An Investigation in Strings Theory to outline Supersymmetric Wormholes Nature

Low-Dimensions vs High-Dimensions Theories

In recent years, significant progress has been made in understanding the Euclidean Path Integral (further information in here, Ref. 3) [1] for low-dimensional quantum gravity theories like Jackiw–Teitelboim (JT) Gravity. These insights often involve summing over Saddle Points with varying Topology(ies). In contrast, for higher-dimensional theories like those involving Einstein - Hilbert Gravity coupled with matter, the situation is less straightforward.

What About Wormholes?

Wormholes (Figure 1), which are Geometrical Structures connecting separate regions of Space - Time, are key to understanding certain quantum gravity aspects. However, their presence in the Euclidean path integral can lead to puzzles regarding Unitarity and Non-Factorization [2] of Correlation Functions. This raises questions about the consistency of the Holographic Duality (Figure 2) [3] in such cases.

Additionally, wormholes suggest the possibility of Baby Universes [4], impacting the Swampland Program's guidelines. To address these issues, a deeper exploration of higher-dimensional Euclidean wormholes within String Theory (further information in here, Section 1) is needed.

Constructing Euclidean Wormholes in Supergravity

Building Euclidean wormhole geometries generally requires a source of negative Euclidean Energy [5]. In string theory, this is achieved using axion (further information in here, Section 2) fields. Analytical continuation of a Lorentzian Theory [6] with axions to the Euclidean Regime can yield a negative energy-momentum tensor (further information in here, Section 1), providing the necessary conditions for wormhole formation.


Figure 1.   A Pictorial Representation of a Wormhole Interior

Figure 3.   Statistical and figurative Entropy Concept: the Order and Combinations Number of a small balls group

Step by Step from a Five-Dimensional Framework

One can start with a five-dimensional metric, which is written in the form:

[math]\LARGE{ds{_5}{^2} = dr^2 + e^{2A(r)} ds^2_{\mathbf{R}^4}}[/math]

[math]\Large{ds{_5}{^2} = dr^2 + e^{2A(r)} ds^2_{\mathbf{R}^4}}[/math]

Equation 1.   The form of a Five-Dimensional Metric

where [math]\small{ds^2_{\mathbf{R}^4}}[/math] is the Flat Metric on Euclidean Space, [math]\small{\mathbf{R}^4}[/math], and the Metric Function, [math]\small{A}[/math] ([math]\small{r}[/math]) depends solely on the radial coordinate, [math]\small{r}[/math]. The Euclidean gravity path integral compactifies [math]\small{\mathbf{R}^4}[/math] to the Normal Space, [math]\small{T^4}[/math].

Supersymmetry is broken from [math]\small{N = 4}[/math] to [math]\small{N = 1}[/math], with distinct Quantum Field Theory(ies) (QFTs) on each side of the wormhole, differentiated by their Yang-Mills coupling constants and other parameters. A Special Point exists where the wormhole's "neck" shrinks to zero, leading to a Singular Metric resembling the well-known GPPZ Solution [7].

... Here is Supersymmetry!

To ensure supersymmetry, the solutions must satisfy the first-order Bogomol'nyi - Prasad - Sommerfield (BPS) Equations (Equation 2 and Equation 3). These equations guide the construction of the supersymmetric Euclidean wormhole solutions. A key point in this analysis is whether the wormhole solutions dominate over corresponding disconnected geometries with the same Boundary Conditions.

[math]\LARGE{\mathcal{E}{_A} \equiv A' + \frac{1}{3} W = 0}[/math]

[math]\Large{\mathcal{E}{_A} \equiv A' + \frac{1}{3} W = 0}[/math]

Equation 2.   First BPS Equation: the form of Total Energy in the Euclidean Space

in which [math]\small{A'}[/math] is the derivative of the metric function and [math]\normalsize{W}[/math] is a real Supepotential.

