by jsendak | Jan 8, 2024 | GR & QC Articles
Random tensor networks (RTNs) have proved to be fruitful tools for modelling
the AdS/CFT correspondence. Due to their flat entanglement spectra, when
discussing a given boundary region $R$ and its complement $bar R$, standard
RTNs are most analogous to fixed-area states of the bulk quantum gravity
theory, in which quantum fluctuations have been suppressed for the area of the
corresponding HRT surface. However, such RTNs have flat entanglement spectra
for all choices of $R, bar R,$ while quantum fluctuations of multiple
HRT-areas can be suppressed only when the corresponding HRT-area operators
mutually commute. We probe the severity of such obstructions in pure AdS$_3$
Einstein-Hilbert gravity by constructing networks whose links are codimension-2
extremal-surfaces and by explicitly computing semiclassical commutators of the
associated link-areas. Since $d=3,$ codimension-2 extremal-surfaces are
geodesics, and codimension-2 `areas’ are lengths. We find a simple 4-link
network defined by an HRT surface and a Chen-Dong-Lewkowycz-Qi constrained HRT
surface for which all link-areas commute. However, the algebra generated by the
link-areas of more general networks tends to be non-Abelian. One such
non-Abelian example is associated with entanglement-wedge cross sections and
may be of more general interest.
Random tensor networks (RTNs) have been valuable in modeling the AdS/CFT correspondence. However, while standard RTNs have flat entanglement spectra for all choices of boundary regions, quantum fluctuations of multiple HRT-areas can only be suppressed if the corresponding HRT-area operators mutually commute. This poses a challenge in constructing networks using extremal-surfaces as links.
Future Roadmap
1. Exploring Pure AdS$_3$ Einstein-Hilbert Gravity
A potential challenge in pure AdS$_3$ Einstein-Hilbert gravity is understanding the severity of obstructions caused by non-commuting link-areas in network construction. By constructing networks using codimension-2 extremal-surfaces as links and calculating the semiclassical commutators of the associated link-areas, we can investigate this problem further.
2. Finding Commuting Link-Areas
In the case of codimension-2 `areas’ being lengths and geodesics in $d=3,$ we discovered a simple 4-link network consisting of an HRT surface and a constrained HRT surface that commute. This finding suggests that it may be possible to identify other specific network configurations where all link-areas commute. This should be explored further to understand the limitations and opportunities associated with such networks.
3. Non-Abelian Algebra and Entanglement-Wedge Cross Sections
While the 4-link network provided a commutative algebra for link-areas, more general networks tend to have non-Abelian algebras. One example of a non-Abelian network is associated with entanglement-wedge cross sections. Investigating these non-Abelian networks and their properties is of interest for a deeper understanding of the AdS/CFT correspondence.
Challenges and Opportunities
- Challenge: The main challenge lies in understanding the severity of obstructions caused by non-commuting link-areas in network construction.
- Opportunity: The discovery of a 4-link network with commuting link-areas suggests that it may be possible to identify other specific configurations for which all link-areas commute.
- Opportunity: Exploring non-Abelian networks, such as the one associated with entanglement-wedge cross sections, can provide valuable insights into the AdS/CFT correspondence.
Key Takeaway: The study of random tensor networks in the context of the AdS/CFT correspondence has shown that while standard RTNs have flat entanglement spectra, non-commuting link-areas pose challenges in network construction. However, exploring specific network configurations and non-Abelian networks can provide opportunities for further understanding and advancements in this area.
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by jsendak | Jan 4, 2024 | GR & QC Articles
Zermelo navigation is not only a fundamental tool in Finsler geometry but
also a fundamental approach to the geometrization of dynamics in physics. In
this paper, we consider the Zermelo navigation problem on optical Riemannian
space and, via Zermelo/Randers/spacetime triangle, explore the generation of
new spacetimes from pre-existing ones. Whether the Randers metric has
reversible geodesics corresponds to the presence of time-reversal symmetry in
the generated spacetime. In cases where the Randers metric has reversible
geodesics, we utilize a radial vector field to generate new static spacetimes
from existing ones. For example, we can generate Schwarzschild, Rindler, de
Sitter, and Schwarzschild-de Sitter spacetimes from flat spacetime. In fact,
the Zermelo navigation method allows for the derivation of a variety of static
spacetimes from flat spacetime. For multi-parameter spacetimes, they can be
generated through various navigation paths. However, for some spacetimes, not
all navigation paths may exist. In the second scenario, when the Randers metric
does not have reversible geodesics, we employ a rotational vector field to
transform non-flat static metrics into slowly rotating spacetimes.
