A mathematical form of force-free magnetosphere equation around Kerr black holes and its application to Meissner effect

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• 作者：Xiao-Bo Gong^a ; ^b ; ^c ; ^{gxbo@ynao.ac.cn" class="auth_mail" title="E-mail the corresponding author} ; Yi Liao^d ; ^{liaoy@mail.sustc.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Zhao-Yi Xu^a ; ^b ; ^c ; ^{zyxu88@ynao.ac.cn" class="auth_mail" title="E-mail the corresponding author}
• 刊名：Physics Letters B
• 出版年：2016
• 出版时间：10 September 2016
• 年：2016
• 卷：760
• 期：Complete
• 页码：112-116
• 全文大小：866 K

文摘

Based on the Lagrangian of the steady axisymmetric force-free magnetosphere (FFM) equation around Kerr black holes (KBHs), we find that the FFM equation can be rewritten in a new form as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316303008&_mathId=si1.gif&_user=111111111&_pii=S0370269316303008&_rdoc=1&_issn=03702693&md5=4df0a600654b778443ddd0d19bf2f53a" title="Click to view the MathML source">f_,rr/(1−μ²)+f_,μμ/Δ+K(f(r,μ),r,μ)=0class="mathContainer hidden">class="mathCode">$f_{,rr}/(1-{\mu }^2)+f_{,\mu \mu }/\mathrm{\Delta }+K(f(r,\mu ),r,\mu )=0$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316303008&_mathId=si2.gif&_user=111111111&_pii=S0370269316303008&_rdoc=1&_issn=03702693&md5=e40ffd5b7ea2103b409a2de7bf8bcfe9" title="Click to view the MathML source">μ=−cos⁡θclass="mathContainer hidden">class="mathCode">$\mu =-\cos \theta$. With coordinate transformation, the above equation can be given as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316303008&_mathId=si3.gif&_user=111111111&_pii=S0370269316303008&_rdoc=1&_issn=03702693&md5=ae6419b00231fc2de4200e4da9e2473e" title="Click to view the MathML source">s_,yy+s_,zz+D(s(y,z),y,z)=0class="mathContainer hidden">class="mathCode">$s_{,yy}+s_{,zz}+D(s(y,z),y,z)=0$. Using this form, we prove that the Meissner effect is not possessed by a KBH–FFM with the condition class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316303008&_mathId=si4.gif&_user=111111111&_pii=S0370269316303008&_rdoc=1&_issn=03702693&md5=1a5c6182ce04928b430285106789fa96" title="Click to view the MathML source">dω/dA_ϕ⩽0class="mathContainer hidden">class="mathCode">$d\omega /d{A}_{\varphi }⩽0$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316303008&_mathId=si5.gif&_user=111111111&_pii=S0370269316303008&_rdoc=1&_issn=03702693&md5=d38059a60e80b5d81e12142393e27a3b" title="Click to view the MathML source">H_ϕ(dH_ϕ/dA_ϕ)⩾0class="mathContainer hidden">class="mathCode">$H_{\varphi }(d{H}_{\varphi }/d{A}_{\varphi })⩾0$, here class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316303008&_mathId=si154.gif&_user=111111111&_pii=S0370269316303008&_rdoc=1&_issn=03702693&md5=d61130826a13dd85fd40e138748510f2" title="Click to view the MathML source">A_ϕclass="mathContainer hidden">class="mathCode">$A_{\varphi }$ is the ϕ   component of the vector potential class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316303008&_mathId=si7.gif&_user=111111111&_pii=S0370269316303008&_rdoc=1&_issn=03702693&md5=145276df6ecbe9b9e1fceb3a099f60a4">class="imgLazyJSB inlineImage" height="15" width="11" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0370269316303008-si7.gif">class="mathContainer hidden">class="mathCode">$\overrightarrow{A}$, ω   is the angular velocity of magnetic fields and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0370269316303008&_mathId=si148.gif&_user=111111111&_pii=S0370269316303008&_rdoc=1&_issn=03702693&md5=50e759a018c70b4be760d0f7176112bb" title="Click to view the MathML source">H_ϕclass="mathContainer hidden">class="mathCode">$H_{\varphi }$ corresponds to twice the poloidal electric current.