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On stimulated emission processes in gravitational fields
Volume 93B, number 1, 2
PHYSICS LETTERS
2 June 1980
ON STIMULATED EMISSION PROCESSES IN GRAVITATIONAL FIELDS Remo RUFFINI and Luigi STELLA Istituto di !~'sica G. Marconi, Universit~tdi Roma, Rome, Italy Received 16 January 1980
We here introduce necessary conditions to be fulfilled in order to have stimulated emission processes in a gravitational field. We also define "regions of constant shift" with respect to an asymptotic observer.
Evidence has been given by a variety of radio observations that processes of microwave amplification by stimulated emission of electromagnetic radiation (MASER) occur on astrophysical scales [1 ]. A detailed map of these sources has been obtained by the use of very large base interferometric techniques [2]. The suggestion has been recently advanced [3] that processes of light amplification by stimulated emission (LASER) occur, as well, on astrophysical scales. In this paper, we establish some necessary conditions for the existence of stimulated emission processes in the presence of gravitational fields. These criteria are all the more relevant since maser emission processes are expected to occur from large scale regions (R 1014 cm) in the presence of fairly massive stars [2] (M ~ 10 M o ) and laser emission processes are expected from the strong gravitational field surrounding a gravitationally collapsed object [ 3 - 6 ] . It is well known that the observation of the frequency of an oscillator depends on the state of motion of the emitter and of the receiver (Doppler effect) as well as on the relative values of the gravitational potenrials at the location o f the emitter and of the receiver (gravitational shift). It is a direct consequence of general relativity that all these effects can be summarized in a simple formula: COem/COobs = (UCXKa)em/(UaKa)obs,
(1)
where K a is the four-velocity o f the photon and U s the four-velocity of the emitter or of the receiver. Quite apart from the laser or maser mechanisms
themselves (inversion of population levels, etc.), we are going to examine constraints imposed on the oscillator on the grounds of kinematical or gravitational considerations alone. A necessary condition, then, in order to have stimulated emission between an oscillator at a point P and an identical one at a point P' is that the two oscillator frequencies, related by eq. (1), fulfill the inequality IWV- cowl ~ ACO ,
(2)
where ACOis the width of the laser or maser. To elucidate the measurement of this constraint, we consider a disk of matter in a circular orbit around a Schwarzschild black hole of mass M. The angular velocity of a particle of the disk at a distance r is then given by g2 = M 1 / 2 r -3/2 .
(3)
Here and in the following, we use geometrical units with G = c = 1. Two identical oscillators at points P and P' in the plane of the disk will have their frequencies shifted by the gravitational and Doppler effects by the amount _ (1 - 3M/r') 1/2 COp/COp, 1 + z (1 - 3M/r) 1/2 =
× 1 -T- [r(1 + tg2~)/M - 21-1/2 1 ~- [r'(1 + t g 2 ~ ' ) / M - 2 ] -1/2 "
(4)
Here r and r' are, respectively, the radius of the orbit of the particle at P and P'; 90 ° +~ and 90 ° +~' are, respectively,
107
Volume 93B, number 1,2
PHYSICS LETTERS Z 0
90
.BH) 50
270
M 4
180
Fig. 1. The dashed re#on is the set of points in a keplerian disk in "frequegcy contact" with a point P orbiting a black hole at a radius r = 50 m. This re#on consists of those points having frequencies within a bandwidth t, to/co = 10 -2 about the frequency co of the point P. Clearly in the computation of the frequency shifts both the gravitational and the Doppler effects of the orbiting material have to be considered. The unit at the bottom of the figure corresponds to 50 m.
