Pseudo-Superluminal Motion 1

Transcription

1 seudo-superluminal Motion 1 On seudo-superluminal Motion Anamitra alit Author /Teaher(free-laner physiist),india,154 Motijheel Aenue,Kolkata: palit.anamitra@gmail.om h: Abstrat: Modern physis onfirms the impossibility of Superluminal Motion through the onsiderations of Speial Relatiity. In General Relatiity we may apply this onstraint rigorously only to the Loal Inertial Frames where Einstein s Field Equations are linear. This artile, inidentally seeks to inestigate the possibility of seudo-superluminal motion in the non loal ontext without iolating Speial Relatiity Keywords: General Relatiity, Loal Inertial Frames, Manifold, Tangent lane AC: q INTRODUCTION Finite speed of signal transmission is one of the greatest disoeries that hae reolutionized modern physis.speial Relatiity 1 through its seond postulate laims that the speed of light is independent of its soure. Though ery muh ounter-intuitie if iewed through the lassial ideas it turns out to be an amazing fat. In ombination with the first postulate of relatiity it leads to the noel aspet of spae and time geing mixed up into a omposite fabri. One of the fundamental outomes of all this is the finite speed of signal transmission. Inidentally all this refers to what we know as Flat Spaetime or Minkowski Spae. General Relatiity is heaily based on the onept of the Loal Inertial frames(lif) whih break up ured spae into a set of small inertial territories. Cured Spaetime is goerned by Einstein s Field equations whih are non-linear in nature. But the Loal Inertial frames offer us the adantage of Speial Relatiity---the Field Equations beome linear. Calulations beome simpler and omfortable in Flat Spaetime whih exists here only in the loal ontext, of ourse. NON-LOCAL CONSIDERATIONS Now we onsider the obseration of an eent at a point from a point suh that they hae a finite separation between them, so that both my not be loated in the same Inertial loal frame. But eah point arries its own LIF with it. In our thought experiment we hae two obserers one at and the other at. A light ray flashes aross an infinitesimally small, spatial interal at. It is obsered from both the points and. The spatial interal noted by both is the same. But the time reorded for the passage is different for the obserers sine their loks run at different rates, the metri oeffiients pertaining to time, generally speaking, are different for the two points.

2 seudo-superluminal Motion FIGURE 1. Loal and Non-Loal obserations Metri 3 : ds g g dx g dy g dz (1) zz yy zz Spatial interal at : g dx g dy g dz xx yy zz Both the obserers reord the same alue for the aboe. Non-Loal time interal obsered from : dt g ( ) () Loal time interal obsered from : dt g (3) Non-Loal Obseration: Speed of light at as obsered from : Loal Obseration: Speed of light at as obsered from : g g ( ) But the speed of light as obsered from is the loal speed of light that is,,where is the standard alue for speed of light in auum as we know it. Therefore, g g ( ) Or, Or, g g ( )

3 seudo-superluminal Motion 3 Non-Loal speed of ligh,, is gien by: g g ( ) (4) [ :Obseration being Loal] Therefore the speed of light for non-loal obseration may be greater than equal to or less than the speed of light as we know it, the standard alue, depending on the alue of g : g the ratio ( ) Now let s onsider a partile moing aross an infinitesimally small spatial interal at (instead of a light ray). Spatial separation: g dx g dy g dz xx yy zz Both the obserers reord the same alue for it. Time interal obsered from : dt g ( ) Time interal obsered from : dt g Non-Loal Obseration: Speed of partile at as obsered from : ( : partile) g ( ) Loal Obseration: Speed of partile at as obsered from : ( : partile) Therefore, ( : partile) g g ( ) ( : partile) ( : partile) ( : artile) g ( ) g g (5) So the non-loal speed of the partile, ( : partile), may exeed the standard loal g : g alue of the speed of light depending on the alue of the ratio ( ) Inidentally the loal speed of the partile is always less than the loal speed of light, that is, ( : partile) Therefore from relation (5) we hae,

4 seudo-superluminal Motion 4 ( : partile) g g ( ) (6) But the Right-hand-side of relation (6) is the non-loal speed of light [see relation (4)] Therefore ( : partile) (7) Thus the non-loal speed of the partile is less than the non loal speed of light, though the non-loal speed of the partile an exeed the loal standard speed of light in auum g : g.the light ray is always ahead of the depending on the alue of the ratio: ( ) partile does not maer whether you are onerned with loal or non-loal obseration. We are not iolating relatiity in any manner. Now the non-loal speed of light or some partile is important in deiding the aerage speed of light oming aross a finite interal of spae Time of non-loal time of trael of a light ray is gien by: dt Time Taken : T A B g ( ) / g ( A) g ( ) / g ( A) The aerage speed of light for non-loal trael aross marosopi distanes: Aerage A A B g ( ) / g ( A) B B A A g ( ) / g ( A) B (8) (9)

