Solving the Temperature Problem under Relativistic Conditions within the Frame of the First Principle of Thermodynamics

Transcription

1 Physs & Astronomy Internatonal Journal Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams Abstrat The frst prnples of thermodynams under relatvst ondtons and wth allowane for surfae tenson were formulated. Two ases were studed - for Mnkowsk and Euldean spaes. Wth the help of these laws the temperature of the system was shown to vary aordng to Ott H at the adabat and nonadabat aeleraton.e. the temperature nreases for an observer n the laboratory referene frame as the veloty of the objet nreases. A Ensten onsdered that the temperature had on the ontrary to derease under these ondtons as the veloty nreased. It was shown where he had made an error. Keywords: Speal Relatvty Relatvst thermodynams Surfae tenson Temperature Volume Issue - 8 Revew Artle Researh and Produton Enterprse Tekhnolazer Mosow Russa *Correspondng author: Eml Vktorovh Wezmann Researh and Produton Enterprse Tekhnolazer Mosow 8 Russa Emal: Reeved: January 7 8 Publshed: February 4 8 Introduton Relatvst thermodynams was reated more than entury ago-n 97 when von Mosengel s artle was publshed []. He was Plank s dsple. Aordng to von Mosengel and Plank s onlusons the temperature T of the system under study has to vary proportonally to under relatvst ondtons and at an adabat aeleraton v / v s the veloty of the system n the referene frame at rest s the speed of lght. Ensten ame to a smlar onluson []. Whle studyng the frst prnple of thermodynams under relatvst ondtons nobody n the th entury took nto onsderaton a surfae tenson meanwhle ths thermodynam parameter s as mportant as the pressure p. Both Plank wth hs dsple and Ensten wrote down the frst prnple of thermodynams under relatvst ondtons negletng the surfae tenson as well. Besdes both of the great sentsts made an error ntegratng Gbbs equaton we shall show t below n seton. Vetsman EV [3-5] was the frst researher who understood an mportane of the surfae tenson for relatvst thermodynams. He showed that the surfae tenson s a Lorentz nvarant [3] at any rate up to a veloty v. He also showed that the nterfae equatons of state were orret under relatvst ondtons provdng the temperature of system under study vared n nverse proporton to [4]. Vetsman [5] also obtaned expressons for spef thermodynam funtons (J m - ) under relatvst ondtons (the nternal energy U the enthalpy H the free energy F and free enthalpy G). These funtons are orret f T ~/. Thus the results obtaned by Vetsman are n a full aordane wth the result obtaned by Ott [6] for the relatvst temperature T.e. T T / (here and below the symbol denotes that ths quantty s at rest). All above results are ompletely orret for an adabat aeleraton of the system. Callen & Horwtz [7] onsder that the temperature T n general s a relatvst nvarant. Fnally Ekart [8] omes to a onluson that temperature θ onstθ. We wll show below n seton 3 that ths assumpton s norret. M. Plank wrtes down the frst prnple of thermodynams As du dq + da () du dq p dv () Where Q s the heat suppled to the system or removed from t (J) A the work done by the system or wth t (J). Under relatvst ondtons the work A s wrtten down by M. Plank as da pdv + v dg (3) U + pv G v (4) Where V s the vetor of the system veloty G the momentum the symbol means vetor multplaton []. G and ( U + pv ) (5) Form the four-dmensonal vetor (4-vetor) of energymomentum whh has an nvarant length equal to ( U + pv ) (6) s the magnary unt. Submt Manusrpt Phys Astron Int J 8 (): 53

