*Astron. Astrophys. 326, 99-107 (1997)*
## Appendix A: expression of *dS*
The elemental area *dS* of a parallelogram defined by 2
vectors can be expressed, in curvilinear
coordinates, in a general form given by differential geometry.
Locally, we can suppose the space to be flat and characterized by 3
vectors . The projections of *dS* on each
plan are given by the antisymetrical tensor
:
In a 3 space, we rather use the vector ,
dual of , defined by:
is the determinant of the space metrics and
is the Levi-Civita tensor. We can now obtain
the expression of the surface *dS* modulus of
:
By setting the 4-tensor *G* which coefficients are defined by:
we can re-write the expression of the elementary surface *dS*
(sum on repeated indices):
## Appendix B: Kerr metric case. Expression of *dS* in the plan
In Kerr metrics, the space metric tensor is diagonal
To obtain the expression of the elementary surface *dS* in the
plan of the disk at radial coordinate
*r*, we have to take in Eq. (A6). One
gets then:
As a matter of interest, the metrics coefficients
we use here, are calculated in the
corresponding locally non-rotating frame . In
the plane of the accretion disk, only the
metrics coefficient differs from the one of
Boyer-Lindquist coordinate frame (Eq. (1)). Thus, in frame
, and
are equal to:
A ring surface is obtain after integration of Eq. (B3) with respect
to , that is:
© European Southern Observatory (ESO) 1997
Online publication: April 20, 1998
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