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Essentials of Classical Brane Dynamics
General Relativity and Quantum Cosmology · Submitted 10 Dec 2000
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The later sections show how -- to lowest order in the thin limit -- the evolution of such a brane worldsheet will always be governed by a simple tensorial equation of motion whose left hand side is the contraction of the relevant surface stress tensor $ bar T^{\mu\nu}$ with the (geometrically defined) second fundamental tensor $K_{\mu\nu}{^\rho}$, while the right hand side will simply vanish in the case of free motion and will otherwise be just the orthogonal projection of any external force density that may happen to act on the brane.
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The abstract, sentence by sentence
This article provides a self contained overview of the geometry and dynamics of relativistic brane models, of the category that includes point particle, string, and membrane representations for phenomena that can be considered as being confined to a worldsheet of the corresponding dimension (respectively one, two, and three) in a thin limit approximation in an ordinary 4 dimensional spacetime background. This category also includes ``brane world'' models that treat the observed universe as a 3-brane in 5 or higher dimensional background. The first sections are concerned with purely kinematic aspects: it is shown how, to second differential order, the geometry (and in particular the inner and outer curvature) of a brane worldsheet of arbitrary dimension is describable in terms of the first, second, and third fundamental tensor. The later sections show how -- to lowest order in the thin limit -- the evolution of such a brane worldsheet will always be governed by a simple tensorial equation of motion whose left hand side is the contraction of the relevant surface stress tensor $ bar T^{\mu\nu}$ with the (geometrically defined) second fundamental tensor $K_{\mu\nu}{^\rho}$, while the right hand side will simply vanish in the case of free motion and will otherwise be just the orthogonal projection of any external force density that may happen to act on the brane.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This article provides a self contained overview of the geometry and dynamics of relativistic brane models, of the category that includes point particle, string, and membrane representations for phenomena that can be considered as being confined to a worldsheet of the corresponding dimension (respectively one, two, and three) in a thin limit approximation in an ordinary 4 dimensional spacetime background. | contribution 0.92 | 0.54 | 0.10 | asserted 0.99 |
| s1 | This category also includes ``brane world'' models that treat the observed universe as a 3-brane in 5 or higher dimensional background. | background 0.90 | 0.10 | 0.04 | asserted 1.00 |
| s2 | The first sections are concerned with purely kinematic aspects: it is shown how, to second differential order, the geometry (and in particular the inner and outer curvature) of a brane worldsheet of arbitrary dimension is describable in terms of the first, second, and third fundamental tensor. | contribution 0.55 | 0.82 | 0.07 | asserted 1.00 |
| s3 | The later sections show how -- to lowest order in the thin limit -- the evolution of such a brane worldsheet will always be governed by a simple tensorial equation of motion whose left hand side is the contraction of the relevant surface stress tensor $ bar T^{\mu\nu}$ with the (geometrically defined) second fundamental tensor $K_{\mu\nu}{^\rho}$, while the right hand side will simply vanish in the case of free motion and will otherwise be just the orthogonal projection of any external force density that may happen to act on the brane. | result 0.89 | 0.79 | 0.68 | asserted 0.99 |
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The paper
Original abstract
This article provides a self contained overview of the geometry and dynamics of relativistic brane models, of the category that includes point particle, string, and membrane representations for phenomena that can be considered as being confined to a worldsheet of the corresponding dimension (respectively one, two, and three) in a thin limit approximation in an ordinary 4 dimensional spacetime background. This category also includes ``brane world'' models that treat the observed universe as a 3-brane in 5 or higher dimensional background. The first sections are concerned with purely kinematic aspects: it is shown how, to second differential order, the geometry (and in particular the inner and outer curvature) of a brane worldsheet of arbitrary dimension is describable in terms of the first, second, and third fundamental tensor. The later sections show how -- to lowest order in the thin limit -- the evolution of such a brane worldsheet will always be governed by a simple tensorial equation of motion whose left hand side is the contraction of the relevant surface stress tensor $ bar T^{\mu\nu}$ with the (geometrically defined) second fundamental tensor $K_{\mu\nu}{^\rho}$, while the right hand side will simply vanish in the case of free motion and will otherwise be just the orthogonal projection of any external force density that may happen to act on the brane.
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References
How the literature uses it
29 arXiv papers cite this in 39 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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In the next part, it is shown how -- to lowest order in the thin limit -- the evolution of such a brane worldsheet will always be governed by a simple tensorial equation of motion whose left hand side is the contraction of the second fundamental tensor with the relevant surface stress tensor, while the right hand side will simply vanish in the case of free motion and will otherwise be just the orthogonal projection of any external force density that may happen to act on the brane.
Brane dynamics for treatment of cosmic strings and vortons · 1.00
Papers citing it · 29 sentences without a confident label
Uses its method or tool 3 papers
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For a thin brane, such as our black ring, this is equivalent to imposing the `equations of motion' $K_μν^ρT^μν=0$, derived by Carter in [ref] , where $K_μν^ρ$ is the extrinsic curvature tensor of the brane embedding.
Black Rings in (Anti)-deSitter space · Equilibrium condition · 0.85
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Then, following [ref] , we will denote by $A,B,..=2, …, D-1$ flat tangent indices in $ℬ$ and by $X,Y,…=0,1$ normal flat indices, and introduce adapted vielbein $_A^μ$ and $λ_X^μ$ (they are particular cases of $E_a^μ$) , such that q^μ_ν= _A^μ^A_ν, h^μ_ν= λ_X^μλ^X_νOne can show in particular that, since $λ_X^μλ_Yμ =η_XY$ ($η$ denotes the flat Minkowski metric), one can make the following identifications λ_0^μ= n^μ-l^μ/√(2), λ_1^μ= n^μ+l^μ/√(2) with reference to the null vectors introduced in the previous Appendix.
Gravitational Chern-Simons Lagrangians and black hole entropy · Entropy formula and spin connection · 0.81
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Following [ref] , it is convenient to introduce the tangential covariant derivative ∇_μ=h_μ^ν∇_ν , so K_μν^ρ=h_ν^σ∇_μh_σ^ρ .
Essentials of Blackfold Dynamics · Extrinsic curvature · 0.90
Compares with it 1 paper
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It is now convenient to compare the normal bundle connection with the one introduced in [ref] , ϖ_μ^ν_ρ= h_σ^νλ^X_ρ∇̅_μλ_X^σ, where ∇̅_μ= q^ν_μ∇_νUsing $∇E_a^μ= -_a^b E_b^μ$ we can rewrite ϖ_μ^ν_ρ= q_μ^σϵ_ρ^ν(_n)_σ^01 Saturating with $ϵ_ν^ρ$ we obtain precisely the RHS of ([ref]).
Gravitational Chern-Simons Lagrangians and black hole entropy · Entropy formula and spin connection · 0.99
Cites it as background 6 papers
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The physical picture is related to the traditional braneworld models [ref] , or the blackfold approaches with higher dimensional embedding [ref] .
Emergent Dark Matter in Late Time Universe on Holographic Screen · Holographic Scenario: Holographic FRW Universe · 0.96
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The general extrinsic dynamics of a brane has been analyzed by Carter in [ref] .
Essentials of Blackfold Dynamics · Blackfold equations · 0.89
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More details can be found in the papers of Carter [ref] .
Deformation of Codimension-2 Surface and Horizon Thermodynamics · Theory of Submanifold and Geometry of Codimension -2 Surface · 0.86
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