- Email: [email protected]
Potential flow around a massless particle in general relativity
Physics LettersA 171 (1992) 3-6 North-Holland
PHYSICS LETTERS A
Potential flow around a massless particle in general relativity A. Feinstein a n d J. Ib~tfiez Departamento de Fisica Teorica, Universidad del Pals Vasco, 48080 Bilbao, Spain Received 15 June 1992; accepted for publication 22 September 1992 Communicatedby J.P. Vigier
We present a time dependent analytic solution for subsonic flow of gas around a masslessparticle in general relativity. The gas obeys a stiff adiabatic equation of state. Unlike the case of a plane infinitely extended gravitational wave, the motion of the gas particles disturbed by a plane fronted gravitational wave produced by a masslesspatlicle does not result in divergent focusing.
Recently Bondi and Pirani [1] have re-analysed the focusing properties of plane gravitational waves by considering the effect of a sandwich plane wave on a set of test particles. They have shown that stationary test panicles, however far apart they are initially, will all meet and collide in a finite time after being disturbed by the wave. This is well in accordance with the earlier studies by Penrose [ 2 ] of the focusing properties of plane gravitational waves. One may visualize this peculiar property of the plane gravitational wave yet in a different way by considering a flow of fluid panicles on the geometry of, say, an impulsive gravitational wave. The focusing will show itself in clustering of the fluid (gas) panicles up to an infinite number density behind the wave front. It is tempting to think that this pathological behaviour, the number density divergence, is the result of an infinite extent of the plane wave in all directions in its plane, and that this phenomenon does not happen with gravitational waves of finite extent. For otherwise our Universe would be a very unpleasant place to live in, full of singularities produced by gravitational waves either colliding among each other or focusing to an infinite density the gas or fluid panicles they encoumer on their way. The main purpose of this Letter is to show, by considering the exact analytic example of a gas flow around a massless particle, described in general relativity by a plane fronted localized gravitational wave, that indeed no such divergent pathologies happen in this situation.
We will exactly solve the hydrodynamical equations of motion for the adiabatic gas flow around a massless particle in the test fluid approximation in which the fluid's own gravitational field is neglected. In this approximation the fluid serves to probe the geometry of the gravitational wave. We will be mainly concerned with the behaviour of the number density of the fluid particles in question which will serve as an indicator of what happens with the fluid. To start with, we consider a gas of particles with the adiabatic equation of state p = yp. The Bianchi identities, T ~ " =0, reduce for such a fluid to ( p + p )u~, + p,~,u~'=O ,
( 1)
and (p+ p ) u ~ u ~ + pa, +p,~u~u~, = 0 ,
(2)
which are the conservation and Euler equations respectively. Restricting ourselves to potential flow it is customary to define the velocity potential a.~,: % Uu = x/_~vtr,P '
(3)
so that the evolution, the sound wave, equation for a becomes tr:uu- 21 7 - l [ln(_a,~tr.~)].,,aa,=0"
(4)
For simplicity, and in order to derive an analytic solution, we shall consider the stifffluid case, p=p. This
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
3
Volume 171, number 1,2
PHYSICS LETTERSA
choice of the adiabatic index 7 simplifies eq. (4) considerably and reduces it to a linear Klein-Gordon equation for the velocity potential,
a:~=0.
(5)
The number density, energy density and the pressure of such a fluid are given by n2 = p = p =
(6)
- a . ~ t r ,~ .
