Classical Thermodynamics as a Consequence of Spacetime Geometry

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1 Classical Thermoynamics as a Consequence of Spaceime Geomery Jay R. Yablon 9 Norhumberlan Drive Schenecay, New York jyablon@nycap.rr.com January 6, 25 Absrac: Jus as Maxwell s magneic equaions emerge enirely from = of exerior calculus applie o a gauge poenial A, so oo oes he secon law of hermoynamics emerge from applying = o a scalar poenial s. If we represen his as s = U =, hen when he Gauss / Sokes heorem is use o obain he inegral formulaion of his equaion, an afer breaking a ime loop ha appears in he inegral equaion, we fin ha U behaves precisely like he inernal energy sae variable, an ha he secon law of hermoynamics for irreversible processes naurally emerges. Conens. Inroucion The Reversible Enropy Equaion The Irreversible Enropy Equaion The Secon Law of Thermoynamics Conclusion... 9 References... 9

2 . Inroucion Nowhere is he power of ifferenial forms geomery o eermine physical resuls in spaceime more apparen han for he magneic monopole equaion of classical elecroynamics. One posulaes a gauge poenial one-form A = A x wih an energy-imensione vecor, ν efines from his a fiel srengh wo-form F = A = A x x, an applies he 2! [ ν ] funamenal resul = of exerior calculus ha he exerior erivaive of an exerior erivaive is zero, o obain F = A = which conains Gauss an Faraay s classical laws for magneism. Then, one applies an open riple inegral o he monopole equaion hree-forms, also applies he Gauss / Sokes heorem H = H is M where H is a generalize p-form an M M he close exerior bounary of a p+-imensional manifol, an hereby obains F = F = ( = ) which are he classical magneic equaions in inegral form. Goo reviews of he unerlying exerior calculus an ifferenial forms are provie, for example, in [] Chaper 4 an [2] Chaper IV.4. The resul ha = oes no, however, sop wih is use o obain A =. I applies o any p-form of any rank. In his paper, we shall emonsrae ha by saring wih a imensionless scalar poenial s which is a zero-form, efining a one form U s, nex obaining he wo-form U = s =, an finally inegraing wih Gauss / Sokes via U = U = ( = ), he energy-imensione vecor U in U = U x urns ou o behave like he inernal energy of a hermoynamic sysem, an he resuling inegral form equaions when suie in eail, urn ou o conain he laws of classical hermoynamics. In a nushell: jus as = when applie o a one-form poenial via F = A = an hen inegrae conains he classical Gauss an Ampere laws for magneism, his same = when applie o a zero-form poenial via U = s = an hen inegrae conains he laws of classical hermoynamics. 2. The Reversible Enropy Equaion As jus inrouce, he equaion from which we will procee is he wo form equaion: U = s = (2.) as well as is inegral formulaion U = U = (2.2) which uses Gauss / Sokes. Le us sar by eveloping (2.). Firs, we may expan he ifferenial forms o exrac he ensor equaion:

3 Uν ν U = ν s ν s =. (2.3) We efine he four componens of he prospecive inernal energy vecor as U σ ( U, ) course he spaceime graien operaor = (, ) wih / u, an of =. We shall work hroughou in fla Minkowski spaceime wih he meric ensor iag ( ) (,,, ) lower inexes. η = use o raise an For he space componens, wih =, ν = 2 we obain U 2 2U = 2τ 2 τ =, an once all hree componens are obaine, his reaily generalizes o: u = τ = (2.4) The laer τ = of course is he mahemaical ieniy ha he curl of he graien of a scalar is zero. The laer conains he physical conen u =, which ells is ha he curl of he prospecive inernal energy hree-vecor u is zero. Wih =, ν = k =, 2,3 we obain U U = τ τ =, which becomes: ν k k k k τ u U = τ =. (2.5) The laer equaion is simply he commuaor ieniy [ ] τ = is he expansion of τ =., τ =, which ogeher wih Puing (2.4) an (2.5) ogeher showing only U σ gives us he pair of ifferenial equaions which analogize via he ifferenial form o B = an B + E =, namely: u =. (2.6) u + U = Now le s urn o he inegral equaion (2.2). Expaning he forms in (2.2) 2! ( U U ) x x ν = U x σ =. (2.7) ν ν σ Separaing space an ime componens an accouning for all inex permuaions we obain: 2