2 Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams 8 Vetsman Evdently we an wrte down the frst prnple of thermodynams n vew of the surfae tenson (normal ase) as du dq p dv + σ dω (7) Where ω s the element of area (m ). Then the relatons (3) and (4) are wrtten down as da pdv + σdω + v dg (8) U + pv σω G v. (9) Consequently the relatons (5) and (6) should have the form ( U + pv σω) () ( U + pv σω ). () Here we have to note that the quantty ω wll transform as v n dfferent ways: dependng on the orentaton of the surfae n spae [39] and Fgure &. Attemptng to obtan the transformaton law of the temperature as v M.Plank used dependenes (3) and (4) but not () and (). Other researhers dd the same takng no aount of a surfae tenson therefore the results obtaned by them were norret. Thus the man goals of ths paper are: a) Obtanng the frst prnple of thermodynams under relatvst ondtons n 3-D and 4-D versons takng nto onsderaton the surfae tenson. b) Obtanng the temperature transformaton under these ondtons from the above mentoned prnple at the adabat and non-adabat aeleraton. Solvng the Problem: 3-D Case Wrte down the frst prnple of thermodynams (normal ondtons Euldean ase) n vew of the surfae tenson as dq du + de + p dv d da v d G () kn σ ω a Where E s the maro knet energy of the objet under kn study A the work done aganst the system aeleraton up to a a veloty v << E A. The momentum G s taken here kn a aordng to (9). We wll adabatally aelerate the objet up to a relatvst veloty v now and wll ntegrate () term by term from state of the system to state for the ase represented n Fgure takng nto aount (8) (9) and the equaltes p p σ σ ω ω t also should be noted that here and below we study not a whole objet but only ts part (a subsystem): dq du + p dv σ d ω vdg (3) dv Q( v) Q U U + pv pv v U + pv σω { } () () () 3 v (4) Now we take the ntegral n (4). dv d v U + pv σω U + pv σω { } { } ( ) v { U + pv σω } { U + pv σω } v/. Aordng to M. Plank [8] U { U + pv } In vew of the surfae tenson we have: U U ( ) pv σω + + (5) (6). (7) Then takng nto onsderaton (6) we an wrte down (4) as { } Q( v) Q U + pv + σω ( ) U + pv () pv { U + pv σω } (8) If we do not nput the heat n an aelerated system from ts soure then the left sde of (8) equals zero and we annot obtan any dependenes of the knd T T( v) at frst sght the dependene (8) does not ontan the temperature. However ths relatonshp ontans t-n a latent form. Show t. Let an objet be e.g. the lqud movng wth a relatvst veloty v. For the observer beng at rest n a laboratory referene frame the temperature of ths lqud depends on the ooperatve veloty v of the mropartles n the objet relatve to ts entre of mass. Then the veloty omponents of the total veloty w of a ertan mropartle n the movng lqud equal []: w + v w (9a) vw + w w w (9b) vw + w 3 3 vw + (9) Where w ( 3 ) are the veloty omponents of the mropartle n the movng referene frame. Takng nto onsderaton the dependenes (9) t s easy to understand that the heat n the movng system nreases as v sne the ooperatve veloty of the mropartles nreases n Ctaton: Vetsman EV (8) Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams. Phys Astron Int J (): 53. DOI:.546/paj.8..53

3 Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams 8 Vetsman the system movng relatve to ts mass entre. If the heat nreases so does the temperature. However S S n our ase ths means dq SdT and (8) takes the form Q ( v) Q () SdT { U + p V σω } { ( )} U + pv + U + pv pv σω ().e. the ntegral n () gves an nrement of enthalpy at the adabat aeleraton of the system from veloty v up to a veloty v for the observer beng n the laboratory referene frame. Equaton () s ompletely onsstent t s orret Q n the range of velotes v for Q ( v). As a result T T / n aordane to Ott [6]. If t were ( ) Q v Q we would сome to absurdty as v. Indeed the left sde of () vanshes but the rght sde of () tends to. M.Plank as well as A. Ensten were mstaken onsderng that the temperature had to transform at adabat aeleraton aordng to the law T T () As v. The reason of ths mstake wll be shown below n Dsusson. Now we study the seond ase (Fgure ). The relaton (4) should now be wrtten down n vew of the relaton As ω ω () dv dq U U + pv pv σω + σω v U + pv σω { } () () () 3 v Sne nstead of the rato (7) we have now U U + pv σω Then formula (8) takes the form: { } (3) ( ) (4) Q( v) Q () { U + pv σω } ( σω ) U + pv U + pv. (5) σω σω pv + Takng nto onsderaton () we have now: { } Q v U pv pv ( ) + ( σω ) + σω. (6) Q Equaton (6) s orret as and n ase f Q ( v) t s norret f ( ) Q v Q. There are more omplated ases than those represented n Fgure & n partular the ases of droplet or bubble however we do not study them n