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Astron. Astrophys. 326, 99-107 (1997)

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Appendix A: expression of dS

The elemental area dS of a parallelogram defined by 2 vectors [FORMULA] 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 [FORMULA]. The projections of dS on each plan [FORMULA] are given by the antisymetrical tensor [FORMULA]:

[EQUATION]

In a 3 space, we rather use the vector [FORMULA], dual of [FORMULA], defined by:

[EQUATION]

[FORMULA] is the determinant of the space metrics and [FORMULA] is the Levi-Civita tensor. We can now obtain the expression of the surface dS modulus of [FORMULA]:

[EQUATION]

By setting the 4-tensor G which coefficients are defined by:

[EQUATION]

we can re-write the expression of the elementary surface dS (sum on repeated indices):

[EQUATION]

Appendix B: Kerr metric case. Expression of dS in the plan [FORMULA]

In Kerr metrics, the space metric tensor is diagonal

[EQUATION]

To obtain the expression of the elementary surface dS in the plan [FORMULA] of the disk at radial coordinate r, we have to take [FORMULA] in Eq. (A6). One gets then:

[EQUATION]

As a matter of interest, the metrics coefficients [FORMULA] we use here, are calculated in the corresponding locally non-rotating frame [FORMULA]. In the plane [FORMULA] of the accretion disk, only the [FORMULA] metrics coefficient differs from the one of Boyer-Lindquist coordinate frame (Eq. (1)). Thus, in frame [FORMULA], [FORMULA] and [FORMULA] are equal to:

[EQUATION]

A ring surface is obtain after integration of Eq. (B3) with respect to [FORMULA], that is:

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: April 20, 1998
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