The possibility of a simple derivation of the Schwarzschild metric

Transcription

1 The possibility of a simple deiation of the Shwazshild meti Jan Czeniawski Institute of Philosophy, Jagiellonian Uniesity, Godzka 5, Kakow, Poland uzzeni@yf-k.edu.pl Abstat. In spite of alleged impossibility poofs, simple deiations of the Shwazshild meti, based solely on Einstein s equialene piniple and Newton s fee fall eloity fomula, ae pesented.. Intodution Some authos laim that no simple deiation of the Shwazshild meti is possible without the expliit use of geneal elatiity. In thei opinion, It is the spatial-distotion aspet of gaity that ensues that too simple a deiation of the Shwazshild meti must fail []. Although they do not state it expliitly, they seem to think that this aspet is so ounteintuitie that it annot be deied in any intuitie way. Howee, it has long been well known that the uatue of spae an be isualized ey easily in tems of an appopiate distotion of measuing ods []. Thus, additional easons seem to be needed. Two othe aguments fo the aboe opinion ae not so tiially inonlusie. Fist, the intepetation of the equialene piniple that is used in suh deiations is laimed to lead to an inoet fomula fo the spae-time meti in the stati paallel gaitational field [3]. Seondly, it is maintained that both the Shwazshild field and its ountepat in Nodstöm s theoy satisfy the equialene piniple; they ae, howee, essentially diffeent, sine only in the fist of them is the empiially onfimed global bending of light pesent [4]. They will be examined subsequently.. Rindle s ounteexample The fist alleged ounteexample against the possibility of a simple deiation of the Shwazshild meti, aised by W. Rindle [3], ests on a deiation of the meti in a stati paallel gaitational field, whih has little in ommon with typial simple deiations [5,7]. Let us theefoe ty to deie it moe in thei spiit. Aoding to Einstein s equialene piniple [8], the influene of gaitation on phenomena in a loal efeene fame that is at est in the field is equialent to the influene of the aeleated motion of a loal efeene fame in whih phenomena ae desibed in the absene of gaitation. It is easy to see that the latte edue to the length ontation, time dilation and simultaneity distotion oesponding to the eloity of the fame aquied as a esult of the aeleation. Thus, if the aeleation is paallel to the axis of the oodinate x, the meti in suh a fame aquies the fom: ds dt' dx' dy dz, () whee the loal oodinates x, t in the aeleated fame ae elated to the oesponding oodinates in the stationay fame by the usual diffeential fom of the Loentz tansfomation fo the alue of the eloity that has been eahed. Now, let us onside a loal fame at est in a stati gaitational field paallel to the axis of the oodinate x, absent fo positie alues of this oodinate. The equialene piniple in the aboe intepetation implies that the meti in suh a fame will be expessed by (), whee x, t ae loal oodinates defined in this fame by means of physial measuing ods and loks. Let us extapolate

2 the oodinate t in a fame that is at est outside the field on the aea of the field in suh a way that its sale emains onstant though spae and it agees with the loal time oodinates in the fames at est in the field with espet to simultaneity. Fo suh a hoie of this oodinate, the esults of the influene of gaitation on physial objets edue to effets analogous to length ontation and time dilation. Thus, they ae gien by the fomulae: dt' dt, () dx dx'. (3) By substituting the aboe expessions into (), we obtain: dx ds ( ) dt dy dz. (4) Let the quantity be gien by the eloity fomula fo fee fall in a field with gaitational aeleation g antipaallel to the axis of the oodinate x, with the initial alue zeo fo x = 0: gx. (5) By substituting this expession into (4), we obtain [0]: gx dx ds ( ) dt dy dz gx. (6) Now, let us intodue new oodinates T, X, Y, Z suh that X is elated to x by the fomula: g x X (7) g and T, Y, Z ae equal to t, y, z, espetiely. We then get [0]: g ds X dt dx dy dz. (8) It is easy to see that, up to a speial hoie of units, this is exatly what we ought to hae [3], ontay to Rindle s opinion. 3. Sexl s ounteexample Let us onside the seond of the alleged ounteexamples, asibed to R. Sexl [4]. Nodstöm s theoy an be fomulated as a meti theoy of gaitation, whih diffes fom the geneal theoy of elatiity with espet to the field equations. As a esult of this diffeene, its meti is onfomally flat, whih means that in its spae-times thee is no global light bending, whih is one of the most impotant peditions of geneal elatiity. On the othe hand, it may seem [] that, as a meti theoy, Nodstöm s theoy should satisfy the equialene piniple. If so, then this piniple should be insuffiient fo deiing the Shwazshild meti athe than its ountepat in that theoy. Cetainly, if Einstein s equialene piniple is undestood as implying only that the theoy of gaitation be a meti theoy, suh a onlusion is unaoidable. Howee, is this all Einstein oiginally meant [8]? It seems lea that any theoy satisfying the equialene piniple must be a meti theoy, but the onese is not as obious. If gaitation has to be loally equialent to (aeleated) motion of the efeene fame, it seems that it should (i) affet the tempoal dimension and the spatial dimension of physial phenomena paallel to the dietion of the field, but (ii) leae This is always possible in a stationay field - see, e.g., [9], p. 0.