[math]\LARGE{\mathcal{E}^{i} \equiv (z^{i})' - \frac{1}{3} K^{i\bar{J}} \partial_{\bar{J}} W = 0}[/math]

[math]\Large{\mathcal{E}^{i} \equiv (z^{i})' - \frac{1}{3} K^{i\bar{J}} \partial_{\bar{J}} W = 0}[/math]

Equation 3.   Second BPS Equation: the Energy for each point on the Euclidean Space

[math]\Large{i}[/math] is a generic Point on Euclidean space; ( [math]\Large{z^{i}}[/math])[math]\Large{'}[/math] is the first derivative of the Scalar Field, [math]\Large{z^{i}}[/math] of each generic point, with respect to the radial coordinate; and [math]\Large{K^{i\bar{J}}}[/math] is the Kähler Metric.

Using a Consistent Truncation7 of maximal five-dimensional Supergravity with an SO(6) Gauge Group, a specific set of BPS equations is derived. A simplified set of field variables and a constant of motion help in solving these equations. A critical aspect is the interpretation of these solutions within the context of the holographic duality and the implications for non-factorization.


Figure 2.   Illustration of the Hologram for a Sphere

Implications of Wormholes Discovery

The existence of Euclidean wormholes and their role in quantum gravity path integrals raise significant questions about unitarity, holographic duality, and the Swampland program. Future work involves uplifting these wormhole solutions to ten dimensions and exploring the implications of imaginary scalars in the Euclidean regime. The factorization puzzle in Holography remains an open question, with possible solutions involving supersymmetry considerations and potential Fermion Zero-Mode [8] effects. This area of research holds great potential for advancing our understanding of quantum gravity and its holographic connections.

  1. Springer. "Euclidean Path Integrals"

  2. Inspire HEP. "Factorization and Non-Factorization of In-Medium Four-Quark Condensates"

  3. "Editorial: New frontiers in holographic duality"

  4. Big Think. "Are we living in a baby universe that looks like a black hole to outsiders?"

  5. Wiley Online Library. "Energy-Efficient Memristive Euclidean Distance Engine for Brain-Inspired Competitive Learning"

  6. Nature. "Strongly enhanced effects of Lorentz symmetry violation in entangled Yb+ ions"

  7. ResearchForLife7 (revisited from arXiv). "Discussion on Consistent Truncations: Uplifting the GPPZ Solutions"

  8. Inspire HEP. "Fermion Zero Modes and Topological-charge on a Domain Wall of the D-brane-like Dot"

A Quantum Gravity problem: Exploring a Continous Early Universe?

The Singularity Role in Cosmological Research

The Big Bang Singularity (Figure 1) poses a significant challenge within standard cosmological and Inflationary Theory(ies). The divergence of tensors in the Einstein Equations near the singularity suggests that classical General Relativity may not be applicable under such extreme conditions. Quantum Gravitational Effects likely become relevant in proximity to the singularity, prompting the exploration of quantum theories of Gravity. Various approaches, including String Theory [1] and Loop Quantum Gravity [2], have been pursued to address this issue. A quantum theory of gravity emerges from the unification of General Relativity and Quantum Mechanics, potentially introducing a fundamental minimal length and a non-zero minimal Uncertainty in position measurements.

Fundation of Quantum World

To extend the Uncertainty Principle (Equation 1) and incorporate a non-zero minimal uncertainty, [math]\normalsize{x_{0}}[/math] in Position, [math]\normalsize{x}[/math] the Commutation relation between Position ([math]\normalsize{\hat{X}}[/math]) and Linear Momentum ([math]\normalsize{\hat{P}}[/math]) Operators is modified. This modification yields a generalized uncertainty relation, leading to novel consequences in the Statistical Mechanics of free particle systems.

[math]\LARGE{\Delta{x}\Delta{p} \geq {{\hbar} \over {2}}}[/math]

[math]\Large{\Delta{x}\Delta{p} \geq {{\hbar} \over {2}}}[/math]

Equation 1.   Classical Heinsenberg Uncertainty Principle expression

in which [math]\normalsize{\Delta{x}}[/math] and [math]\normalsize{\Delta{p}}[/math] are, in the order, the Position and Linear Momentum Operators uncertainties; while [math]\normalsize{\hbar}[/math] is the reduced Planck Constant.

