Classical Thermodynamics Entropy Laws as a Consequence of Spacetime Geometry

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1 Classical Thermoynamics Enropy Laws as a Consequence of Spaceime Geomery Jay R. Yablon 9 Norhumberlan Drive Schenecay, New York jyablon@nycap.rr.com January 7, 25 Absrac: Jus as Maxwell s magneic equaions emerge enirely from applying = of exerior calculus 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 he enropy of irreversible processes naurally emerges. PACS: 5.7.-a, 2.4.Hw Conens. Inroucion The Reversible Enropy Equaion The Irreversible Enropy Equaion The Secon Law of Thermoynamics Summary an Conclusion... References... 2

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 be synonymous wih he secon (enropy) law of classical hermoynamics, for reversible an irreversible processes. In a nushell: jus as = when applie o a one-form poenial via F = A = an hen inegrae conains he classical Gauss an Faraay laws for magneism, his same = when applie o a zero-form poenial via U = s = an hen inegrae conains he secon law 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.).

3 Firs, we may expan he ifferenial forms o exrac he ensor equaion: 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 former 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 a pair of ifferenial equaions which analogize via he ifferenial forms o Maxwell s 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) we obain: 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 yiels: 2

4 ( ) k U k ku x x k Ukx ( ) ( ) ( ) + U U x x + U U x x + U U x x. (2.8) = U x + = The covarian (lower-inexe) U ( U, ) k = 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 remins us 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 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 represen 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 were 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. To o so we replace inegral. Wha can we say abou U an U U wih an irreversible ime U in relaion o an o one anoher? If U represens an inernal energy which is always posiive or zero, hen ( ) 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 U = U = irrespecive of he energy, because of he close reversible ime loop. On he oher han, if he efinie ime a he upper boun in if, hen so oo, U. Therefore: U is greaer han or equal o zero, i.e., 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) Above, we have also mainaine he equaliy beween he firs expression which inclues enropy erms ( ( T ( s + S )) l) T s S an he secon U l u which inclues he ime loop broken ino an arrow of ime. Consequenly, he inequaliy now naurally applies o hese enropy erms as well. Thus, i will now be imporan o see if he secon law relaions TS δ Q an / or S are inclue in (3.2) in some clear form. In his regar, i is imporan o poin ou ha we eliberaely waie o break he ime loop unil we ha arrive a (2.9) which conains enropy, even hough i woul have been 5

7 possible o o so earlier a (2.4) which conains hea or (2.9) which conains inernal energy. This is because he secon law in he form TS δ Q ells us ha when an inequaliy arises, i oes so beween he enropy an he hea. Therefore, if we ha broken he ime loop a (2.4), ( δ + ) q δ we woul also have ha o isconnec he firs expression which inclues ( Q) l from he secon expression U u l. By waiing unil (2.9) o esablish he arrow of ime raher han oing so earlier, we have alreay effecively embee a varian of TS δ Q ino (3.2). We shall review his in more eail shorly, bu before we o so, i is beer o reuce (3.2) ino a simpler an clearer form. Firs, as is one a his sage of eveloping Maxwell s inegral equaions 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) 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 evaluae over an open pah. This oes no mach up, so wha o we o? This same siuaion is encounere in Maxwell s equaions. For example, he bounary for Gauss Law for magneism B S = is a close surface. Bu if one acually evelops Faraay s law from he ifferenial forms F = F =, he equaion firs arrive a is E l = ( / ) B S conaining he same B S. Bu here oo 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 siuaion also occurs when eveloping Ampere s law. So in (3.4), we nee o mach up he s δ w S wih a close loop in T s + S δ w δ W l, jus as is one for Maxwell s equaions. Here, we nee o urn perimeer of he open bounary in ( T ) ( ( ) ) he open line inegral ino a close line inegral. Making his bounary change, (3.4) is now: 6

8 δ δ = l. (3.5) ( T ( s + S ) δw W ) l ( T s w) S U u Following his change, here are now close loop inegrals on boh sies of he above, so we nee o see if any erms migh muually cancel. 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) Surely enough, he ime-epenen ( ) equivalen o he erm ( / ) l / u l erm now on he lef of he equaliy is u on he righ of he equaliy. Bu because of he inequaliy, we nee o be careful how we work wih his equivalence. The bes approach is o now separae (3.6) ino is wo inequaliies, an hen o isolae his maching erm in each, hus: l u s q U l u ( T ( S ) δ δq U ) l ( T s δ w) Then, we may again recombine hese wo inequaliies, an also compac ( ) S. (3.7) / u u, hus: U u l ( T ( s + S ) δ q Q + U ) l + ( T ) s w S δ δ. (3.8) Nex, from (2.2) we may separae he ime-epenen relaionship u = δq δw from he space-epenen U = δq δ W. Insering his ino (3.8), cancelling a δq δ Q =, an isribuing he minus sign from ousie he laer loop inegral yiels: U ( δq δw) l ( T s T S + δq + W ) l + ( T ) s w S δ δ. (3.9) s δ l o all sies yiels: Then, aing ( T q) U + ( T s δq) l ( T s δw) l ( T S + W ) l + ( T ) s w S δ δ. (3.) 7