3 the two othe spatial dimensions unaffeted. This is expessed by the appoximate fom () of the meti in a efeene fame that is at est in a gaitational field. On the othe hand, the spae-time of Nodstöm s theoy has a meti of the fom [4]: / ds e ( dt dx dy dz ), (9) whee is position-dependent. It is easy to see that this implies that gaitation affets all spatiotempoal dimensions of phenomena in the same way. Thus, the foms () and (9) oinide, up to a onstant fato, only in the tiial ase when gaitation is absent. Consequently, the meti field (9) does not satisfy the equialene piniple, as Einstein seems to hae undestood it. 4. The Lenz-Shiff deiation Let us onside a loal efeene fame at est in a stati spheially symmeti gaitational field. Let be the adial oodinate defined in the standad way and t the standad time oodinate of an infinitely distant obsee (see, e.g., [9], pp. 0 and 36). By an agument simila to that of Se., in onsequene of the equialene piniple, the meti in the aboe-mentioned fame will hae the fom: ds dt' dx' ( d sin d ), (0) whee the loal spatial oodinate x in the adial dietion and the loal time oodinate t ae elated to, t by the fomulae [5,7]: d dx', () dt' dt, () whee is the eloity that a adially aeleated fame in the absene of gaitational field must hae in ode to be equialent to the stationay fame in question fo a gien alue of the oodinate (it is easy to see that the fomula () is idential with ()). Afte the appopiate substitutions, we get a meti of the fom: d ds ( ) dt ( d sin d ). (3) Let us assume that is gien by the Newtonian fomula fo fee fall with the initial eloity zeo at infinity, i.e., it is equal to the esape eloity in Newton s gaitation theoy: km, (4) whee k is the gaitational onstant and M is the mass of the soue of the gaitational field. Finally, by substituting (4) to (3), we get the fom of the Shwazshild meti: km d ds ( ) dt d d km ( sin ). (5) 5. The ie model deiation What might make one a bit uneasy about the Lenz-Shiff deiation is the unlea status of. It annot, as one might think, be intepeted as the eloity of a fame falling in a gaitational field, undestood as the fist deiatie of the oodinate with espet to the time oodinate t. As we hae obseed in the peious setion, it is athe the eloity of a fame in the absene of gaitation. 3

