by jsendak | Sep 23, 2024 | GR & QC Articles
arXiv:2409.13069v1 Announce Type: new
Abstract: We generalize the targeted $mathcal{B}$-statistic for continuous gravitational waves by modeling the $h_0$-prior as a half-Gaussian distribution with scale parameter $H$. This approach retains analytic tractability for two of the four amplitude marginalization integrals and recovers the standard $mathcal{B}$-statistic in the strong-signal limit ($Hrightarrowinfty$). By Taylor-expanding the weak-signal regime ($Hrightarrow0$), the new prior enables fully analytic amplitude marginalization, resulting in a simple, explicit statistic that is as computationally efficient as the maximum-likelihood $mathcal{F}$-statistic, but significantly more robust. Numerical tests show that for day-long coherent searches, the weak-signal Bayes factor achieves sensitivities comparable to the $mathcal{F}$-statistic, though marginally lower than the standard $mathcal{B}$-statistic (and the Bero-Whelan approximation). In semi-coherent searches over short (compared to a day) segments, this approximation matches or outperforms the weighted dominant-response $mathcal{F}_{mathrm{ABw}}$-statistic and returns to the sensitivity of the (weighted) $mathcal{F}_{mathrm{w}}$-statistic for longer segments. Overall the new Bayes-factor approximation demonstrates state-of-the-art or improved sensitivity across a wide range of segment lengths we tested (from 900s to 10days).
Future Roadmap: Challenges and Opportunities
Based on the conclusions of the article, there are several potential challenges and opportunities on the horizon for readers.
1. Incorporation of the Generalized $mathcal{B}$-statistic
The article proposes a generalized $mathcal{B}$-statistic for continuous gravitational waves. This statistic models the $h_0$-prior as a half-Gaussian distribution with scale parameter $H$. Readers should explore the potential of incorporating this new statistical approach into their own research or projects.
2. Analytic Tractability and Computation Efficiency
The new prior enables fully analytic amplitude marginalization, resulting in a simple and explicit statistic. This makes the computation efficient, similar to the maximum-likelihood $mathcal{F}$-statistic. Readers should investigate how this analytic tractability and computational efficiency can improve their current methods and algorithms.
3. Sensitivity of Weak-Signal Bayes Factor
Numerical tests show that the weak-signal Bayes factor achieves sensitivities comparable to the $mathcal{F}$-statistic, though marginally lower than the standard $mathcal{B}$-statistic. Readers should explore the sensitivity of the weak-signal Bayes factor in their specific applications and evaluate its performance against existing methods.
4. Performance in Semi-Coherent Searches
In semi-coherent searches, the new Bayes-factor approximation matches or outperforms the weighted dominant-response $mathcal{F}_{mathrm{ABw}}$-statistic for short segments. However, for longer segments, it returns to the sensitivity of the (weighted) $mathcal{F}_{mathrm{w}}$-statistic. Readers should analyze the performance of this new approximation in their specific semi-coherent searches and compare it with other existing statistics.
5. Wide Range of Segment Lengths
The new Bayes-factor approximation demonstrates state-of-the-art or improved sensitivity across a wide range of segment lengths tested (from 900s to 10 days). This opens up the opportunity for readers to explore different segment lengths and assess the performance and efficiency of the new approximation in their particular scenarios.
In conclusion, readers should consider adopting the generalized $mathcal{B}$-statistic, explore its analytic tractability and computational efficiency, evaluate the sensitivity of the weak-signal Bayes factor, compare the performance in semi-coherent searches, and assess the applicability of the new approximation across various segment lengths. By addressing these challenges and leveraging the opportunities presented, researchers can advance the field of continuous gravitational waves detection and analysis.
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by jsendak | Sep 21, 2024 | GR & QC Articles
arXiv:2409.12204v1 Announce Type: new
Abstract: We revisit the dynamics of the post-Newtonian (PN) two-body problem for two inspiraling compact bodies. Starting from a matter-only reduced Hamiltonian, we present an adapted framework based on the Lie series approach, enabling the derivation of complete perturbative solutions within the conservative sector. Our framework supports both circular and eccentric orbits, and is applicable to any perturbation respecting rotational invariance and time-independence. In the context of the Arnowitt-Deser-Misner (ADM) canonical formalism, this includes up to at least 3PN order and local terms beyond. We provide an example application at 2PN, recovering classical periapsis advance and orbital period corrections, alongside the full orbital evolution in time coordinates. We discuss eventual extension to spinning and time-dependent systems.
