Nuclear Physics B (Proc. Suppl.) 6 (1989) 367-369 367 North-Holland, Amsterdam

KINETIC THEORY IN MAXIMAL-ACCELERATION INVARIANT PHASE SPACE

Howard E. Brandt

Harry Diamond Laboratories, Adelphi, Maryland 20783, and University of Maryland, College Park, Maryland 20742

A vanishing directional derivative of a scalar field along particle trajectories in maximal accel- eration invariant phase space is identical in form to the ordinary covariant Vlasov equation in curved spacetime in the presence of both gravitational and nongravitational forces. A natural foundation is thereby provided for a covariant kinetic theory of particles in maximal-acceleration invariant phase space.

1. INTRODUCTION

The differential geometry and gauge structure

of maximal-acceleration invariant phase space

was recently developed in terms of a simple an-

holonomic basis in which the line element splits

naturally into block-diagonal form with the or-

dinary spacetime metric, g~9, in both the base

manifold and the fiber, and the gauge structure

on the fiber bundle is made manifest. 1-4 The

anholonomic basis and dual basis vectors are

given by

~uV 3/3v, p~13/3vU}

and

{~A} H {~e, ~a} H {dx H, O0(dv u + rUsvadx8)} .

Here lower case Greek and Latin indices refer to

horizontal vectors in the spacetime submanifold

and vertical vectors in the fiber submanifold,

respectively. The symmetric spacetime affine

connection is FUe8. The phase space coordinates

are defined by the set {xA; A=O,I,..7} = {xU,xm;

~=O,i,2,3; m=4,5,6,7} E (x~,00v~; ~=O,i,2,3}.

Here the Einstein sum convention for repeated in-

dices is used throughout, Greek indices range

from 0 to 3, lower case Latin indices range from

4 to 7, and upper case Latin indices range from H

0 to 7. Also x are the ordinary spacetime coor-

dinates, v H = dxH/ds are the four-velocity coor-

dinates, ds is the infinitesimal spacetime inter-

c21a0 is the minimum radius of val, O0 curva- /

ture of world lines, a0 = 2~e(c7/MG) " ~ is the

maximal proper acceleration relative to the vac- 5 uum , c is the velocity of light in vacuum, G is

the universal gravitational constant, M is

Planck's constant divided by 27 , and a is a di-

mensionless number of order unity and determined

by the equation of state of radiation at the ex-

treme density of one quantum per Planck volume

The explicit line element of maximal-accelera-

tion invariant phase space is

dg 2 = g~ dx~dx ~ (1

2 + O0gu~(dv + rUasvadxB)(dv~ + F~y6vYdx6 ).

In the present work we show that the manifestly

covariant general relativistic Vlasov equation

follows naturally from the vanishing of the di-

rectional derivative df/do of a scalar field f,

the distribution function, along the particle

trajectories in maximal-acceleration invariant

phase space.

2. COVARIANT VLASOV EQUATION

Consider a vanishing value of the gradient

one-form ~f on the tangent vector d/dg, namely,

df/d~ = <~f,d/da> = 0. (2)

In terms of the stated basis vectors, one has

dg<~f,d/dg> = ~A'~Af = [dx~(3/3x H - r8 v~3/3v ~)

+ (dv ~ + rU 8vadxS)3/3v~]f, (3)

0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

368 H.E. Brandt / Kinetic theory in maximal-acceleration invariant phase space

or

<~f,d/do> = (ds/do)(v~/3x ~ + dv~/ds ~/$vU)f.(4)

In the presence of nongravitational forces F D,

proper acceleration is nonvanishing, and one has,

for the covariant four-acceleration of a parti-

cle of rest mass m,

+ rUBvav6 = F;/m . (5) a ~ dvU/ds E

Substituting Eq.(5) in Eq.(4) yields

<df,d/do> = (ds/mdo)[p~/ax ~

+ (mF ~ - F~ ~p~pB)3/3p~]f, (6)

where pB = my ~ is the particle four-momentum.