Alternatively, using a mixed vector field, we generate slowly rotating
spacetimes starting from flat spacetime. We provide examples of generating Kerr
spacetimes and Kerr-de Sitter spacetimes.
Zermelo Navigation and the Geometrization of Dynamics in Physics
In this paper, we have explored the concept of Zermelo navigation problem in the context of optical Riemannian space and its implications in the generation of new spacetimes from pre-existing ones. The Zermelo/Randers/spacetime triangle provides us with a framework to understand this generation process.
- The presence of reversible geodesics in the Randers metric indicates the existence of time-reversal symmetry in the generated spacetime.
- When the Randers metric has reversible geodesics, we can utilize a radial vector field to generate new static spacetimes from existing flat spacetime. Examples include Schwarzschild, Rindler, de Sitter, and Schwarzschild-de Sitter spacetimes.
- For multi-parameter spacetimes, various navigation paths can be utilized to generate them.
- Not all navigation paths may exist for some spacetimes.
- In scenarios where the Randers metric does not have reversible geodesics, we can employ a rotational vector field to transform non-flat static metrics into slowly rotating spacetimes.
- Alternatively, using a mixed vector field, we can generate slowly rotating spacetimes starting from flat spacetime.
- We provide examples of generating Kerr spacetimes and Kerr-de Sitter spacetimes using these techniques.
Roadmap for Future Exploration
The findings presented in this paper open up several avenues for future research and exploration:
- Further investigation into the relationship between Zermelo navigation and Finsler geometry, and how it can be applied to other areas of physics beyond optics.
- Exploration of the limitations and constraints of generating spacetimes through Zermelo navigation. Understanding which spacetimes can be generated and which cannot.
- Developing more comprehensive methods for generating multi-parameter spacetimes using various navigation paths.
- Investigation into the physical properties and implications of the generated spacetimes. How do they compare to known spacetimes? What are their unique characteristics?
- Extending the application of Zermelo navigation to other mathematical frameworks and theories, such as general relativity or quantum mechanics.
Challenges and Opportunities
While the concept of Zermelo navigation in the generation of new spacetimes presents exciting opportunities, there are also challenges to be addressed:
- The mathematical complexity involved in understanding and calculating the Randers metric and its geodesics.
- The identification of navigation paths for generating specific spacetimes may require advanced mathematical techniques and computations.
- Limited availability of known spacetimes with reversible geodesics, which may restrict the range of generated spacetimes.
- Interpreting and understanding the physical significance of the generated spacetimes and their implications in real-world dynamics.
- Potential conflicts or inconsistencies with existing theories or frameworks in physics, which may need to be resolved or reconciled.
In conclusion, the Zermelo navigation method offers a promising approach to generating new spacetimes from existing ones, extending our understanding of dynamics in physics. Further research and exploration in this field can lead to significant advancements and insights in various areas of theoretical and applied physics.
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by jsendak | Jan 1, 2024 | GR & QC Articles
We describe the dynamical formation of the shadow of a collapsing star in a
Hayward spacetime in terms of an observer far away from the center and a free
falling observer. By solving the time-like and light-like radial geodesics we
determine the angular size of the shadow as a function of time. We found that
the formation of the shadow is a finite process for both observers and its size
is affected by the Hayward spacetime parameters. We consider several scenarios,
from the Schwarzschild limit to an extreme Hayward black hole.
Examining the Conclusions of the Text:
In the given text, the authors describe the dynamical formation of the shadow of a collapsing star in a Hayward spacetime. By studying the behaviors of an observer far away from the center and a free-falling observer, the authors determine the angular size of the shadow as a function of time. They find that the formation of the shadow is a finite process for both observers and that its size is influenced by the parameters of the Hayward spacetime.