For instance, the Phase Space measure is adjusted accordingly for particles adhering to the generalized uncertainty principle. Several such generalizations exist, each with its own characteristics. For instance, the Kempf - Mangano - Mann (K.M.M.) [3] Deformation Commutation relation is represented as:

[math]\LARGE{[\hat{X}, \hat{P}] = i(1 + {\lambda^2}{\hat{P}^2})}[/math]

[math]\Large{[\hat{X}, \hat{P}] = i(1 + {\lambda^2}{\hat{P}^2})}[/math]

Equation 2.   Kempf - Mangano - Mann Commutation relation

Another example is Maggiore's generalization [4], as follows.

[math]\LARGE{[\hat{X}, \hat{P}] = i(1 + {\lambda^2}{\hat{P}^2} + {m^2{c^2}})}[/math]

[math]\Large{[\hat{X}, \hat{P}] = i(1 + {\lambda^2}{\hat{P}^2} + {m^2{c^2}})}[/math]

Equation 3.   Maggiore's Generalized Commutation relation

where [math]\normalsize{\lambda}[/math] denotes an extremely small Length Parameter; [math]\normalsize{m}[/math] is the Mass and [math]\normalsize{c}[/math] is the Light Speed.

Both commutation relations yield the generalized uncertainty relation,

[math]\LARGE{\Delta{x}\Delta{p} \geq {1 + \lambda^2{\Delta{p}^2}}}[/math]

[math]\Large{\Delta{x}\Delta{p} \geq {1 + \lambda^2{\Delta{p}^2}}}[/math]

Equation 4.   Maggiore's and K.M.M. Commutation Relations combination

with the second relation converging to this form in a suitable limit. A three-dimensional extension preserving rotational symmetry is proposed, leading to noncommutative geometry.

The Statistical Approach

The Statistical Mechanics of free ultra-relativistic particles subject to the Kempf-Mangano-Mann deformation are investigated within the Grand Canonical Ensemble approach. In traditional quantum statistical mechanics, where the parameter [math]\normalsize{\lambda}[/math] equals zero, the Heisenberg Uncertainty Principle partitions phase space into cells of volume [math]\normalsize{h^3}[/math], with [math]\normalsize{h}[/math] representing the Planck constant. Calculating the Grand Canonical Partition Function1 for a non-relativistic Ideal Gas2 in conventional quantum mechanics involves rewriting the sum over one-particle states in terms of integrals. In the high energy - temperature limit, characterized by negligible particle mass and inter - particle forces, particles behave akin to free ultra-relativistic particles, obeying Maxwell - Boltzmann Statistics. Consequently, the grand canonical partition function takes the following form

[math]\LARGE{ln{Z} = \beta{P}{V} = \int{\frac{e^{-{\beta{E}}{d^3{x}}{d^3{p}}}} {h^3(1 + {\lambda^2}{p^2})^3}}}[/math]

[math]\Large{ln{Z} = \beta{P}{V} = \int{{e^{-{\beta{E}}{d^3{x}}{d^3{p}}}} \over {h^3(1 + {\lambda^2}{p^2})^3}}}[/math]

Equation 5.   Grand Canonical Partition Function at High Energy - Temperature conditions

  • [math]\Large{Z}[/math]         is the Partition Function(1);

  • [math]\Large{\beta}[/math]          is the Thermodynamic Beta(2);

  • [math]\Large{P}[/math]         is the Pressure(2) of the system;

  • [math]\Large{V}[/math]        is the Volume(2) of the system;

  • [math]\Large{E}[/math]        is the Total Energy(2) of the system;

  • [math]\Large{x}[/math]         is the Position of the particle ([math]\Large{d^3}[/math] is the total differential cube);

  • [math]\Large{p}[/math]         is the Linear Momentum of the particle;

  • [math]\Large{\lambda}[/math]        is the Lenght Parameter of the uncertainty principle;

Contrary to conventional quantum statistical mechanics, this analysis reveals that the entropy and internal energy of the system attain finite values as temperature approaches infinity. Furthermore, the possibility of negative temperatures and pressures in such systems is explored, elucidating differences from Spin systems.