5 seudo-superluminal Motion 5 So the aerage speed of light may be different from the loal speed (whih orresponds to the known alue---the speed of light in auum) When a light ray is oming towards an obserer aross some interal of spae he would be more interested in the aerage speed of light oer the interal than the loal speed of light loal speed of light for arious points traersed by the light ray.. SYNCHRONIZATION OF CLOCKS: For the purpose of synhronization 4 of loks we take the speed of light onstant oer large marosopi distanes. It it really justified in iew of the fat that the speed of light may hange in the non-loal sense espeially when we are onsidering sensitie experiments like the OERA 5 or ICARUS 6?. It would be an interesting reminder for us that the OERA experiment failed(due to able fault: loose able onnetion) with the ondition that the speed of light was taken to be onstant with respet to obseration stations in disregard of the fat that the light ray traeled oer large marosopi distanes in the proess of synhronization. The ICARUS experiment sueeded on the basis of the same aspet ---the non-loal ariation of the speed of light was not gien a due onsideration.. Sample Calulations FIGURE.Transmission of light ray from a satellite to two earth stations at A and B The aboe figure shows a non-rotating earth-like planet with obseration stations at A and B. S is a satellite from where light signals are being sent. These are being reeied at

6 seudo-superluminal Motion 6 the earth stations A and B. O is taken to be the z:axis.o=r;so=. OS=d,a fixed oordinate distane.os=, a fixed/onstant angle. N is perpendiular to OS. Now, ON=rCos SN=OS-ON=d-rCos N rsin tan SN d rcos d tan r tancos rsin (10) Taking differentials from (10) we hae: dr( Sin tan Cos ) rd ( Cos tan Sin ) (11) Agein from relation (10) we obtain: d tan r (1) Sin tancos Shwarzshild s Metri: 1 ds (1 GM / r) (1 GM / r) dr r ( d Sin d ) The spatial element on the line SB, say at, is gien by: 1 GM r ( Cos tan Sin ) 1 d r d r ( Sin tan Cos ) 1 GM ( Cos tan Sin ) Or, 1 1 rd r ( Sin tan Cos ) Spatial element on AS is gien by: 1/ GM 1 dr r [Sine both and are zero on AS] Time of trael of light ray from B to S: T S 1 GM ( Cos tan Sin ) 1 1 r ( Sin tan Cos ) r may be taken from (1) Time of light ray from S to A: GM 1 r( ) GM 1 r( S) rd (13) (14) (15)

7 seudo-superluminal Motion 7 1 GM r 1 r T dr (16) r S GM 1 r( ) GM 1 r( S) [Inidentally, for this path and d are both zero. SO we hae onsidered integration wrt dr] These alulations take are of the tik rate at eah point on the path of the light ray while in they GS they onsider the tik rates at the point of transmission and reeption only. NON-LINEARITY OF EINSTEIN S FIELD EUATIONS The fat that Einstein s field equations are non linear is a well known fat in physis. But in the inertial frames of referene the Christoffel symbols 7 ealuate to zero alue and the field equations are no more non linear. They beome linear. So if you are working in a laboratory you are enjoying the priilege of linearity whih is not there outside your laboratory if it(lab) happens to be a loal inertial frame. For non loal obserations the non linearity of the field equations are supposed to play a ery big role as in our ase of pseudo superluminal motion. One issue beomes important in this respet : To what extent is our lab fixed on the earth s surfae is an inertial frame of referene? Lab Fixed on the Earth s Surfae You are working in your small laboratory room fixed on the earth so that you may all it a loal inertial frame[and you are working for a suitably small interal of time]. Now you may think of a freely falling lift in front of you. That lift is a beer approximation of a LIF. Your Lab room does not orrespond to the beer approximation. The ontrast should would beome glaringly onspiuous if you imagine the graity to be a million times stronger---that is if you onsider your lab room to be in a region of strong spaetime urature. The freely falling lift is a LIF while your lab room in this example may be termed as a Loal Non Inertial Frame The basi adantage proided by the Loal Inertial Frames is the Speial Relatiity ontext. The point that naturally arises is that to what extent do we expet deiations from SR in the loal non-inertial frame? The Tangent lane to the Manifold Let us onsider the tangent plane 8 at the point of ontat with the ured spaetime surfae. The tangent surfae offers the adantage of the Speial Relatiity ontext.sine time goes on hanging in both the tangent plane and the ured surfae(though differently), our laboratory, its spaetime, loation(oordinates) at the most an be at a momentary ontat with the point.then the spae-time point of the laboratory will moe along the ured surfae unless we make some tehnologial arrangement of