Future Roadmap and Potential Challenges:
- Further Investigation of the Shadow Formation: Building on the findings of this study, future research could focus on understanding the intricacies of the shadow formation process in a Hayward spacetime. This could involve studying different collapsing star scenarios and varying the parameters of the spacetime to explore their impact on the size and dynamics of the shadow.
- Comparative Analysis with Other Spacetimes: To gain a comprehensive understanding of shadow formation, comparative analysis with other types of spacetimes, such as Schwarzschild or Kerr, may be necessary. This would help establish similarities and differences in the behavior of shadows under various gravitational influences.
- Experimental Validation: The theoretical predictions derived from this study could be tested through observational data, such as astrophysical observations or simulations. Experimental validation would provide concrete evidence for the formation process and allow for further refinement of theoretical models.
- Gravitational Wave Effects: Investigating the influence of gravitational waves on shadow formation in a Hayward spacetime could be an interesting area for future research. Understanding how these waves interact with collapsing stars and affect the size and dynamics of the resulting shadows could deepen our knowledge of both gravitational wave physics and black hole astrophysics.
Opportunities on the Horizon:
The findings presented in this study open up several opportunities for future research and exploration in the field of black hole astrophysics. The following opportunities could be pursued:
- Advancing our Understanding of Black Hole Dynamics: By further investigating the formation and evolution of black hole shadows in different spacetimes, we can enhance our understanding of the complex dynamics involved in these extreme gravitational systems. This knowledge could contribute to our overall comprehension of black hole astrophysics.
- Contributions to Fundamental Physics Theories: The study of shadow formation in different spacetimes, including the Hayward spacetime considered here, may have implications for fundamental physics theories such as general relativity, quantum gravity, and black hole thermodynamics. Insights gained from these studies may offer invaluable contributions to these fields.
- Potential Applications in Astrophysical Observations: Understanding the behavior of shadows in collapsing star scenarios and under different spacetime parameters can have practical applications in astrophysical observations. By analyzing the observed shadows, scientists could gain insights into the properties of the central collapsed objects, providing valuable information about their mass, spin, and other characteristics.
In conclusion, the study of shadow formation in a Hayward spacetime has revealed important insights into the dynamics of collapsing stars and their resulting shadows. Building on these findings, future research directions include further investigating the shadow formation process, comparative analysis with other spacetimes, experimental validation, and studying the influence of gravitational waves. These opportunities hold the potential to significantly advance our understanding of black hole astrophysics, contribute to fundamental physics theories, and find applications in astrophysical observations.
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by jsendak | Dec 31, 2023 | GR & QC Articles
We consider a congruence of null geodesics in the presence of a quantized
spacetime metric. The coupling to a quantum metric induces fluctuations in the
congruence; we calculate the change in the area of a pencil of geodesics
induced by such fluctuations. For the gravitational field in its vacuum state,
we find that quantum gravity contributes a correction to the null Raychaudhuri
equation which is of the same sign as the classical terms. We thus derive a
quantum-gravitational focusing theorem valid for linearized quantum gravity.
Recent research has explored the behavior of null geodesics in the presence of a quantized spacetime metric. By coupling the geodesics to a quantum metric, researchers have observed fluctuations in the congruence of the geodesics. In particular, they have calculated the change in the area of a pencil of geodesics caused by these fluctuations.
Notably, for the gravitational field when it is in a vacuum state, the study reveals that quantum gravity introduces a correction to the null Raychaudhuri equation. Importantly, this correction is of the same sign as the classical terms. Thus, a quantum-gravitational focusing theorem can be derived that is valid for linearized quantum gravity.
Roadmap for the Future
1. Further Study and Understanding
To advance our knowledge in this field, further study and research are needed. It is crucial to better comprehend the behavior and implications of null geodesics under a quantized spacetime metric. Researchers should focus on investigating different scenarios and explore the impact of various conditions on these geodesic fluctuations. This could involve studying different quantum metrics and their effects on the congruence of null geodesics.
2. Experimental Validation
One of the challenges ahead lies in experimentally verifying the findings from theoretical calculations. Designing and conducting experiments that can observe and measure the fluctuations induced by quantum gravity will be crucial in validating the derived quantum-gravitational focusing theorem. Experimental setups should aim to test the predictions made and provide empirical evidence for the effects of quantized spacetime metrics on null geodesics.