The background space time produced by a massless particle [3,4], a "photon", moving in the negative z direction and situated at r = 0 , may be thought of as composed by two different fiat regions, d s 2 - d u d v + d r 2 + r 2 d~ 2 ,
for u < 0 ,
cr.rr U tr ~ 4tT'uv+ (1 - - 4 p u / r 2 ) 2 --64p2 r 2 l -- 16p2u2/r 4 1 -- 1 6 p u / r 2 - 16p2u2/r 4 + r(l_4pu/r2)2(l_16p2uE/r4
2 ,
a = ½E( - v + u ) + ~ ( r , u ) ,
(7)
Here p is a constant associated with the "photon" linear momentum. We have chosen the coordinates so that the metric is continuous across the shock surface u = 0 [ 5 ]. Note that unlike the plane waves considered by Bondi and Pirani [ 1 ] the plane fronted ( p - p ) gravitational wave described by eq. (7) is of finite extent and localized. We shall choose our initial conditions for the fluid in such a way that the fluid is at rest before the shock wave produced by the particle strikes it. This in turn corresponds to the following choice of the velocity potential in the region u < 0, tr=-½Et=½E(-v+u)
,
(8)
where the constant E represents the initial number density of the fluid. We are now interested in what happens with the fluid afterwards (u > 0). To do so we shall solve the evolution equation (5) in the region u > 0 and demand the continuity of the velocity potential across the shock surface. Since for large r the metric is again explicitly Minkowskian we shall additionally impose relaxation of the fluid flow for r>> 1 and constant u > 0. In other words the fluid should have the same velocity potential at large r as for u < 0. Assuming no dependence on the angular coordinate 0, the evolution equation (5) for the velocity potential in the region u > 0 is
(9)
(10)
with the boundary conditions given by .(2(oo, u ) = O ,
~(r,O)=O.
(11)
Substituting eq. ( 1O) into eq. (9) and introducing a new variable z - u / r 2, we get the following equation for £2, 1 - z 2 + 2 z _,
for u > 0 .
) o'r=0
We shall consider solutions to eq. (9) in the following form,
= d u dr-t- ( 1 - 4 p u / r 2 ) 2 dr 2 + ( 1 +4pu/r2)2r 2 d O
30 November 1992
a"+
(12)
2Ep(l-z) =
z(l+z)
Here a prime denotes the derivative with respect to the variable z. Equation (12) can be integrated immediately and by adjusting the integration constant to the boundary conditions ( l 1 ) we get f2=-2Ep[21n(l+4pu/r2)-4pu/r
2] .
(13)
The velocity potential is then a=½E(-v+u)
+ 2 E p [ - 2 In( 1 + 4 p u / r 2) + 4 p u / r 2 ]
(14)
and the number density of the fluid particles is given by n 2 = p = E 2 ( r 2 + 4 p u + 4 p r) ( r 2 + 4 p u - 4 p r ) (r2+4pu) 2
(15)
We first observe that the solution becomes unphysical in the region 2p(l - x / 1 - u / p ) u
< r < 2p( 1 + x / l - u / p ) , (16)
where the energy density of the gas particles becomes negative. To see that one should not be preoccupied by this behaviour of the fluid we shall insert the physical constants omitted in the presentation. Inserting the fundamental constants we obtain for the parameter p:
Volume 171, number 1,2 Ge
Gh
PHYSICS LETTERS A
l~
/ 7 = C4 -- C3~" ~ T '
(17)
where e and 2 are the " p h o t o n " energy and the wavelength respectively. G, c, h and lp are the gravity constant, the speed o f light, the Planck constant and the Planck length. Thus the unphysical behaviour o f the test fluid under consideration is confined to the region with a length-scale At. Ar < 4lg / 2 < 41p ~ lp .
(18)
Since one does not expect from classical general relativity to work properly at Planck scales, it is easy to discard this region. It is amusing, however, that even at the classical level the fluid flow ceases to be a potential flow, i.e. irrotational, in the vicinity o f the massless particle in a region with Planck length dimensions. One may only speculate as to what happens with our solution on these scales. From the hydrodynamical point o f view it could be that the rotational turbulence develops introducing a confined vorticity into the fluid behaviour. However, at this point it is better to discard this region from further considerations.
