From: Snapshot-Content-Location: https://arxiv.org/abs/2603.25171 Subject: [2603.25171] Sign Errors in "The Four Laws of Black Hole Mechanics" Date: Fri, 27 Mar 2026 03:57:00 -0700 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----MultipartBoundary--F1yBeK8uKWA7BkN9gkVIZ3RHfxbJbR9RwRnHc6wsMe----" ------MultipartBoundary--F1yBeK8uKWA7BkN9gkVIZ3RHfxbJbR9RwRnHc6wsMe---- Content-Type: text/html Content-ID: Content-Transfer-Encoding: quoted-printable Content-Location: https://arxiv.org/abs/2603.25171 [2603.25171] Sign Errors in "The Four Laws of Black Hole Mechanics" =20 =20 =20 =20 =20 =20 =20
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General Relativity and Quantum Cosmology

arXiv:2603.25171 (gr-qc)
[Submitted on 26 Mar 2026]

Title:Sig= n Errors in "The Four Laws of Black Hole Mechanics"

Authors:Richard P. Behiel
=20 View PDF
Abstract:In 1973, Bardeen, Ca= ter, and Hawking published "The Four Laws of Black Hole Mechanics", establi= shing the mathematical framework that would later be understood as the ther= modynamics of black holes. Central to the paper is equation (33), which wri= tes the variation of the total energy-momentum integral in terms of physica= lly meaningful quantities: angular momentum, particle number, and entropy. = Equation (33) feeds into the differential mass formula, equation (34), whic= h is the first law of black hole mechanics. This note identifies two compen= sating sign errors in the BCH paper. The first error, demonstrated by a der= ivation from equation (32), is that equations (33) and (34) should carry mi= nus signs rather than plus signs on the last two integrals, those involving= the redshifted chemical potential and the redshifted temperature. he secon= d error is that the definitions of total particle number N and total entrop= y S given after equation (20) are missing minus signs that are required for= these quantities to be positive. These two errors cancel, in that reversin= g the signs in the definitions of N and S to ensure positive quantities mak= es equations (33) and (34) correct.
All conclusions of the BCH paper remain valid. This note is intended me= rely as a guide for readers who, in working through the derivation step by = step, might otherwise be puzzled by the sign discrepancies. Numbered equati= ons refer to the BCH paper; lettered equations are introduced in this note.
Comments: 5 pages
Subjects: General Relativity and Quantum = Cosmology (gr-qc)
Cite as: arXiv:2603.25171 [gr-qc]
  (or arXiv:2603.251= 71v1 [gr-qc] for this version)
  https://doi.org/1= 0.48550/arXiv.2603.25171
arXiv-= issued DOI via DataCite (pending registration)

Submission history

From: Richard Behiel [view email] =
[v1] Thu, 26 Mar 2026 08:44:54 UTC (275 KB)
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margin-left: 6px; margin-right: 3= px; } #pwc-data-output { padding: 7px; margin-bottom: 10px; margin-top: 20px; } #labstabs #pwc-data-output > p.pwc-provided { color: rgb(102, 102, 102); fo= nt-size: 0.75rem; margin-top: 1.5rem; margin-bottom: 0px; } #labstabs #pwc-data-output > p.pwc-provided > a { color: rgb(102, 102, 102)= ; } #labstabs #pwc-data-output .pwc-icon-primary { width: 14px; height: 14px; t= op: 3px; } #pwc-data-output > h3 { margin-top: 5px; margin-bottom: 15px; } #pwc-data-output > .pwc-data-card { margin-top: 20px; } #pwc-data-output > .pwc-data-card .pwc-data-name { font-size: 0.95rem; } #pwc-data-output > .pwc-data-card .pwc-data-card-image { display: inline-bl= ock; vertical-align: top; margin-right: 5px; margin-bottom: 5px; width: 34p= x; height: 30px; } #pwc-data-output > .pwc-data-card .pwc-data-card-image img { height: 25px; = width: 25px; border-radius: 50%; margin: 0.25rem; } #pwc-data-output > .pwc-data-card .pwc-data-card-image .pwc-data-card-image= -placeholder { height: 25px; width: 