Finally substituting Eq.(6) in Eq.(2), we obtain

[ p ~ 8 / a x ~ + (mF ~ - F ~ a s p a p B ) a / a p ~ ] f = 0 , (7) which i s i d e n t i c a l i n form to t h e m a n i f e s t l y c o -

v a r i a n t g e n e r a l r e l a t i v i s t i c V lasov e q u a t i o n in

t h e p r e s e n c e o f an e x t e r n a l n o n g r a v i t a t i o n a l

f o r c e F ~ . 6 A c o n g r u e n c e i s formed in max imal -

a c c e l e r a t i o n i n v a r i a n t phase s p a c e by t h e c u r v e s

{ x e ( o ) , P 0 v a ( o ) } c o r r e s p o n d i n g to t h e phase f l ow

g e n e r a t e d by t h e h i o u v i l l e v e c t o r f o r t h e com-

b i n e d g r a v i t a t i o n a l and n o n g r a v i t a t i o n a l f i e l d s .

No tewor thy i s t h a t t h e c o o r d i n a t e s x ~ and

P0v ~ of maximal-acceleration invariant phase

space, appearing in Eq.(3) and labeling the man-

ifold in which the kinetic equation is embedded,

are universal and do not refer to a particular

particle mass• The spacetime-four-velocity man-

ifold is universal• Also, the universal parame-

ter P0 makes no explicit appearance in the kinet-

ic equation (7) because of the cancellation oc-

curring in Eq.(3).

The four-force F ~ and spacetime affine connec-

tion F~ appearing in Eq.(7) must be consistent-

ly determined in terms of action principles

based on the differential geometry of maximal

acceleration invariant phase space. A challenge

also presents itself to explicitly incorporate

within the present formalism other generally

known quantum corrections to the Einstein-

Vlasov equations resulting from covariant exten-

sion of the Wigner transformation to curved 7

spacetime.

3. DIFFERENTIAL GEOMETRY

In the anholonomic basis, the connection co-

efficients I for the bundle manifold can be re-

written in terms of the gauge fields and sub- 2

manifold connections as follows:

( 8 ) r~ ~ = { ~ } , (8)

(8 ) r~ = I (F ~ + I b~), (9) ~b = (8) tUba ~ ba + ff~ba (8)FUab = -½T(aUb) , (10)

(8)FmaB ½(Fma6 (a B) ) (11)

(8)Fmab = ½(Tb m + Tm b) , (12)

(8)Fmba = {mbe } + }(Tba m - Tm b) , (13)

= ~m (8)Fmab ab (14)

Here F a i s t h e gauge c u r v a t u r e f i e l d ,

Fauu = D[~ Aav] ' (15)

indices have been raised and lowered with the

spacetime metric tensor, A ~ is the gauge poten-

tial

A s (16) = PovAF~k~ ,

and the operator D i s

D = 3/3x ~ - p~iA~ 3/3vB (17)

which corresponds to the horizontal basis vec-

tors ~ . Noteworthy :is that the contraction of

the four-momentum with the operator D is a

Liouville vector field. The notations

T ' [ . ( U . . ~ ) . . . . . . E T ' ' . i j . . v " + T ' [ . v . . ~ . . and

T ' " . . . . . . [ ; . . ~ ] ~ T ' . ~ . v. - T ' . . v . . ~ . . a r e

employed , and any lower c a s e L a t i n i n d e x , a , a p -

p e a r i n g e x p l i c i t l y in a s p a c e t i m e t e n s o r or

c o n n e c t i o n i s d e f i n e d to be a - 4 i m p l i c i t l y . AI-

fie so i s t h e C h r i s t o f f e l c o n n e c t i o n d e f i n e d on U~ f o u r - v e l o c i t y s p a c e

