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Text of the page (random words):
and current flows as we know given the electromagnetic four potential a a displaystyle a_ a the anti symmetric electromagnetic field f a b displaystyle f_ ab and the trace free stress energy tensor t a b displaystyle t_ ab t g a b t a b 0 displaystyle t g ab t_ ab 0 will be respectively determined by f a b a b a a a b a b a a a b displaystyle f_ ab a_ b a a_ a b a_ b a a_ a b 3 t a b 1 4 π f a c f b c 1 4 g a b f c d f c d displaystyle t_ ab frac 1 4 pi left f_ ac f_ b c frac 1 4 g_ ab f_ cd f cd right 4 which respects the source free covariant maxwell equations f a b b 0 f a b c 0 displaystyle big f ab big _ b 0 quad f_ ab c 0 5 a eq 5 a can be simplified to g f a b b 0 f a b c 0 displaystyle left sqrt g f ab right _ b 0 quad f_ ab c 0 5 b in the calculations as γ b c a γ c b a displaystyle gamma _ bc a gamma _ cb a also since r 8 π t 0 displaystyle r 8 pi t 0 for electrovacuum eq 2 reduces to r a b 8 π t a b displaystyle r_ ab 8 pi t_ ab 6 now suppose the weyl type axisymmetric electrostatic potential is a a φ ρ z d t a displaystyle a_ a phi rho z dt _ a the component φ displaystyle phi is actually the electromagnetic scalar potential and together with the weyl metric eq 1 eqs 3 4 5 6 imply that 2 ψ ψ 2 γ ρ ρ γ z z displaystyle nabla 2 psi nabla psi 2 gamma _ rho rho gamma _ zz 7 a 2 ψ e 2 ψ φ 2 displaystyle nabla 2 psi e 2 psi nabla phi 2 7 b 1 ρ γ ρ ψ ρ 2 ψ z 2 e 2 ψ φ ρ 2 φ z 2 displaystyle frac 1 rho gamma _ rho psi _ rho 2 psi _ z 2 e 2 psi big phi _ rho 2 phi _ z 2 big 7 c 1 ρ γ z 2 ψ ρ ψ z 2 e 2 ψ φ ρ φ z displaystyle frac 1 rho gamma _ z 2 psi _ rho psi _ z 2e 2 psi phi _ rho phi _ z 7 d 2 φ 2 ψ φ displaystyle nabla 2 phi 2 nabla psi nabla phi 7 e where r 0 displaystyle r 0 yields eq 7 a r t t 8 π t t t displaystyle r_ tt 8 pi t_ tt or r φ φ 8 π t φ φ displaystyle r_ varphi varphi 8 pi t_ varphi varphi yields eq 7 b r ρ ρ 8 π t ρ ρ displaystyle r_ rho rho 8 pi t_ rho rho or r z z 8 π t z z displaystyle r_ zz 8 pi t_ zz yields eq 7 c r ρ z 8 π t ρ z displaystyle r_ rho z 8 pi t_ rho z yields eq 7 d and eq 5 b yields eq 7 e here 2 ρ ρ 1 ρ ρ z z displaystyle nabla 2 partial _ rho rho frac 1 rho partial _ rho partial _ zz and ρ e ρ z e z displaystyle nabla partial _ rho hat e _ rho partial _ z hat e _ z are respectively the laplace and gradient operators moreover if we suppose ψ ψ φ displaystyle psi psi phi in the sense of matter geometry interplay and assume asymptotic flatness we will find that eqs 7 a e implies a characteristic relation that e ψ φ 2 2 c φ 1 displaystyle e psi phi 2 2c phi 1 7 f specifically in the simplest vacuum case with φ 0 displaystyle phi 0 and t a b 0 displaystyle t_ ab 0 eqs 7 a 7 e reduce to 4 γ ρ ρ γ z z ψ 2 displaystyle gamma _ rho rho gamma _ zz nabla psi 2 8 a 2 ψ 0 displaystyle nabla 2 psi 0 8 b γ ρ ρ ψ ρ 2 ψ z 2 displaystyle gamma _ rho rho big psi _ rho 2 psi _ z 2 big 8 c γ z 2 ρ ψ ρ ψ z displaystyle gamma _ z 2 rho psi _ rho psi _ z 8 d we can firstly obtain ψ ρ z displaystyle psi rho z by solving eq 8 b and then integrate eq 8 c and eq 8 d for γ ρ z displaystyle gamma rho z practically eq 8 a arising from r 0 displaystyle r 0 just works as a consistency relation or integrability condition unlike the nonlinear poisson s equation eq 7 b eq 8 b is the linear laplace equation that is to say superposition of given vacuum solutions to eq 8 b is still a solution this fact has a widely application such as to analytically distort a schwarzschild black hole 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 ψ ρ ρ 1 ρ ψ ρ ψ z z ψ ρ 2 ψ z 2 γ ρ ρ γ z z displaystyle psi _ rho rho frac 1 rho psi _ rho psi _ zz psi _ rho 2 psi _ z 2 gamma _ rho rho gamma _ zz a 1 a ψ ρ ρ 1 ρ ψ ρ ψ z z e 2 ψ φ ρ 2 φ z 2 displaystyle psi _ rho rho frac 1 rho psi _ rho psi _ zz e 2 psi big phi _ rho 2 phi _ z 2 big a 1 b 1 ρ γ ρ ψ ρ 2 ψ z 2 e 2 ψ φ ρ 2 φ z 2 displaystyle frac 1 rho gamma _ rho psi _ rho 2 psi _ z 2 e 2 psi big phi _ rho 2 phi _ z 2 big a 1 c 1 ρ γ z 2 ψ ρ ψ z 2 e 2 ψ φ ρ φ z displaystyle frac 1 rho gamma _ z 2 psi _ rho psi _ z 2e 2 psi phi _ rho phi _ z a 1 d φ ρ ρ 1 ρ φ ρ φ z z 2 ψ ρ φ ρ 2 ψ z φ z displaystyle phi _ rho rho frac 1 rho phi _ rho phi _ zz 2 psi _ rho phi _ rho 2 psi _ z phi _ z a 1 e and ψ ρ 2 ψ z 2 γ ρ ρ γ z z displaystyle psi _ rho 2 psi _ z 2 gamma _ rho rho gamma _ zz a 2 a ψ ρ ρ 1 ρ ψ ρ ψ z z 0 displaystyle psi _ rho rho frac 1 rho psi _ rho psi _ zz 0 a 2 b γ ρ ρ ψ ρ 2 ψ z 2 displaystyle gamma _ rho rho big psi _ rho 2 psi _ z 2 big a 2 c γ z 2 ρ ψ ρ ψ z displaystyle gamma _ z 2 rho psi _ rho psi _ z a 2 d 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 ψ ρ z displaystyle psi rho z relates with the electrostatic scalar potential φ ρ z displaystyle phi rho z via a function ψ ψ φ displaystyle psi psi phi which means geometry depends on energy and it follows that ψ i ψ φ φ i ψ ψ φ φ 2 ψ ψ φ 2 φ ψ φ φ φ 2 displaystyle psi _ i psi _ phi cdot phi _ i quad quad nabla psi psi _ phi cdot nabla phi quad quad nabla 2 psi psi _ phi cdot nabla 2 phi psi _ phi phi cdot nabla phi 2 b 1 eq b 1 immediately turns eq 7 b and eq 7 e respectively into ψ φ 2 φ e 2 ψ ψ φ φ φ 2 displaystyle psi _ phi cdot nabla 2 phi big e 2 psi psi _ phi phi big cdot nabla phi 2 b 2 2 φ 2 ψ φ φ 2 displaystyle nabla 2 phi 2 psi _ phi cdot nabla phi 2 b 3 which give rise to ψ φ φ 2 ψ φ 2 e 2 ψ 0 displaystyle psi _ phi phi 2 big psi _ phi big 2 e 2 psi 0 b 4 now replace the variable ψ displaystyle psi by ζ e 2 ψ displaystyle zeta e 2 psi and eq b 4 is simplified to ζ φ φ 2 0 displaystyle zeta _ phi phi 2 0 b 5 direct quadrature of eq b 5 yields ζ e 2 ψ φ 2 c φ b displaystyle zeta e 2 psi phi 2 tilde c phi b with b c displaystyle b tilde c being integral constants to resume asymptotic flatness at spatial infinity we need lim ρ z φ 0 displaystyle lim _ rho z to infty phi 0 and lim ρ z e 2 ψ 1 displaystyle lim _ rho z to infty e 2 psi 1 so there should be b 1 displaystyle b 1 also rewrite the