{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"ce8ae14f-8e9e-580e-b04a-e949d8748382","task_key":"train--Challenge~5f53~5fmain","task_revision_id":"2","upstream_id":"Challenge_53_main","short_description":"The ambient metric is defined as follows:","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nd = sp.symbols('d')\\n\\ndef answer(d):\\n    r\\\"\\\"\\\"\\n    Return the expressions of the coefficients in SymPy format.\\n\\n    Inputs\\n    ----------\\n    d: sympy.Symbol, variable, $d$\\n\\n    Output\\n    ----------\\n    coeff1: sympy.Expr, the coefficient in front of $P^{k}{}_{i} P_{kj}$\\n    coeff2: sympy.Expr, the coefficient in front of $B_{k(i}P^k{}_{j)}$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULT BELOW ------------------\\n    coeff1 = ...   # a SymPy expression of inputs\\n    coeff2 = ...   # a SymPy expression of inputs\\n    # ---------------------------------------------------------------\\n\\n    return coeff1, coeff2\",\"problem_description\":\"\\n\\n# Problem setup:\\nThe ambient metric is defined as follows:\\n\\\\begin{equation}\\nds^2 = 2\\\\rho dt^2 + 2t dt d\\\\rho + t^2 \\\\gamma_{ij}(x,\\\\rho)dx^{i}dx^{j}\\\\,,\\n\\\\end{equation}\\nwhere $\\\\gamma_{ij}(x,\\\\rho)$ can be expanded asymptotically in terms of $\\\\rho$ as\\n\\\\begin{align}\\n\\\\gamma_{ij}(x,\\\\rho)&= \\\\gamma^{(0)}_{ij}(x)+ \\\\gamma^{(1)}_{ij}(x)\\\\rho +\\\\gamma^{(2)}_{ij}(x)\\\\rho^2 + \\\\cdots\\\\,.\\n\\\\end{align}\\nGiven $\\\\gamma^{(0)}_{ij}$, the higher-order terms in the expansion can be solved order by order from the Ricci-flatness condition of the ambient metric. For $k\\\\geqslant 2$, each $\\\\gamma^{(k)}_{ij}(x)$ has a pole at $d=2k$.\\n\\nOn the other hand, the Riemann tensor of the ambient metric defines the $k$th extended obstruction tensor $\\\\Omega^{(k)}_{ij}$ in the following way:\\n\\\\begin{equation}\\n\\\\Omega^{(k)}_{ij}= R_{\\\\rho ij\\\\rho;\\\\underbrace{\\\\scriptstyle \\\\rho\\\\cdots \\\\rho}_{k-1}}|_{\\\\rho=0,t=1}.\\n\\\\end{equation}\\nThis tensor also has a pole at $d=2k+2$. The residue of $\\\\gamma^{(k)}_{ij}(x)$ is proportional to that of $\\\\Omega^{(k-1)}_{ij}$.\\n\\n# Main problem:\\n\\n Suppose $\\\\text{Res}_{d=2k}\\\\gamma^{(k)}_{ij}=A_k\\\\text{Res}_{d=2k}\\\\Omega^{(k-1)}_{ij}$. Show that for $k=2,3$, $\\\\gamma^{(k)}_{ij}-A_k\\\\Omega^{(k-1)}_{ij}$ is proportional to $P^{k}{}_{i} P_{kj}$ and $B_{k(i}P^k{}_{j)}$, respectively. Determine these two coefficients.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}