Figure 1.   A geometric horizontal Section of a Space - Time Fabric (white grid-like lattice) deformation with a Singularity (black part at the bottom)

No more Big Bang Singularities

The modified Equations of State due to the generalized uncertainty principle are considered in the context of the early Universe dynamics [5] (Figure 2). Additionally, the study suggests that negative temperatures may lead to alternative solutions for the Friedmann Equations (Equation 6), potentially altering the history of the Universe [6]. It is proposed that these modifications alone can potentially resolve the big bang singularity by ensuring a constant Entropy, thus avoiding singularities in the dynamical equations. This relation implies constant entropy, consistent with Reversibility assumptions [7].

                                                a)     [math]\LARGE{{\dot{a} \over a^2} = {1 \over 3}{\rho}}[/math]

                                                b)     [math]\LARGE{{\ddot{a} \over a} = - {1 \over 6}(\rho + 3P)}[/math]

       a)     [math]\large{{\dot{a} \over a^2} = {1 \over 3}{\rho}}[/math]

       b)     [math]\large{{\ddot{a} \over a} = - {1 \over 6}(\rho + 3P)}[/math]

Equation 6.   The two (a and b label) Friedmann Equations

[math]\Large{a}[/math] is a scalar factor (single and double upper dots represent, in the order, the first and second derivatives with respect to proper time); [math]\Large{\rho}[/math] is the Mass Density of Universe and [math]\Large{P}[/math] is the Pressure.

Substituting the equations of state in the Friedmann equations, it leads to following result.

[math]\LARGE{a^3({1 - {3 \over 2}{x^2}}) = \mathbf{Constant}}[/math]

[math]\Large{a^3({1 - {3 \over 2}{x^2}}) = constant}[/math]

Equation 7.   The reformuled Friedmann Equation

Importantly, the minimum scale factor is greater than zero, contrary to conventional cases. The singularity occurs at infinite temperature or [math]\normalsize{x = 0}[/math], but the term [math]\normalsize{a^3}[/math] does not tend to zero as [math]\normalsize{x \rightarrow 0}[/math], indicating finite Entropy at [math]\normalsize{x = 0}[/math], a non-trivial consequence of the minimal uncertainty principle.

craiyon_185234_Growing_Virtual_Universe (1)

Figure 2.   Pictorial representation of a Growing Newborn Universe

A very Important Principle

So the potential of the generalized uncertainty principle to address fundamental cosmological singularities, such as the big bang singularity, has been highlighted. By integrating this principle into the statistical mechanics of particle systems and considering its implications for the early universe, novel insights into the nature of Space - Time and the dynamics of cosmic evolution emerge. Further exploration of these concepts promises to deepen our understanding of the fundamental nature of the Universe.

  1. American Scientist. "Is String Theory Even Wrong?"

  2. Nature. "Experimental simulation of loop quantum gravity on a photonic chip"

  3. ResearchForLife7 (revisited from arXiv). "K.M.M. Overview on Minimal Uncertainty Length"

  4. ResearchForLife7 (revisited from ScienceDirect). "A Depth on the Maggiore's Commutation Relations"

  5. IOPscience. "Critical dynamics in the early universe"

  6. Scientific American. "Origin of the Universe"

  7. ResearchGate. "Time Reversibility and the Logical Structure of the Universe "


The Effect of Dark Matter and Dark Energy on Gravitational Time Advancement

Nowadays, the effects of Dark Matter (Figure 1) and Dark Energy on Gravitational Time Advancement, on local and large distance scales, are quite known. Some research consider various models of dark matter and dark energy, including the Cosmological Constant, Λ and compares their impact on gravitational phenomena with the standard Schwarzschild Geometry (Equation 1).

Beyond the Standard Theories...

Postulating the acceleration of the Universe's expansion, the latest advancements led to the inclusion of dark energy, characterized by a negative pressure, into the Energy - Momentum Tensor, Tik of the Universe. Dark matter, a non-luminous component dominating galaxies, is supported by observations such as rotation curve surveys, cosmic microwave background (Figure 2) measurements [1], and baryon acoustic oscillations [2]. 

