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| In [[general relativity]], the '''Weyl metrics''' (named after the German-American mathematician [[Hermann Weyl]]) refer to the class of ''static'' and ''axisymmetric'' solutions to [[Einstein's field equation]]. Three members in the renowned [[Kerr-Newman metric|Kerr-Newman]] family solutions, namely the [[Schwarzschild metric|Schwarzschild]], nonextremal [[Reissner-Nordström metric|Reissner-Nordström]] and extremal Reissner-Nordström metrics, can be identified as Weyl-type metrics.
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| ==Standard Weyl metrics==
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| The Weyl class of solutions has the generic form<ref name=Weyl1>Jeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Chapter 10.</ref><ref name=Weyl2>Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt. ''Exact Solutions of Einstein's Field Equations''. Cambridge: Cambridge University Press, 2003. Chapter 20.</ref>
| | Potential buyers of Lancel see reviving the loss-making leather goods maker as a high-risk gamble that could take at least six or eight years to pay off, sources close to the matter say.<br><br>Facing a struggle to offload the business, Swiss parent Richemont (CFR.VX) would be ready to pay for two years of losses - up to an estimated 20 million euros ($28 million) - to entice bidders, the sources say.<br><br>So far private equity firms Change Capital and Lion Capital have expressed interest while Asian group Swire is looking to team up with a private equity firm to make a bid, according to the sources.<br>Another potential buyer, who has not made up his mind yet, is Jacques Veyrat, the former CEO of private French trading group Louis Dreyfus who used to run its Neuf Cegetel telecoms unit.<br><br>Veyrat left Louis Dreyfus in 2011 with a golden parachute of 270 million euros, according to the French press. He has since set up the Impala group, which invests in companies in distress.<br>Veyrat, Swire, Lion Capital and Change Capital declined to comment, as did Lancel and Richemont.<br>People close to the talks say the bidding process for Lancel, which had already been going on for months, could last several more weeks.<br><br>Lancel is run part-time by [http://Www.Wikipedia.org/wiki/Fabrizio Fabrizio] Cardinali, CEO of Dunhill, another Richemont brand.<br>Lancel has hired a new designer, Nicole Stulman, who was previously at Reed Krakoff, Hermes (HRMS.PA) and [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine Bags Outlet] and does not have a head of merchandising to work with since that person has left for Dunhill, sources close to the sale process said.<br><br>Other key people at the company are either on sick leave or have not been replaced, the sources said.<br>Lancel's difficulties come as the luxury sector as a whole remains resilient despite a slowdown in China, continuing to grow, although at a slower pace than in previous years.<br>PRIVATE EQUITY<br>The relationship between private equity and fashion brands can be difficult because the typical three- to five-year investment horizon may be too short to turn a label around.<br><br>Private equity deals are usually financed by debt, which puts pressure on the fashion brand when it needs to invest in marketing and new shops.<br>Lanvin, acquired by Taiwanese media magnate Shaw-Lan Wang from L'Oreal in 2001, took more than seven years to become profitable. TPG struggled for nine years to restructure the leather goods brand Bally before selling it in 2008.<br><br>There have been few successful revamps of struggling brands that are still going concerns. Carven, Balenciaga and Lanvin lay dormant for decades before they were acquired, allowing their new owners to make a fresh start.<br>Lancel was founded by accessories designer Ang�le Lancel in 1876, remaining in the hands of her descendants until it was acquired in the late 1970s by the Zorbibe brothers, who with then chief executive Sidney Toledano, now Dior CEO, launched the hit bucket-shaped Elsa bag and made Lancel a profitable business.<br><br><br>Industry executives say Lancel was making margins higher than Louis Vuitton - today's market leader - when Richemont bought it in 1997.<br>"All my girlfriends in the 1990s dreamt of having a Lancel bag," said Serge Carreira, who lectures on luxury at the Institut d'Etudes Politiques in Paris, known as "Sciences Po".<br><br>Lancel had not expanded internationally or creatively as its rivals did, he said.