4 ( ) k U k ku x x k Ukx ( ) ( ) ( ) + U U x x + U U x x + U U x x = U x + = The covarian (lower-inexe) U ( U, ) k ν. (2.8) = u, an of course he ifferenial elemens ν ν anicommue x x = x x. So separaing he ime inegral from he space inegral in he op line an being careful wih he signs, his may be wrien as: ( ( ) ) ( ) u + U l u S = U u l =. (2.9) Now, le us spen a momen on he erm U, which is somehing of an oiy because i represens a close loop line inegral over ime of he prospecive sae variable for inernal energy. We of course know ha he abiliy o ravel a close ime loop is ficive in he naural worl, bu so oo is a reversible hermoynamic process. So le s follow his hrough: The inegral U says ha we sar wih he prospecive inernal energy U a ime =, hen move forwar in ime, bu hen evenually loop aroun an come back o =. So whaever we o beween he firs ime an he secon ime will be reverse, because we arrive righ back a ime. So he inegral U is, in many ways, he very efiniion of a reversible process. An his, of course, is ficive, because ime is only experience in one irecion. This is he firs inicaion we have of some hermoynamic possibiliies. Shorly, we shall break his ime loop o esablish Eingon s arrow of ime, bu before we o so, we will wan o make a connecion o enropy S while (2.9) sill represens a reversible process, because in a reversible process, TS = δq is an equaliy raher han an inequaliy, where T is he emperaure, Q is he hea, an δ which operaes on hea is an inexac ifferenial which means ha he hea upon which i operaes is no a hermoynamic sae funcion. Once he process becomes irreversible, hen he enropy law becomes TS δ Q, bu his inequaliy shoul be naurally supplie by he spaceime geomery, no insere by han. We sar wih he firs law of hermoynamics which we shall wrie as U = δ Q δw. (2.) The exac ifferenial form for inernal energy is 2! ( ν ν ) ν U = U U x x which we can compac using a commuaor o U = U x x ν, while δ Q an δ W are he inexac 2 [ ν ] ifferenial forms for hea an work an herefore are no sae funcions. We can wrie hese in expane form as δq = δ Q x x ν an δw = δ W x x ν. By puing a negaive sign in 2 [ ν ] 2 [ ν ] fron of he work ifferenial in (2.) we are represening sysems which gain hea, bu perform work on (lose work energy o) he environmen. To represen he componens of he 3

5 conravarian hea an work four-vecors we may employ Q ( Q, q ) an W ( W, ) exac ifferenial as alreay noe is (, ). An we shall use δ ( δ, ) w. The δ o represen he componens of he inexac ifferenials. So expaning U = δ Q δw an using he foregoing, we may wrie he firs law in ensor forma as: U = δ Q δ W. (2.) [ ν ] [ ν ] [ ν ] The k componens of he above, conras (2.5), are: ( ) U U = δ Q δ Q δ W δ W = u U = δ q δq + δ w + δ W. (2.2) k k k k ν k The 2 componens of (2.) are U 2 2U = δq2 δ2q δ2w + δw 2 an his generalizes o: u = δ q + δ w. (2.3) We hen use (2.2) an (2.3) in (2.9) o obain: ( ( + ) ) ( ) δ q δq δ w δw l δ q δ w S = U u l =. (2.4) Now ha we have a reversible equaion which conains δ q + δ Q we urn o enropy. As alreay noe afer (2.9), whenever a process is reversible as is (2.4) because of he close ime loop in U, he enropy is relae o hea an emperaure by he ifferenial forms: TS = δq. (2.5) Here, he enropy S = S x is also a one-form wih four-vecor componens ha we shall wrie as S ( S, ) = s. From he above we exrac he ensor expression: T S = δ Q. (2.6) [ ν ] [ ν ] The k relaionship is hen: ( ) ( s ) T S S = δ Q δ Q = T + S = δ q δ Q, (2.7) k k k k while he 2 inex equaion ( ) T S S = δ Q δ Q generalizes for all space inexes o: T s = δ q. (2.8) We hen use (2.7) an (2.8) o replace all he hea in (2.4) wih enropy, hus avancing o: 4

