Special Theory of Relativity on Curved Space-time

Transcription

1 Special Theory of Relativity on Curved Space-time Moninder Sinh Modil Department of Physics, Indian Institute of Technoloy, Kanpur, India Date: 6 Auust 4

2 Special Theory of Relativity on Curved Space-time Abstract Space-time measurements, of edanken experiments of special relativity need modification in curved spaces-times. It is found that in a space-time with metric, the special relativistic factor, has to be replaced by, where V µ (,v,,), V V is the 4-velocity, and v the relative velocity between the two frames Examples are iven for Schwarzschild metric, Freidmann-Robertson-Walker metric, and the Gödel metric. Amon the novelties are paradoxical tachyonic states, with becomin imainary, for velocities less than that of liht, due to space-time curvature. Relativistic mass becomes a function of space-time curvature, m P P, where P µ (E,p) is the 4-momentum, sinalin a new form of Mach s principle, in which a lobal object namely the metric tensor, is effectin interia. PACS: 3.3.+p, 4

3 Special Theory of Relativity on Curved Space-time. Introduction It is enerally considered that the laws of physics can locally be described in a flat space-time settin. This is described by the statement that locally any space-time is Minkowskian. However, space-time curvature can sinificantly effect local physics. One example of this is the Hawkin radiation []. Another example is the effect of acceleration on readin of particle detectors in curved space-times []. Space-time curvature effects local structure of liht cone, and the behaviour of null eodesics. Since edanken experiments of Special Theory of relativity (STR) are based upon propaation of liht sinals, therefore. it may be enquired, how would the various results of STR et modified, when the space-time is curved. Special theory of relativity (STR) is based upon the Minkowski metric, ds η dx µ dx ν, µ, ν,,, 3 () In a curved space-time with the line element, ds dx dx, () effect of local curvature, in the form of components of metric tensor, may be expected to show up in the local measurements. The line element of a null eodesic in both flat and curved space-times is characterized by, ds. (3) 3

4 This equations encodes the fundamental principle of STR, namely the constancy of speed of liht in all inertial frames. It turns out that this is the key equation, for derivin equations for STR in both flat as well as curved space-times derived in this paper. In this paper we investiate this issue, and derive time dilation, lenth contraction, enerymass relations for STR in a curved space-time settin. We use Schwarzschild metric, Robertson-Walker metric, and Gödel metric, as examples for this. Now, consider the familiar edanken experiment of special relativity in which a beam of liht is sent from ceilin to floor of a train movin with velocity v, with respect to the round. Let F be the frame movin alon with train, and F round, be the frame fixed to round. Let heiht of the train carriae be denoted by h. Let x direction be horizontal alon the round, and y direction, vertical, while t stands for the time coordinate in F round. In F the time co-ordinate is represented by t. The z direction is bein inored for the present. In a time interval dt, in frame F round the train travels a distance, dx vdt. (4) In this much time interval, a ray of liht, covers the distance h, from the ceilin to the floor of the train carriae, dy h. (5) In the frame F movin alon with train, time taken for this journey is dt. 4

5 Let the metric be diaonal. However, the result derived below can remains unchaned, when the metric has an off diaonal element such as. Equatin line element of a null eodesic in the co-movin frame F, to zero, one ets, ds dt dy. (6) Line element for null eodesic as seen in the round fixed movin frame F ruond, is, ds ( v ) dt dy (7) From above two equations it follows that, v d t dt, (8) which is the formula for special relativistic time dilation in a curved space-time. This may be re-written as, V V d t dt (9) where V µ is the 4-velocity, µ V (, v,,). () In a eneral relativistic frame work, usin conformal time, can be equated to, ivin, dt dt. () V V Thus we can define a eneral relativistic,, () V V 5

6 for special relativistic transformations in a curved space-time with metric. We note that the usual,, (3) v of Minkowski metric, can be expressed as,. (4) V V η Next consider the edanken experiment in which the ray of liht is sent from one end of carriae to another in a horizontal direction. Let dx be the lenth of carriae in F and dx the lenth in F round.. A mirror is mounted on the other end of carriae, which reflects the ray back. Time for round trip in F is dt, while in F round it is dt. Equatin line elements of null eodesics in F and F round to zero we have, dt dx dt dx. (5) Usin expression eq.() for time dilation, this ives the required formula for lenth contraction, which for, may be re-written as, dx dx, (6) V V dx. (7) d x V V. Lorentz transformations 6

7 µ X by, Lorentz transformations in a curved space-time for a 4-vector ( x, x, x, x 3 ), may be obtained from those for a flat space-time, by replacin x ' ( x vx ), x ' ( x vx ), x ' x, x 3 ' x 3. (8) Proper time τ in GTR is essentially defined by the equation of line element, d τ dx µ dx ν, (9) which obscures effect of relative velocity between inertial frames. However, usin a 4- velocity, µ V and puttin co-ordinate differentials, ( v, v, v, v 3 ), () dx i i v dt, () equation for proper time may be re-written as, dτ dt V V dt. () Proper velocity may be defined as, η µ µ v. (3) Effect of space-time curvature on relativistic mass m can be calculated, usin the usual collision experiment between two particles. Assumin the usual non-relativistic definition of momentum, as product of mass times velocity, alon with law of conservation of (rest) 7