2 June 1980
the angles between the line connecting the two points P and P' and the line connecting the points P and P ' to the center o f the disk. The ~- (+) signs in eq. (4) apply when the angular m o m e n t u m o f the photon going from P to P' is parallel (antiparallel) to the angular m o m e n t u m o f the disk. In fig. 1 we consider a point P corotating with the disk at a distance r = 50 m, and in the dashed region all the points P', also in corotation, which are in "frequency contact" with the point P, namely, all points with an oscillator frequency differing from the one at P b y an amount which has been chosen for the sake o f example to satisfy Aw/co <, 10 - 2 . This qualitative behavior, for a keplerian disk-like motion, is essentially independent o f the numerical value of the width or of the location of the point P. Furthermore, eq. (4) and all o f the above considerations are not modified if at the center of the configuration there is a star o f mass M and radius R instead of the black hole. Every point P with a distance r > R from the center of the configuration can be in "frequency contact" with an entire ring-like region in the disk and with two almost radially pointing columns. The lasing or masing process can only occur in these regions o f "frequency contact" and in directions in which a critical size for the emitting region is reached. This size is dictated by the laser or maser mechanism (radiation flux, density o f partixles, etc.). A further necessary condition must be fulfilled in order that amplified stimulated emission generated in the system be observed by an external observer at point O. All the emitting points need to have constant
6
Fig. 2. The angles ~ and ~ def'med in eq. (6) are represented here. The plane ,r is determined by the line of sight and the black hole and the plane O by the motion of the keplerian disk. 108
Volume 93B, number 1, 2
PHYSICS LETTERS
2 June 1980
shifts, w i t h the balancing o f Doppler and gravitational effects, w i t h respect to the observer at the p o i n t O [clearly this equality has to be fulfilled only within the b a n d w i d t h o f the laser or the maser, as in eq. (2)], We define the "regionsof constant shift" w i t h respect to an observer O at infinity to be the sets of points P satisfying the following condition:
90
OapIoa 0 =
( Uc~K~)p/( U,~K'~)O =
const.
(5)
For the sake o f e x a m p l e , we again consider a keplerian ring around a Schwarzschild black hole. For an observer at rest at infinity, we have
180
COp/W0 =
1+ z =
l+_cos8 [r(tg2~ + 1)/M-2] - 1 / 2 (1 - 3M/r) 1/2 (6)
Fig. 3. Lines of constant shift for a keplerian disk seen edge-on (h = 90 °) with respect to the line of sight. The disk is rotating counter-clockwise. Selected numerical values axe indicated on ~1 the red-shifted constant shift lines (z > 0) and on the blueshifted lines (z < 0). The line of sight coincides with the dashed line.
270
90
60
1
'
30 •
'
'
•
Ij
I . I,
~p I.!
O
TI"
110
270
III
3()0
Z V
Fig. 4. This diagram represents the intersection of the lines of constant shift defined in fig. 3 with the regions in "frequency contact" gith the point P defined in fig. 1. The shadowed region indicates the maximal intersection which fulfills, within the bandwidth ~to/~o = 10-2, both necessary conditions indicated in the text in order to generate amplified radiation visible to an asymptotic ~bserver. This region, corresponding to qJ = 0° , gives rise to a red-shifted signal, while the region at 9 = 180 ° produces a blue-shifted ;ignal. A third region of maximal intersection corresponds to a line with z ~ 0 (gravitational and Doppler shifts balancing each other) ~orresponding to the almost radial columns of fig. 1. The relevance of this structure for the Doppler SS433 is analyzed elsewhere :6,7]. 109
Volume 93B, number 1, 2
PHYSICS LETTERS
2 June 1980
~qere 8 (see fig. 2) is the angle between the plane of the tisk and the plane determined by the black hole center and the line of sight, while ~ + 90 ° indicates the angle )etween the line of sight and the line connecting the 9oint of emission P to the center of the black hole. 3oth angles ~ and ~ change as a function of the position )f the emitting point P on the disk and as a function )f the position of the observer with respect to the disk 9lane. If we now indicate by X the angle between the ine o f sight and the angular momentum I o f the disk md by ff a polar angle to identify the emitting point ? on the disk (the line ~k = 0 is orthogonal to the line )f sight), we have
with the maximum depth in the direction of the observer. It is likely that only intersecting regions of large enough depth in the direction of sight can lead to observable phenomena of amplified stimulated emission. The astrophysical settings for the simultaneous fulfillment of the two necessary conditions given above will only occur along selected directions. It is also a matter of course that observations o f the same astrophysical system from different directions of sight will lead to different laser of maser patterns. The implications of this analysis for the known maser sources [1,2] and for the Doppler SS433 will be presented elsewhere [ 6 - 8 ] .