ths paper. Fgure : The movng flat nterfae the veloty v of ts moton s perpendular to the flat one. L s the thkness of the surfae. Fgure : The movng flat nterfae the veloty v of ts moton s parallel to the flat one. L s the thkness of the surfae. It should be added that we ome to the absurdty onsderng the temperature T to be a Lorentz-nvarant. Now determne how the temperature and heat have to transform under relatvst ondtons when the heat s nput nto the system from the outsde. To solve ths problem we should take nto onsderaton that the thermodynamal state of the system s ndependent of the way of ts transton to t. Let there be two states of the system ( and ) and two ts ntermedate ones ( and ). In state the system s at rest and ontans some quantty of heat whh s equal to Q. We nput nto the system an addtonal quantty of the heat Q now the system s n state. The heat n the system equals Q + Q. Aelerate adabat the system up to veloty v. Now the system s n state the heat of the system s Ctaton: Vetsman EV (8) Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams. Phys Astron Int J (): 53. DOI:.546/paj.8..53

4 Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams 8 Vetsman Q Q + Q. (7) However the system an go to state otherwse: through state. To do that we must aelerate adabatally the system beng n state up to the veloty v. Then the system wll be n state. Its heat equals Q Q. (8) Further we nput a quantty of heat nto the system Q and transfer t to state. Then the quantty of heat n the system Q equals Q Q Q + Q + Q. (9) Evdently Q + Q Q + Q (3) From equatons (3) t follows that Q Q (3) Now we an formulate the frst prnple of thermodynams under relatvst ondtons (3-D formalsm): Where ( ) de dq + dq + da + v dg (3) s T d E U + E kn s the energy of the system Q the s heat hangng the entropy of the system (entrop heat)..e. dq TdS Q the heat not hangng the entropy of the system T (relatvst heat) for the observer beng at rest n the referene frame.e. dq SdT v dg A the work of deformaton e.g. s d da pdv + σdω. d v dg the nrement of the work expended on the s nreasng of mropartles veloty of the system movng (the mropartles) relatve to ts mass entre v dg the nrement of the work expended on the nreasng of the system veloty from zero up to v. If v the law (3) wll be transformed n the law (). Takng nto onsderaton the relatons (9) - () we an wrte down: ( U + pv σω ) ( U + pv σω) G s v (33) The quanttes n braes of (33) form n Mnkowsk spae a vetor of energy-momentum havng nvarant lengths equal to ( + pv σω) U (34) MØller [] represents a quantty Q as s ( h) Q { G Q } (35) ( h) Q Where G v Q s the amount of the heat transferred to the system durng the proess n fat. Q dq (see above the s law (3)). The relaton (35) was obtaned by MØller [] wth takng nto aount the nfluene of the vessel walls ontanng our substane.e. for a system onsstng of two subsystems: substane and the vessel wth ts walls. It s mportant to note that the law (3) was obtaned wthout aountng for the nfluene of the walls of the vessel. The law (3) s orret n Euldean spae up to the veloty of system moton v max. If v > v the mro partle velotes max begn to preval n the dreton X over those n the dretons X and X 3 for the observer n the laboratory referene frame n aordane to the relatvst law of the veloty omposton. Now both the heat and temperature are not already salars they are vetors. Then we an wrte down the law (3) as de d Q k + dq k + da + v d G (36) k( s) k( T) d Where k (37) Is the dmensonless vetor Q and Q are the vetors k( s) k( T) k 3. Solvng the Problem: 4D-Case Solve now the problem for 4-D ase to obtan the frst prnple of thermodynams n Mnkowsk spae. Frst of all t should be noted that there were attempts to solve the problem for 4-D ase e.g.. by Ekart [8]. He wrote the frst prnple of thermodynams as md ε + ( / )[( q / x + q Du )] + w ( u / x ) (38) / m ( g m m ) (39) Where g s the fundamental tensor m ( m ) a four-vetor whh has the unts g m -3 and depends only on the moleular weght and the moton of moleules u m / m D the unque dfferental operator orrespondng losely to the lassal operator D/Dt ε the nternal