30 November 1992
Returning now to the behaviour o f the fluid number density outside this region we see that the particle number density of the gas given by eq. ( 15 ) is a bounded function in space and time with the bound given by n 2 < E 2. In fig. I we have displayed the evolution o f the number density of the fluid. We can see that near r = 0, just after the massless particle strikes the fluid, the n u m b e r density o f the fluid particles drops and then tends to the initial value with time. Far away from r = 0 the number density tends rapidly towards the initial value. At no t i m e the number density o f the particles diverges indicating that unlike for the case o f infinitely extended plane gravitational waves the divergent focusing does not occur. Before closing, we should like to mention that recently Shapiro [ 6 ] has studied exact steady-state solutions for the flow of gas with a stiff equation o f state around the Schwarzschild black hole. Since the mettic for the massless particle given by eq. (7) is the limiting behaviour o f the Schwarzschild geometry in the case where the black hole is boosted with the speed o f light [ 3-5 ] it would be intex'esting in future to study as to whether the exact time-dependent solutions for the gas flow around the black hole tend
.8 .6
2
Fig. 1. The behaviour of the gas particle denshy n 2 after being struck by a gravitational wave produced by a m a ~ e ~ particle in terms of
the Cartesian coordinates x and z for y= 0 and t= - 3. The particle is m o v i n g in the negative z direction and is situated at z= -- 3, x= 0 (u = 0, r= 0). The ranges of the coordinales x and z are - 10 < x < 10, - - 3 < z < 1 and the parameter/7= 1. The initial density number is taken to be I.
Volume 171, number 1,2
PHYSICS LETTERS A
30 November 1992
to the s o l u t i o n d e s c r i b e d in this paper.
References
T h i s w o r k was s u p p o r t e d by the C I C Y T grant no. PS90-0093.
[ 1] H. Bondi and F.A.E. Pirani, Proc. R. Soc. A 421 (1989) 395. [2] R. Penrose, Rev. Mod. Phys. 37 (1965) 215. [ 3 ] P.C. Aichelburg and R.U. Sex1,Gen. Relativ. Gravit. 2 ( 1971 ) 303. [4] W.B. Bonnor, Commun. Math. Phys. 13 (1969) 163. [5] P.D. d'Eath, Phys. Rev. D 18 (1978) 990. [6] S.L. Shapiro, Phys. Rev. D 39 (1989) 2839.
PHYSICS LETTERS A
Potential flow around a massless particle in general relativity A. Feinstein a n d J. Ib~tfiez Departamento de Fisica Teorica, Universidad del Pals Vasco, 48080 Bilbao, Spain Received 15 June 1992; accepted for publication 22 September 1992 Communicatedby J.P. Vigier
We present a time dependent analytic solution for subsonic flow of gas around a masslessparticle in general relativity. The gas obeys a stiff adiabatic equation of state. Unlike the case of a plane infinitely extended gravitational wave, the motion of the gas particles disturbed by a plane fronted gravitational wave produced by a masslesspatlicle does not result in divergent focusing.
Recently Bondi and Pirani [1] have re-analysed the focusing properties of plane gravitational waves by considering the effect of a sandwich plane wave on a set of test particles. They have shown that stationary test panicles, however far apart they are initially, will all meet and collide in a finite time after being disturbed by the wave. This is well in accordance with the earlier studies by Penrose [ 2 ] of the focusing properties of plane gravitational waves. One may visualize this peculiar property of the plane gravitational wave yet in a different way by considering a flow of fluid panicles on the geometry of, say, an impulsive gravitational wave. The focusing will show itself in clustering of the fluid (gas) panicles up to an infinite number density behind the wave front. It is tempting to think that this pathological behaviour, the number density divergence, is the result of an infinite extent of the plane wave in all directions in its plane, and that this phenomenon does not happen with gravitational waves of finite extent. For otherwise our Universe would be a very unpleasant place to live in, full of singularities produced by gravitational waves either colliding among each other or focusing to an infinite density the gas or fluid panicles they encoumer on their way. The main purpose of this Letter is to show, by considering the exact analytic example of a gas flow around a massless particle, described in general relativity by a plane fronted localized gravitational wave, that indeed no such divergent pathologies happen in this situation.
