Hilbet s fogotten equation of velocity dependent acceleation in a gavitational field David L Bekahn, James M Chappell, and Deek Abbott School of Electical and Electonic Engineeing, Univesity of Adelaide, SA 5005 Austalia Dated: August 7, 07 The pinciple of equivalence is used to ague that the known law of deceasing acceleation fo high speed motion, in a low acceleation egime, poduces the same esult as found fo a weak gavitational field, with subsequent implications fo stonge fields This esult coincides with Hilbet s little exploed equation of 97, egading the velocity dependence of acceleation unde gavity We deive this esult, fom fist pinciples exploiting the pinciple of equivalence, without need fo the full geneal theoy of elativity I INTRODUCTION While it is accepted that geneal elativity povides the coect theoy fo macoscopic elativistic motion including gavity, nevetheless the special theoy emains viable fo descibing acceleated motion as well as acceleated fames in flat space Fo example, Einstein easoned that light would bend unde gavity based on the pinciple of equivalence and acceleation aguments alone He then calculated the moe pecise esult using geneal elativity, taking into account the effect of cuved spacetime Using a simila appoach we deduce that acceleation unde gavity is velocity dependent using acceleating fames unde special elativity, which we then confim with the geneal theoy, poducing a esult that coincides with one by Hilbet We also find a simila discepancy of a facto of two between the esult using flat space acceleations and the full geneal elativistic analysis We can begin by defining a spacetime coodinate diffeential with a fou-vecto dx µ = c, dx, dy, dz, with contibution fom thee spatial dimensions and t is the time in a paticula efeence fame and c is the invaiant speed of light In this pape we ae able to focus exclusively on one-dimensional motion and so we can suppess two of the space dimensions witing a spacetime vecto dx µ = c, dx We have the metic tenso g µν = that defines the covaiant vecto 0 0 dx µ = g µν dx ν = c, dx In the co-moving fame we have dx = 0 and so dx µ = c, 0, which defines τ the local pope time We define the pope velocity v µ = dxµ whee v = dx/ and = dx µ = γc, γv, γ = = 3 v c We then have the magnitude of the spacetime velocity vµ v µ = γ c γ v = c 4 that is a Loentz invaiant, whee we have used the Einstein summation convention We also have the pope acceleation a µ = dvµ = γ4 va/c, γ 4 a, 5 whee we have poduced the special case of onedimensional motion in which v is paallel to a We then find the magnitude of the spacetime acceleation aµ a µ = γ 8 v a /c γ 8 a = γ 3 a 6 Now, in the momentaily co-moving fame MCF we have v = 0 giving the acceleation vecto a µ = 0, α and the velocity v µ = c, 0, which gives a µ a µ = α and the expected othogonality v µ a µ = 0 Hence, compaing the magnitudes of the pope acceleation in Eq 6 with the magnitude in the MCF we find α = γ 3 a so that in an altenate non-comoving fame we obseve an acceleation a = α/γ 3 7 An altenative path to this esult is to apply a Loentz boost to the MCF pope acceleation a µ = 0, α, with the tansfomation t = γt + vx/c and x = γx + vt This poduces a µ = γvα/c, γα and so compaing this with Eq 5 we have γα = γ 4 a o α = γ 3 a, confiming Eq 7 We now conside how a ocket s acceleation appeas when viewed fom diffeent inetial efeence fames each with diffeent initial velocities Then, using the pinciple of equivalence, we tansfe ou esults to a gavitational setting A Thought expeiment Conside a ocket out in space fa fom the effects of any gavitational influences Within this, effectively flat egion of space, we place small fames of efeence that
individually can measue the acceleation of passing objects We will call these types of fames PG fo paticle goup The PG fames ae cuently at est elative to the ocket and also with espect to each othe and they ae spead thoughout the space suounding the ocket The ocket also has a hole at the top and bottom so that the PG can pass staight though allowing them to measue the acceleation of the ocket The ocket also has an inbuilt mechanism so that, when the ocket is acceleating, it will op a second goup of paticles, labeled PG, fom the top of the ocket, at pedetemined fixed time intevals as measued by the ocket Thus, PG can also measue the ocket s acceleation Now, fo the sake of agument, let the ocket be acceleated at 98 ms and as specified, PG will be opping fom the top of the ocket The ocket now acceleates away fom the PG fames with acceleation α = T/m = 98 ms, whee m is the mass of the ocket and assuming T is an applied thust in ode to maintain a constant pope acceleation The PG, once eleased, compise inetial objects not pataking in the ocket s acceleation Additionally, as the ocket continues its acceleation it will encounte PG lying in its path that will ente the hole at the top of the ocket and while passing though measue the acceleation of the ocket Now, as the ocket is maintaining a steady acceleation, clealy the velocity of the ocket will be steadily inceasing Hence the ocket will be encounteing the PG at highe and highe elative velocities The question we now wish to conside is: Will PG and PG measue the same acceleation fo the ocket? Based on standad theoy, we expect the answe to be in the negative This is because