energy (J kg - erg g - ) q heat flow (J m - s - ) w the stress tensor (N m - J m -3 ). In fat the relaton (38) s an attempt to obtan the frst prnple of thermodynams n a dfferental 4D-form. It should be noted that the term ( / ) q Du does not appear n the lassal ase. Ths term s very small n all ordnary ases and may be nterpreted as a work done by a heat flow through aelerated matter n the dreton opposte to the aeleraton. It may be explaned as due to the nerta of energy. The expresson (38) s norret. The dfferent terms n t have dfferent unts. If the relaton (38) s wrtten down as Ctaton: Vetsman EV (8) Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams. Phys Astron Int J (): 53. DOI:.546/paj.8..53

5 Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams 8 Vetsman 3 md ε + [ q / x + w ( u / x )] + ( / ) q Du (4) Then all terms wll have the same unts. They have to transform dentally under relatvst ondtons and we do not ome to absurdty. We have to obtan the frst prnple of thermodynams n Mnkowsk spae under relatvst ondtons havng taken relaton (3) as a bass. To solve the problem eah term on the rght sde of (3) should be represented n a 4D-form. Havng done t we automatally represent n turn the term on the left sde of (3).e. nternal energy as 4-D physal objet. Suh representaton of nternal energy follows from the prnple of two observers we formulated. Begn the representaton wth the work of aeleraton.e. v dg. In our ase the veloty omponents v (3). Then we have for omponents and 4: v mv u dp d (4) And after the subsequent ntegraton we have v mv mv IÑ Ñ ( ) ( ) (4) Where C s the normalzaton oeffent u the omponent of the 4-veloty of our objet n the dreton X. The essene of ths oeffent presented as follows. There are some of thermodynamal parameters n Relatvst Thermodynams that equally transform n Euldean and Mnkowsk spaes e.g. the nterval of tme t or element of length l whh s parallel to X. Aordng to the energy onservaton law the energy has to vary n Euldean and Mnkowsk spaes equelly.e. n nverse proporton to. We an fnd ths oeffent from a ondton: when the veloty mv of the system equals v then ts knet energy I. In ths ase C. For v << we obtan the well-known expresson for the knet energy of a movng body for ths ase: mv I. For the omponent 4 () we have: m u dp d (43) 4 4 And after the subsequent ntegraton we have n turn: m m m Cm 4 ( ) ( ) I C C (44) If the ntal state s the state at. C s the normalzaton onstant for ths ase. It an be found from the ondton: m I f the veloty of our system equals v 4. Then C. At v I rato. m 4.e. we have the well known We an also represent omponents and 4 usng other representaton of them n Mnkowsk spae: ( ) u u m u u () ()4 u dg C d I C () ()4 ( ( ) ( 4 ) ) m u u ( ) m ( u + u ()4 ()_ ) u + u ()4 () u dg C d 4 4 Where v /. I C ( ( ) + 4 ( ) ) m u u ( ) 4 (45) (46) (47) (48) Now wrte down the expresson for the work done by the pressure when the omponent v of the veloty v muh exeeds the omponents v (3) and we annot already onsder that the pressure s a relatvst nvarant. In ase v << the nrement of the work dw s wrtten down n the real spae as dw pdv (49) V x x x () () ()3. For the relatvst ase we have n turn: dw p Vd ε j 34 (5) j Where p s the tensor of pressure ε the tensor of j j deformaton (%). t V x x x t x x x V 3 () () ()3 (5) Where t s a fxed tme nterval whh s the element of a 4D-hyperube. Now takng nto onsderaton (49)-(5) we wll represent all terms nluded n the expresson of the work onneted wth pressure. mñ ()44 m 44 p dv µ V dε µ (5) V V j ()44 ()44 Ctaton: Vetsman EV (8) Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams. Phys Astron Int J (): 53. DOI:.546/paj.8..53