8 mass, does not lead to, momentum conservation in the movin frame. Therefore the momentum is re-defined usin proper velocity. Let subscripts A and B denote incomin particles, and C and D outoin particles. Conservation of momentum between the stationary and the movin frames, ives, m η + m η m η + m η, (4) A A B B C C D D m η + m η m η + m η, (5) A A B B C C D D where, the superscript on η indicates stationary frame. The law of conservation of momentum holds, provided the conserved quantity is the relativistic mass, m m, (6) where m is the rest mass. We have the followin definitions for relativistic momentum and enery, i i i p mη mv, (7) E. (7) m η m Enery-mass relation would be iven by, m P P, (8) 3 where P µ ( p, p, p, p ), is the 4-momentum, and p E, is enery as usual. Above equation indicates that relativistic mass is a function of local curvature, which is a new form of Mach s principles [3], in the sense that inertia is now a function of local value of a lobal object, namely the space-time curvature tensor. Note that enery-mass expression in STR, E p m, (9) can be written as, 8

9 m η P µ P ν. (3) 3. Some examples For Schwarszchild line element [4], ds GM GM dt dr r dθ r sin θdϕ r r, (3) let the frame F be fixed with respect to the black-hole, and frame F be movin with relative velocity v between the frames F. (This corresponds to radial motion) We have, the Schwarszchild special relativistic factor, Sch. (3) GM v r We note that for lare values of r, Sch, the Minkowski factor. This also happens for small values of Black hole mass M, as well as the ravitational constant G. For Friedmann-Robertson-Walker (FRW) [4], with the line element, dr ds dt R ( t) + r dθ + r sin θdϕ, (33) kr we have, we have the FRW special relativistic factor, FRW, (34) R ( t) v kr which aain reduces to the Minkowski factor for lare values of r. 9

10 For metrics with an off diaonal element equation for turns out to be same as eq.(). For Gödel universe [4], with the line element, ds a dt x x e dtdy dx + e dy dz, (35) special relativistic factor Godel is same as Minkowski factor, for 4-velocity V µ (,v,,), which is relative motion alon x direction. However, for V µ (,,v,), which represents relative motion alon y axis, the special relativistic factors ets modified, Godel. (36) x x e v e v Both frames F and F are inertial and rotatin with compass of inertia. 4. New Horizons From eq.() it follows that becomes imainary for, V <, (37) V sinalin paradoxical tachyonic states, in the sense that, due to space-time curvature, mass is becomin imainary, for velocities less than that of liht. For instance in the case of Schwarszchild black hole Sch becomes imainary for, GM r > v. (38) For FRW universe, FRW, becomes imainary for, R ( t) v r >. (39) k

11 While for Gödel universe, Godel, becomes imainary (for V µ (,,v,), i.e. relative motion v alon y direction), for, 6 x > ln. (4) v 5. Further work It will be interestin to investiate classical and quantum fields, with modified STR in a curved space-time settin. Quantum field theory replaces enery-momemtum relations of STR by relations between enery-momentum operators. On a curved spacetime, modified STR enery-momentum relations would have to be implemented. Horizons in various space-times, such as FRW, black-holes, Gödel universe, AdS universe would have to be redefined for various inertial frames, takin into account their velocities. It would also be interestin to investiate effect on horizons, for acceleratin frames. Closed Time-like Curves (CTCs) occur in a variety of situations in General relativity, and are a subject of an onoin debate [5,6]. Some of them such as the worm hole CTC [7], and Gott CTC [8], contain velocity appearin in components of metric tensor, i.e., the CTC construction is based upon time dilation due to relative velocity. Effect of STR in a curved space-time settin, on these CTCs needs to be considered. Certain other CTCs, such as the Gödel universe CTC [4], the Kerr black hole CTC [9], the Tomimatsu-Sato solution s CTC [], von-stockum universe s CTC [], are linked

12 to rotation. Rotation in STR has also been a subject of extensive debate [], to which discussions in this paper may contribute, via definition of. References. Hawkin, S.W.: Comm. Math. Phys., 43, 99 (975).. Birrell, N.D., and Davies, P.C.W.: Quantum Fields in Curved Space, Cambride University Press, (98). 3. Misner, C.W., Thorne, K.S., and Wheeler J.A.: Gravitation, W.H.Freeman and Company (973). 4. Gödel, Kurt: Rev. Mod. Phys.,, 447, (949). 5. Hawkin, S.W.: Phys. Rev., D 46, 63 (99). 6. Krasnikov, S.V.: On the Quantum Stability of the Time Machine, p3, in, Quantum Field Theory under the Influence of External Conditions, ed. Borda M., B.G.Teubner Verlasesellschaft, (996). 7. Moris, M.S, Thorne, K.S., and Yurtsever U.: Phys. Rev. Lett., 6, 446 (988). 8.Gott III, J.R.: Phys. Rev. Lett., 66, 6 (99). 9. Carter B.: Phys. Rev., 74, 559 (968).. Gibbons, G., and Russel-Clark, R.: Phys. Rev. Lett., 3, 398 (973). Tipler F.J.: Phys. Rev., D 9, 3 (974).. von Stockum, W.J.: Proc. R. Soc. Edinb., 57, 35 (937).. Grøn, Ø.: Space eometry in rotatin reference frames: A historical appraisal, DRAFT, Relativity in Rotatin Frames,

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