Revisiting the Dynamics of the Post-Newtonian Two-Body Problem
In this article, we revisit the dynamics of the post-Newtonian (PN) two-body problem for inspiraling compact bodies. We present an adapted framework based on the Lie series approach, allowing for the derivation of complete perturbative solutions within the conservative sector.
Framework for Circular and Eccentric Orbits
Our framework supports both circular and eccentric orbits and is applicable to any perturbation respecting rotational invariance and time-independence. This framework encompasses up to at least 3PN order and local terms beyond, in the context of the Arnowitt-Deser-Misner (ADM) canonical formalism.
Example Application at 2PN
To demonstrate the effectiveness of our framework, we provide an example application at 2PN. We were able to recover classical periapsis advance and orbital period corrections, along with the complete orbital evolution in time coordinates.
Potential Challenges and Opportunities
Although our current framework is limited to non-spinning and time-independent systems, there is potential for future extension to include spinning and time-dependent systems. This would allow for a more comprehensive understanding of the dynamics of inspiraling compact bodies.
Roadmap for Readers
- Introduction to the post-Newtonian two-body problem.
- Overview of the Lie series approach and its adaptation for the problem.
- Explanation of the framework’s applicability to circular and eccentric orbits.
- Discussion of the inclusion of perturbations respecting rotational invariance and time-independence.
- Illustration of an example application at 2PN, showcasing the recovery of periapsis advance, orbital period corrections, and orbital evolution.
- Potential challenges and opportunities for future extension to spinning and time-dependent systems.
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by jsendak | Sep 20, 2024 | GR & QC Articles
arXiv:2409.11425v1 Announce Type: new
Abstract: In a recent paper: “On the time dependency of $a_0$” the authors claim that they have tested “one of the predictions of the Scale Invariant Vacuum (SIV) theory on MOND” by studying the dependence of the Modified Newtonian Dynamics (MOND) acceleration at two data sets, low-$z$ ($3.2times10^{-4}le zle 3.2times10^{-2}$) and high-$z$ ($0.5le zle 2.5$). They claim “both samples show a dependency of $a_0$ from $z$”. Here, the work mentioned above is revisited. The explicit analytic expression for the $z$-dependence of the $a_0$ within the SIV theory is given. Furthermore, the first estimates of the $Omega_m$ within SIV theory give $Omega_{m}=0.28pm 0.04$ using the low-z data only, while a value of $Omega_{m}=0.055$ is obtained using both data sets. This much lower $Omega_m$ leaves no room for non-baryonic matter! Unlike in the mentioned paper above, the slope in the $z$-dependence of $A_0=log_{10}(a_0)$ is estimated to be consistent with zero Z-slope for the two data sets. Finally, the statistics of the data are consistent with the SIV predictions; in particular, the possibility of change in the sign of the slopes for the two data sets is explainable within the SIV paradigm; however, the uncertainty in the data is too big for the clear demonstration of a $z$-dependence yet.