special elativity assets that, as viewed by PG, the ocket s velocity will convege to the light speed uppe bound, and so the acceleation will appea to decease Since, this physical setting is descibed by Eq 7, the one-dimensional elativistic equation fo acceleation a, as measued in the PG fames, can be witten as a = α γ 3 = T 3/ v m c, 8 whee α is the acceleation measued in the co-moving fames PG, v is the velocity of the ocket elative to PG Now, given this esult, we can ask a pivotal question with espect to the physics of the situation: Given the pinciple of equivalence will this esult fo acceleating obseves be eplicated unde gavity? We pesume fo appopiately small egions of the field, based on the pinciple of equivalence, the answe must be in the affimative B Gavitational fields The cental ole played by the equivalence pinciple in the geneal theoy was stated by Einstein in 907: we [] assume the complete physical equivalence of a gavitational field and a coesponding acceleation of the efeence system Einstein s equivalence pinciple is based pimaily on the well established equivalence of gavitational and inetial mass, also called the weak equivalence pinciple, which has been confimed by expeiment to an accuacy bette than 0 5 It is now geneally accepted that the full Einstein equivalence pinciple equies a cuved spacetime metic theoy of gavity in which paticles follow geodesics within this space as descibed by Einstein in his geneal theoy 3 Hence, incopoating the equivalence pinciple, ou cuent poposition is that since Eq 8 petains to a efeence fame descibed above with an acceleating ocket then we also must have in a gavitational field 3/ a = g v, 9 whee g is the acceleation due to gavity, which when stationay in gavity is a pope acceleation analogous to α This shows that fo gavity the ate of acceleation fo fee falling obseves equivalent to PG is velocity dependent We now confim this conclusion by deiving a compaable esult using the Schwazschild solution of geneal elativity II c SCHWARZSCHILD SOLUTION Fo a static, non-otating, spheical mass the field equations of geneal elativity give the Schwazschild solution 3 with the metic c = c dθ cos θdφ, 0 whee µ = GM/c and is measued fom the cente and outside the mass 3 We theefoe have g = and g tt = µ Now, we have the geodesic equation a α = dvα = Γ α µνv µ v ν that can also be witten in an equivalent fom d dx g ν αν = g µν dx µ dx ν x α This less common fom of the geodesic equation can be convenient as the Chistoffel symbols Γ α µν do not need to be explicitly computed So, setting the index α to the coodinate, we poduce d g = g + g tt,
3 utilizing the fact that we have a diagonal metic and the angula tems ae zeo fo adial motion We fistly calculate g = µ, g tt = µ 3 Also, dividing Eq 0 though by c and emoving the angula tems we have c = 4 Substituting these esults into Eq, and afte cancellations we find the well known esult d = GM = a 5 An altenate, pehaps moe diect deivation of this esult is also shown in Appendix A Note that a is the acceleation equied to emain at est at adius and coesponds to the magnitude of the fou-acceleation α calculated ealie This shows a constant acceleation as assumed fo the ocket fame as measued by PG, efeed to ealie as pope acceleation This thus coesponds with Eq 8 when v = 0 This implies the magnitude of the fou-acceleation is gµν a µ a ν = GM GM g = µ/ 6 We can wite Eq 5 as and so c d GM c GM = 0 7 = constant = E 0 m, 8 c4 whee we assume a paticle with initial enegy E 0 Hence µ = c + E 0 m, 9 c4 whee γv 0 as, if we assume fo lage that mc E 0 = v 0 /c Now = and so we find = c and so using Eq 4 we detemine = E 0 mc µ 0 m c 4 E 0, whee v 0 as Diffeentiating with espect to coodinate time, using the chain ule, d = d/, we find d = µc µ 3 µ m c 4 E0 Fo a paticle appoaching a gavitational potential at a speed v 0 we have d = µc In the weak field we have µ 3 0 and so v 0 c 3 d = µc 3v 0 c, 4 a esult fist deived by Hilbet 4 6 in 97, fo paticles moving adially in a gavitational potential Theefoe we can see that the Schwazschild solution also gives a velocity dependent acceleation fo obseves at est with espect to the gavitational field coodinates This implies an appaent weakening of the field stength in gavity, fo adially moving objects, elative to stationay obseves in weak gavitational fields Indeed, to a fist appoximation, we have a velocity dependence fom special elativity given in Eq 8 of 3v c compaed with a Schwazschild dependence, shown in Eq 3 of 3v c This appoximate confimation of the esult using the Schwazschild solution suggests the basic pinciple to be sound enough to waant expeimental testing This might be achieved in an eath bound fame, if thee ae accuate enough clocks to measue such deviations fom cuent expected acceleations III EXPERIMENTAL TESTS Integating the expession in Eq, we can find the pope time taken between two heights as τ = 0 5 µ c + E 0 m c 4 This allows us to calculate the expected time diffeence fo a falling paticle based on velocity dependence v 0, and so allowing an expeimental test of this pinciple 7 Also, due to the ocket s mild acceleation ate, then inside the ocket fame itself, thee will be extemely mino time dilation effects This allows the stationay fame in gavity, to be the fame of efeence to measue faily accuately the ates of acceleation of PG and PG It is theefoe poposed that this should be the efeence fame fo an expeimental test of the pinciple The maximum effect pedicted in Eq 8 will be fo paticles falling in the Eath s gavitational field at velocities appoaching the speed of light