6 ( T (( + S ) W ) l ) ( T ) s δ w δ s δ w S = U u l =. (2.9) Now ha we have inclue he reversible enropy relaionship in his reversible equaion, i is ime o see wha happens when we make he above irreversible. 3. The Irreversible Enropy Equaion As we observe a (2.9), he inegral U which sill appears in (2.9) informs us ha his equaion is for a ficive reversible process. This is why we able o properly uilize he reversible enropy relaionship TS = δq of (2.5) in (2.9). Now le us break his ime loop, an esablish an arrow of ime. Specifically, le us now replace U U wih an irreversible ime inegral. Wha can we say abou U an U in relaion o an o one anoher? If U represens inernal energy, hen given ha energies are always represene as posiive or zero, U ( ) a all imes. However, in U we are saring a a given ime =, moving somewhere else in ime, an hen ficively reurning o he same ime = a which we sare. So he ime loop inegral han, if he efinie ime a he upper boun in, hen so oo, U. Therefore: U = U =, irrespecive of he energy. On he oher U is greaer han or equal o zero, i.e., if U U =. (3.) So if we now subsiue U U wih ino (2.9), hen he erm on he righ will become greaer han or equal o, an o capure his, we nee o simulaneously replace he final = wih a. Doing so, we obain: ( ( T ( S ) δ ) l ) ( w) s + w δw T s δ S = U u l. (3.2) Now, we have an expression which expressly inclues enropy erms, an by breaking he ime loop an ino an arrow of ime, we have some expressions involving hese enropy erms being. So we nee o see if he secon law relaions TS δ Q an / or S are inclue in (3.2) in some clear form. Firs, as is one a his sage of eveloping for Maxwell s inegral from ifferenial forms, le us muliply hrough all of (3.2) by /, hus: ( ( ( ) δ W ) l) ( T ) S T s + S w δ = U l s δ w u. (3.3) 5

7 In wha is now ( / ) U we have an offseing / =. An we may also apply = o boh his an he firs erm. So we hen have: δ s δ w = l. (3.4) ( T ( s + S ) δw W ) l ( T ) S U u Now le s look a he erms o he lef of he equal sign. On he righ we have an inegral ( T s δ w) S over an open surface. The open surface is boune by a close loop, ye he line inegral ( ) ( + δw ) T s S δ W l on he lef is also aken over an open pah. So his oes no mach up. Wha o we o? The same siuaion is encounere in Maxwell s equaions. For example, Gauss Law for magneism B S = is efine over a close surface. Bu if one acually evelops Faraay s law from he ifferenial forms F = F =, he equaion one firs arrives a is E l = ( / ) B S conaining he same B S. Bu he bounaries are mismache. So o mach hem up we conver he close surface o an open surface, an hereby obain E l = ( / ) B S which is Faraay s law. Then, he pah of he close line inegral can be ienifie wih he bounary of he open wo-imensional surface hrough which he magneic fiel is flowing. The same exercise occurs when eveloping Ampere s law. So in (3.4), we nee o mach up he perimeer of he open bounary in ( T s δ w) S wih a close loop in ( T ( s + S ) δw δ W ) l. Here, we nee o urn he open line inegral ino a close line inegral. Making his bounary change, (3.4) now becomes: δ δ = l. (3.5) ( T ( s + S ) δw W ) l ( T s w) S U u Because here are now close loop inegrals on boh sies of he above equaliy following he change in (3.5), we nee o see if any erms migh muually cancel. A his poin, le us backrack a bi. Wriing (2.2) as δ w δw = δ q δq + u + U, we replace he work erms in he loop inegral an use = / o obain: T ( + S ) δ Q + + U l ( T ) U l s q δ = u s δ w S u. (3.6) / u l erm now on he lef of he equaliy is he same as he erm ( / ) u l on he righ of he equaliy. Bu because of he inequaliy, we nee o be careful how we work wih his equivalence. Surely enough, he ime-epenen ( ) The bes approach is o now separae (3.6) ino is wo inequaliies wih he imeerivaive ousie each inegral, an hen o isolae his maching erm in each, hus: 6