8 seudo-superluminal Motion 8 ontaining our laboratory on the tangent plane by arranging a freely falling frame.to materialize the loal transformation from 4D ured spae to Minkowski spae we hae to arrange a freely falling frame--the falling lift in the simple this ase of the earth. Let the oordinates of the ured 4D surfae be and the loal oordinates on the tangent surfae at : ( 0, 1,, 3). The first oordinate in parenthesis represents time in eah system. If we want to keep the on the tangent surfae in order to enjoy the adantage of Speial Relatiity, the lift should aelerate wrt to the ured surfae[generally speaking] Transformations (Equation Set 17) Our the tangent plane is atually an inertial frame of referene. Consider a world line on it through the point of ontat, a short world line of ourse. Let us denote the world line by: F(,,, ) 0 (18) 0 1 For the transformed alues the quantities,, in general will be non zero. To understand the situation at the point of ontat we onsider a simpler analogy. We dy take the parabola gien by: y x. Gradient: a The tangent to it at the point dx M(a,b) is gien by: y b a x a At the point M(point of ontat) the alue of dy is idential for both the parabola and the dx tangent whih is a straight line in this ase. But what about the seond order d y deriatie, dx? For the parabola: d y d y dx. For the straight line: 0 dx. The M M seond order deriaties differ een at the point of ontat. You may translate this example to higher dimensions. oints to Obsere: 1. The point of ontat on the manifold and the tangent plane are not idential in so far as the seond order deriaties are onsidered. The first order deriaties on the two planes at the point of ontat are idential. But they are not idential(in general) at other neighboring points.. To stay on the tangent plane[lif],een at the point of ontat,, some aeleration is neessary. We need a freely falling frame to stay on the said tangent plane.

9 seudo-superluminal Motion 9 Speed of Light in Loal Inertial Frames Indeed, we may write the metri: ds g g dx (19) xx ds dt (0) In the aboe metri, that is in (19), the x-axis has been oriented along the infinitesimal path of a light ray. Now ds 0 for the null geodesi. Therefore from relation (0) we hae, Mod 1 dt Inidentally =1 in the natural units and we hae the same inariable speed of light in auum proided we define physial time interal as: dt ( physial) g is the oordinate time interal. We are geing the speed of light [standard alue] with respet to the tangent plane. Inidentally, equation (0) orresponds to the tangent plane, the LIF.What about equation (19)?It represents ured spaetime. At the point of ontat we,of ourse, get the same alue for the first order deriatie for both the surfaes of whih the speed of light is an example. But een for short distanes this fails the piture is so triky een in the ontest of the loal inertial frames. seudo super luminal speed of light in the non-loal ontext is oming into piture! We may try to alulate the aeleration of the light ray at the point of ontat of the tangent plane with the manifold(wrt to the manifold).any deiation from an LIF due to absene of orret amount of aeleration required to stay on the tangent plane will neessitate suh an inestigation ACKNOWLEDGMENTS I find this an opportunity to express my indebtedness to authors whose works hae always inspired me to maintain an atie interest in the area in the area of physis. My heartfelt respets go to Dr Timur Kamalo and other persons of the ICT ommiee for proiding me the opportunity to present the forgoing artile at ICT, 01, Mosow. REFERENCES 1. J B Hartle, Graity:An introdution to Einstein s General Relatiity earson Eduation,In,,003,, pp J B Hartle, Graity:An introdution to Einstein s General Relatiity earson Eduation,In,,003,, pp S. Weinberg,Graitation and Cosmology,John Wiley and Sons(Asia)te Ltd, p 8-10, J B Hartle, Graity:An introdution to Einstein s General Relatiity earson Eduation,In,,003,, pp Wikipedia, OERA Experiment, hp://en.wikipedia.org/wiki/oera_experiment 6 Wikipedia,ICARUS(Experiment)hp://en.wikipedia.org/wiki/ICARUS_%8experiment%9 7 Steen Weinberg,Graitation and Cosmology,John Wiley and Sons(Asia)te Ltd, pp R M Wald, General Relatiity, Oerseas ress(india) t. Ltd,006,pp14-15

10 seudo-superluminal Motion 10