<br>Lancel makes more than 80 percent of its sales in France and in the year to end June, it had an operating loss of 10 million euros on revenues of 135 million euros, sources close to the matter said.<br>Richemont, the world's second biggest luxury group, does not publish separate sales and [http://www.Bing.com/search?q=profit+figures&form=MSNNWS&mkt=en-us&pq=profit+figures profit figures] for its brands, which include Cartier and Van Cleef & Arpels.<br><br>Longchamp, founded in 1948 and still owned by the founding Cassegrain family, has annual sales of 500 million euros.<br>"ELECTRIC SHOCK"<br>"What Lancel needs in an electric shock in terms of design, marketing, store concept and image," says Boston Consulting Group senior partner Jean-Marc Bellaiche.<br>Analysts say Lancel needs to enter the accessible segment of the luxury market, alongside Coach (COH.N), Michael Kors (KORS.N) and Longchamp with handbags costing 200-800 euros.<br><br>Lancel lost credibility at the high end of the luxury market by offering excessive discounts and putting out staid, old-fashioned designs, such as the Brigitte Bardot bag in 2010 - a 1970s style version of the bucket bag - and the document holder Isabelle Adjani bag in 2008.<br><br>"Lancel's products, while perfectly fine, look a bit dated and are not a must-buy," said Hugh Devlin, a consultant at legal firm Withers LLP, who specializes in luxury and fashion brands.<br><br>"With the right creative team and strategy, that could definitely change but whoever buys it will need to be prepared to take a risk."<br>Lancel clearly thinks it can make it as a luxury brand. A few months ago, it introduced an ostrich leather "L" bag costing 4,500 euros, emulating Gucci (PRTP.PA) and Louis Vuitton's (LVMH.PA) recent efforts to move upmarket.<br><br>Carreira disagrees. "You do not become a luxury brand because you sell a bag for 4,000 euros," he said.<br>To rebuild its name in the crowded accessories market, experts say Lancel should do some ready-to-wear, as Longchamp did, to stimulate traffic in shops and rejuvenate itself.<br>Another potential investor who looked at Lancel said it had suffered from under-investment and excessive management turnover.<br><br>Having had about a dozen chief executives since 1997, the company will now need to be given time by its new owners to implement long-term strategy, industry observers said.<br>(Editing by Giles Elgood) |
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| <math>(1)\quad ds^2=-e^{2\psi(\rho,z)}dt^2+e^{2\gamma(\rho,z)-2\psi(\rho,z)}(d\rho^2+dz^2)+e^{-2\psi(\rho,z)}\rho^2 d\phi^2\,, | |
| </math> | |
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| where <math>\psi(\rho,z)</math> and <math>\gamma(\rho,z)</math> are two metric potentials dependent on ''Weyl's canonical coordinates'' <math>\{\rho\,,z \}</math>. The coordinate system <math>\{t,\rho,z,\phi\}</math> serves best for symmetries of Weyl's spacetime (with two [[Killing vector fields]] being <math>\xi^t=\partial_t</math> and <math>\xi^\phi=\partial_\phi</math>) and often acts like [[cylindrical coordinates]],<ref name="Weyl1" /> but is ''incomplete'' when describing a [[black hole]] as <math>\{\rho\,,z \}</math> only cover the [[Event horizon|horizon]] and its exteriors.
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| Hence, to determine a static axisymmetric solution corresponding to a specific [[stress-energy tensor]] <math>T_{ab}</math>, we just need to substitute the Weyl metric Eq(1) into Einstein's equation (with c=G=1):
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| <math>(2)\quad R_{ab}-\frac{1}{2}Rg_{ab}=8\pi T_{ab}\,,</math> | |
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| and work out the two functions <math>\psi(\rho,z)</math> and <math>\gamma(\rho,z)</math>.
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| ==Reduced field equations for electrovac Weyl solutions==
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| One of the best investigated and most useful Weyl solutions is the electrovac case, where <math>T_{ab}</math> comes from the existence of (Weyl-type) electromagnetic field (without matter and current flows). As we know, given the electromagnetic four-potential <math>A_a</math>, the anti-symmetric electromagnetic field <math>F_{ab}</math> and the trace-free stress-energy tensor <math>T_{ab}</math> <math>(T=g^{ab}T_{ab}=0)</math> will be respectively determined by
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| <math>(3)\quad F_{ab}=A_{b\,;\,a}-A_{a\,;\,b}=A_{b\,,\,a}-A_{a\,,\,b}</math><br />