constant c displaystyle tilde c as 2 c displaystyle 2c for mathematical convenience in subsequent calculations and one finally obtains the characteristic relation implied by eqs 7 a 7 e that e 2 ψ φ 2 2 c φ 1 displaystyle e 2 psi phi 2 2c phi 1 7 f this relation is important in linearize the eqs 7 a 7 f and superpose electrovac weyl solutions newtonian analogue of metric potential ψ ρ z edit in weyl s metric eq 1 e 2 ψ n 0 2 ψ n n textstyle e pm 2 psi sum _ n 0 infty frac pm 2 psi n n thus in the approximation for weak field limit ψ 0 displaystyle psi to 0 one has g t t 1 2 ψ o ψ 2 g ϕ ϕ 1 2 ψ o ψ 2 displaystyle g_ tt 1 2 psi mathcal o psi 2 quad g_ phi phi 1 2 psi mathcal o psi 2 9 and therefore d s 2 1 2 ψ ρ z d t 2 1 2 ψ ρ z e 2 γ d ρ 2 d z 2 ρ 2 d ϕ 2 displaystyle ds 2 approx big 1 2 psi rho z big dt 2 big 1 2 psi rho z big left e 2 gamma d rho 2 dz 2 rho 2 d phi 2 right 10 this is pretty analogous to the well known approximate metric for static and weak gravitational fields generated by low mass celestial bodies like the sun and earth 5 d s 2 1 2 φ n ρ z d t 2 1 2 φ n ρ z d ρ 2 d z 2 ρ 2 d ϕ 2 displaystyle ds 2 big 1 2 phi _ n rho z big dt 2 big 1 2 phi _ n rho z big left d rho 2 dz 2 rho 2 d phi 2 right 11 where φ n ρ z displaystyle phi _ n rho z is the usual newtonian potential satisfying poisson s equation l 2 φ n 4 π ϱ n displaystyle nabla _ l 2 phi _ n 4 pi varrho _ n just like eq 3 a or eq 4 a for the weyl metric potential ψ ρ z displaystyle psi rho z the similarities between ψ ρ z displaystyle psi rho z and φ n ρ z displaystyle phi _ n rho z inspire people to find out the newtonian analogue of ψ ρ z displaystyle psi rho z when studying weyl class of solutions that is to reproduce ψ ρ z displaystyle psi rho z nonrelativistically by certain type of newtonian sources the newtonian analogue of ψ ρ z displaystyle psi rho z proves quite helpful in specifying particular weyl type solutions and extending existing weyl type solutions 2 schwarzschild solution edit the weyl potentials generating schwarzschild s metric as solutions to the vacuum equations eq 8 are given by 2 3 4 ψ s s 1 2 ln l m l m γ s s 1 2 ln l 2 m 2 l l displaystyle 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_ 12 where l 1 2 l l l ρ 2 z m 2 l ρ 2 z m 2 displaystyle l frac 1 2 big l_ l_ big quad l_ sqrt rho 2 z m 2 quad l_ sqrt rho 2 z m 2 13 from the perspective of newtonian analogue ψ s s displaystyle psi _ ss equals the gravitational potential produced by a rod of mass m displaystyle m and length 2 m displaystyle 2m placed symmetrically on the z displaystyle z axis that is by a line mass of uniform density σ 1 2 displaystyle sigma 1 2 embedded the interval z m m displaystyle z in m m note based on this analogue important extensions of the schwarzschild metric have been developed as discussed in ref 2 given ψ s s displaystyle psi _ ss and γ s s displaystyle gamma _ ss weyl s metric eq 1 becomes d s 2 l m l m d t 2 l m 2 l l d ρ 2 d z 2 l m l m ρ 2 d ϕ 2 displaystyle 