Equivalence of Noether charge and Hilbert action boundary term formulas for the black hole entropy in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>R</

By working with the covariant phase space formalism, we have shown that not only can the Hamiltonian conjugate to a Killing vector field <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>ξ</a:mi></a:math> be expressed as the sum of the associated Noether charge and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>ξ</c:mi></c:math> contracted with the Hilbert action boundary term for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>F</e:mi><e:mo stretchy="false">(</e:mo><e:msub><e:mi>R</e:mi><e:mrow><e:mi>a</e:mi><e:mi>b</e:mi><e:mi>c</e:mi><e:mi>d</e:mi></e:mrow></e:msub><e:mo stretchy="false">)</e:mo></e:math> gravity, but it can also be written as its contraction with another <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>ξ</i:mi></i:math> independent tensor field. With this, we have proven the equivalence of Noether charge and Hilbert action boundary term formulas for the stationary black hole entropy in <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>F</k:mi><k:mo stretchy="false">(</k:mo><k:msub><k:mi>R</k:mi><k:mrow><k:mi>a</k:mi><k:mi>b</k:mi><k:mi>c</k:mi><k:mi>d</k:mi></k:mrow></k:msub><k:mo stretchy="false">)</k:mo></k:math> gravity, which is further substantiated by our explicit computation using both formulas. Published by the American Physical Society 2024