We will exactly solve the hydrodynamical equations of motion for the adiabatic gas flow around a massless particle in the test fluid approximation in which the fluid's own gravitational field is neglected. In this approximation the fluid serves to probe the geometry of the gravitational wave. We will be mainly concerned with the behaviour of the number density of the fluid particles in question which will serve as an indicator of what happens with the fluid. To start with, we consider a gas of particles with the adiabatic equation of state p = yp. The Bianchi identities, T ~ " =0, reduce for such a fluid to ( p + p )u~, + p,~,u~'=O ,
( 1)
and (p+ p ) u ~ u ~ + pa, +p,~u~u~, = 0 ,
(2)
which are the conservation and Euler equations respectively. Restricting ourselves to potential flow it is customary to define the velocity potential a.~,: % Uu = x/_~vtr,P '
(3)
so that the evolution, the sound wave, equation for a becomes tr:uu- 21 7 - l [ln(_a,~tr.~)].,,aa,=0"
(4)
For simplicity, and in order to derive an analytic solution, we shall consider the stifffluid case, p=p. This
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
3
Volume 171, number 1,2
PHYSICS LETTERSA
choice of the adiabatic index 7 simplifies eq. (4) considerably and reduces it to a linear Klein-Gordon equation for the velocity potential,
a:~=0.
(5)
The number density, energy density and the pressure of such a fluid are given by n2 = p = p =
(6)
- a . ~ t r ,~ .
The background space time produced by a massless particle [3,4], a "photon", moving in the negative z direction and situated at r = 0 , may be thought of as composed by two different fiat regions, d s 2 - d u d v + d r 2 + r 2 d~ 2 ,
for u < 0 ,
cr.rr U tr ~ 4tT'uv+ (1 - - 4 p u / r 2 ) 2 --64p2 r 2 l -- 16p2u2/r 4 1 -- 1 6 p u / r 2 - 16p2u2/r 4 + r(l_4pu/r2)2(l_16p2uE/r4
2 ,
a = ½E( - v + u ) + ~ ( r , u ) ,
(7)
Here p is a constant associated with the "photon" linear momentum. We have chosen the coordinates so that the metric is continuous across the shock surface u = 0 [ 5 ]. Note that unlike the plane waves considered by Bondi and Pirani [ 1 ] the plane fronted ( p - p ) gravitational wave described by eq. (7) is of finite extent and localized. We shall choose our initial conditions for the fluid in such a way that the fluid is at rest before the shock wave produced by the particle strikes it. This in turn corresponds to the following choice of the velocity potential in the region u < 0, tr=-½Et=½E(-v+u)
,
(8)
where the constant E represents the initial number density of the fluid. We are now interested in what happens with the fluid afterwards (u > 0). To do so we shall solve the evolution equation (5) in the region u > 0 and demand the continuity of the velocity potential across the shock surface. Since for large r the metric is again explicitly Minkowskian we shall additionally impose relaxation of the fluid flow for r>> 1 and constant u > 0. In other words the fluid should have the same velocity potential at large r as for u < 0. Assuming no dependence on the angular coordinate 0, the evolution equation (5) for the velocity potential in the region u > 0 is
(9)
(10)
with the boundary conditions given by .(2(oo, u ) = O ,
~(r,O)=O.
(11)
Substituting eq. ( 1O) into eq. (9) and introducing a new variable z - u / r 2, we get the following equation for £2, 1 - z 2 + 2 z _,
for u > 0 .
) o'r=0
We shall consider solutions to eq. (9) in the following form,
= d u dr-t- ( 1 - 4 p u / r 2 ) 2 dr 2 + ( 1 +4pu/r2)2r 2 d O
30 November 1992
a"+
(12)
2Ep(l-z) =
z(l+z)
Here a prime denotes the derivative with respect to the variable z. Equation (12) can be integrated immediately and by adjusting the integration constant to the boundary conditions ( l 1 ) we get f2=-2Ep[21n(l+4pu/r2)-4pu/r
2] .
