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Massless particles and losing clocks
Volume 140, numberS
PHYSICS LETFERS A
25 September 1989
MASSLESS PARTICLES AND LOSING CLOCKS Thomas SCHUCKER Department of Mathematics, University of California, Berkeley, CA 94720, USA Received 9 June 1989; accepted for publication 28 July 1989 Communicated by J.P. Vigier
The general relativistic time delay induced by massless particles travelling near a clock is recalculated.
Recently Dray and ‘t Hooft [1] have remarked a striking prediction of general relativity: The gravitational field of a massless particle [2], which is concentrated on a degenerate Machian cone, slows down the ticking of clocks in its vicinity. The same phenomenon should occur in the presence of a particle whose rest mass is small compared to its kinetic energy. Dray and ‘t Hooft base their analysis on the properties of null geodesics in the metric generated by a massless particle [2]. A possible objection to this point of view might be that realistic clocks are rather massive and their geodesics do not exhibit the described phenomenon. To avoid any such ambiguity we consider an interferometer and calculate directly the expected phase shift from the wave equation in the above metric [2]. Imagine a massless particle, the source, flying on the positive z-axis with energy E. For the time being let us assume an extended, cylindrical source of longitudinal dimension L and transverse radius a and with constant energy density E/~J~ Lita 2• Following Bonnor [2] it generates the metric (I) d,s2=—dx2—dy2+2dudv+2A(x,y, u)du2,
~
where u and v are light cone coordinates: t—z
t+z
u:~-—~-, v:=—~-, and
(2)
GE
ifr~
:__4~j=_~-+8~T_ln(r/a)~ ifr?~aandO~u~L, 0,
otherwise,
(3)
with G Newton’s constant and r:= .,.,,/x2 +y2. The point particle limit with energy density Eô(x)ô(y)ô(z— t) is obtained as L and a tend to zero independently and yields (4) where ~sis an arbitrary constant carrying the dimension of length. Consider an interferometer with light of wavelength A. and two beams pointing initially in the negative z0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
209
Volume 140, number 5
PHYSICS LETTERS A
25 September 1989
direction separated from the z-axis by a distance r1 and r1 respectively. Their deviations O~,,j= 1, 2 from the z-direction after having met the source are calculated from the null geodesics in the metric (1) [3]: 8GE/r~ tanOJ=lI(8GE,)7~
r1>~a,
(5)
and can safely be neglected for realistic energies E and distances r1. In this approximation the two beams are described by plane wave solutions of the Klein—Gordon equation in the metric (1), which reads 2~ a2~~ a2~, 8(0 0 (6 a 8x2 82 ~r, u, ~ Au8v —
—
—
—
For finite pulse length L of the source the two initial plane waves are q(r~,u, v)=exp[i(2x/1)~J~v]
u~0.
,
(7)
Imposing continuity of Q~for u =0 we obtain the two plane waves in the interaction region: 9~(r~, u, v)=exp{i(2~t/2)~,/~[v+A(r~, ~L)u]},
0~
(8)
and imposing continuity at u = L the final plane waves are 9~(r~, u, v)=exp{i(2it/2)~h[v+A(r 3, IL)L]},
L~
(9)
Note that the phase shift A (reinserting the speed of light for convenience), A=A(r,, ~L) —A(r2, ~L)
~,
(10)
is independent of the pulse length L of the source and independent of the wave length A. of the interferometer. This phase shift agrees with Dray and ‘t Hooft’s retarding of clocks. If weAtake the source to be a laserany beam 4m then the phase shift is about 10 m, beyond exof typical energy E= 1 J and beam an radius a = 10 question remains: For pointlike photons, a tending to zero, perimental verification. However, intriguing will we get an observable phase shift in any future theory of quantum gravity?
References [l)T. Dray andG. ‘t Hooft, Nuci. Phys. B 253 (1985) 173. [2] R. Penrose, in: Batelle rencontres 1967, eds. CM. DeWitt and J.A. Wheeler (Benjamin, New York, 1968) p. 198; W.B. Bonnor, Commun. Math. Phys. 13 (1969) 163; P.C. Aichelburg and R.U. Se~d,Gen. Rel. Gray. 2 (1971) 203. [3] T. SchOcker, The gravitational cross section of massless particles, Class. Quantum Gray., to be published.
