insight - Scientific Computing - # Carrollian Amplitudes

Defining Carrollian Amplitudes in General Dimensions Using Bulk Reduction

Q: How does the concept of a "local Cartesian frame" at the boundary of spacetime relate to the observer's perspective and the potential for a holographic description of quantum gravity?

The concept of a "local Cartesian frame" at the boundary of spacetime, as introduced in the context of Carrollian amplitudes, has profound implications for our understanding of the observer's perspective and the potential for a holographic description of quantum gravity: Observer Dependence and the Nature of Locality: The introduction of a local Cartesian frame at each point on the celestial sphere highlights the observer-dependent nature of Carrollian holography. This frame is tied to the specific choice of null direction associated with a particular observer at null infinity. This observer dependence is a fundamental aspect of gravity in asymptotically flat spacetimes, where there is no preferred notion of time or spatial infinity. Holographic Encoding of Information: The local Cartesian frame provides a natural basis for defining local degrees of freedom on the celestial sphere. These degrees of freedom are expected to be holographically dual to the bulk gravitational degrees of freedom. The observer dependence of the local frame suggests that different observers may have access to different, but complementary, information about the bulk spacetime. This is reminiscent of the idea of complementarity in black hole physics, where different observers may have different descriptions of the same physical reality. Quantum Gravity and the Emergence of Spacetime: The observer-dependent nature of the local Cartesian frame and the holographic encoding of information suggest a picture where spacetime itself may be an emergent concept. In this picture, the fundamental degrees of freedom live on the celestial sphere, and the familiar notion of spacetime emerges from the entanglement structure of these degrees of freedom. This is in line with several approaches to quantum gravity, such as holographic duality and quantum foam models, where spacetime is not a fundamental entity but rather arises from a more fundamental, pre-geometric structure.

Core Concepts

This paper proposes a new method for defining Carrollian amplitudes in general dimensions using the bulk reduction formalism, addressing limitations of previous definitions reliant on stereographic coordinates and offering a framework for exploring Carrollian holography in higher dimensions.

Abstract

Bibliographic Information

Liu, W.-B., Long, J., Xiao, H.-Y., & Yang, J.-L. (2024). On the definition of Carrollian amplitudes in general dimensions. arXiv preprint arXiv:2407.20816v2.

Research Objective

This paper aims to address the challenge of defining spinning Carrollian amplitudes from bulk reduction in general dimensions, independent of coordinate choice on the celestial sphere.

Methodology

The authors employ the bulk reduction formalism, where the boundary field is obtained through asymptotic expansion of the bulk field. They introduce a vielbein field on the unit sphere to define the spinning fundamental field in a local Cartesian frame, enabling a more straightforward definition of Carrollian amplitudes.

Key Findings

The Carrollian amplitude in general dimensions is related to the momentum space scattering matrix by a modified Fourier transform.
The proposed definition of the Carrollian amplitude exhibits well-defined Poincaré transformation properties.
An isomorphism exists between the local rotation of the vielbein field and the superduality transformation.

Main Conclusions

The paper provides a consistent framework for defining Carrollian amplitudes in general dimensions, potentially advancing the understanding of Carrollian holography and its applications in higher dimensions.

Significance

This research contributes to the growing field of flat holography, particularly Carrollian holography, by offering a more general and robust definition of Carrollian amplitudes, a crucial element in studying scattering amplitudes and exploring holographic dualities in asymptotically flat spacetimes.

Limitations and Future Research

The paper primarily focuses on the theoretical framework. Further research could explore specific examples and applications of this framework in various dimensions and investigate the role of soft modes in the completeness relation and their implications for Carrollian amplitudes.

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Quotes

"Carrollian amplitude is the natural object that defines the correlator of the boundary Carrollian field theory."
"In this paper, we will try to solve these problems in the framework of bulk reduction [18, 45]."
"Interestingly, the ambiguity of the vielbein field in the Cartesian frame matches exactly with the superduality transformation."

Key Insights Distilled From

On the definition of Carrollian amplitudes in general dimensions

by Wen-Bin Liu,... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2407.20816.pdf

On the definition of Carrollian amplitudes in general dimensions

Deeper Inquiries

How does this new definition of Carrollian amplitudes contribute to the understanding of specific physical phenomena in flat spacetimes, particularly in the context of black hole physics and gravitational waves?

Carrollian amplitudes, defined through this new approach using vielbeins and local Cartesian frames, offer a promising avenue for understanding physical phenomena in flat spacetimes, particularly in the context of black hole physics and gravitational waves. Here's how:

Relationship to Scattering Amplitudes: Carrollian amplitudes are directly related to scattering amplitudes in momentum space through integral transforms. This connection allows us to leverage the well-established tools and techniques of scattering amplitudes to study the dynamics of gravity in asymptotically flat spacetimes. This is particularly relevant for understanding the scattering of gravitons, which are the fundamental quanta of gravitational waves.