-1 aX (U ffa ~P0 g ( 3 / 3 v g'v'a ) - 3/3VXguv) " (18) E ov

In t h e p r e s e n t work and in Re f . 2 i t i s n o t a s -

sumed t h a t t h e symmet r i c s p a c e t i m e a f f i n e

H.E. Brandt / Kinetic theory in maximal-acceleration invariant phase space 369

connection F p is simply the ordinary Christof-

fel connectionB{UaB}, ~ however in the more re-

strictive case in which they are equal, then the

expressions given here reduce directly to those

of Ref. I. The field T8 is given by

T 8 { B } _ (19)

where ~B is one of the nonvanishing commuta-

tor coefficient fields given by

~B a : ool3/~v~AB u . (20)

The operator pol~/3v~ corresponds to the verti-

cal basis vectors ~ . a The scalar curvature of the bundle reduces

to 1

(8)R = R + VR - !~BY= 4 , ~~8Y (21)

D I IT(aBY)~4 *(aBy) - TaYaTBY8 - 2-D~x~T 8a

!~(~87)~4 . . . . 2 D ~ la (aBy) H~YaffByB DP0v B B~

where R is the ordinary Riemann curvature sca-

lar of the spacetime submanifold, expressed in

terms of the Christoffel connection; and VR is

the scalar curvature of the fiber submanifold

defined on four-velocity space, namely

VR e gU~(o~l~/~v[~Ha \~] + I~ [UlB B]) . (22)

Also in Eq. (21), the spacetime covariant deriv-

ative is

DDx B T ~ : (~l~xBT~pa - {IuB}T~I~) , (23)

and analogously in the four-velocity subspace

D = (P~l~/~v~ (24) B u~ - R X u B I a ~ ) " Dpov

A possible action for maximal-acceleration

invariant phase space with metric determinant G

is then given by

I = f d4xO~d4vcrG - (8)R . (25)

Possible field equations based on Eq.(25), and

particle equations of motion based on an action

determined by the line element Eq.(1) would

self-consistently determine the connection coef-

ficients F~ B and nongravitational four-force F ~

appearing in the covariant Vlasov equation.

4. CONCLUSIONS

The ordinary covariant Vlasov equation in

curved spacetime, in the presence of both gravi-

tational and nongravitational forces, follows

naturally by considering the vanishing direction-

al derivative of a scalar field along particle

trajectories in maximal-acceleration invariant

phase space. A natural foundation is thereby

provided for a covariant kinetic theory of parti-

cles in maximal-acceleration invariant phase

space. This formalism would have possible appli-

cations to problems involving the use of kinetic

theory to describe phenomena near black hole

event horizons or in the very early universe.

REFERENCES

i. H.E. Brandt, Differential Geometry and Gauge Structure of Maximal-Acceleration Invariant Phase Space, in: XVth International Colloqui- um on Group Theoretical Methods in Physics, ed. R. Gilmore (World Scientific, Singapore, 1987) pp. 569-577.

2. H.E. Brandt, Maximal Proper Acceleration and the Structure of Spacetime, invited paper to be presented at Fifth Marcel Grossmann Meet- ing on Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories, Perth, West Australia, 8-12 August, 1988.

3. H.E. Brandt, The Maximal-Acceleration Group, in; XIIIth International Colloquium on Group Theoretical Methods in Physics, ed. W.W. Zachary (World Scientific, Singapore, 1984) pp. 519-522.

4. H.E. Brandt, Maximal-Acceleration Invariant Phase Space, in: The Physics of Phase Space, eds. Y.S. Kim and W.W. Zachary (Springer Verlag, Berlin, 1987) pp. 414-416.

5. H.E. Brandt, Maximal Proper Acceleration Relative to the Vacuum, Lett. Nuovo Cimento 38 (1983) 522; 39 (1984) 192.

6. J.M. Stewart, Non-Equilibrium Relativistic Kinetic Theory (Springer Verlag, Berlin, 1971).

7. J. Winter, Wigner Transformation in Curved Spacetime and the Curvature Correction of the Vlasov Equation for Semiclassical Gravitating Systems, Phys. Rev. D 32 (1985) 1871.