;in ~ = sin ~, sin ff,
References
(7)
:os ~ = sin X cos qs/(1 - sin2X sin 2 qs)l/2 . Regions of constant shift for X = 90 ° are shown in fig. 3. It is then clear that in order to have a mechanism o f ~timulated emission working and being observable, both aecessary conditions given above must be fulfiUed. ~timulated emission will be observed only from the inLersection of the two sets: namely, regions of "constant ~hift" and regions in "frequency contact". For the sake of example, we again consider the case of the disk around the black hole. In fig. 4 we show the intersection o f the two sets of points defined by the above necessary conditions. The dashed region show the intersection of the two sets with the same bandwidth Aw/w = 10 - 2 and
110
[1 ] See e.g.J.M. Moran, Radio observation of galactic masers, in: Frontiers of astrophysics, ed. E.H. Avrett (Harvard U.P., 1976). [2] See e.g.R. Genzd et al., Astron. Astrophys. 66 (1978) 13, and references therein. [3] R. Ruffini, Gravitationally collapsed objects, in: Proc. second M. Grossmann meeting (July 1979), to be published. [4] R. Ruffini, Nuovo Cimento Lett. 26 (1979) 239. [5] Fang Li Zhi and R. Ruffmi, Phys. Lett. 86B (1979) 193. [6] R. Ruffini and L. SteUa, On the emission region of the Dopplar SS433, Nuovo Cimento Lett., to be published; and in preparation. [7] R. Ruffmi, Black holes in the universe, Invited talk at the annual meeting of the AAAS (San Francisco, January 1980), to be published in Science. [8] R. Rosner, R. Ruff'mi and G. Vaiana, in preparation.
PHYSICS LETTERS
2 June 1980
ON STIMULATED EMISSION PROCESSES IN GRAVITATIONAL FIELDS Remo RUFFINI and Luigi STELLA Istituto di !~'sica G. Marconi, Universit~tdi Roma, Rome, Italy Received 16 January 1980
We here introduce necessary conditions to be fulfilled in order to have stimulated emission processes in a gravitational field. We also define "regions of constant shift" with respect to an asymptotic observer.
Evidence has been given by a variety of radio observations that processes of microwave amplification by stimulated emission of electromagnetic radiation (MASER) occur on astrophysical scales [1 ]. A detailed map of these sources has been obtained by the use of very large base interferometric techniques [2]. The suggestion has been recently advanced [3] that processes of light amplification by stimulated emission (LASER) occur, as well, on astrophysical scales. In this paper, we establish some necessary conditions for the existence of stimulated emission processes in the presence of gravitational fields. These criteria are all the more relevant since maser emission processes are expected to occur from large scale regions (R 1014 cm) in the presence of fairly massive stars [2] (M ~ 10 M o ) and laser emission processes are expected from the strong gravitational field surrounding a gravitationally collapsed object [ 3 - 6 ] . It is well known that the observation of the frequency of an oscillator depends on the state of motion of the emitter and of the receiver (Doppler effect) as well as on the relative values of the gravitational potenrials at the location o f the emitter and of the receiver (gravitational shift). It is a direct consequence of general relativity that all these effects can be summarized in a simple formula: COem/COobs = (UCXKa)em/(UaKa)obs,
(1)
where K a is the four-velocity o f the photon and U s the four-velocity of the emitter or of the receiver. Quite apart from the laser or maser mechanisms
themselves (inversion of population levels, etc.), we are going to examine constraints imposed on the oscillator on the grounds of kinematical or gravitational considerations alone. A necessary condition, then, in order to have stimulated emission between an oscillator at a point P and an identical one at a point P' is that the two oscillator frequencies, related by eq. (1), fulfill the inequality IWV- cowl ~ ACO ,
(2)
where ACOis the width of the laser or maser. To elucidate the measurement of this constraint, we consider a disk of matter in a circular orbit around a Schwarzschild black hole of mass M. The angular velocity of a particle of the disk at a distance r is then given by g2 = M 1 / 2 r -3/2 .