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| <math>(4)\quad T_{ab}=\frac{1}{4\pi}\,\Big(\, F_{ac}F_b^{\;c} -\frac{1}{4}g_{ab}F_{cd}F^{cd} \Big)\,,</math> | |
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| which respects the source-free covariant Maxwell equations:
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| <math>(5.a)\quad \big(F^{ab}\big)_{;\,b}=0\,,\quad F_{[ab\,;\,c]}=0\,.</math>
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| Eq(5.a) can be simplified to:
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| <math>(5.b)\quad \big(\sqrt{-g}\,F^{ab}\big)_{,\,b}=0\,,\quad F_{[ab\,,\,c]}=0</math>
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| in the calculations as <math>\Gamma^a_{bc}=\Gamma^a_{cb}</math>. Also, since <math>R=-8\pi T=0</math> for electrovacuum, Eq(2) reduces to | |
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| <math>(6)\quad R_{ab}=8\pi T_{ab}\,.</math><br /> | |
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| Now, suppose the Weyl-type axisymmetric electrostatic potential is <math>A_a=\Phi(\rho,z)[dt]_a</math> (the component <math>\Phi</math> is actually the [[Electromagnetic four-potential|electromagnetic scalar potential]]), and together with the Weyl metric Eq(1), Eqs(3)(4)(5)(6) imply that
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| <math>(7.a)\quad \nabla^2 \psi =\,(\nabla\psi)^2 +\gamma_{,\,\rho\rho}+\gamma_{,\,zz}</math><br />
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| <math>(7.b)\quad \nabla^2\psi =\,e^{-2\psi} (\nabla\Phi)^2 </math><br />
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| <math>(7.c)\quad \frac{1}{\rho}\,\gamma_{,\,\rho} =\,\psi^2_{,\,\rho}-\psi^2_{,\,z}-e^{-2\psi}\big(\Phi^2_{,\,\rho}-\Phi^2_{,\,z}\big) </math><br />
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| <math>(7.d)\quad \frac{1}{\rho}\,\gamma_{,\,z} =\,2\psi_{,\,\rho}\psi_{,\,z}- 2e^{-2\psi}\Phi_{,\,\rho}\Phi_{,\,z} </math><br /> | |
| <math>(7.e)\quad \nabla^2\Phi =\,2\nabla\psi \nabla\Phi\,,</math><br />
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| where <math>R=0</math> yields Eq(7.a), <math>R_{tt}=8\pi T_{tt}</math> or <math>R_{\varphi\varphi}=8\pi T_{\varphi\varphi}</math> yields Eq(7.b), <math>R_{\rho\rho}=8\pi T_{\rho\rho}</math> or <math>R_{zz}=8\pi T_{zz}</math> yields Eq(7.c), <math>R_{\rho z}=8\pi T_{\rho z}</math> yields Eq(7.d), and Eq(5.b) yields Eq(7.e). Here <math>\nabla^2 = \partial_{\rho\rho}+\frac{1}{\rho}\,\partial_\rho +\partial_{zz}</math> and <math>\nabla=\partial_\rho\, \hat{e}_\rho +\partial_z\, \hat{e}_z </math> are respectively the [[Laplace operator|Laplace]] and [[Gradient Operator|gradient]] operators. Moreover, if we suppose <math>\psi=\psi(\Phi)</math> in the sense of matter-geometry interplay and assume asymptotic flatness, we will find that Eqs(7.a-e) implies a characteristic relation that
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| <math>(7.f)\quad e^\psi =\,\Phi^2-2C\Phi+1\,.</math><br />
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| Specifically in the simplest vacuum case with <math>\Phi=0</math> and <math>T_{ab}=0</math>, Eqs(7.a-7.e) reduce to<ref name=Weyl4>R Gautreau, R B Hoffman, A Armenti. ''Static multiparticle systems in general relativity''. IL NUOVO CIMENTO B, 1972, '''7'''(1): 71-98.</ref>
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| <br />
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| <math>(8.a)\quad \gamma_{,\,\rho\rho}+\gamma_{,\,zz}=-(\nabla\psi)^2 </math><br />
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| <math>(8.b)\quad \nabla^2 \psi =0 </math><br />
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| <math>(8.c)\quad \gamma_{,\,\rho}=\rho\,\Big(\psi^2_{,\,\rho}-\psi^2_{,\,z} \Big) </math><br />
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| <math>(8.d)\quad \gamma_{,\,z}=2\,\rho\,\psi_{,\,\rho}\psi_{,\,z} \,.</math>
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| We can firstly obtain <math>\psi(\rho,z)</math> by solving Eq(8.b), and then integrate Eq(8.c) and Eq(8.d) for <math>\gamma(\rho,z)</math>. Practically, Eq(8.a) arising from <math>R=0</math> just works as a consistency relation or [[integrability condition]].
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| Unlike the nonlinear [[Poisson equation|Poisson's equation]] Eq(7.b), Eq(8.a) is the linear [[Laplace equation]]; that is to say, superposition of given vacuum solutions to Eq(8.a) is still a solution. This fact has a widely application, such as to analytically [[Distorted Schwarzschild metric|distort a Schwarzschild black hole]].