6 Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams 8 Vetsman 4 ( ) ()44 ()44 dv d dζ ()44 44 V()44 m m (53) I 44 p V Ζ m ()44 (54) As seen the relaton (54) defnes the energy of substane n the whole objet under study mnus the energy of substane n the surfae layer of the objet. ζ () p dv p Vdε µ Vdε µ V Vε () V() (55) ( ) ζ () ζ () dv d dζ () V() p dv Ζ ζ ( ) (56) (57) Where ζ s the energy expended on the nrement of the () hyperube volume n the dreton X at rest. We an formally represent the expresson (54) n another form (4) usng the 4D-fore F (N) and ths fore aton dstane. Then we an n turn wrte down v F F τ (4) p dv F S dx S dx (58) () Where F s the omponent of the fore F n the dreton X n the movng referene frame F τ the omponent of the fore (4) F n the dreton 4. However t s not onvenent to use the expressons of the type (58) for solvng our problems therefore further we wll use the expressons of the type (5)-(56). ζ () p dv p Vdε µ Vdε µ (59) V ( ) ζ () ζ () dv d () V() () (4) p dv Ζ ζ () (6) (6) The analogous expressons wll be obtaned for the ase p dv p dv p dv p dv p dv and so on. If our system s losed one then all ts omponents of the knd p dv p dv p dv p dv p dv p V wll be equal to zero. Now we should obtan the expressons of the work onneted wth surfae tenson.e. wth W σ ds. (6) As we have seen above the work of the surfae tenson s a funton of the observaton angle of the objet under study (Fgure & ). However the ases represented on the fgures are the smplest. A more omplated ase s represented on Fgure 3 []. As seen from Fgure 3 the work done by the surfae tenson depends (the work) on the loaton of the observer n the laboratory referene frame. If the oordnate axs Y s an observaton lne at the moment t the observaton angle θ π / and the observer s removed to very long dstane Y from the objet then ( BD) ( B) ( D) + as vñ and we have two areas: the frst area s parallel to the veloty of the objet movement v the seond one s perpendular to t. Evdently the frst area depends on the veloty aordng to the law S S where S s the area at v. The seond one s ndependent of v. The frst area shows what part of the area of the flat objet ABCD depends on v. Fgure 3: Objet under nvestgatons n ase veloty v s not parallel and not perpendular to the nterfaal regon. A B DC denoted by the dashed lne s a transton regon between ontatng phases as v ( L ) a and b are the length and wdth of the surfae for v b s the wdth of the same surfae for v. Evdently f we turn our objet horzontally by an angle then we shall already have three projetons of the objet on the planes formed by the oordnate axes. The areas of two of them wll vary aordng to the above law the thrd area wll be ndependent of the veloty v for our observer. However n the most ommon ases the objet under study has a bent surfae. What to do? In ths ase we an break down the surfae of the objet nto Ctaton: Vetsman EV (8) Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams. Phys Astron Int J (): 53. DOI:.546/paj.8..53

7 Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams 8 Vetsman 5 nfntely great numbers of nfntely small flat elements. The area of eah element has projetons on the planes formed by the oordnate axes. Takng that nto onsderaton we an solve the problem onneted wth the work of surfae tenson when the surfae of the objet s bent. In ths ase the work an be represented as done by surfae tenson on three planes whh are mutually perpendular. Two planes are parallel to v the thrd one s perpendular. Therefore the areas lyng n the parallel planes wll vary n proporton to as vñ n turn the areas lyng n perpendular plane wll not vary under these ondtons. Now havng gven the all needed explanatons we an represent the expressons for the work of the surfae tenson fores havng begun wth the omponent σ beng n the planes formed by the 44 oordnate axes. m σ ρ ε (63) ds S d ρ ( ) S () mñ () S d S d 4 () 4 m (64) m σ 44 () I ds Μ m m S x x () ()3 t (65a) (65b) Where S S are the values of the 3D-spae-tme objet () () the spatal area of whh s perpendular to the veloty v << and as v ñ ρ() s the spef densty per tme unt (g m - s - ) of substane n ths objet m the substane mass n the surfae layer loated perpendularly to the veloty v. In turn we obtan the expressons for the work of the surfae tenson fores for the planes parallel to the veloty v: ρ ()// σ ds ρ ds S dε (66) 44 ()// // ()// ()// 44 ρ // m S // ()// // d S d m 4 ()// 4 // ( m / ) S ()// ( / ) (67) m // 3 σ 44 ()// // // // I ds Μ m m S x x t S And // () () ()// (68) (69a) S x x t S (69b) // () ()3 ()// where S S are the volumes of