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 an everywhere zero in all Lorenz frames by τ =. We coul have zeroe his ou back a (2.9), bu kep his in place so ha we woul be able o obain his ieniy (3.) an properly mach up all he inegral bounaries as we i earlier. So (3.) simplifies o: ( T s q) l ( s w) l ( T ) δ. (3.2) U + δ T δ S + W l The above is a precise resaemen of he original reversible U = U = 8 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, an (3.2) above back o equaliies, hen his woul be compleely he same as U = U = woul hen escribe a ficive reversible process. 4. The Secon Law of Thermoynamics A his poin, le us a ( ) very righ. Thus: T S δ W l o all sies of he above so ha a zero is on he ( T s T S δ q W ) l ( T s T S δ w W ) U + + δ + δ l. (4.) This now separaes ino wo inequaliies, namely: U an ( δ δ w) q l. (4.2) ( T + T S δ w W ) l s δ. (4.3) Now, le us reurn spaceime inexes so we can examine covarian behaviors in spaceime iag η =,,, o make sure he signs are correc. For (4.2) we obain: using ( ) ( ) ν k ( δ Qk + δ Wk ) δ ( W ) δ ( Q). (4.4) U x = Q x = W

10 Here we have use he fac ha ( ) loop inegral ( δ Q δ ) δq + δ W x = for wha is once again a close ime + W = which mus always be zero no maer wha he values of δq + δw may be. The above compacs ino a ifferenial form via ( ) Q W = Q W x, an i saes ha he ime componen of he inernal energy will always be greaer han or equal U δ Q. o he inexac ime ifferenial of work minus hea aken over a close loop, ( ) Nex, for (4.3) we have: W k ( T Sk + T ks + δwk δkw ) x = ( T [ S] δ[ W] ) x. (4.5) Le us also now compac his ino a ifferenial form. The expression S [ ] is a ime componen (an really, he ime bivecor) of a secon rank anisymmeric ensor [ Sν ]. Similarly for δ W [ ], excep his conains an inexac ifferenial. In general, if we only conrac one inex of such a ensor Sν = Sν, hen S S x ν = ν an herefore S = S x is he ime componen of a fourvecor of ifferenial forms. So in view of his, (4.5) compacs fully o: ( TS δ W ). (4.6) When he inexac work ifferenial δ W =, his reuces o: S. (4.7) This is he secon law of hermoynamics for an irreversible process. Imporanly, 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 forms equaion U = s = (2.) in he inegral form U = U = arrow of ime, of (2.2) an convering he emergen reversible ime loop o an U U. Were we o urn U back o U, everyhing woul again. become reversible, an (4.7) woul become he ficive S = Finally, having foun he secon law (4.7), le us reurn o where we lef off afer (3.2) an look for some form of is varian TS δ Q. I will be appreciae ha (3.2) is enirely equivalen o (3.2); i is simply (3.2) afer complee reucion of erms. In urn, (3.2) is he irreversible version of he reversible (2.9), an for a reversible process (2.9) is equivalen o (2.4) an o (2.9). An, o ge from (2.4) o (2.9) we use he reversible enropy relaionship (2.5). Therefore, if we now reurn o (2.4) an (2.9) an break he ime loop 9 U in hese equaions raher han waiing for (2.9), we can embe he secon law varian TS U δ Q

11 if we isconnec wha is hen U l u from he equaions on he lef sie of he (2.4) an (2.9) equaliies. Tha is, merging ogeher (3.2) an (2.4) an (2.9) bu keeping he laer wo equal o zero, hen afer breaking he ime loop U ( T (( s + S ) δw δw ) l ) ( T s δ w) ( ( δq + δq δw δw ) l ) ( q w) S ( ( u U ) l) ( u) S S U, we may wrie: δ δ. (4.8) = + = We know from (4.6) ha he inequaliy in he above makes he op line synonymous wih ( TS δ W ) afer all reucions are carrie ou. Bu by keeping he boom wo lines equal o zero as hey originally were, even afer form of TS δ Q ha we shall now make explici. U U, we are effecively embeing a If we now o all of he same reucions we use o ge from (2.4) or (2.9) o (4.3) bu keep everyhing in erms of hea or inernal energy raher han enropy, he counerpar we obain o (4.3) is: ( δ q + Q δ w W ) l = ( u + U ) = δ δ. (4.9) l Comparing he laer equaliy o (4.3), his means ha: ( s + S ) l ( δ q + Q) T T δ l. (4.) Resoring spaceime inexes as before hen yiels: k k T [ ks] x δ[ kq] x, (4.) an compacing o ifferenial forms prouces: TS δq. (4.2) This is how he secon law varian equaions. TS δ Q becomes embee in he ifferenial forms If we hen compac (4.9) ino: ( δq δw ) = ( δ[ Q] δ[ W] ) x = U = [ U] x =, (4.3)

12 we may combine (4.2) wih (4.3) o ie all of his ogeher in he relaionship: ( δ ) ( δ δ ) TS W Q W = U =. (4.4) In he above, U = is a conservaion law for he inernal energy, ( TS δ W ) is one varian of he secon law which becomes S when work is being neiher expene nor applie, an TS δ Q is anoher varian of he secon law. 5. Summary an Conclusion Summarizing he enire evelopmen, if he inernal energy U = U x σ σ is aken o be he one form U s obaine from a scalar poenial s hence U s U σ U, u = =, hen wih ( ) (he form U has he same symbol as he ime componen U which are muually isinguishe by conex), an afer breaking he ime loop o creae an arrow of ime via U = U, he resuling ifferenial forms equaion U = U = of ifferenial forms equaions: is equivalen in all respecs o he pair ( TS δw ) ( δq δw ) = U = U δ ( W Q) wih a free ime inex, as obaine in (4.4) an (4.4). U = s = is equivalen o u = u + U = (5.) Likewise, he form equaion (2.6) obaine in (2.6). These are he analogues of Gauss law of magneism an Faraay s law, in inegral an ifferenial represenaions, respecively. In conclusion, jus as Maxwell s classical magneic equaions emerge enirely from applying = of exerior calculus o a gauge poenial A, so oo oes he secon law of classical hermoynamics ogeher wih oher relae relaionships 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

13 irreversible sysem, an are also able o embe a secon law relaionship TS δq beween enropy an hea. 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) 2