2000 character limit reached
Carrollian Origins of Bjorken Flow (2302.03053v2)
Published 6 Feb 2023 in hep-th, hep-ph, and nucl-th
Abstract: Bjorken flow is among the simplest models of fluids moving near the speed of light ($c$) while Carroll symmetry arises as a contraction of Poincar\'{e} group when $c \to 0$. We show that Bjorken flow and its phenomenological approximations are completely captured by Carrollian fluids. Carrollian symmetries arise on generic null surfaces and a fluid moving at $c$ is restricted to such a surface, thereby naturally inheriting the symmetries. Carrollian hydrodynamics is thus not exotic, but rather ubiquitous, and provides a concrete framework for fluids moving at or near the speed of light.
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- When we speak of Bjorken flow, we will work in the natural units c=ℏ=1𝑐Planck-constant-over-2-pi1c=\hbar=1italic_c = roman_ℏ = 1. We will make the factors of c𝑐citalic_c explicit when we take the Carroll limit below.
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- The attentive reader may point out that we have not shown that ε=ε(τ)𝜀𝜀𝜏\varepsilon=\varepsilon(\tau)italic_ε = italic_ε ( italic_τ ), as required by Bjorken’s approximation. This is implicit. The Carroll fluid will have an equation of state p=p(ε)𝑝𝑝𝜀p=p(\varepsilon)italic_p = italic_p ( italic_ε ). We have shown p=p(τ)𝑝𝑝𝜏p=p(\tau)italic_p = italic_p ( italic_τ ). Hence ε=ε(τ)𝜀𝜀𝜏\varepsilon=\varepsilon(\tau)italic_ε = italic_ε ( italic_τ ).
- This acts as ∇^iVj=∂^iVj+γ^ikjVksubscript^∇𝑖superscript𝑉𝑗subscript^𝑖superscript𝑉𝑗subscriptsuperscript^𝛾𝑗𝑖𝑘superscript𝑉𝑘\hat{\nabla}_{i}V^{j}=\hat{\partial}_{i}V^{j}+\hat{\gamma}^{j}_{ik}V^{k}over^ start_ARG ∇ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_V start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = over^ start_ARG ∂ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_V start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT + over^ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT italic_V start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT. The Levi-Civita-Carroll connection coefficients are given via γ^jki=ail2(∂^jakl+∂^kajl−∂^lajk).subscriptsuperscript^𝛾𝑖𝑗𝑘superscript𝑎𝑖𝑙2subscript^𝑗subscript𝑎𝑘𝑙subscript^𝑘subscript𝑎𝑗𝑙subscript^𝑙subscript𝑎𝑗𝑘\hat{\gamma}^{i}_{jk}=\frac{a^{il}}{2}\left(\hat{\partial}_{j}a_{kl}+\hat{% \partial}_{k}a_{jl}-\hat{\partial}_{l}a_{jk}\right).over^ start_ARG italic_γ end_ARG start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT = divide start_ARG italic_a start_POSTSUPERSCRIPT italic_i italic_l end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG ( over^ start_ARG ∂ end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_k italic_l end_POSTSUBSCRIPT + over^ start_ARG ∂ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_j italic_l end_POSTSUBSCRIPT - over^ start_ARG ∂ end_ARG start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT ) . For more details, see [25].
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