8 l u s q U l u ( T ( S ) δ δq U ) l ( T s δ w) Then, we may once again recombine hese wo inequaliies as such: U l ( T ( S ) Q U ) l ( T ) u s + δ q + + s w S S. (3.7) δ δ. (3.8) Then, we use ( ) T s + S = δ q + δ Q from (2.7) o backrack an furher cancel erms, hus: U l U l ( T ) u + s w S δ. (3.9) s δ w S above originae in an is equal o he S in (2.9). Bu in (2.4) we foun ha by mahemaical ieniy, Nex, he erm conaining ( T ) u erm ( ) u = τ =. So his erm S = u S = δ S = S = (3.) ( T s δ w) ( ) ( T s w) ( u) is always zero by τ =, in all places a all imes. We coul have zeroe his ou back a (2.9), bu kep his in place so ha we woul be able o obain hese enropy an work ieniies an properly mach up all he inegral bounaries. So reurning he remaining o insie he line inegral, his furher simplifies (3.9) o:. (3.) U u l U l Finally, we forwar rack again. From (2.2) we may separae he ime-epenen relaionship u = δq δw from he space-epenen U = δq + δ W o rewrie he above: ( δ q δ w) l ( + W ) δ δ. (3.2) U Q l An from (2.7) we may also separae he ime-epenen δ q = T s from he space epenen δq = T S an reurn he enropy erms back ino (3.2), hus: ( T s δ w) ( T S + W ) δ. (3.3) U l l 7

9 I will be seen ha all of hese erms were originally in (3.5) before we backracke an hen reurne. Inee, wih he excepion of U which sill remains, all hese erms were in ( T ( s + S ) δw δ W ) l, an everyhing else has roppe ou while in he process he irreversible inequaliy has been resiuae. The above is a precise resaemen of he original reversible U = U = of (2.2) following full evelopmen an cancellaion of erms, an afer replacing he ime loop inegral wih an arrow of ime inegral via U U. If one were o rever he inequaliies in he above back o equaliies, hen his woul be compleely he same as U = U =, an (3.3) woul hen escribe a ficive reversible process. 4. The Secon Law of Thermoynamics A his poin, le us focus on he laer inequaliy in (3.3), an move everyhing o he lef sie, hus: ( T + T S δ w W ) l s δ. (4.) Also, le s reurn spaceime inexes so we can examine covarian behaviors in spaceime an iag η =,,, o make sure he signs are correc. This yiels: using ( ) ( ) ν k k ( T Sk + T ks + δwk δkw ) x = ( T [ ks] δ[ kw] ) x = ( T [ S ] δ[ W] ) x.(4.2) Now le s compac his ino he ifferenial one-form equaion. This is a bi ifferen from a usual ifferenial form, because here is one loose (unconrace) ime inex. The expression S [ ] is a ime componen (an really, he ime bivecor) of a secon rank anisymmeric ensor. Similarly for δ W [ ], excep his conains an inexac ifferenial. In general, if we only conrac one inex of such a ensor S = S, hen S = S x an ν ν ν ν herefore S = S x is he ime componen of a four-vecor of ifferenial forms. So wih ha, (4.2) compacs fully o: ( TS δ W ). (4.3) When he inexac work ifferenial δ W =, his reuces o: S. (4.4) An his is he secon law of hermoynamics for an irreversible process. A no poin along he way i we have o pu his inequaliy ino his equaion by han. This inequaliy was a naural resul of eveloping he ifferenial form equaion U = s = (2.) in he inegral form 8

10 U = U = of (2.2) an convering he resuling reversible ime loop ha emerge o an arrow of ime, U U. Were we o urn U back o become reversible, an (4.4) woul become he ficive S = 5. Conclusion. U, everyhing woul again Jus as Maxwell s classical magneic equaions emerge enirely from = of exerior calculus applie o a gauge poenial A, so oo oes classical hermoynamics emerge from applying = o a scalar poenial s. If we represen his as s = U =, hen U behaves precisely as he inernal energy sae variable, an afer breaking a ime loop ha appears in he inegral equaion U = U = o make he his equaion irreversible, we obain he secon law of hermoynamics in he form S, which governs he enropy sae variable S for an irreversible sysem. References [] Misner, C. W., Thorne, K. S., an Wheeler, J. A., Graviaion, Freeman (973) [2] Zee, A., Quanum Fiel Theory in a Nushell, Princeon (23) 9