3. Applications to Cosmology
The understanding gained from studying the behavior of null geodesics in the presence of a quantized spacetime metric can have significant implications for cosmology. By incorporating the effects of quantum gravity into cosmological models, we may gain new insights into the behavior and evolution of the universe. This could potentially lead to advances in our understanding of the early universe, dark matter, and other cosmological phenomena.
4. Challenges and Limitations
While this research provides valuable insights, there are challenges and limitations that need to be addressed. The complexity of quantized spacetime metrics and the calculations involved make this field highly theoretical and mathematically intensive. Collaborative efforts between physicists, mathematicians, and computer scientists will be necessary to overcome these challenges and make further progress.
Overall, this research on null geodesics in the presence of a quantized spacetime metric opens up new avenues for the study of quantum gravity. The derived quantum-gravitational focusing theorem provides a framework for understanding the behavior of linearized quantum gravity on null geodesics. The future roadmap includes further study, experimental validation, applications to cosmology, and addressing the challenges and limitations in this field.
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by jsendak | Dec 30, 2023 | GR & QC Articles
In this paper, we investigate quasinormal modes of scalar and electromagnetic
fields in the background of Einstein–scalar–Gauss–Bonnet (EsGB) black holes.
Using the scalar and electromagnetic field equations in the vicinity of the
EsGB black hole, we study nature of the effective potentials. The dependence of
real and imaginary parts of the fundamental quasinormal modes on parameter $p$
(which is related to the Gauss–Bonnet coupling parameter $alpha$) for
different values of multipole numbers $l$ are studied. We analyzed the effects
of massive scalar fields on the EsGB black hole, which tells us the existence
of quasi–resonances. In the eikonal regime, we find the analytical expression
for the quasinormal frequency and show that the correspondence between the
eikonal quasinormal modes and null geodesics is valid in the EsGB theory for
the test fields. Finally, we study grey-body factors of the electromagnetic
fields for different multipole numbers $l$, which deviates from Schwarzschild’s
black hole.
Conclusion:
- The paper investigated the quasinormal modes of scalar and electromagnetic fields in the background of Einstein-scalar-Gauss-Bonnet (EsGB) black holes.
- The study analyzed the effects of massive scalar fields on the EsGB black hole, revealing the existence of quasi-resonances.
- In the eikonal regime, an analytical expression for the quasinormal frequency was derived, showing the validity of the correspondence between eikonal quasinormal modes and null geodesics in the EsGB theory for test fields.
- The grey-body factors of the electromagnetic fields for different multipole numbers deviated from Schwarzschild’s black hole.
Future Roadmap
Potential Challenges:
- Further investigation is needed to explore the nature of the effective potentials for both scalar and electromagnetic fields in the vicinity of EsGB black holes.
- The dependence of the real and imaginary parts of the fundamental quasinormal modes on the parameter p (related to the Gauss-Bonnet coupling parameter α) should be further studied for a wider range of multipole numbers l.
- More research is required to fully understand the effects of massive scalar fields on the EsGB black hole and its implications for quasi-resonances.
- Validation and verification of the analytical expression for the quasinormal frequency in the eikonal regime using experimental or observational data is necessary to establish its practicality.
- Further exploration is needed to understand the deviation of grey-body factors of electromagnetic fields in EsGB black holes compared to Schwarzschild’s black hole, and its potential implications.
Opportunities on the Horizon:
- This research opens up opportunities for studying complex physics phenomena in EsGB black holes and their implications for the nature of gravitational waves and singularities.
- In-depth understanding of the effects of massive scalar fields on EsGB black holes and the existence of quasi-resonances can contribute to advancements in theoretical physics and astrophysics.
- The analytical expression for the quasinormal frequency in the eikonal regime and its validation could provide a valuable tool for future research in black hole physics.
- The exploration of deviations in grey-body factors of electromagnetic fields in EsGB black holes can lead to new insights into the behavior of matter and radiation near these exotic objects.
Overall, this research paves the way for further investigation into the properties and behaviors of EsGB black holes, presenting both challenges and opportunities for advancements in theoretical physics and astrophysics.
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