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| <div style="clear:both;width:65%;" class="NavFrame collapsed">
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| <div class="NavHead" style="background-color:#FFFFFF; text-align:left; font-size:larger;">Box A: Remarks on the electrovac field equation</div>
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| <div class="NavContent" style="text-align:left;">
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| We employed the axisymmetric Laplace and gradient operators to write Eqs(7.a-7.e) and Eqs(8.a-8.d) in a compact way, which is very useful in the derivation of the characteristic relation Eq(7.f). In the literature, Eqs(7.a-7.e) and Eqs(8.a-8.d) are often written in the following forms as well:
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| <math>(A.1.a)\quad \psi_{,\,\rho\rho}+\frac{1}{\rho}\psi_{,\,\rho}+\psi_{,\,zz}=\,(\psi_{,\,\rho})^2+(\psi_{,\,z})^2 +\gamma_{,\,\rho\rho}+\gamma_{,\,zz}</math><br />
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| <math>(A.1.b)\quad \psi_{,\,\rho\rho}+\frac{1}{\rho}\psi_{,\,\rho}+\psi_{,\,zz}=e^{-2\psi}\big(\Phi^2_{,\,\rho}+\Phi^2_{,\,z}\big)</math><br />
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| <math>(A.1.c)\quad \frac{1}{\rho}\,\gamma_{,\,\rho} =\,\psi^2_{,\,\rho}-\psi^2_{,\,z}-e^{-2\psi}\big(\Phi^2_{,\,\rho}-\Phi^2_{,\,z}\big) </math><br />
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| <math>(A.1.d)\quad \frac{1}{\rho}\,\gamma_{,\,z} =\,2\psi_{,\,\rho}\psi_{,\,z}- 2e^{-2\psi}\Phi_{,\,\rho}\Phi_{,\,z} </math><br />
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| <math>(A.1.e)\quad \Phi_{,\,\rho\rho}+\frac{1}{\rho}\Phi_{,\,\rho}+\Phi_{,\,zz} =\,2\psi_{,\,\rho}\Phi_{,\,\rho} +2\psi_{,\,z}\Phi_{,\,z} </math>
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| and
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| <br />
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| <math>(A.2.a)\quad (\psi_{,\,\rho})^2+(\psi_{,\,z})^2=-\gamma_{,\,\rho\rho}-\gamma_{,\,zz} </math><br />
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| <math>(A.2.b)\quad \psi_{,\,\rho\rho}+\frac{1}{\rho}\psi_{,\,\rho}+\psi_{,\,zz} =0 </math><br />
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| <math>(A.2.c)\quad \gamma_{,\,\rho}=\rho\,\Big(\psi^2_{,\,\rho}-\psi^2_{,\,z} \Big) </math><br />
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| <math>(A.2.d)\quad \gamma_{,\,z}=2\,\rho\,\psi_{,\,\rho}\psi_{,\,z} \,.</math>
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| </div>
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| </div>
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| <div style="clear:both;width:65%;" class="NavFrame collapsed">
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| <div class="NavHead" style="background-color:#FFFFFF; text-align:left; font-size:larger;">Box B: Derivation of the Weyl electrovac <math>\psi\sim\Phi</math> characteristic relation</div>
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| <div class="NavContent" style="text-align:left;">
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| Considering the interplay between spacetime geometry and energy-matter distributions, it is natural to assume that in Eqs(7.a-7.e) the metric function <math>\psi(\rho,z)</math> relates with the electrostatic scalar potential <math>\Phi(\rho,z)</math> via a function <math>\psi=\psi(\Phi)</math> (which means geometry depends on energy), and it follows that
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| <math>
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| (B.1)\quad \psi_{,\,i}=\psi_{,\,\Phi}\cdot \Phi_{,\,i} \quad,\quad \nabla\psi=\psi_{,\,\Phi}\cdot \nabla \Phi \quad,\quad
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| \nabla^2\psi=\psi_{,\,\Phi}\cdot \nabla^2 \Phi+\psi_{,\,\Phi\Phi}\cdot (\nabla \Phi)^2 ,
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| </math>
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| Eq(B.1) immediately turns Eq(7.b) and Eq(7.e) respectively into
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| <math>
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| (B.2)\quad \Psi_{,\,\Phi}\cdot \nabla^2\Phi\,=\,\big(e^{-2\psi}-\psi_{,\,\Phi\Phi} \big)\cdot (\nabla\Phi)^2,
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| </math><br />
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| <math>
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| (B.3)\quad \nabla^2\Phi\,=\,2\psi_{,\,\Phi}\cdot (\nabla\Phi)^2,
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| </math>
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| which give rise to
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| <math>
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| (B.4)\quad \psi_{,\,\Phi\Phi}+2 \,\big(\psi_{,\,\Phi}\big)^2-e^{-2\psi}=0.
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| </math>
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| Now replace the variable <math>\psi</math> by <math>\zeta:= e^{2\psi}</math>, and Eq(B.4) is simplified to
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| <math>
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| (B.5)\quad \zeta_{,\,\Phi\Phi}-2=0.
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| </math>
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| Direct quadrature of Eq(B.5) yields <math> \zeta=e^{2\psi}=\Phi^2+\tilde{C}\Phi+B</math>, with <math>\{B, \tilde{C}\}</math> being integral constants. To resume asymptotic flatness at spatial infinity, we need <math> \lim_{\rho,z\to\infty}\Phi=0</math> and <math> \lim_{\rho,z\to\infty}e^{2\psi}=1</math>, so there should be <math>B=1</math>. Also, rewrite the constant <math>\tilde{C}</math> as <math>-2C</math> for mathematical convenience in subsequent calculations, and one finally obtains the characteristic relation implied by Eqs(7.a-7.e) that
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| <math>
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| (7.f)\quad e^{2\psi}=\Phi^2-2C\Phi+1\,.