Future Roadmap for Readers: Challenges and Opportunities on the Horizon
The recent paper “On the time dependency of $a_0$” claims to have tested a prediction of the Scale Invariant Vacuum (SIV) theory on Modified Newtonian Dynamics (MOND) by studying the dependence of MOND acceleration at two data sets: low-z (.2times10^{-4}le zle 3.2times10^{-2}$) and high-z ([openai_gpt model=”gpt-3.5-turbo-16k” max_tokens=”3000″ temperature=”1″ prompt=”Examine the conclusions of the following text and outline a future roadmap for readers, indicating potential challenges and opportunities on the horizon. The article should be formatted as a standalone HTML content block, suitable for embedding in a WordPress post. Use only the following HTML tags:
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. Exclude all other HTML tags, including those for page structure: arXiv:2409.11425v1 Announce Type: new
Abstract: In a recent paper: “On the time dependency of $a_0$” the authors claim that they have tested “one of the predictions of the Scale Invariant Vacuum (SIV) theory on MOND” by studying the dependence of the Modified Newtonian Dynamics (MOND) acceleration at two data sets, low-$z$ ($3.2times10^{-4}le zle 3.2times10^{-2}$) and high-$z$ ($0.5le zle 2.5$). They claim “both samples show a dependency of $a_0$ from $z$”. Here, the work mentioned above is revisited. The explicit analytic expression for the $z$-dependence of the $a_0$ within the SIV theory is given. Furthermore, the first estimates of the $Omega_m$ within SIV theory give $Omega_{m}=0.28pm 0.04$ using the low-z data only, while a value of $Omega_{m}=0.055$ is obtained using both data sets. This much lower $Omega_m$ leaves no room for non-baryonic matter! Unlike in the mentioned paper above, the slope in the $z$-dependence of $A_0=log_{10}(a_0)$ is estimated to be consistent with zero Z-slope for the two data sets. Finally, the statistics of the data are consistent with the SIV predictions; in particular, the possibility of change in the sign of the slopes for the two data sets is explainable within the SIV paradigm; however, the uncertainty in the data is too big for the clear demonstration of a $z$-dependence yet.”].5le zle 2.5$). The authors find a dependency of $a_0$ on $z$ in both data sets, which prompts a revisit of their work. The aim of this roadmap is to outline potential challenges and opportunities in understanding the implications of this research.
Challenges
- Data Uncertainty: The uncertainty in the data is currently too large to clearly demonstrate a significant $z$-dependence of $a_0$. Further analysis and data collection with reduced uncertainties are required to validate this dependency.
- Non-Baryonic Matter: The low value of $Omega_m=0.055$ obtained using both data sets leaves no room for non-baryonic matter. This challenges current cosmological models that rely on the presence of non-baryonic matter to explain certain phenomena.
Opportunities
- SIV Theory: The explicit analytic expression for the $z$-dependence of $a_0$ within the SIV theory is provided, offering a potential explanation for the observed dependency. Further exploration of the SIV theory may lead to new insights into the relationship between MOND and cosmological dynamics.
- Z-Slope Consistency: Unlike the previous paper, the estimate for the slope in the $z$-dependence of $A_0=log_{10}(a_0)$ is found to be consistent with a zero Z-slope for both data sets. This finding supports the SIV paradigm and suggests that MOND may indeed be influenced by cosmological factors.
- Estimates of $Omega_m$: The first estimates of $Omega_m$ within the SIV theory are provided, giving values of $Omega_{m}=0.28pm 0.04$ using the low-z data and $Omega_{m}=0.055$ using both data sets. These estimates offer valuable insights into the cosmological matter content and can inform future research in this area.
In conclusion, while the mentioned paper provides intriguing evidence for a $z$-dependence of $a_0$ in MOND, further investigation is needed to overcome the challenges posed by data uncertainties and the absence of non-baryonic matter. Opportunities lie in exploring the implications of the SIV theory, understanding the consistent Z-slope estimate, and refining the estimates of $Omega_m$. These avenues of research offer promising prospects for advancing our understanding of the fundamental nature of MOND and its connection to cosmological dynamics.
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by jsendak | Sep 19, 2024 | GR & QC Articles
arXiv:2409.10602v1 Announce Type: new
Abstract: We derive the equations of motion for relativistic elastic membranes, that is, two-dimensional elastic bodies whose internal energy depends only on their stretching, starting from a variational principle. We show how to obtain conserved quantities for the membrane’s motion in the presence of spacetime symmetries, determine the membrane’s longitudinal and transverse speeds of sound in isotropic states, and compute the coefficients of linear elasticity with respect to the relaxed configuration. We then use this formalism to discuss two physically interesting systems: a rigidly rotating elastic disk, widely discussed in the context of Ehrenfest’s paradox, and a Dyson sphere, that is, a spherical membrane in equilibrium in Schwarzschild’s spacetime, with the isotropic tangential pressure balancing the gravitational attraction. Surprisingly, although spherically symmetric perturbations of this system are linearly stable, the axi-symmetric dipolar mode is already unstable. This may be taken as a cautionary tale against misconstruing radial stability as true stability.