the 3D-spae-tme objet ()// // the spatal areas of whh s parallel to the veloty v << and as vñ ρ s the spef densty per tme unt (g m - s - ) of ()// substane n ths objet m the substane mass n the surfae // layer loated parallel to the veloty v at rest the values S // n (69a) and (69b) are not equal to one another n ommon ase. Evdently the nrements of the knd (63) and (66) are already n the dependene (43) therefore they wll not further be taken nto onsderaton. For the real work of the surfae tenson fores we have the followng relatons (the plane s perpendular to the veloty v): S() σ ds σ ds ρ dε (7) ρ S ()3 S () ()3 3 S () ()3 ds d d 3 S S () ()3 σ ds 3 3 ()3 (7) (7) Ξ (73) t S S x x () ()3 () F L + F L σ L L L L S (74) Where σ σ s the surfae tenson per tme unt on the real 3 3 surfae (X X 3 ) the whose 3D-spae-tme objet (the surfae) s perpendular to the veloty v S S () are the volumes of the 3D-spae-tme objet the spatal area of whh s perpendular to the veloty v << and as vñ s the ()3 ()3 Ctaton: Vetsman EV (8) Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams. Phys Astron Int J (): 53. DOI:.546/paj.8..53

8 Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams 8 Vetsman 6 energy expended on the nrement of the area perpendular to the veloty v F F are the surfae tenson fores n dreton 3 and 3 atng (the fores) n the plane perpendular to the veloty v L L dstanes n dretons and 3 on whh the fores 3 and 3 are atng. σ σ ρ ε (75) ds ds S d // ()// // () ρ // // S S () () S () () () ds () d () () σ ds () () (76) (77) Ξ (78) S x x t S (79) // () () ()// F L + F L σ L L L L S (8) Where σ σ s the surfae tenson per tme unt on the real surfae (X X ) the whose 3D-spae-tme objet s parallel to the veloty v S S () are the volumes of the 3D-spae-tme objet the spatal area of whh s parallel to the veloty v << and as vñ ζ ζ s the energy expended on the () () nrement of the area parallel to the veloty v F F are the surfae tenson fores n dretons and 3 atng (the fores) n the plane parallel to the veloty v L L dstanes n dretons and 3 on whh the fores and are atng. The ase σ σ does not fundamentally dfferent from the 3 3 prevous one. Now we are gong to onvert the heat. Frst of all we should take nto onsderaton that n ths ase we use 4D-veloty nludng the velotes of mropartles. In 3D-formalsm the energy of aeleraton vares n nverse proporton to and ths energy has also to onvert n heat n the nverse proporton to for the observer beng n the laboratory referene frame at rest. Therefore we have for the relatvst heat κλ ( T) ( T) Q κλ T Q dq g κ λ 34 κ λ (8) κλ Where s the tensor of the relatvst heat g ( ) κλ the fundamental tensor of rank. Q( T ) Q (8) ( T ) Where Q s the Q at rest. ( T ) ( T ) To some extent the quantty Q ( T ) s smlar to the term ( / ) q Du n the above expresson (38). Suh terms an be only n relatvst thermodynams. From the aforesad on the thermo dynamal heat Q nludng ( s ) the expressons (7)-(3) we an represent ths quantty as And κλ ( s) ( s) Q dq g κ λ 34 κ λ (83) κλ Q ( s ) Q ( s ). (84) Can we obtan expressons smlar to the relatons (8) - (84) for the nternal energy not takng nto onsderaton the expressons obtaned above for 4D-work and heat? Evdently no sne we annot determne the nrement of nternal energy wthout the alulaton of the energy put nto the system or removed from t by means of work and heat. The nternal energy annot aurately be determned by physal deves. Takng nto onsderaton the all above expressons for the transformaton of heat and work n 4D formalsm we an onlude that the nternal energy U has to transform under relatvst ondtons as U U (85) And the frst prnple of thermodynams n 4D-formalsm an be wrtten as Q Q ( s) ( T) m ζ () κλ κλ 3 E D D + D + D + D ( / ) + mv D ( / ) + D. () κλ κλ.3 Where E( U E ) kn (86) + s the energy of the system D s dfferental sgn n 4D-formalsm. Obtanng (86) we have taken nto onsderaton the relatons () (4) (43) (55) (59) (7) (75) and the equalty I I + I + I The relaton (86) s qute orret n the veloty nterval -v of the objet under study. The both sdes of ts transform dentally under relatvst ondtons - we do not ome to absurdty as v sne