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| </math>
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| This relation is important in linearize the Eqs(7.a-7.f) and superpose electrovac Weyl solutions.
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| </div>
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| </div>
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| ==Newtonian analogue of metric potential Ψ(ρ,z)==
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| In Weyl's metric Eq(1), <math>e^{\pm2\psi}=\sum_{n=0}^{\infty} \frac{(\pm2\psi)^n}{n!}</math>; thus in the approximation for weak field limit <math>\psi\to 0</math>, one has
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| <br />
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| <math>(9)\quad g_{tt}=-(1+2\psi)-\mathcal {O}(\psi^2)\,,\quad g_{\phi\phi}=1-2\psi+\mathcal {O}(\psi^2)\,,
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| </math>
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| and therefore
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| <br />
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| <math>(10)\quad ds^2\approx-\Big(1+2\psi(\rho,z)\Big)\,dt^2+\Big(1-2\psi(\rho,z)\Big)\Big[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\phi^2\Big]\,.</math>
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| This is pretty analogous to the well-known approximate metric for static and weak [[gravitational field]]s generated by low-mass celestial bodies like the Sun and Earth,<ref>James B Hartle. Gravity: An Introduction To Einstein's General Relativity. San Francisco: Addison Wesley, 2003. Eq(6.20) transformed into Lorentzian cylindrical coordinates</ref>
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| <br />
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| <math>(11)\quad ds^2=-\Big(1+2\Phi_{N}(\rho,z)\Big)\,dt^2+\Big(1-2\Phi_{N}(\rho,z)\Big)\,\Big[d\rho^2+dz^2+\rho^2d\phi^2\Big]\,.</math>
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| where <math>\Phi_{N}(\rho,z)</math> is the usual [[Gravitational potential|''Newtonian'' potential]] satisfying Poisson's equation <math>\nabla^2_{L}\Phi_{N}=4\pi\varrho_{N}</math>, just like Eq(3.a) or Eq(4.a) for the Weyl metric potential <math>\psi(\rho,z)</math>. The similarities between <math>\psi(\rho,z)</math> and <math>\Phi_{N}(\rho,z)</math> inspire people to find out the ''Newtonian analogue'' of <math>\psi(\rho,z)</math> when studying Weyl class of solutions; that is, to reproduce <math>\psi(\rho,z)</math> nonrelativistically by certain type of Newtonian sources. The Newtonian analogue of <math>\psi(\rho,z)</math> proves quite helpful in specifying particular Weyl-type solutions and extending existing Weyl-type solutions.<ref name="Weyl1" />
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| ==Schwarzschild solution==
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| The Weyl potentials generating [[Schwarzschild metric|Schwarzschild's metric]] as solutions to the vacuum equations Eq(8) are given by<ref name="Weyl1" /><ref name="Weyl2" /><ref name="Weyl4" />
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| <br />
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| <math>(12)\quad \psi_{SS}=\frac{1}{2}\ln\frac{L-M}{L+M}\,,\quad \gamma_{SS}=\frac{1}{2}\ln\frac{L^2-M^2}{l_+ l_-}\,,</math>
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| where
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| <br />
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| <math>(13)\quad L=\frac{1}{2}\big(l_+ + l_- \big)\,,\quad l_+ =\sqrt{\rho^2+(z+M)^2}\,,\quad l_- =\sqrt{\rho^2+(z-M)^2}\,.</math>
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| From the perspective of Newtonian analogue, <math>\psi_{SS}</math> equals the gravitational potential produced by a rod of mass <math>M</math> and length <math>2M</math> placed symmetrically on the <math>z</math>-axis; that is, by a line mass of uniform density <math>\sigma=1/2</math> embedded the interval <math>z\in[-M,M]</math>. (Note: Based on this analogue, important extensions of the Schwarzschild metric have been developed, as discussed in ref.<ref name="Weyl1" />)
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| Given <math>\psi_{SS}</math> and <math>\gamma_{SS}</math>, Weyl's metric Eq(\ref{Weyl metric in canonical coordinates}) becomes
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| <br />
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| <math>(14)\quad ds^2=-\frac{L-M}{L+M}dt^2+\frac{(L+M)^2}{l_+ l_-}(d\rho^2+dz^2)+\frac{L+M}{L-M}\,\rho^2 d\phi^2\,,</math>
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| and after substituting the following mutually consistent relations
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| <br />
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| <math>(15)\quad L+M=r\,,\quad l_+ + l_- =2M\cos\theta\,,\quad z=(r-M)\cos\theta\,,</math><br />
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| <math>\;\;\quad \rho=\sqrt{r^2-2Mr}\,\sin\theta\,,\quad l_+ l_-=(r-M)^2-M^2\cos^2\theta\,,</math>
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| one can obtain the common form of Schwarzschild metric in the usual <math>\{t,r,\theta,\phi\}</math> coordinates,
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| <br />
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| <math>(16)\quad ds^2=-\Big(1-\frac{2M}{r} \Big)\,dt^2+\Big(1-\frac{2M}{r} \Big)^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta\, d\phi^2\,.</math>
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| The metric Eq(14) cannot be directly transformed into Eq(16) by performing the standard cylindrical-spherical transformation <math>(t,\rho,z,\phi)=(t,r\sin\theta,r\cos\theta,\phi)</math>, because <math>\{t,r,\theta,\phi\}</math> is complete while <math>(t,\rho,z,\phi)</math> is incomplete. This is why we call <math>\{t,\rho,z,\phi\}</math> in Eq(1) as Weyl's canonical coordinates rather than cylindrical coordinates, although they have a lot in common; for example, the Laplacian <math>\nabla^2:= \partial_{\rho\rho}+\frac{1}{\rho}\partial_\rho +\partial_{zz}</math> in Eq(7) is exactly the two-dimensional geometric Laplacian in cylindrical coordinates.
