5.2 Mass Formulae5 Applications of the Coset 5 Applications of the Coset

5.1 The Mazur Identity 

In the presence of a second Killing field, the EM equations (32Popup Equation) experience further, considerable simplifications, which will be discussed later. In this section we will not yet require the existence of an additional Killing symmetry. The Mazur identity [133Jump To The Next Citation Point In The Article], which is the key to the uniqueness theorem for the Kerr-Newman metric [131Jump To The Next Citation Point In The Article], [132], is a consequence of the coset structure of the field equations, which only requires the existence of one Killing field. Popup Footnote

In order to obtain the Mazur identity, one considers two arbitrary hermitian matrices, tex2html_wrap_inline4247 and tex2html_wrap_inline4249 . The aim is to compute the Laplacian (with respect to an arbitrary metric tex2html_wrap_inline3837) of the relative difference tex2html_wrap_inline4253, say, between tex2html_wrap_inline4249 and tex2html_wrap_inline4247,

  equation847

It turns out to be convenient to introduce the current matrices tex2html_wrap_inline4259 and tex2html_wrap_inline4261, and their difference tex2html_wrap_inline4263, where tex2html_wrap_inline4265 denotes the covariant derivative with respect to the metric under consideration. Using tex2html_wrap_inline4267, the Laplacian of tex2html_wrap_inline4253 becomes

displaymath878

For hermitian matrices one has tex2html_wrap_inline4271 and tex2html_wrap_inline4273, which can be used to combine the trace of the first two terms on the RHS of the above expression. One easily finds

  equation913

The above expression is an identity for the relative difference of two arbitrary hermitian matrices. If the latter are solutions of a non-linear tex2html_wrap_inline3625 -model with action tex2html_wrap_inline4277, then their currents are conserved [see Eq. (32Popup Equation)], implying that the second term on the RHS vanishes. Moreover, if the tex2html_wrap_inline3625 -model describes a mapping with coset space tex2html_wrap_inline4281, then this is parametrized by positive hermitian matrices of the form tex2html_wrap_inline4283 . Popup Footnote Hence, the ``on-shell'' restriction of the Mazur identity to tex2html_wrap_inline3625 -models with coset tex2html_wrap_inline4281 becomes

  equation942

where tex2html_wrap_inline4289 .

Of decisive importance to the uniqueness proof for the Kerr-Newman metric is the fact that the RHS of the above relation is non-negative. In order to achieve this one needs two Killing fields: The requirement that tex2html_wrap_inline4215 be represented in the form tex2html_wrap_inline4293 forces the reduction of the EM system with respect to a space-like Killing field; otherwise the coset is tex2html_wrap_inline4295, which is not of the desired form. As a consequence of the space-like reduction, the three-metric tex2html_wrap_inline3837 is not Riemannian, and the RHS of Eq. (35Popup Equation) is indefinite, unless the matrix valued one-form tex2html_wrap_inline4299 is space-like. This is the case if there exists a time-like Killing field with tex2html_wrap_inline4301, implying that the currents are orthogonal to k : tex2html_wrap_inline4305 . The reduction of Eq. (35Popup Equation) with respect to the second Killing field and the integration of the resulting expression will be discussed in Sect. 6 .



5.2 Mass Formulae5 Applications of the Coset 5 Applications of the Coset

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
http://www.livingreviews.org/lrr-1998-6
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