the both sdes of these relatons beome nfne. The law (86) s the frst 4D-prnple of thermodynams wthout takng nto onsderaton of the work of the eletromagnet fores and n ase the vetor v of the system moton s parallel to the oordnate axs X. The relaton (86) s the fundamental one ts left and rght sdes transform dentally under relatvst ondtons but we an determne the transformaton of the left sde of the relaton only by means of the transformaton of the rght sde. In the general ase the fundamental nature of ths or that dependene an be determned by means of the prnple Ctaton: Vetsman EV (8) Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams. Phys Astron Int J (): 53. DOI:.546/paj.8..53

9 Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams 8 Vetsman 7 of two observers. The prnple s formulated below antpatng by an example. Let a proess of heat transfer take plae n our system. In the smplest ase the proess an be desrbed n several ways: frst by the formula of heat transfer I (J m - s - ) q Fourer seond by the formula of the heat transfer taken from rreversble thermodynams [3] thrd by Vetsman s formula []. Here they: T I λ (87) q x I I q T a (88) q T q D 3 (89) x Where λ s the thermal ondutvty oeffent (J m - s - grad - salar) a q the phenomenologal thermal ondutvty oeffent (J m - s - salar) D the oeffent of thermal dffuson (m s - tensor of rank ) q the spef densty of heat (J m -3 ). Let two observers be n a laboratory referene frame. The frst observer s measurng the quanttes beng on the left sde of the expressons (87)-(89) by deves under relatvst ondtons. In turn the seond observer s measurng the parameters beng on the rght sde of these relatons under the same ones. If these measured quanttes are substtuted n the above expressons then we do not lead to absurdty only n the ase (89) []. In the ases (87) and (88) the left sdes of these formulae wll transform dfferently from ther rght ones. It means that relatons (87) and (88) are not the fundamental physal dependenes. Now we an formulate the prnple of two observers: A physal dependene wll be truly fundamental one only n ase the parameters obtaned by the frst and the seond ndependent observers for the left and rght sdes of ths dependene do not lead to absurdty and ontradtons under relatvst ondtons by beng aordngly substtuted n the left and rght sdes of the physal formula. 4D-spae omponent ( U + Ζ Ξ () () κλ ) 3 κλ 3 G u κ λ (9) sp ñ Β And 4D-tme omponent ( U κλ ) 3 κλ 3 G + Ζ Ξ κ λ (9) t Forms an nvarant quantty ( U () () () κλ ) 3 κλ 3 G + Ζ Ξ κ λ (9) Where Β u/ u s the 4-veloty vetor n real spae. Dsusson As seen above the temperature of the system s present n the frst prnple of thermodynams n a latent form - by means of the seond prnple of thermodynams Q S (93) T Where S s the entropy whh s Lorentz-nvarant aordng to Plank.e. S S. (94) If we wrte down the transformaton of heat Q under relatvst ondtons as Q Q (95) Then the temperature has to transform under these ondtons aordng to ().e. aordng to M.Plank. However f we wrte down the transformaton of Q as Q Q (96) Then the temperature T has to transform under these ondtons aordng to the followng formula T T (97).e. aordng to Ott H [6]. As we have seen above the relatons (96) and (97) do not lead to the absurdty the relatons () and (4) on the ontrary do. Where were Plank and Ensten mstaken? We make an attempt to show t startng from (). Ensten proeeds from Gbb s equaton taken n the form (normal state): Where E s the nternal energy. TdS de + pdv vdg (98) Aelerate adabatally our system up to the veloty v. Further ntegrate (98) term by term from state (v) up to state (v ): TdS de + pdv vdg. (99) Sne S S then n vew of (6) E E + pv pv ( U + pv ). () However Ensten meanwhle operates wth TdS when there s no any nput of heat n the system! We annot use the above Gbbs rato here we are to use only the frst prnple of thermodynams under ondton that dq S dt (see above)! If we nput the heat n our system aelerated up to a veloty v then the nrement of the heat dq wll ontan the term TdS however the heart of the matter has not to hange. As before we ome to absurdty onsderng that the temperature wll be transformed aordng to Plank-Ensten s law. Takng the transformaton of Ctaton: Vetsman EV (8) Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams. Phys Astron Int J (): 53. DOI:.546/paj.8..53