(3)
Here and in the following, we use geometrical units with G = c = 1. Two identical oscillators at points P and P' in the plane of the disk will have their frequencies shifted by the gravitational and Doppler effects by the amount _ (1 - 3M/r') 1/2 COp/COp, 1 + z (1 - 3M/r) 1/2 =
× 1 -T- [r(1 + tg2~)/M - 21-1/2 1 ~- [r'(1 + t g 2 ~ ' ) / M - 2 ] -1/2 "
(4)
Here r and r' are, respectively, the radius of the orbit of the particle at P and P'; 90 ° +~ and 90 ° +~' are, respectively,
107
Volume 93B, number 1,2
PHYSICS LETTERS Z 0
90
.BH) 50
270
M 4
180
Fig. 1. The dashed re#on is the set of points in a keplerian disk in "frequegcy contact" with a point P orbiting a black hole at a radius r = 50 m. This re#on consists of those points having frequencies within a bandwidth t, to/co = 10 -2 about the frequency co of the point P. Clearly in the computation of the frequency shifts both the gravitational and the Doppler effects of the orbiting material have to be considered. The unit at the bottom of the figure corresponds to 50 m.
2 June 1980
the angles between the line connecting the two points P and P' and the line connecting the points P and P ' to the center o f the disk. The ~- (+) signs in eq. (4) apply when the angular m o m e n t u m o f the photon going from P to P' is parallel (antiparallel) to the angular m o m e n t u m o f the disk. In fig. 1 we consider a point P corotating with the disk at a distance r = 50 m, and in the dashed region all the points P', also in corotation, which are in "frequency contact" with the point P, namely, all points with an oscillator frequency differing from the one at P b y an amount which has been chosen for the sake o f example to satisfy Aw/co <, 10 - 2 . This qualitative behavior, for a keplerian disk-like motion, is essentially independent o f the numerical value of the width or of the location of the point P. Furthermore, eq. (4) and all o f the above considerations are not modified if at the center of the configuration there is a star o f mass M and radius R instead of the black hole. Every point P with a distance r > R from the center of the configuration can be in "frequency contact" with an entire ring-like region in the disk and with two almost radially pointing columns. The lasing or masing process can only occur in these regions o f "frequency contact" and in directions in which a critical size for the emitting region is reached. This size is dictated by the laser or maser mechanism (radiation flux, density o f partixles, etc.). A further necessary condition must be fulfilled in order that amplified stimulated emission generated in the system be observed by an external observer at point O. All the emitting points need to have constant
6
Fig. 2. The angles ~ and ~ def'med in eq. (6) are represented here. The plane ,r is determined by the line of sight and the black hole and the plane O by the motion of the keplerian disk. 108
Volume 93B, number 1, 2
PHYSICS LETTERS
2 June 1980
shifts, w i t h the balancing o f Doppler and gravitational effects, w i t h respect to the observer at the p o i n t O [clearly this equality has to be fulfilled only within the b a n d w i d t h o f the laser or the maser, as in eq. (2)], We define the "regionsof constant shift" w i t h respect to an observer O at infinity to be the sets of points P satisfying the following condition:
90
OapIoa 0 =
( Uc~K~)p/( U,~K'~)O =
const.
(5)
For the sake o f e x a m p l e , we again consider a keplerian ring around a Schwarzschild black hole. For an observer at rest at infinity, we have
180
COp/W0 =
1+ z =
l+_cos8 [r(tg2~ + 1)/M-2] - 1 / 2 (1 - 3M/r) 1/2 (6)
Fig. 3. Lines of constant shift for a keplerian disk seen edge-on (h = 90 °) with respect to the line of sight. The disk is rotating counter-clockwise. Selected numerical values axe indicated on ~1 the red-shifted constant shift lines (z > 0) and on the blueshifted lines (z < 0). The line of sight coincides with the dashed line.