4 Thus, its meaning is speified in iumstanes diffeent fom those in whih the meti is deied. One might theefoe egad the suess of the deiation as a mee oinidene. Fotunately, thee is anothe deiation, in whih the meaning of is lea. Let us assume that gaitation edues to some motion, in a Eulidean spae, of a substatum, in whih speial elatiity holds loally. This means that in a esting fame eeything goes on as though it moed and the substatum ested. Thus, in suh a ie model [] of gaitation, Einstein s equialene piniple is tiially satisfied. Let in a spheially symmeti field the substatum fall adially, obeying Newton s law of fee fall, and let be the eloity of its falling. Finally, let in omoing fames the spatiotempoal dimensions of physial phenomena emain the same as if the substatum ested. This means, in patiula, that a omoing lok synhonized with a lok esting at infinity will etain synhony with that lok duing its falling. Thus, a time oodinate T an be defined by suh loks synhonized in this way. Moeoe, the distanes between simultaneous eents measued by omoing ods ae just the distanes in Eulidean spae. It is easy to see that, in onsequene of the aboe assumptions, the spae-time meti in a omoing fame is loally appoximated by (0), in whih is intepeted as the usual adial oodinate and the loal spatial oodinate in the adial dietion x and the time oodinate t ae elated to, T by the appoximate fomulae of the diffeential fom of the Galilean tansfomations: d' d dt, (6) dt' dt, (7) whee, intepeted as the eloity of the oigin of the fame, is a funtion of its time-dependent oodinate 0. By substituting (6) and (7) into (0), we get: ds dt ( d dt ) ( d sin d ), (8) o, equialently: ds ( ) dt dtd d ( d sin d ). (9) In this way, the loal oodinates x, t hae been eliminated. The appoximate expession (9) is the moe auate the less diffes fom 0. Thus, if we substitute fo 0 in, this expession fo the meti beomes exat. Now, let us assume that the eloity satisfies the Newtonian fee fall law, expessed by (4). We obtain the following fom of the meti: km km ds ( ) dt dtd d ( d sin d ), (0) found fo the fist time by A. Gullstand [3] and known as the Painleé-Gullstand meti [4], o the Painleé-Gullstand-Lemaite meti (f.. [5] and elated efeenes theein), although it seems that in the woks of Painleé and Lemaite this peise fom does not appea. In the ie model the meti (0) was deied fo the fist time by A. Tautman [6]. It might seem ey diffeent fom the Shwazshild meti. Howee, it is well-known to be just the Shwazshild meti in non-standad oodinates. It an be tuned into the usual fom by the following tansfomation of the time oodinate [4,5]: km dt dt d km. () ( ) It is woth noting that the meti in a onstant field gien by the expession (6) was oiginally deied [0] in the Painleé-Gullstand fom: gx ds ( ) dt gxdtdx dx dy dz () and then tansfomed into (6) by a tansfomation gien by the fomula: 4

5 analogous to (). dt dt gx dx gx ( ), (3) 6. Disussion One might think that the fomula (0) eeals the tue fom of the Shwazshild meti. Yet suh a onlusion would be pematue. Let us etun to (4). It has two solutions with diffeent signs, whih apply to fee fall with the initial alue zeo at infinity and to fee esape with the final alue zeo. The solution with a negatie sign was used in the deiation of Se. 5. Why not, then, assume that the substatum, instead of flowing inwads, flows adially outwads, obeying the Newtonian law of fee esape? Supisingly, this also woks. Following steps ompletely analogous to that deiation, we now get the meti with the fom: km km ds ( ) dt dtd d ( d sin d ). (4) This is just the othe esion of the Painleé-Gullstand meti [4,5], disoeed independently by A. Gullstand [3] and P. Painleé [7], and edisoeed by G. Lemaite [8]. Thee is yet anothe ie model deiation, found by W. Lashkaew [0]. Instead of the time oodinate T, one an intodue the oodinate t in the standad way appliable in stati fields (f. Se. aboe). Then all the steps of the deiation of Se. 4 follow, with the only diffeene that this time the expessions () and () hae a ey simple intepetation, as expessing the length ontation and time dilation, espetiely, whih esult fom the stationay fame s motion elatie to the substatum. The ie model may seem attatie as it is ey intuitie. Neetheless, it has some weak points. One of them is the aboe-mentioned duality of the ie model deiations of the Shwazshild meti. Next, one an obsee that, if the assumptions of the ie model ae adopted, then the effets expessed by () and () ae to be expeted, but the eese is not tue. Thus, the assumption that thee is some eal substatum flow seems unneessay fo deiing the geneal elatiisti meti. What is moe, an exessiely ealisti appoah to the ie model enountes seious diffiulties. Fist, if the homogeneity of the substatum is assumed, then its flow does not seem to satisfy the ontinuity equation [4]. Othewise, one would hae to explain why its possible inhomogeneities do not affet physial ods and loks. Next, although the model has poed suessful in deiing some inteesting speial ases of geneal elatiisti metis [0,6], the most supising being the Ke meti [], its appliability to all physially easonable solutions of the Einstein equations emains to be shown. Thus, some eseations with espet to it seem in ode. 5

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