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| ==Nonextremal Reissner-Nordström solution==
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| The Weyl potentials generating the nonextremal [[Reissner-Nordström metric|Reissner-Nordström]] solution (<math>M>|Q|</math>) as solutions to Eqs(7} are given by<ref name="Weyl1" /><ref name="Weyl2" /><ref name="Weyl4" />
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| <br />
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| <math>(17)\quad \psi_{RN}=\frac{1}{2}\ln\frac{L^2-(M^2-Q^2)}{(L+M)^2} \,, \quad \gamma_{RN}=\frac{1}{2}\ln\frac{L^2-(M^2-Q^2)}{l_+ l_-}\,,
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| </math>
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| where
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| <br />
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| <math>(18)\quad L=\frac{1}{2}\big(l_+ + l_- \big)\,,\quad l_+ =\sqrt{\rho^2+(z+ \sqrt{M^2-Q^2})^2}\,,\quad l_- =\sqrt{\rho^2+(z-\sqrt{M^2-Q^2})^2}\,.</math>
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| Thus, given <math>\psi_{RN}</math> and <math>\gamma_{RN}</math>, Weyl's metric becomes
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| <br />
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| <math>(19)\quad ds^2=-\frac{L^2-(M^2-Q^2)}{(L+M)^2}dt^2+\frac{(L+M)^2}{l_+ l_-}(d\rho^2+dz^2)+\frac{(L+M)^2}{L^2-(M^2-Q^2)}\rho^2 d\phi^2\,,</math>
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| and employing the following transformations
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| <br />
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| <math>(20)\quad L+M=r\,,\quad l_+ + l_- =2\sqrt{M^2-Q^2}\,\cos\theta\,,\quad z=(r-M)\cos\theta\,,</math><br />
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| <math>\;\;\quad \rho=\sqrt{r^2-2Mr+Q^2}\,\sin\theta\,,\quad l_+ l_-=(r-M)^2-(M^2-Q^2)\cos^2\theta\,,</math>
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| one can obtain the common form of non-extremal Reissner-Nordström metric in the usual <math>\{t,r,\theta,\phi\}</math> coordinates,
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| <br />
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| <math>(21)\quad ds^2=-\Big(1-\frac{2M}{r}+\frac{Q^2}{r^2} \Big)\,dt^2+\Big(1-\frac{2M}{r}+\frac{Q^2}{r^2} \Big)^{-1}dr^2+r^2d\theta^2+r^2\sin^2\theta\, d\phi^2\,.</math>
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| ==Extremal Reissner-Nordström solution==
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| The potentials generating the [[Extremal black hole|extremal]] Reissner-Nordström solution (<math>M=|Q|</math>) as solutions to Eqs(7} are given by<ref name="Weyl4" /> (Note: We treat the [[Extremal black hole|extremal]] solution separately because it is much more than the degenerate state of the nonextremal counterpart.)