210
PHYSICS LETFERS A
25 September 1989
MASSLESS PARTICLES AND LOSING CLOCKS Thomas SCHUCKER Department of Mathematics, University of California, Berkeley, CA 94720, USA Received 9 June 1989; accepted for publication 28 July 1989 Communicated by J.P. Vigier
The general relativistic time delay induced by massless particles travelling near a clock is recalculated.
Recently Dray and ‘t Hooft [1] have remarked a striking prediction of general relativity: The gravitational field of a massless particle [2], which is concentrated on a degenerate Machian cone, slows down the ticking of clocks in its vicinity. The same phenomenon should occur in the presence of a particle whose rest mass is small compared to its kinetic energy. Dray and ‘t Hooft base their analysis on the properties of null geodesics in the metric generated by a massless particle [2]. A possible objection to this point of view might be that realistic clocks are rather massive and their geodesics do not exhibit the described phenomenon. To avoid any such ambiguity we consider an interferometer and calculate directly the expected phase shift from the wave equation in the above metric [2]. Imagine a massless particle, the source, flying on the positive z-axis with energy E. For the time being let us assume an extended, cylindrical source of longitudinal dimension L and transverse radius a and with constant energy density E/~J~ Lita 2• Following Bonnor [2] it generates the metric (I) d,s2=—dx2—dy2+2dudv+2A(x,y, u)du2,
~
where u and v are light cone coordinates: t—z
t+z
u:~-—~-, v:=—~-, and
(2)
GE
ifr~
:__4~j=_~-+8~T_ln(r/a)~ ifr?~aandO~u~L, 0,
otherwise,
(3)
with G Newton’s constant and r:= .,.,,/x2 +y2. The point particle limit with energy density Eô(x)ô(y)ô(z— t) is obtained as L and a tend to zero independently and yields (4) where ~sis an arbitrary constant carrying the dimension of length. Consider an interferometer with light of wavelength A. and two beams pointing initially in the negative z0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
209
Volume 140, number 5
PHYSICS LETTERS A
25 September 1989
direction separated from the z-axis by a distance r1 and r1 respectively. Their deviations O~,,j= 1, 2 from the z-direction after having met the source are calculated from the null geodesics in the metric (1) [3]: 8GE/r~ tanOJ=lI(8GE,)7~
r1>~a,
(5)
and can safely be neglected for realistic energies E and distances r1. In this approximation the two beams are described by plane wave solutions of the Klein—Gordon equation in the metric (1), which reads 2~ a2~~ a2~, 8(0 0 (6 a 8x2 82 ~r, u, ~ Au8v —
—
—
—
For finite pulse length L of the source the two initial plane waves are q(r~,u, v)=exp[i(2x/1)~J~v]
u~0.
,
(7)
Imposing continuity of Q~for u =0 we obtain the two plane waves in the interaction region: 9~(r~, u, v)=exp{i(2~t/2)~,/~[v+A(r~, ~L)u]},
0~
(8)
and imposing continuity at u = L the final plane waves are 9~(r~, u, v)=exp{i(2it/2)~h[v+A(r 3, IL)L]},
L~
(9)
Note that the phase shift A (reinserting the speed of light for convenience), A=A(r,, ~L) —A(r2, ~L)
~,
(10)
is independent of the pulse length L of the source and independent of the wave length A. of the interferometer. This phase shift agrees with Dray and ‘t Hooft’s retarding of clocks. If weAtake the source to be a laserany beam 4m then the phase shift is about 10 m, beyond exof typical energy E= 1 J and beam an radius a = 10 question remains: For pointlike photons, a tending to zero, perimental verification. However, intriguing will we get an observable phase shift in any future theory of quantum gravity?
References [l)T. Dray andG. ‘t Hooft, Nuci. Phys. B 253 (1985) 173. [2] R. Penrose, in: Batelle rencontres 1967, eds. CM. DeWitt and J.A. Wheeler (Benjamin, New York, 1968) p. 198; W.B. Bonnor, Commun. Math. Phys. 13 (1969) 163; P.C. Aichelburg and R.U. Se~d,Gen. Rel. Gray. 2 (1971) 203. [3] T. SchOcker, The gravitational cross section of massless particles, Class. Quantum Gray., to be published.
210