10 Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams 8 Vetsman 8 the temperature aordng to H.Ott under these ondtons we have no ontradtons. The transformaton of heat and temperature under relatvst ondtons s onsdered to depend on the veloty of the soure of heat puttng nto the system [4]. In ths regard the author of [4] examnes two ases: () when the objet under study and the soure of heat are movng wth the same veloty () when they have dfferent velotes. The problem s solved n Mnkowsk spae under some assumptons. In partular van Kampen proeeds from the assumpton that the nternal energy U wll not transform under relatvst ondtons of the system by formula (6) but as U U. () Then n the frst ase the heat had to transform n an adabat aeleraton of the system aordng to (96) and the temperature had to transform aordng to (97) f entropy S S. In the seond ase the heat transforms under relatvst ondtons aordng to omplated laws. In order to fnd them van Kampen uses an magnary model. There are two blak bodes a and b separated by a thn metall sheet. Relatve to the laboratory frame a and b have velotes u a b and u parallel to the sheet. The heat may be leakng from the subsystem a to the subsystem b. Of ourse suh a system does not exst n nature and annot be reated artfally. Aordng to van Kampen the heat Q n the subsystems a and b has to transform under relatvst ondtons as a dq ρ + γρ () b a b b dq ρ + γρ (3) Where ρ and ρ are the energy densty n the subsystems a b ( ) / multpled by A t γ u A s the area of a small 4 hole through whh the heat goes from the subsystem a to the subsystem b t the nterval of tme when the hole s open uv/ s the veloty of lght adopted equal to. Then we have for the whole system ( )( ) dq + dq γ ρ + ρ > (4) a b a b It should be noted that usng the above model van Kampen has obtaned a omplated law of the heat transformaton under relatvst ondtons. However usng of dfferent magnary models we an obtan some speal relatons of no fundamental mportane. Conluson a. The frst prnple of thermodynams was obtaned under relatvst ondtons n vew of the surfae tenson. a Usng the frst prnple of thermodynams t was shown that the temperature vared under relatvst ondtons n adabat aeleraton aordng to Ott H [6].e. n nverse proporton to. It was shown where A. Ensten made the mstake whh led afterwards to the norret dependene T T. Aknowledgement None. Conflt of Interest Author delares there s no onflt of nterest. Referenes. Mosengel KV (97) Theore der statonären Strahlung n enem glehförmg bewegten Hohlraum. Annalen der Physs 37(5): Ensten A (97) Über das Relatvtätsprnzp und de aus demselben gezogenen Folgerungen. Jahrbuh deutshe Radoaktvtaet und Elektronk 4: Vetsman EV (3) On relatvst surfae tenson. Journal of Collod and Interfae Sene 65(): Vetsman EV (5) Corrgendum to Relatvst state equatons for the nterfae. Journal of Collod and Interfae Sene 95(): Vetsman EV (9) bd 337: Ott H (963) Lorentz-Transformaton der Wärme und der Temperatur. Zetshrft für Physk 75(): Callen H Horwtz G (97) Relatvst Thermodynams. Ameran Journal of Physs 39: Ekart C (94) The Thermodynams of Irreversble Proesses. III. Relatvst Theory of the Smple Flud. Physal Revew 58: Vetsman EV (7) Some problems n relatvst thermodynams. Journal of Expermental and Theoretal Physs 5(5): Levh VG (96) The Course of Theoretal Physs. Fzmatgz Mosow Russa.. MØller C (967) Relatvst Thermodynams. A strange Indent n Hstory of Physs. Munksgaard Publsher Denmark p Madelung E (957) De matematshe Hlfsmttel des Physkers. Sprnger-Verlag Berln Germany. 3. Haase R (963) Thermodynamk der rreversblen prozesse. Dr. Detrh Stenkopff Verlag Darmstadt Germany. 4. Kampen NGV (968) Relatvst Thermodynams of Movng Systems. Physal Revew Journals Arheve 73(): Ctaton: Vetsman EV (8) Solvng the Temperature Problem under Relatvst Condtons wthn the Frame of the Frst Prnple of Thermodynams. Phys Astron Int J (): 53. DOI:.546/paj.8..53