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Single Kerr-Schild Metric for Taub-NUT Instanton (2405.09518v4)

Published 15 May 2024 in hep-th and gr-qc

Abstract: It is shown that a complex coordinate transformation maps the Taub-NUT instanton metric to a Kerr-Schild metric. This metric involves a semi-infinite line defect as the gravitational analog of the Dirac string, much like the original metric. Moreover, it facilitates three versions of classical double copy correspondence with the self-dual dyon in electromagnetism, one of which involving a nonlocal operator. The relevance to the Newman-Janis algorithm is briefly noted.

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  79. Given the GSF condition, the Kerr-Schild “graviton field” decomposes as gμ⁢ν−ημ⁢ν=ϕ⁢ℓμ⁢ℓν=hμ⁢νL+ 2⁢m⁢∂(μξν)g_{\mu\nu}{\,-\,}\eta_{\mu\nu}=\phi\hskip 1.00006pt\ell_{\mu}\ell_{\nu}=\smash% {{}^{\text{L}}\kern-0.50003pth_{\mu\nu}}{\,+\,}2m\hskip 1.00006pt\partial_{% \smash{(\mu}\vphantom{\mu}}\xi_{\smash{\nu)}\vphantom{\nu}}italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT - italic_η start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = italic_ϕ roman_ℓ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = start_FLOATSUPERSCRIPT L end_FLOATSUPERSCRIPT italic_h start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT + 2 italic_m ∂ start_POSTSUBSCRIPT ( italic_μ end_POSTSUBSCRIPT italic_ξ start_POSTSUBSCRIPT italic_ν ) end_POSTSUBSCRIPT with h¯μ⁢νL=2⁢u(μ⁢(Aν)L)\smash{{}^{\text{L}}\kern-0.50003pt\bar{h}^{\mu\nu}}=\smash{2u^{(\mu}(\smash{{% }^{\text{L}}\kern-0.50003ptA^{\nu)}})}start_FLOATSUPERSCRIPT L end_FLOATSUPERSCRIPT over¯ start_ARG italic_h end_ARG start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT = 2 italic_u start_POSTSUPERSCRIPT ( italic_μ end_POSTSUPERSCRIPT ( start_FLOATSUPERSCRIPT L end_FLOATSUPERSCRIPT italic_A start_POSTSUPERSCRIPT italic_ν ) end_POSTSUPERSCRIPT ) and ξμ=χ⁢uμ−(ℓμ−uμ)subscript𝜉𝜇𝜒subscript𝑢𝜇subscriptℓ𝜇subscript𝑢𝜇\xi_{\mu}=\chi\hskip 1.00006ptu_{\mu}-(\ell_{\mu}{\,-\,}u_{\mu})italic_ξ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = italic_χ italic_u start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT - ( roman_ℓ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT - italic_u start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ), provided that the single copy gauge potential is given in Kerr-Schild and Lorenz gauges as ϕ⁢ℓμ=AμL+q⁢∂μχitalic-ϕsubscriptℓ𝜇superscriptsubscript𝐴𝜇L𝑞subscript𝜇𝜒\phi\hskip 1.00006pt\ell_{\mu}=\smash{{}^{\text{L}}\kern-0.50003ptA_{\mu}}+q% \hskip 1.00006pt\partial_{\mu}\chiitalic_ϕ roman_ℓ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT = start_FLOATSUPERSCRIPT L end_FLOATSUPERSCRIPT italic_A start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT + italic_q ∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_χ. Interestingly, the harmonic (Lorenz) gauge condition ∂ν(h¯μ⁢νL)=0subscript𝜈superscriptsuperscript¯ℎ𝜇𝜈L0\partial_{\nu}(\smash{{}^{\text{L}}\kern-0.50003pt\bar{h}^{\mu\nu}})=0∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ( start_FLOATSUPERSCRIPT L end_FLOATSUPERSCRIPT over¯ start_ARG italic_h end_ARG start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ) = 0 is satisfied, so ϕ⁢ℓμ⁢ℓνitalic-ϕsubscriptℓ𝜇subscriptℓ𝜈\phi\hskip 1.00006pt\ell_{\mu}\ell_{\nu}italic_ϕ roman_ℓ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT solves linearized Einstein’s equations with source Tμ⁢νL:=12⁢□⁢(h¯μ⁢νL)=u(μ⁢(□⁢Aν)L)=u(μ⁢Jν)\smash{{}^{\text{L}}\kern-0.50003ptT^{\mu\nu}}:={\textstyle\frac{1}{2}}\hskip 1% .00006pt\square(\smash{{}^{\text{L}}\kern-0.50003pt\bar{h}^{\mu\nu}})=\smash{u% ^{(\mu}(\square\smash{{}^{\text{L}}\kern-0.50003ptA}^{\nu)})}=\smash{u^{(\mu}J% ^{\nu)}}start_FLOATSUPERSCRIPT L end_FLOATSUPERSCRIPT italic_T start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT := divide start_ARG 1 end_ARG start_ARG 2 end_ARG □ ( start_FLOATSUPERSCRIPT L end_FLOATSUPERSCRIPT over¯ start_ARG italic_h end_ARG start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ) = italic_u start_POSTSUPERSCRIPT ( italic_μ end_POSTSUPERSCRIPT ( □ start_FLOATSUPERSCRIPT L