[math]\Large{ds^2 = - \Biggl( - 1 - {2\mu \over r}\Biggr) dt^2 + \Biggl(1 - {2\mu \over r}\Biggr)dr^2 + {r^2 \over d\Omega^2}}[/math]

[math]\small{ds^2 = - \Biggl( - 1 - {2\mu \over r}\Biggr) dt^2 + \Biggl(1 - {2\mu \over r}\Biggr)dr^2 + {r^2 \over d\Omega^2}}[/math]

Equation 1.   The Schwarzschild Geometry equation


[math]\Large{\mu = {{G \cdot M} \over {{c_0}^2}}}[/math]


[math]\large{\mu = {{G \cdot M} \over {{c_0}^2}}}[/math]

  • [math]\Large{\textit{s}}[/math]        is a spherical Surface;

  • [math]\Large{\mu}[/math]       is the Schwarzschild Radius;

  • [math]\Large{G}[/math]      is the Gravitational Constant;

  • [math]\Large{M}[/math]    is the body Mass;

  • [math]\Large{c_{0}}[/math]     is the Light Speed in Vacuum;

  • [math]\Large{\textit{r}}[/math]        is the Radial Coordinate;

  • [math]\Large{\textit{t}}[/math]         is the Time Coordinate;

  • [math]\Large{\Omega}[/math]      is a Point on the two sphere [math]\Large{{\textit{S}} \hspace{0.1cm} ^2}[/math];

Dark Energy and Dark Matter Models

The first and important candidate for dark energy is the cosmological constant, characterized by a constant Energy Density with an Equation-Of-State Parameter, w  equal to -1. However, its theoretical problem lies in its size, significantly lower than the expected Vacuum Energy Density. Various alternative theories, including Quintessence [3]  and Chaplygin Gas Models, (CG) [4] have been proposed to address this issue. Modifications to General Relativity, such as scalar tensor theories and [math]\Large{f(R)}[/math] gravity models, have also been suggested to explain late-time accelerated expansion with no reference to dark energy.


Figure 1.   A figurative representation of Cosmos objects (white lights) surrounded
by Dark Matter lattice (red and blue structure)

Similarly, dark matter candidates include WIMPs [5], Axions, and sterile Neutrinos. Theories like MOND propose modifications to Newtonian Dynamics, while Conformal Gravitational Theory [6], based on Weyl Symmetry, aims to explain flat rotation curves of Galaxies without the need for dark matter.

What is the Gravitational Time Advancement?

Gravitational Time Advancement occurs when an observer is in a stronger gravitational field compared to the photon's trajectory. A study finds that dark energy leads to a gravitational time delay, reducing the gravitational time advancement effect. In contrast, the conformal theory suggests a large time advancement effect, particularly at distances beyond approximately 30 kpc (Kiloparsec).


Figure 2.   An instance of small ground station satellite antenna to pick up
the Milky Way Microwave Background

Experimental Verification and Future Missions

This article emphasizes the challenging nature of experimentally verifying the gravitational time advancement effect. BEACON [7] and GRACE-FO [8] missions are expected to provide a higher accuracy in probing the Earth's gravitational field, offering opportunities to measure gravitational time advancement effects at large distances.

Gravitational Time Advancement as a Proof of Dark Matter and Dark Energy

However, analytical expressions are derived, revealing that dark energy induces a gravitational time delay, while the Schwarzschild Metric results in both time delay and time advancement effects. The findings suggest that measuring gravitational time advancement at large distances could serve as a means to prove the dark matter and certain dark energy models. Also parameters, on upper bound conditions, will be affected by this gravitational phenomenon.

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  2. Nasa (.gov). "Baryon Acoustic Oscillations - Roman Space Telescope"

  3. "The quintessence of cosmology"

  4. Academic Accelerator. "Chaplygin Gas: Most Up-to-Date Encyclopedia, News & Reviews"

  5. Arstechnica. "No WIMPS! Heavy particles don’t explain gravitational lensing oddities"

  6. Physics.wustl. "Quantum Conformal Gravity"

  7. ResearchGate. "A Search for New Physics with the BEACON Mission"

  8. Forbes. "NASA's GRACE-FO Mission Will Study How Earth's Climate Is Evolving"

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