{"id":528,"date":"2009-02-07T21:50:40","date_gmt":"2009-02-07T21:50:40","guid":{"rendered":"http:\/\/spinor.info\/weblog\/?p=528"},"modified":"2009-02-07T21:53:41","modified_gmt":"2009-02-07T21:53:41","slug":"more-on-black-hole-mechanics","status":"publish","type":"post","link":"https:\/\/spinor.info\/weblog\/?p=528","title":{"rendered":"More on black hole mechanics"},"content":{"rendered":"

I remain troubled by this business with black holes<\/a>.<\/p>\n

In particular, the zeroth law. Many authors, such as Wald, say that the zeroth law states that a body’s temperature is constant at equilibrium. I find this formulation less than satisfactory. Thermodynamics is about equilibrium systems to begin with, so it’s not like you have a choice to measure temperatures in a non-equilibrium system; temperature is not even defined there! A proper formulation for the zeroth law is between systems: the idea that an equilibrium exists between systems 1 and 2 expressed in the form of a function f<\/em>(p<\/em>1<\/sub>, V<\/em>1<\/sub>, p<\/em>2<\/sub>, V<\/em>2<\/sub>) being zero. Between systems 2 and 3, we have g<\/em>(p<\/em>2<\/sub>, V<\/em>2<\/sub>, p<\/em>3<\/sub>, V<\/em>3<\/sub>) = 0, and between systems 3 and 1, we have h<\/em>(p<\/em>3<\/sub>, V<\/em>3<\/sub>, p<\/em>1<\/sub>, V<\/em>1<\/sub>) = 0. The zeroth law says that if f<\/em>(p<\/em>1<\/sub>, V<\/em>1<\/sub>, p<\/em>2<\/sub>, V<\/em>2<\/sub>) = 0 and g<\/em>(p<\/em>2<\/sub>, V<\/em>2<\/sub>, p<\/em>3<\/sub>, V<\/em>3<\/sub>) = 0, then h<\/em>(p<\/em>3<\/sub>, V<\/em>3<\/sub>, p<\/em>1<\/sub>, V<\/em>1<\/sub>) = 0. From this, the concept of empirical temperature can be obtained. I don’t see the analog of this for black holes… can we compare two black holes on the basis of J<\/em> and \u03a9 (which take the place of V<\/em> and p<\/em>) and say that they are in “equilibrium”? That makes no sense to me.<\/p>\n

On the other hand, if you have a Pfaffian in the form of dA<\/em> + B<\/em> dC<\/em>, there always exists an integrating denominator X<\/em> (in this simple case, one doesn’t even need Carath\u00e9odory’s principle and assume the existence of irreversible processes) such that X<\/em> dY<\/em> = dA<\/em> + B<\/em> dC<\/em>. So simply writing down dM<\/em> \u2013 \u03a9 dJ<\/em> already gives rise to an equation in the form X<\/em> dY<\/em> = dM<\/em> \u2013 \u03a9 dJ<\/em>. That \u03ba and A<\/em> serve nicely as X<\/em> and Y<\/em> may be no more than an interesting coincidence.<\/p>\n

But then there is the area theorem such that dA<\/em> > 0 (just like dS<\/em> > 0). Is that another coincidence?<\/p>\n

And then there is Hawking radiation. The temperature of which is proportional to the surface gravity, T<\/em> = \u03ba\/2\u03c0, which is what leads to the identification S<\/em> = A<\/em>\/4. Too many coincidences?<\/p>\n

I don’t know. I can see why this black hole thermodynamics business is not outright stupid, but I remain troubled.<\/p>\n<\/fb:like>","protected":false},"excerpt":{"rendered":"

I remain troubled by this business with black holes. In particular, the zeroth law. Many authors, such as Wald, say that the zeroth law states that a body’s temperature is constant at equilibrium. I find this formulation less than satisfactory. Thermodynamics is about equilibrium systems to begin with, so it’s not like you have a […]<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-528","post","type-post","status-publish","format-standard","hentry","category-physics","category-3-id","post-seq-1","post-parity-odd","meta-position-corners","fix"],"_links":{"self":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/528","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=528"}],"version-history":[{"count":4,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/528\/revisions"}],"predecessor-version":[{"id":530,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/528\/revisions\/530"}],"wp:attachment":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=528"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=528"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=528"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}