end_FLOATSUPERSCRIPT italic_A start_POSTSUPERSCRIPT italic_ν ) end_POSTSUPERSCRIPT ) = italic_u start_POSTSUPERSCRIPT ( italic_μ end_POSTSUPERSCRIPT italic_J start_POSTSUPERSCRIPT italic_ν ) end_POSTSUPERSCRIPT. In this sense Tμ⁢νL=u(μ⁢Jν)\smash{{}^{\text{L}}\kern-0.50003ptT^{\mu\nu}}=\smash{u^{(\mu}J^{\nu)}}start_FLOATSUPERSCRIPT L end_FLOATSUPERSCRIPT italic_T start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT = italic_u start_POSTSUPERSCRIPT ( italic_μ end_POSTSUPERSCRIPT italic_J start_POSTSUPERSCRIPT italic_ν ) end_POSTSUPERSCRIPT could be identified as a source for the solution. We have been presuming stationary solutions with Killing vector uμ=δμ0u^{\mu}=\delta^{\mu}{}_{0}italic_u start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT = italic_δ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_FLOATSUBSCRIPT 0 end_FLOATSUBSCRIPT and also assumed ℓμ⁢uμ=−1subscriptℓ𝜇superscript𝑢𝜇1\ell_{\mu}u^{\mu}=-1roman_ℓ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_u start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT = - 1, while overbar has denoted trace reversion. Note that this stress-energy tensor is consistent with the Kerr-Schild double copy, which states Jμ=−2T¯μ0J^{\mu}{\,=\,}-2\bar{T}^{\mu}{}_{0}italic_J start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT = - 2 over¯ start_ARG italic_T end_ARG start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_FLOATSUBSCRIPT 0 end_FLOATSUBSCRIPT [10, 11].
  80. in the sense that T00superscript𝑇00T^{00}italic_T start_POSTSUPERSCRIPT 00 end_POSTSUPERSCRIPT vanishes
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  87. This can be explicitly checked by noting that the ∂μℓνsubscript𝜇subscriptℓ𝜈\partial_{\mu}\ell_{\nu}∂ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT roman_ℓ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT equation in the previous page is reproduced by a spinor equation ∂γ⁢γ˙o~α˙=(1/r)⁢δγ⁢δ˙⁢o~δ˙⁢o~γ˙⁢ι¯α˙subscript𝛾˙𝛾superscript~𝑜˙𝛼1𝑟subscript𝛿𝛾˙𝛿superscript~𝑜˙𝛿subscript~𝑜˙𝛾superscript¯𝜄˙𝛼\partial_{\smash{{\gamma}{\dot{{\gamma}}}}\vphantom{\beta}}\hskip 1.00006pt% \tilde{o}^{{\dot{\alpha}}}=(1/r)\hskip 1.00006pt\smash{\delta_{\smash{{\gamma}% {\dot{{\delta}}}}\vphantom{\beta}}\tilde{o}^{{\dot{{\delta}}}}\hskip 0.50003pt% \tilde{o}_{{\dot{{\gamma}}}}}\hskip 1.00006pt{\bar{\iota}}^{{\dot{\alpha}}}∂ start_POSTSUBSCRIPT italic_γ over˙ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT over~ start_ARG italic_o end_ARG start_POSTSUPERSCRIPT over˙ start_ARG italic_α end_ARG end_POSTSUPERSCRIPT = ( 1 / italic_r ) italic_δ start_POSTSUBSCRIPT italic_γ over˙ start_ARG italic_δ end_ARG end_POSTSUBSCRIPT over~ start_ARG italic_o end_ARG start_POSTSUPERSCRIPT over˙ start_ARG italic_δ end_ARG end_POSTSUPERSCRIPT over~ start_ARG italic_o end_ARG start_POSTSUBSCRIPT over˙ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT over¯ start_ARG italic_ι end_ARG start_POSTSUPERSCRIPT over˙ start_ARG italic_α end_ARG end_POSTSUPERSCRIPT. Note also that the Goldberg-Sachs theorem [135, 136] implies they are GSF in the curved background as well.
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  91. This function can be written as i⁢Z~A⁢uμ⁢(γμ)A⁢I¯CBC⁢Z~B𝑖superscript~𝑍Asubscript𝑢𝜇subscriptsuperscript𝛾𝜇Asuperscriptsubscript¯𝐼CBCsuperscript~𝑍Bi\hskip 1.00006pt\smash{\tilde{Z}}^{\mathrm{A}}\hskip 1.00006ptu_{\mu}(\gamma^% {\mu})_{\mathrm{A}}{}^{\mathrm{C}}\hskip 1.00006pt\bar{I}_{{\mathrm{C}}{% \mathrm{B}}}\hskip 1.00006pt\smash{\tilde{Z}}^{\mathrm{B}}italic_i over~ start_ARG italic_Z end_ARG start_POSTSUPERSCRIPT roman_A end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_γ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT roman_A end_POSTSUBSCRIPT start_FLOATSUPERSCRIPT roman_C end_FLOATSUPERSCRIPT over¯ start_ARG italic_I end_ARG start_POSTSUBSCRIPT roman_CB end_POSTSUBSCRIPT over~ start_ARG italic_Z end_ARG start_POSTSUPERSCRIPT roman_B end_POSTSUPERSCRIPT, where I¯ABsubscript¯𝐼AB\bar{I}_{{\mathrm{A}}{\mathrm{B}}}over¯ start_ARG italic_I end_ARG start_POSTSUBSCRIPT roman_AB end_POSTSUBSCRIPT denotes infinity twistor.
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