25px; border-radius: 50%; margin: 0.25r= em; text-align: center; opacity: 0.3; } #pwc-data-output > .pwc-data-card .pwc-data-card-image .pwc-data-card-image= -placeholder span { position: relative; top: 4px; text-align: center; color= : rgb(255, 255, 255); font-size: 13px; } #pwc-data-output > .pwc-data-card .pwc-data-card-name { display: inline-blo= ck; vertical-align: top; } #pwc-data-output > .pwc-data-card .pwc-data-card-name .pwc-data-name-introd= uced { margin-top: 4px; font-size: 11px; } #pwc-data-output > .pwc-data-card .pwc-data-card-name .pwc-data-name-meta {= margin-top: 4px; font-size: 11px; color: rgb(102, 102, 102); } .replicate-model { padding: 15px 0px; display: flex; } .replicate-model-image { width: 100px; height: 100px; object-position: cent= er center; object-fit: cover; flex-grow: 0; flex-shrink: 0; } .replicate-model-details { padding: 10px 15px; vertical-align: top; } .replicate-model-details-heading { margin: 0px; padding: 0px; } .replicate-model-prediction-count { opacity: 0.5; } .replicate-model-prediction-icon { height: 0.9rem; width: 0.9rem; margin-ri= ght: 0px; margin-top: -0.15rem; } .sciencecast-cast { margin-bottom: 2rem; } .sciencecast-thumbnail { border-radius: 0.5rem; border: 1px solid rgb(238, = 238, 238); } .gotitpub-logo { margin-left: 0pt; font-weight: 500; letter-spacing: normal= ; line-height: 100%; font-family: "EB Garamond", ui-serif, Georgia, Cambria= , "Times New Roman", Times, serif; } .alphaXiv-logo { margin-left: 0pt; font-weight: 500; letter-spacing: normal= ; line-height: 100%; font-family: "EB Garamond", ui-serif, Georgia, Cambria= , "Times New Roman", Times, serif; } .spaces-summary { margin-bottom: 0.5em; } .spaces-model { padding: 15px 0px; display: flex; } .spaces-thumbnail { height: 5.5rem; border-radius: 0.5rem; } .spaces-model-details { padding: 0px 15px; vertical-align: top; } .spaces-model-details-heading { margin: 0px; padding: 0px; } .spaces-model-subheader { color: gray; } .spaces-load-all-link { cursor: pointer; padding: 0.75em; background-color:= rgb(238, 245, 249); border: none; border-radius: 0.5rem; margin: 0.5em 0px= ; display: inline-block; } #labstabs { clear: both; margin: 1em 1.5em; } #labstabs .labstabs { display: flex; flex-wrap: wrap; white-space: normal; = justify-content: flex-start; } #labstabs .labstabs > label { order: 1; display: block; padding: 0.75em 1.5= em; font-size: 0.85em; color: rgb(74, 74, 74); margin-right: 0.2rem; cursor= : pointer; font-weight: bold; transition: background 0.2s; border-radius: 6= px 6px 0px 0px; background-color: rgb(238, 245, 249); } #abs #labstabs .labstabs > label { font-size: 0.88em; } #abs-outer #abs #labstabs blockquote.abstract { margin-left: 0.5em; margin-= right: 0.5em; } #labstabs .labstabs .tab { order: 99; flex-grow: 1; width: 100%; display: n= one; padding: 1rem; border: 1px solid rgb(130, 130, 130); z-index: 1; top: = -1px; position: relative; } #labstabs .labstabs .tab h1 { font-size: 1.25em; font-weight: normal; } #labstabs .labstabs .tab p { font-size: 0.85rem; } #labstabs .labstabs .tab #coreRecommenderOutput { padding: 0px; margin: 0px= ; } #labstabs .labstabs .tab .icon-labs { text-align: center; } #labstabs .labstabs .tab .icon-labs svg { width: 130px; } #labstabs .labstabs input[type=3D"radio"] { display: none; } #labstabs .labstabs input[type=3D"radio"]:checked + label { background: rgb= (255, 255, 255); border-width: 1px 1px 0px; border-top-style: solid; border= -right-style: solid; border-left-style: solid; border-top-color: rgb(130, 1= 30, 130); border-right-color: rgb(130, 130, 130); border-left-color: rgb(13= 0, 130, 130); border-image: initial; border-bottom-style: initial; border-b= ottom-color: initial; z-index: 2; color: black; } #labstabs .labstabs input[type=3D"radio"]:checked + label + .tab { display:= block; } @media