The Future Roadmap for Relativistic Elastic Membranes
Introduction
In this article, we explore the equations of motion for relativistic elastic membranes and discuss their properties and applications. We derive these equations from a variational principle and investigate the conserved quantities and speeds of sound associated with the membrane’s motion. Additionally, we analyze two interesting systems: a rigidly rotating elastic disk and a Dyson sphere in equilibrium.
Challenges
- Understanding the mathematical derivation of the equations of motion for relativistic elastic membranes.
- Determining the conserved quantities and speeds of sound for membranes in the presence of spacetime symmetries.
- Calculating the coefficients of linear elasticity with respect to the relaxed configuration of the membrane.
- Exploring the stability of spherically symmetric perturbations in the Dyson sphere system.
Opportunities
- Advancing our understanding of the behavior of relativistic elastic membranes in various scenarios.
- Potential applications in physics and engineering fields, such as modeling the behavior of materials under extreme conditions.
- Developing new theoretical frameworks for studying the dynamics of spacetime symmetries and their effects on elastic membranes.
Conclusion
Studying relativistic elastic membranes provides insights into the behavior of two-dimensional elastic bodies and their interactions with spacetime symmetries. By deriving the equations of motion, determining conserved quantities, and exploring specific systems, we can deepen our understanding of the intricate dynamics involved. However, challenges remain in comprehending the mathematics and analyzing stability, offering opportunities for further research and potential applications in diverse fields.
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by jsendak | Sep 18, 2024 | GR & QC Articles
arXiv:2409.09094v1 Announce Type: new
Abstract: The Kerr-star spacetime is the extension over the horizons and in the negative radial region of the Kerr spacetime. Despite the presence of closed timelike curves below the inner horizon, we prove that the timelike geodesics cannot be closed in the Kerr-star spacetime. Since the existence of closed null geodesics was ruled out by the author in Sanzeni [arXiv:2308.09631v3 (2024)], this result shows the absence of closed causal geodesics in the Kerr-star spacetime.
The Future of Causal Geodesics in the Kerr-star Spacetime
In a recent study, researchers have examined the properties of the Kerr-star spacetime, an extension of the well-known Kerr spacetime. The Kerr-star spacetime includes regions beyond the horizons and in the negative radial region.
Exploring Timelike Geodesics
One intriguing aspect of the Kerr-star spacetime is the presence of closed timelike curves below the inner horizon. These closed timelike curves have been a subject of interest due to their potential for time travel. However, the researchers have made a fascinating discovery – they have proven that timelike geodesics cannot be closed in the Kerr-star spacetime.
Absence of Closed Causal Geodesics
Moreover, a previous study by author Sanzeni has already ruled out the existence of closed null geodesics in the Kerr-star spacetime. This recent result complements the previous findings by demonstrating the absence of closed causal geodesics as well.
Roadmap for the Future
While the current research sheds light on the topology of the Kerr-star spacetime, there are several avenues for further exploration and challenges to overcome:
- Understanding the nature of closed timelike curves: Despite the inability to form closed timelike geodesics in the Kerr-star spacetime, the presence of closed timelike curves below the inner horizon remains intriguing. Future studies should delve deeper into the properties and implications of these curves.
- Extending the analysis to other spacetimes: The results obtained in this study are specific to the Kerr-star spacetime. It would be beneficial to investigate the presence or absence of closed causal geodesics in other spacetimes as well.
- Examining the consequences of these findings: The absence of closed causal geodesics in the Kerr-star spacetime has implications for our understanding of the behavior of particles and light in this exotic region. Further research should focus on unraveling the consequences of this absence and its potential impact on theoretical models.
- Experimental validation: While theoretical studies offer deep insights, experimental validation is crucial to confirm the findings. Scientists could design experimental setups or observations to test the predictions and conclusions derived from the Kerr-star spacetime.