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| <br />
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| <math>(22)\quad \psi_{ERN}=\frac{1}{2}\ln\frac{L^2}{(L+M)^2}\,,\quad \gamma_{ERN}=0\,,\quad\text{with}\quad L=\sqrt{\rho^2+z^2}\,.</math>
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| Thus, the extremal Reissner-Nordström metric reads
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| <br />
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| <math>(23)\quad ds^2=-\frac{L^2}{(L+M)^2}dt^2+\frac{(L+M)^2}{L^2}(d\rho^2+dz^2+\rho^2d\phi^2)\,,</math>
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| and by substituting
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| <br />
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| <math>(24)\quad L+M=r\,,\quad z=L\cos\theta\,,\quad \rho=L\sin\theta\,,</math>
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| we obtain the extremal Reissner-Nordström metric in the usual <math>\{t,r,\theta,\phi\}</math> coordinates,
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| <br />
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| <math>(25)\quad ds^2=-\Big(1-\frac{M}{r} \Big)^2 dt^2+\Big(1-\frac{M}{r} \Big)^{-2}dr^2+r^2d\theta^2+r^2\sin^2\theta\, d\phi^2\,.</math>
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| Mathematically, the extremal Reissner-Nordström can be obtained by taking the limit <math>Q\to M</math> of the corresponding nonextremal equation, and in the meantime we need to use the [[L'Hospital rule]] sometimes.<br />
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|
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| Remarks: Weyl's metrics Eq(1) with the vanishing potential <math>\gamma(\rho,z)</math> (like the extremal Reissner-Nordström metric) constitute a special subclass which have only one metric potential <math>\psi(\rho,z)</math> to be identified. Extending this subclass by canceling the restriction of axisymmetry, one obtains another useful class of solutions (still using Weyl's coordinates), namely the ''conformastatic'' metrics,<ref>Guillermo A Gonzalez, Antonio C Gutierrez-Pineres, Paolo A Ospina. ''Finite axisymmetric charged dust disks in conformastatic spacetimes''. Physical Review D, 2008, '''78'''(6): 064058. [http://arxiv.org/abs/0806.4285 arXiv:0806.4285v1]</ref><ref>Antonio C Gutierrez-Pineres, Guillermo A Gonzalez, Hernando Quevedo. ''Conformastatic disk-haloes in Einstein-Maxwell gravity''. Physical Review D, 2013, '''87'''(4): 044010. [http://arxiv.org/abs/1211.4941v2]</ref>
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| <br />
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| <math>(26)\quad ds^2\,=-e^{2\lambda(\rho,z,\phi)}dt^2+e^{-2\lambda(\rho,z,\phi)}\Big(d\rho^2+dz^2+\rho^2 d\phi^2 \Big)\,,</math>
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| where we use <math>\lambda</math> in Eq(22) as the single metric function in place of <math>\psi</math> in Eq(1) to emphasize that they are different by axial symmetry (<math>\phi</math>-dependence).
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| == Weyl vacuum solutions in spherical coordinates ==
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| Weyl's metric can also be expressed in [[spherical coordinates]] that
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| <br />
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| <math>(27)\quad ds^2\,=-e^{2\psi(r,\theta)}dt^2+e^{2\gamma(r,\theta)-2\psi(r,\theta)}(dr^2+r^2d\theta^2)+e^{-2\psi(r,\theta)}\rho^2 d\phi^2\,,</math>
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| which equals Eq(1) via the coordinate transformation <math>(t,\rho,z,\phi)\mapsto(t,r\sin\theta,r\cos\theta,\phi)</math> (Note: As shown by Eqs(15)(21)(24), this transformation is not always applicable.) In the vacuum case, Eq(8.b) for <math>\psi(r,\theta)</math> becomes
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| <br />
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| <math>(28)\quad r^2\psi_{,\,rr}+2r\,\psi_{,\,r}+\psi_{,\,\theta\theta}+\cot\theta\cdot\psi_{,\,\theta}\,=\,0\,.</math>
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| The [[asymptotically flat]] solutions to Eq(28) is<ref name="Weyl1" />
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| <br />
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| <math>(29)\quad \psi(r,\theta)\,=-\sum_{n=0}^\infty a_n \frac{P_n(\cos\theta)}{r^{n+1}}\,, </math>
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| where <math>P_n(\cos\theta)</math> represent [[Legendre polynomials]], and <math>a_n</math> are [[Multipole moment|multipole]] coefficients. The other metric potential <math>\gamma(r,\theta)</math>is given by<ref name="Weyl1" />
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| <br />
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| <math>(30)\quad \gamma(r,\theta)\,=-\sum_{l=0}^\infty \sum_{m=0}^\infty a_l a_m</math> <math>\frac{(l+1)(m+1)}{l+m+2}</math> <math>\frac{P_l P_m-P_{l+1}P_{m+1}}{r^{l+m+2}}\,.</math>
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| ==See also==
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| * [[Schwarzschild metric]]
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| * [[Reissner–Nordström metric]]
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| * [[Distorted Schwarzschild metric]]
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| ==References==
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| {{reflist}}
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| [[Category:Black holes]]
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| [[Category:General relativity]]
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| [[Category:Exact solutions in general relativity]]
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Potential buyers of Lancel see reviving the loss-making leather goods maker as a high-risk gamble that could take at least six or eight years to pay off, sources close to the matter say.