screen and (max-width: 768px) { .column.is-full-mobile { flex: 0 0 auto !important; width: 100% !importan= t; } } @media (max-width: 45em) { #labstabs { padding-bottom: 1em; margin: 1em; } #labstabs .labstabs .tab, #labstabs .labstabs label { order: initial; 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} #labstabs .column.lab-switch { flex-grow: 0; } #labstabs .toggle .lab-switch label.switch { position: relative; display: i= nline-block; width: 38px; height: 20px; padding: 0px; background-color: tra= nsparent; margin: 0px; } #labstabs .toggle .lab-switch .switch input { opacity: 0; width: 0px; heigh= t: 0px; } #labstabs .toggle .lab-switch .slider { position: absolute; cursor: pointer= ; inset: 0px; border-radius: 24px; background-color: rgb(204, 204, 204); tr= ansition: 0.4s; } #labstabs .toggle .lab-switch .slider::before { position: absolute; content= : ""; height: 16px; width: 16px; left: 2px; bottom: 2px; border-radius: 50%= ; background-color: white; transition: 0.4s; } #labstabs .toggle .lab-switch input.enabled + .slider { background-color: r= gb(28, 139, 214); } #labstabs .toggle .lab-switch input.enabled + .slider { box-shadow: rgb(33,= 150, 243) 0px 0px 1px; } #labstabs .toggle .lab-switch input.enabled + .slider::before { transform: = translateX(18px); } #abs-outer .subheader { background-color: rgb(238, 238, 238); color: rgb(0,= 0, 0); padding: 0.25em 0px; border-bottom: 1px solid rgb(204, 204, 204); } #abs-outer .subheader h1 { margin: 0px; font-size: 1.1em; padding: 0px 0px = 0.2em 20px; font-weight: bold; font-style: normal; } #abs .descriptor { display: none; } #abs-outer .extra-services .descriptor { display: none; } #abs-outer #coreRecommenderOutput { max-width: 100%; } #abs h1.title { margin: 0.5em 0px 0.5em 20px; font-size: x-large; font-weig= ht: bold; line-height: 120%; } #abs .authors { margin: 0.5em 0px 0.5em 20px; font-size: medium; line-heigh= t: 150%; } #abs .authors a { font-size: medium; } div#long-author-list { margin: 0px; display: none; } #abs .dateline { margin: 0.5em 0px 0.5em 20px; font-style: italic; font-siz= e: small; } #abs blockquote.abstract { line-height: 1.55; font-size: 1.05em; margin-bot= tom: 1.5em; } #abs .comments { } #abs .metatable { font-size: 0.92em; line-height: 1.5; margin: 0px 0px 1.5e= m 20px; } #abs .tablecell { padding: 0.1em 0.5em 0em 0em; vertical-align: top; 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padding: 0p= x; } #abs-outer .endorsers { margin: 1em 0px 1.5em 20px; font-size: small; font-= style: italic; clear: both; } #abs-outer .header-breadcrumbs-mobile { display: none; } #abs-outer #abs .dateline { margin-top: 15px; margin-bottom: 0px; } #abs-outer #abs h1.title { margin-top: 0.25em; } .endorsers .help { display: inline-block; } @media screen and (max-width: 768px) { #abs-outer .header-breadcrumbs-mobile { display: block; color: black; fon= t-size: 0.85em; margin: 0.25em 0px 0.5em 1em; } #abs-outer #abs a.mobile-submission-download { display: flex; justify-con= tent: center; margin: 0.7em 0.25em; border-radius: 10px; background-color: = rgb(64, 139, 208); padding: 0.25em 1em; font-weight: 800; color: white; tex= t-decoration: none; font-size: 20px; text-align: center; } #abs-outer #abs a.mobile-download-grey { background-color: rgb(238, 238, = 238); color: rgb(86, 86, 86); } #abs-outer .extra-services h2 { font-size: 15px; margin-bottom: 0.5em; } #abs-outer .extra-services h3 { font-size: 14px; 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color: rgb(4, 107, 175) !important; } .abs-button-small { font-size: 1em; padding: 0.25em 1em; margin: 0.75em 0= .5em 0px 0px; } .beta-link { font-size: x-small; color: rgb(179, 27, 27); } .abs-button-grey { border: 1px solid rgb(102, 102, 102); color: rgb(102, = 102, 102) !important; } #abs-outer .extra-services .bib-sidebar { border: 0px !important; } #abs-outer .extra-services .bib-sidebar a { display: inline-block; border= -radius: 5px; background: rgb(230, 230, 230); border: 1px solid rgb(102, 10= 2, 102); padding: 0.25em 1em; margin: 0px 0.5em 0px 0px; color: rgb(102, 10= 2, 102) !important; font-size: 1em !important; } #abs-outer .extra-services .bib-sidebar .bib-sidebar-paper .bib-sidebar-a= uthors a, #abs-outer .extra-services .bib-sidebar .bib-sidebar-paper .bib-s= idebar-title a { display: inline; background: none; border: 0px; color: rgb= (0, 0, 0); padding: 0.25em 0px; margin: 0px; font-size: 1em !important; } .bib-sidebar-authors { list-style-type: none; } #abs-outer .extra-services .bib-sidebar .bib-sidebar-paper { margin: 1em = 0px; 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visibility: visible= ; } .mobile-header { background-color: rgb(179, 27, 27); } .mobile-header .columns { height: 65px; align-items: center; } .mobile-header .column { border-left: 1px solid rgb(252, 85, 84); border-ri= ght: 1px solid rgb(115, 21, 21); height: 65px; padding: 0px 1em; display: f= lex; align-items: center; } .mobile-header .column:first-child { border-left: 0px; } .mobile-header .column:last-child { border-right: 0px; } .mobile-header .column.logo-arxiv { width: 100px; flex: 0 0 auto; } .mobile-header .column.logo-arxiv img { height: 25px; } .mobile-header .column.logo-cornell { display: flex; } .mobile-header .column.logo-cornell img { height: 45px; } .mobile-header .column.nav { justify-content: flex-end; align-self: flex-en= d; } .mobile-header #toggle-container button.toggle-control { background-color: = transparent; border-radius: 0px; border: 0px; font-size: 25px; padding: 3px= ; margin-left: 0.5em; } .mobile-header #toggle-container button.toggle-control svg.icon { width: 1.= 25rem; margin: 0px; } .mobile-header .mobile-toggle-block { margin: 0px; width: 100%; flex: 0 0 a= uto; transition: max-height 3s; position: absolute; top: 64px; left: 0px; b= ackground-color: rgb(179, 26, 26); overflow: hidden; } .mobile-header .mobile-toggle-block form.mobile-search-form { margin: 0.5em= 1em 1em; } .mobile-header .mobile-toggle-block form.mobile-search-form input::-webkit-= input-placeholder { color: rgb(148, 144, 140); } .mobile-header .mobile-toggle-block form.mobile-search-form input::placehol= der { color: rgb(148, 144, 140); } .mobile-header .mobile-toggle-block form.mobile-search-form input { width: = calc(100% - 50px); background: rgb(254, 246, 246); border-radius: 10px 0px = 0px 10px; border: 0px; padding: 5px; color: black; font-size: 1.5em; } .mobile-header .mobile-toggle-block form.mobile-search-form button.button {= width: 50px; background: rgb(252, 85, 84); border-width: 0px 0px 0px 1px; = border-top-style: initial; border-right-style: initial; border-bottom-style= : initial; 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padding: 0.5em; border: 1px so= lid black; margin: 0px; font-weight: normal; font-style: normal; text-align= : left; text-indent: 0px; text-transform: none; line-height: normal; letter= -spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space:= nowrap; float: none; box-sizing: content-box; box-shadow: rgb(170, 170, 17= 0) 5px 5px 15px; } #MathJax_ZoomOverlay { position: absolute; left: 0px; top: 0px; z-index: 30= 0; display: inline-block; width: 100%; height: 100%; border: 0px; padding: = 0px; margin: 0px; background-color: white; opacity: 0; } #MathJax_ZoomFrame { position: relative; display: inline-block; height: 0px= ; width: 0px; } #MathJax_ZoomEventTrap { position: absolute; left: 0px; top: 0px; z-index: = 302; display: inline-block; border: 0px; padding: 0px; margin: 0px; backgro= und-color: white; opacity: 0; } ------MultipartBoundary--F1yBeK8uKWA7BkN9gkVIZ3RHfxbJbR9RwRnHc6wsMe---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-7dd5d68e-818a-42a7-b48b-8dda7089ed8b@mhtml.blink @charset "utf-8"; #MathJax_About { position: fixed; left: 50%; width: auto; text-align: cente= r; border: 3px outset; padding: 1em 2em; background-color: rgb(221, 221, 22= 1); color: black; cursor: default; font-family: message-box; font-size: 120= %; font-style: normal; text-indent: 0px; text-transform: none; line-height:= normal; letter-spacing: normal; word-spacing: normal; overflow-wrap: norma= l; white-space: nowrap; float: none; z-index: 201; border-radius: 15px; box= -shadow: rgb(128, 128, 128) 0px 10px 20px; } #MathJax_About.MathJax_MousePost { outline: none; } .MathJax_Menu { position: absolute; background-color: white; color: black; = width: auto; padding: 5px 0px; border: 1px solid rgb(204, 204, 204); margin= : 0px; cursor: default; font-style: ; font-variant: normal; font-weight: ; = font-stretch: ; font-size: ; font-family: ; font-optical-sizing: ; font-siz= e-adjust: ; font-kerning: ; font-feature-settings: ; font-variation-setting= s: ; font-language-override: ; text-align: left; text-indent: 0px; text-tra= nsform: none; line-height: normal; letter-spacing: normal; word-spacing: no= rmal; overflow-wrap: normal; white-space: nowrap; float: none; z-index: 201= ; border-radius: 5px; box-shadow: rgb(128, 128, 128) 0px 10px 20px; } .MathJax_MenuItem { padding: 1px 2em; background: transparent; } .MathJax_MenuArrow { position: absolute; right: 0.5em; padding-top: 0.25em;= color: rgb(102, 102, 102); font-size: 0.75em; } .MathJax_MenuActive .MathJax_MenuArrow { color: white; } .MathJax_MenuArrow.RTL { left: 0.5em; right: auto; } .MathJax_MenuCheck { position: absolute; left: 0.7em; } .MathJax_MenuCheck.RTL { right: 0.7em; left: auto; } .MathJax_MenuRadioCheck { position: absolute; left: 0.7em; } .MathJax_MenuRadioCheck.RTL { right: 0.7em; left: auto; } .MathJax_MenuLabel { padding: 1px 2em 3px 1.33em; font-style: italic; } .MathJax_MenuRule { border-top: 1px solid rgb(221, 221, 221); margin: 4px 3= px; } .MathJax_MenuDisabled { color: graytext; } .MathJax_MenuActive { background-color: rgb(96, 104, 114); color: white; } .MathJax_MenuDisabled:focus, .MathJax_MenuLabel:focus { background-color: r= gb(232, 232, 232); } .MathJax_ContextMenu:focus { outline: none; } .MathJax_ContextMenu .MathJax_MenuItem:focus { outline: none; } #MathJax_AboutClose { top: 0.2em; right: 0.2em; } .MathJax_Menu .MathJax_MenuClose { top: -10px; left: -10px; } .MathJax_MenuClose { position: absolute; cursor: pointer; display: inline-b= lock; border: 2px solid rgb(170, 170, 170); border-radius: 18px; font-famil= y: "Courier New", Courier; font-size: 24px; color: rgb(240, 240, 240); } .MathJax_MenuClose span { display: block; background-color: rgb(170, 170, 1= 70); border: 1.5px solid; border-radius: 18px; line-height: 0; padding: 8px= 0px 6px; } .MathJax_MenuClose:hover { color: white !important; border: 2px solid rgb(2= 04, 204, 204) !important; } .MathJax_MenuClose:hover span { background-color: rgb(204, 204, 204) !impor= tant; } .MathJax_MenuClose:hover:focus { outline: none; } ------MultipartBoundary--F1yBeK8uKWA7BkN9gkVIZ3RHfxbJbR9RwRnHc6wsMe---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-503f3f34-7294-4ee0-9d7a-889eba5c77d7@mhtml.blink @charset "utf-8"; .MathJax_Preview { color: rgb(136, 136, 136); } #MathJax_Message { position: fixed; left: 1em; bottom: 1.5em; background-co= lor: rgb(230, 230, 230); border: 1px solid rgb(149, 149, 149); margin: 0px;= padding: 2px 8px; z-index: 102; color: black; font-size: 80%; width: auto;= white-space: nowrap; } #MathJax_MSIE_Frame { position: absolute; top: 0px; left: 0px; width: 0px; = z-index: 101; border: 0px; margin: 0px; padding: 0px; } .MathJax_Error { color: rgb(204, 0, 0); font-style: italic; } ------MultipartBoundary--F1yBeK8uKWA7BkN9gkVIZ3RHfxbJbR9RwRnHc6wsMe------