(13)
The velocity potential is then a=½E(-v+u)
+ 2 E p [ - 2 In( 1 + 4 p u / r 2) + 4 p u / r 2 ]
(14)
and the number density of the fluid particles is given by n 2 = p = E 2 ( r 2 + 4 p u + 4 p r) ( r 2 + 4 p u - 4 p r ) (r2+4pu) 2
(15)
We first observe that the solution becomes unphysical in the region 2p(l - x / 1 - u / p ) u
< r < 2p( 1 + x / l - u / p ) , (16)
where the energy density of the gas particles becomes negative. To see that one should not be preoccupied by this behaviour of the fluid we shall insert the physical constants omitted in the presentation. Inserting the fundamental constants we obtain for the parameter p:
Volume 171, number 1,2 Ge
Gh
PHYSICS LETTERS A
l~
/ 7 = C4 -- C3~" ~ T '
(17)
where e and 2 are the " p h o t o n " energy and the wavelength respectively. G, c, h and lp are the gravity constant, the speed o f light, the Planck constant and the Planck length. Thus the unphysical behaviour o f the test fluid under consideration is confined to the region with a length-scale At. Ar < 4lg / 2 < 41p ~ lp .
(18)
Since one does not expect from classical general relativity to work properly at Planck scales, it is easy to discard this region. It is amusing, however, that even at the classical level the fluid flow ceases to be a potential flow, i.e. irrotational, in the vicinity o f the massless particle in a region with Planck length dimensions. One may only speculate as to what happens with our solution on these scales. From the hydrodynamical point o f view it could be that the rotational turbulence develops introducing a confined vorticity into the fluid behaviour. However, at this point it is better to discard this region from further considerations.
30 November 1992
Returning now to the behaviour o f the fluid number density outside this region we see that the particle number density of the gas given by eq. ( 15 ) is a bounded function in space and time with the bound given by n 2 < E 2. In fig. I we have displayed the evolution o f the number density of the fluid. We can see that near r = 0, just after the massless particle strikes the fluid, the n u m b e r density o f the fluid particles drops and then tends to the initial value with time. Far away from r = 0 the number density tends rapidly towards the initial value. At no t i m e the number density o f the particles diverges indicating that unlike for the case o f infinitely extended plane gravitational waves the divergent focusing does not occur. Before closing, we should like to mention that recently Shapiro [ 6 ] has studied exact steady-state solutions for the flow of gas with a stiff equation o f state around the Schwarzschild black hole. Since the mettic for the massless particle given by eq. (7) is the limiting behaviour o f the Schwarzschild geometry in the case where the black hole is boosted with the speed o f light [ 3-5 ] it would be intex'esting in future to study as to whether the exact time-dependent solutions for the gas flow around the black hole tend
.8 .6
2
Fig. 1. The behaviour of the gas particle denshy n 2 after being struck by a gravitational wave produced by a m a ~ e ~ particle in terms of
the Cartesian coordinates x and z for y= 0 and t= - 3. The particle is m o v i n g in the negative z direction and is situated at z= -- 3, x= 0 (u = 0, r= 0). The ranges of the coordinales x and z are - 10 < x < 10, - - 3 < z < 1 and the parameter/7= 1. The initial density number is taken to be I.
Volume 171, number 1,2
PHYSICS LETTERS A
30 November 1992
to the s o l u t i o n d e s c r i b e d in this paper.
References
T h i s w o r k was s u p p o r t e d by the C I C Y T grant no. PS90-0093.
[ 1] H. Bondi and F.A.E. Pirani, Proc. R. Soc. A 421 (1989) 395. [2] R. Penrose, Rev. Mod. Phys. 37 (1965) 215. [ 3 ] P.C. Aichelburg and R.U. Sex1,Gen. Relativ. Gravit. 2 ( 1971 ) 303. [4] W.B. Bonnor, Commun. Math. Phys. 13 (1969) 163. [5] P.D. d'Eath, Phys. Rev. D 18 (1978) 990. [6] S.L. Shapiro, Phys. Rev. D 39 (1989) 2839.