Conclusion
The recent research on the Kerr-star spacetime has unveiled the absence of closed causal geodesics, complementing the earlier findings that ruled out closed null geodesics. This opens up new avenues of research into the nature of closed timelike curves, the exploration of other spacetimes, understanding the consequences of these findings, and experimental validation. By delving into these areas, scientists can continue to push the boundaries of our understanding of the Kerr-star spacetime and its implications in the broader context of general relativity and theoretical physics.
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by jsendak | Sep 17, 2024 | GR & QC Articles
arXiv:2409.08344v1 Announce Type: new
Abstract: We study the exterior solution for a static, spherically symmetric source in Weyl conformal gravity in terms of the Newman–Penrose formalism. We first show that both the static, uncharged black hole solution of Mannheim and Kazanas and the static, charged Reissner–Nordstr”{o}m-like solution can be found more easily in this formalism than in the traditional coordinate-basis approach, where the metric tensor components are taken as the basic variables. Second, we show that the Newman-Penrose formalism offers a particularly convenient framework that is well suited for the discussion of conformal gravity solutions corresponding to Petrov ”type-D” spacetimes. This is illustrated with a two-parameter class of wormhole solutions that includes the Ellis–Bronnikov wormhole solution of Einstein’s gravity as a limiting case. Other salient issues, such as the gauge equivalence of solutions and the inclusion of the electromagnetic field are also discussed.
Introduction
In this article, we explore the exterior solution for a static, spherically symmetric source in Weyl conformal gravity using the Newman-Penrose formalism. We highlight the advantages of this formalism over the traditional coordinate-based approach and discuss its applications in the study of conformal gravity solutions.
Advantages of the Newman-Penrose Formalism
We demonstrate that the Newman-Penrose formalism provides a more straightforward method for finding both the static, uncharged black hole solution and the static, charged Reissner-Nordström-like solution as compared to the traditional coordinate-basis approach. By utilizing the metric tensor components as basic variables, we simplify the computation process.
Applications in Conformal Gravity
We illustrate how the Newman-Penrose formalism offers a convenient framework for analyzing conformal gravity solutions corresponding to Petrov “type-D” spacetimes. We present a specific class of wormhole solutions that includes the Ellis-Bronnikov wormhole solution of Einstein’s gravity as a limiting case. This demonstrates the potential for utilizing conformal gravity to achieve wormhole solutions with interesting properties.
Other Salient Issues
We also address additional significant topics in this article. We discuss the gauge equivalence of solutions in the Newman-Penrose formalism, highlighting the importance of considering different gauge choices to obtain a complete understanding of the physics involved. Additionally, we explore the inclusion of the electromagnetic field and its impact on the conformal gravity solutions.
Future Roadmap, Challenges, and Opportunities
Roadmap
- Further explore the Newman-Penrose formalism for other types of solutions in Weyl conformal gravity
- Investigate the physical implications and potential applications of the two-parameter class of wormhole solutions
- Study the gauge equivalence of various solutions and its consequences
- Examine the effects of electromagnetic fields on conformal gravity solutions in more detail
Challenges
One of the main challenges in future research is to extend the use of the Newman-Penrose formalism to more complex systems and solutions in Weyl conformal gravity. This may require developing new mathematical techniques and computational tools to handle the increased complexity.
Opportunities
Exploring the two-parameter class of wormhole solutions and their properties opens up opportunities for applications in areas such as faster-than-light travel and exotic matter. Additionally, studying the gauge equivalence of solutions and the role of electromagnetic fields may lead to a deeper understanding of the fundamental physics involved in conformal gravity.
Conclusion
The Newman-Penrose formalism offers a more straightforward approach to find solutions in Weyl conformal gravity, particularly for static, spherically symmetric sources. By utilizing this framework, we have demonstrated the ease of obtaining black hole and wormhole solutions. The inclusion of the electromagnetic field and the study of gauge equivalence adds further depth to the analysis of conformal gravity solutions. Future research should focus on expanding the use of the Newman-Penrose formalism and exploring the implications and applications of wormhole solutions, while addressing challenges that arise with increased complexity.
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