4 IV DISCUSSION Appendix A: Lagangian appoach to geodesics We show in this pape that by consideing acceleating objects within the context of special elativity and using the equivalence pinciple, the behavio of weak unifom gavitational fields ae pedicted Specifically, we have shown that acceleation due to gavity, is a function of adial paticle velocity as shown in Eq 8, a esult fist deived by Hilbet This can also be intepeted as a weakening of the field One way to intuitively undestand this effect is that paticles moving at high velocities in a gavitational field have clocks that ae slowed and so effectively spend less time in the gavitational field and so expeience lowe acceleation than slow moving objects It could be claimed that this esult shown in Eq 4 of the velocity dependence of gavity is pehaps an atifact of the paticula coodinates chosen, shown in Eq 0 Howeve, if we ty the othe common vaiants of the Schwazschild metic, such as isotopic coodinates, Billouin coodinates o indeed Schwazschild s oiginal metic, then the same esult as shown in Eq 4 is found Refe to Appendix B fo a list of these fou common metics We have shown thee is no violation of the pinciple of equivalence since velocity dependance holds unde both flat space acceleations and geneal elativity Thee is also the issue though of the facto of two discepancy between the esult using acceleation unde special elativity and that using the full geneal theoy Howeve Einstein also found that the bending of stalight was twice the effect in GR when compaed to using acceleation and the pinciple of equivalence This diffeence is because the pinciple of equivalence holds fo local egions of space and time which coincide with acceleating fames Hence, fo the bending of stalight we need to take account of the additional effect fom the space cuvatue along the tajectoy, which is pesent unde gavity Wheeas, unde acceleation the inetial obseve obseves the acceleation deceasing because the time ove which the foce acts appeas to take longe and longe fom his fame, theefoe the entie effect is due changes in time not space In an attempt confim this, we edo ou calculations fo the Schwazschild solution but set the spatial cuvatue to zeo, then we find that we obtain a facto of two as opposed to thee, which is much close to ou esult in Eq 4 and so appeas to explain this discepancy As noted, ou esult based on acceleating fames, leads to an expected effect about half that pedicted by geneal elativity, as shown in Eq Hence it would make an inteesting expeiment to pecisely measue this effect, and to veify the discepancy between the two types of analysis and povide futhe confimation of geneal elativity This test would also thus allow a futhe veification of the Einstein pinciple of equivalence and Hilbet s equation A Lagangian appoach can also be used as an altenative to the geodesic equation and may be cleae fo those less familia with tenso algeba It is also poduces a shote deivation of Eq 5 Now, we can eaange the metic in Eq 0 to define a Lagangian L = ṫ c ṙ =, A whee ṫ = and ṙ = and fo puely adial motion we have assumed that the angula tems ae zeo We then have the action S = = L and so we can fistly maximize the action using Lagange s equations fo t, namely d L ṫ L t = 0, giving d µ c ṫ = dl = 0 A Hence we have a constant of the motion µ ṫ = E 0 mc, A3 whee E 0 can be shown to be the total enegy fo motion in a Schwazschild metic Substituting Eq A3 back into the metic we find = c E 0 m c 4, A4 in ageement with ou pevious esult in Eq 9 Also, d = d = µc, A5 in ageement with Eq 5 Now, multiplying the Lagangian though by, and solving fo we find = c µ A6 µ Theefoe, using Eq A3 we find = c m c 4 E 0, A7 as shown in Eq This can be eaanged to give E 0 = mc fo the enegy of the paticle, c A8
5 Now, diffeentiating Eq A7 with espect to time gives the coodinate acceleation shown in Eq Substituting fo E 0 we find d = µc 3 c µ A9 Now if a paticle at est slowly entes the field with then the paticles enegy E 0 is appoximately its est enegy mc, howeve if we wish to inject the paticle into the field with velocity v 0 then E 0 = γmc mc = v 0 /c This gives = c v 0 /c A0 We can see that as then v 0 as equied Appendix B: Common foms of the Schwazschild metic A geneal fom of the Schwazschild metic can be witten as c = µ c C µ B C C C dθ C cos θdφ The fou common vaiants, which ae time independent, ae: Schwazschild s oiginal metic with C = 3 + 8µ 3 /3, isotopic coodinates with C = + µ, Billouin coodinates with C = +µ and the moe common fom of the metic with C = jameschappell@adelaideeduau A P Fench, Special Relativity Van Nostand Reinhold, Bekshie, England, 987 S Schlamminge, K-Y Choi, T A Wagne, J H Gundlach, and E G Adelbege, Phys Rev Lett 00, 040 008 3 C W Misne, K S Thone, and J A Wheele, Gavitation Feeman and Company, San Fancisco, 973 4 D Hilbet, Nachichten von de Gesellschaft de Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 97, 53 97 5 D Hilbet, Math Ann 9 94, 0007/BF044847 6 C H McGude III, Physical Review D 5, 39 98 7 V Baginsky, C Caves, and K Thone, Physical Review D 5, 047 977