Proceedings of the XXIII Spanish Relativity Meeting on Reference Frames and Gravitomagnetism: Valladolid, Spain, 6-9 September 2000 9812810021, 9789812810021

Table of contents :
COMMUNICATIONS ON REFERENCE FRAMES AND GRAVITOMAGNETISM ..............164
INVITED LECTURE AND COMMUNICATIONS ON OTHER TOPICS ..............225

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l!i7rT«awTmTaJ«iT

n T m i l tty R u m i ' J u [tr4(u,U)]

= P(U)-~[-P(U,u)}

= 7^ ( f w , u , c / ) A r ( f w , u , [ / )

(6)

which naturally defines the "relative centrifugal force". Both the relative velocity unit vector fields, i>(U,u) and —0{u,U), can be used in turn to introduce a "relative Frenet-Serret" procedure to define spatial frames (-z>(u,LO,^(fw,u,tf),#(fw,u,[/)) inLRSu and (P(t/,iO,^(fw,tf,u)>/%w,£/,u)) in LRSU satisfying the following relations dru

•P(fW,t7)

dru

= ll'[lC(fYr,u,U)Hu>

-N{fw,u,U)

U) + 7(fw,ii,C/)S(fw,u,l/)] ,

#(fw,u,l7) = - l ^ f w . u . t O - ^ f w . u . t / )

,

and D(ivi,U,u)

dru

V(fw,U,u) = lv[k(fWlU,u)HU,u)

D((v,,U,u) -j P(fw,[/,u) =

+ T(f Wi (/ iU )/3(f Wit / iU )] ,

-7^T(fw,C/,u)»7(fw,C/,u)

Using these frames to decompose the force equation a(U) = f{U) for the test particle U along the transverse directions J\({w 0; one has ei = cosxe? + sin\e- e , e 2 = dU/da = s i n h a n + coshae? = (sgnv)V(n,u) i e3 = —dei/dx = sinxe^ - cosxe^ .

(21)

The spacetime torsions are given by half the sign-reversed arclength derivatives along the polar coordinate directions in the acceleration plane, leading to an alternative expression for the Frenet-Serret angular velocity n=--dK/da

,

T2 = --Kdx/da

, w (FS ) = ^

Xu

~da

'

^

showing that the latter is orthogonal to the tangent to the a-parametrized acceleration hyperbola. 5.2.

Relative Frenet-Serret description of circular orbits

The relative Frenet-Serret curvatures and torsions reduce to

r

_ IkFsll

*-(fw,n,£/) — ~^~T

v

'

'

^ i

_n

/(fw,n,C/) — U, ,

1 cosh3 a d , a(U) x 2 s m h a da cosh2 a

T(fw,£/,n) = 0 •

(23)

The magnitude of the relative generalized centrifugal force is then 4w!n,a) = I t a n h a | | | W F S | | .

(24)

Apart from a reordering of its elements, the comoving relative FrenetSerret frame {[—i>(n,U)], Jv(fw,n,u)> #(fw,n,c/)} a t a given point on a circular orbit consists of the same vectors which belong to the Frenet-Serret frame defined along the acceleration hyperbola a(U) = a^(a), provided one identifies tangent vectors to the tangent space itself with tangent vectors by translation to the origin. Analogously the relative Frenet-Serret frame {v{U, n), jj(fw,[/,n), /%w,u,n)} consists of the same vectors which belong to the Frenet-Serret frame defined along the rescaled acceleration curve a(U)/ cosh (a), corresponding to the second derivative proper time gamma factor rescaling.

D. Bini, R. T. Jantzen

20

0 -1.57 -3.14

1-0.5

0

0.5

1

0T5

1

d

0.5

-1 - 0 . 5

0

-0.5

0.05

-0.05

-1 - 0 . 5

0

0.5

1

0.6

1

1.4

Figure 2: The following quantities are plotted versus the velocity v for the family of all circular orbits at the Boyer-Lindquist radial coordinate values r = 4 and 9 = 7r/3 for a Kerr black hole with a/Ai = 0.5: the polar coordinates in the acceleration plane: K (a) and x (b), the two torsions: n (c) and T2 (d), the magnitude of the Frenet-Serret angular velocity II^FSjII ( e ) a n d the acceleration curve (f).

6

Circular orbits in stationary axisymmetric spacetimes: Kerr spacetime, equatorial orbits

Now specialize the results for general circular orbits to the equatorial plane of the Kerr spacetime where3 9 = IT/2. One can introduce a unit vector e^ = —e^ mapping the Boyer-Lindquist coordinates directions (t,r,9,(j>) to those of the associated flat spacetime cylindrical coordinates (t,r,