270
90
60
1
'
30 •
'
'
•
Ij
I . I,
~p I.!
O
TI"
110
270
III
3()0
Z V
Fig. 4. This diagram represents the intersection of the lines of constant shift defined in fig. 3 with the regions in "frequency contact" gith the point P defined in fig. 1. The shadowed region indicates the maximal intersection which fulfills, within the bandwidth ~to/~o = 10-2, both necessary conditions indicated in the text in order to generate amplified radiation visible to an asymptotic ~bserver. This region, corresponding to qJ = 0° , gives rise to a red-shifted signal, while the region at 9 = 180 ° produces a blue-shifted ;ignal. A third region of maximal intersection corresponds to a line with z ~ 0 (gravitational and Doppler shifts balancing each other) ~orresponding to the almost radial columns of fig. 1. The relevance of this structure for the Doppler SS433 is analyzed elsewhere :6,7]. 109
Volume 93B, number 1, 2
PHYSICS LETTERS
2 June 1980
~qere 8 (see fig. 2) is the angle between the plane of the tisk and the plane determined by the black hole center and the line of sight, while ~ + 90 ° indicates the angle )etween the line of sight and the line connecting the 9oint of emission P to the center of the black hole. 3oth angles ~ and ~ change as a function of the position )f the emitting point P on the disk and as a function )f the position of the observer with respect to the disk 9lane. If we now indicate by X the angle between the ine o f sight and the angular momentum I o f the disk md by ff a polar angle to identify the emitting point ? on the disk (the line ~k = 0 is orthogonal to the line )f sight), we have
with the maximum depth in the direction of the observer. It is likely that only intersecting regions of large enough depth in the direction of sight can lead to observable phenomena of amplified stimulated emission. The astrophysical settings for the simultaneous fulfillment of the two necessary conditions given above will only occur along selected directions. It is also a matter of course that observations o f the same astrophysical system from different directions of sight will lead to different laser of maser patterns. The implications of this analysis for the known maser sources [1,2] and for the Doppler SS433 will be presented elsewhere [ 6 - 8 ] .
;in ~ = sin ~, sin ff,
References
(7)
:os ~ = sin X cos qs/(1 - sin2X sin 2 qs)l/2 . Regions of constant shift for X = 90 ° are shown in fig. 3. It is then clear that in order to have a mechanism o f ~timulated emission working and being observable, both aecessary conditions given above must be fulfiUed. ~timulated emission will be observed only from the inLersection of the two sets: namely, regions of "constant ~hift" and regions in "frequency contact". For the sake of example, we again consider the case of the disk around the black hole. In fig. 4 we show the intersection o f the two sets of points defined by the above necessary conditions. The dashed region show the intersection of the two sets with the same bandwidth Aw/w = 10 - 2 and
110
[1 ] See e.g.J.M. Moran, Radio observation of galactic masers, in: Frontiers of astrophysics, ed. E.H. Avrett (Harvard U.P., 1976). [2] See e.g.R. Genzd et al., Astron. Astrophys. 66 (1978) 13, and references therein. [3] R. Ruffini, Gravitationally collapsed objects, in: Proc. second M. Grossmann meeting (July 1979), to be published. [4] R. Ruffini, Nuovo Cimento Lett. 26 (1979) 239. [5] Fang Li Zhi and R. Ruffmi, Phys. Lett. 86B (1979) 193. [6] R. Ruffini and L. SteUa, On the emission region of the Dopplar SS433, Nuovo Cimento Lett., to be published; and in preparation. [7] R. Ruffmi, Black holes in the universe, Invited talk at the annual meeting of the AAAS (San Francisco, January 1980), to be published in Science. [8] R. Rosner, R. Ruff'mi and G. Vaiana, in preparation.