Facing a struggle to offload the business, Swiss parent Richemont (CFR.VX) would be ready to pay for two years of losses - up to an estimated 20 million euros ($28 million) - to entice bidders, the sources say.
So far private equity firms Change Capital and Lion Capital have expressed interest while Asian group Swire is looking to team up with a private equity firm to make a bid, according to the sources.
Another potential buyer, who has not made up his mind yet, is Jacques Veyrat, the former CEO of private French trading group Louis Dreyfus who used to run its Neuf Cegetel telecoms unit.
Veyrat left Louis Dreyfus in 2011 with a golden parachute of 270 million euros, according to the French press. He has since set up the Impala group, which invests in companies in distress.
Veyrat, Swire, Lion Capital and Change Capital declined to comment, as did Lancel and Richemont.
People close to the talks say the bidding process for Lancel, which had already been going on for months, could last several more weeks.
Lancel is run part-time by Fabrizio Cardinali, CEO of Dunhill, another Richemont brand.
Lancel has hired a new designer, Nicole Stulman, who was previously at Reed Krakoff, Hermes (HRMS.PA) and Celine Bags Outlet and does not have a head of merchandising to work with since that person has left for Dunhill, sources close to the sale process said.
Other key people at the company are either on sick leave or have not been replaced, the sources said.
Lancel's difficulties come as the luxury sector as a whole remains resilient despite a slowdown in China, continuing to grow, although at a slower pace than in previous years.
PRIVATE EQUITY
The relationship between private equity and fashion brands can be difficult because the typical three- to five-year investment horizon may be too short to turn a label around.
Private equity deals are usually financed by debt, which puts pressure on the fashion brand when it needs to invest in marketing and new shops.
Lanvin, acquired by Taiwanese media magnate Shaw-Lan Wang from L'Oreal in 2001, took more than seven years to become profitable. TPG struggled for nine years to restructure the leather goods brand Bally before selling it in 2008.
There have been few successful revamps of struggling brands that are still going concerns. Carven, Balenciaga and Lanvin lay dormant for decades before they were acquired, allowing their new owners to make a fresh start.
Lancel was founded by accessories designer Ang�le Lancel in 1876, remaining in the hands of her descendants until it was acquired in the late 1970s by the Zorbibe brothers, who with then chief executive Sidney Toledano, now Dior CEO, launched the hit bucket-shaped Elsa bag and made Lancel a profitable business.
Industry executives say Lancel was making margins higher than Louis Vuitton - today's market leader - when Richemont bought it in 1997.
"All my girlfriends in the 1990s dreamt of having a Lancel bag," said Serge Carreira, who lectures on luxury at the Institut d'Etudes Politiques in Paris, known as "Sciences Po".
Lancel had not expanded internationally or creatively as its rivals did, he said.
Lancel makes more than 80 percent of its sales in France and in the year to end June, it had an operating loss of 10 million euros on revenues of 135 million euros, sources close to the matter said.
Richemont, the world's second biggest luxury group, does not publish separate sales and profit figures for its brands, which include Cartier and Van Cleef & Arpels.
Longchamp, founded in 1948 and still owned by the founding Cassegrain family, has annual sales of 500 million euros.
"ELECTRIC SHOCK"
"What Lancel needs in an electric shock in terms of design, marketing, store concept and image," says Boston Consulting Group senior partner Jean-Marc Bellaiche.
Analysts say Lancel needs to enter the accessible segment of the luxury market, alongside Coach (COH.N), Michael Kors (KORS.N) and Longchamp with handbags costing 200-800 euros.
Lancel lost credibility at the high end of the luxury market by offering excessive discounts and putting out staid, old-fashioned designs, such as the Brigitte Bardot bag in 2010 - a 1970s style version of the bucket bag - and the document holder Isabelle Adjani bag in 2008.
"Lancel's products, while perfectly fine, look a bit dated and are not a must-buy," said Hugh Devlin, a consultant at legal firm Withers LLP, who specializes in luxury and fashion brands.
"With the right creative team and strategy, that could definitely change but whoever buys it will need to be prepared to take a risk."
Lancel clearly thinks it can make it as a luxury brand. A few months ago, it introduced an ostrich leather "L" bag costing 4,500 euros, emulating Gucci (PRTP.PA) and Louis Vuitton's (LVMH.PA) recent efforts to move upmarket.
Carreira disagrees. "You do not become a luxury brand because you sell a bag for 4,000 euros," he said.
To rebuild its name in the crowded accessories market, experts say Lancel should do some ready-to-wear, as Longchamp did, to stimulate traffic in shops and rejuvenate itself.
Another potential investor who looked at Lancel said it had suffered from under-investment and excessive management turnover.
Having had about a dozen chief executives since 1997, the company will now need to be given time by its new owners to implement long-term strategy, industry observers said.
(Editing by Giles Elgood)