BMS Symmetry and Memory Effects: The definition of Carrollian amplitudes in this paper explicitly incorporates the boundary metric and its transformation properties under the  BMS group. This group, the asymptotic symmetry group of asymptotically flat spacetimes, plays a crucial role in understanding memory effects. These effects refer to the permanent displacement of test particles due to the passage of gravitational waves. By studying the transformation properties of Carrollian amplitudes under the BMS group, we can gain insights into the nature of these memory effects and their implications for gravitational wave astronomy.

Black Hole Information Paradox: Carrollian holography, which uses Carrollian amplitudes as fundamental building blocks, offers a potential framework for addressing the black hole information paradox. This paradox arises from the apparent conflict between quantum mechanics and general relativity regarding the fate of information that falls into a black hole. By studying the holographic dual theory living on the null boundary of spacetime, we may gain new insights into the information flow and the resolution of this paradox.

Higher Dimensions and Quantum Gravity: The extension of Carrollian amplitudes to higher dimensions, as presented in this paper, is a significant step towards a more complete understanding of quantum gravity. Higher-dimensional gravity theories, such as string theory, are believed to be necessary for a consistent description of quantum gravity. By studying Carrollian amplitudes in these theories, we can explore the non-perturbative aspects of quantum gravity and its implications for the early universe and black hole physics.

Could alternative approaches, such as twistor theory or embedding formalism, provide complementary insights into defining and computing Carrollian amplitudes in higher dimensions?

Yes, alternative approaches like twistor theory and the embedding formalism could offer valuable complementary insights into defining and computing Carrollian amplitudes, especially in higher dimensions:

Twistor Theory: Twistor theory provides a geometric framework for describing massless particles and their interactions. It has been remarkably successful in simplifying calculations in gauge theory and gravity in four dimensions. In the context of Carrollian amplitudes, twistor methods could potentially:

Simplify the computation of amplitudes:  Twistor space often reveals hidden symmetries and structures that can significantly simplify scattering amplitude calculations. This could be particularly beneficial in the context of Carrollian amplitudes, which are currently quite challenging to compute beyond low-point examples.
Offer a geometric interpretation of the local Cartesian frame: Twistor space naturally encodes the conformal structure of spacetime, which is closely related to the Carrollian structure at null infinity. This could provide a deeper geometric understanding of the local Cartesian frame introduced in the paper.
Facilitate the generalization to higher dimensions: While traditional twistor theory is most naturally formulated in four dimensions, there have been significant advances in extending it to higher dimensions. These extensions could pave the way for defining and computing Carrollian amplitudes in higher-dimensional theories.

Embedding Formalism: The embedding formalism involves embedding the physical spacetime into a higher-dimensional flat space. This approach has been fruitful in studying conformal field theories and their holographic duals. In the context of Carrollian amplitudes, the embedding formalism could:

Simplify the treatment of conformal transformations: The BMS group, which plays a crucial role in Carrollian holography, can be viewed as a subgroup of the conformal group of the celestial sphere. The embedding formalism naturally incorporates conformal transformations, potentially simplifying the analysis of BMS symmetry and its implications for Carrollian amplitudes.
Provide a geometric interpretation of the Carrollian limit: The Carrollian limit, which corresponds to taking the speed of light to zero, can be viewed as a specific limit of the embedding space geometry. This could offer a more geometric understanding of the Carrollian limit and its relation to flat holography.

How does the concept of a "local Cartesian frame" at the boundary of spacetime relate to the observer's perspective and the potential for a holographic description of quantum gravity?

The concept of a "local Cartesian frame" at the boundary of spacetime, as introduced in the context of Carrollian amplitudes, has profound implications for our understanding of the observer's perspective and the potential for a holographic description of quantum gravity:

Observer Dependence and the Nature of Locality: The introduction of a local Cartesian frame at each point on the celestial sphere highlights the observer-dependent nature of Carrollian holography. This frame is tied to the specific choice of null direction associated with a particular observer at null infinity. This observer dependence is a fundamental aspect of gravity in asymptotically flat spacetimes, where there is no preferred notion of time or spatial infinity.

Holographic Encoding of Information: The local Cartesian frame provides a natural basis for defining local degrees of freedom on the celestial sphere. These degrees of freedom are expected to be holographically dual to the bulk gravitational degrees of freedom. The observer dependence of the local frame suggests that different observers may have access to different, but complementary, information about the bulk spacetime. This is reminiscent of the idea of complementarity in black hole physics, where different observers may have different descriptions of the same physical reality.

Quantum Gravity and the Emergence of Spacetime: The observer-dependent nature of the local Cartesian frame and the holographic encoding of information suggest a picture where spacetime itself may be an emergent concept. In this picture, the fundamental degrees of freedom live on the celestial sphere, and the familiar notion of spacetime emerges from the entanglement structure of these degrees of freedom. This is in line with several approaches to quantum gravity, such as holographic duality and quantum foam models, where spacetime is not a fundamental entity but rather arises from a more fundamental, pre-geometric structure.