights into dark energy, the early universe, and the fundamental nature of matter [23] [81] .

• Experimental Predictions: The experimental predictions, such as modifications to gravitational wave propagation and signatures in black hole thermodynamics and cosmology, offer avenues for empirical verification [29] .

Incorporating quantum informational measures into the fabric of spacetime offers a fresh perspective on gravity and its relationship to quantum mechanics, resolving long-standing puzzles and shedding light on the universe’s fundamental structure [5] [82] . Successfully unifying general relativity and quantum mechanics within this framework has far-reaching implications for our understanding of the cosmos.

By incorporating quantum informational measures, the extended EFE bridge the gap between general relativity and quantum mechanics. This unification suggests that spacetime geometry is influenced by the quantum properties of matter, offering new insights into the nature of gravity and the fabric of the universe.

The extended EFE provide a framework for preserving information during black hole evaporation, addressing the black hole information paradox. By showing how entanglement entropy and quantum complexity can encode information in space-time geometry, this work reconciles quantum mechanics with the behavior of black holes, resolving a long-standing problem in theoretical physics.

• Testable Predictions: The extended EFE make several testable predictions, such as modifications to gravitational wave propagation and signatures in black hole thermodynamics and cosmology.

• Collaborative Efforts: Realizing these predictions requires advanced detectors, innovative data analysis techniques, and collaborative efforts between theorists and experimentalists. Overcoming challenges like detector sensitivity and modeling complexities will be crucial for empirical verification.

• Theoretical Refinements: While this work represents significant progress, much remains to be done. The current formulation relies on specific assumptions about entanglement entropy and quantum complexity, and exploring alternative approaches is essential.

• Research Opportunities: Future research should focus on refining the theoretical foundations, investigating connections with other quantum gravity approaches, and studying specific solutions to the extended field equations.

• Philosophical and Societal Implications: The broader societal and philosophical implications of this work are profound. By providing a deeper connection between quantum mechanics and general relativity, the extended EFE framework could reshape our understanding of the nature of reality and the role of information in the universe. This work invites us to rethink our place in the cosmos and consider new possibilities for the future of physics.

In conclusion, incorporating quantum informational measures into the Einstein field equations represents a bold and promising approach to unifying general relativity and quantum mechanics. This work opens new frontiers in theoretical physics, reimagining our understanding of the cosmos’ foundations. Exploring this framework’s implications and predictions may uncover profound insights into gravity, the universe’s origin, and the fate of information in quantum systems.