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 14 and after substituting the following mutually consistent relations l m r l l 2 m cos θ z r m cos θ ρ r 2 2 m r sin θ l l r m 2 m 2 cos 2 θ displaystyle begin aligned l m r quad l_ l_ 2m cos theta quad z r m cos theta rho sqrt r 2 2mr sin theta quad l_ l_ r m 2 m 2 cos 2 theta end aligned 15 one can obtain the common form of schwarzschild metric in the usual t r θ ϕ displaystyle t r theta phi coordinates d s 2 1 2 m r d t 2 1 2 m r 1 d r 2 r 2 d θ 2 r 2 sin 2 θ d ϕ 2 displaystyle ds 2 left 1 frac 2m r right dt 2 left 1 frac 2m r right 1 dr 2 r 2 d theta 2 r 2 sin 2 theta d phi 2 16 the metric eq 14 cannot be directly transformed into eq 16 by performing the standard cylindrical spherical transformation t ρ z ϕ t r sin θ r cos θ ϕ displaystyle t rho z phi t r sin theta r cos theta phi because t r θ ϕ displaystyle t r theta phi is complete while t ρ z ϕ displaystyle t rho z phi is incomplete this is why we call t ρ z ϕ displaystyle t rho z phi in eq 1 as weyl s canonical coordinates rather than cylindrical coordinates although they have a lot in common for example the laplacian 2 ρ ρ 1 ρ ρ z z displaystyle nabla 2 partial _ rho rho frac 1 rho partial _ rho partial _ zz in eq 7 is exactly the two dimensional geometric laplacian in cylindrical coordinates nonextremal reissner nordström solution edit the weyl potentials generating the nonextremal reissner nordström solution m q displaystyle m q as solutions to eqs 7 are given by 2 3 4 ψ r n 1 2 ln l 2 m 2 q 2 l m 2 γ r n 1 2 ln l 2 m 2 q 2 l l displaystyle psi _ rn frac 1 2 ln frac l 2 left m 2 q 2 right left l m right 2 quad gamma _ rn frac 1 2 ln frac l 2 left m 2 q 2 right l_ l_ 17 where l 1 2 l l l ρ 2 z m 2 q 2 2 l ρ 2 z m 2 q 2 2 displaystyle l frac 1 2 big l_ l_ big quad l_ sqrt rho 2 left z sqrt m 2 q 2 right 2 quad l_ sqrt rho 2 left z sqrt m 2 q 2 right 2 18 thus given ψ r n displaystyle psi _ rn and γ r n displaystyle gamma _ rn weyl s metric becomes d s 2 l 2 m 2 q 2 l m 2 d t 2 l m 2 l l d ρ 2 d z 2 l m 2 l 2 m 2 q 2 ρ 2 d ϕ 2 displaystyle ds 2 frac l 2 left m 2 q 2 right left l m right 2 dt 2 frac left l m right 2 l_ l_ d rho 2 dz 2 frac l m 2 l 2 m 2 q 2 rho 2 d phi 2 19 and employing the following transformations l m r l l 2 m 2 q 2 cos θ z r m cos θ ρ r 2 2 m r q 2 sin θ l l r m 2 m 2 q 2 cos 2 θ displaystyle begin aligned l m r quad l_ l_ 2 sqrt m 2 q 2 cos theta quad z r m cos theta rho sqrt r 2 2mr q 2 sin theta quad l_ l_ r m 2 m 2 q 2 cos 2 theta end aligned 20 one can obtain the common form of non extremal reissner nordström metric in the usual t r θ ϕ displaystyle t r theta phi coordinates d s 2 1 2 m r q 2 r 2 d t 2 1 2 m r q 2 r 2 1 d r 2 r 2 d θ 2 r 2 sin 2 θ d ϕ 2 displaystyle ds 2 left 1 frac 2m r frac q 2 r 2 right dt 2 left 1 frac 2m r frac q 2 r 2 right 1 dr 2 r 2 d theta 2 r 2 sin 2 theta d phi 2 21 extremal reissner nordström solution edit the potentials generating the extremal reissner nordström solution m q displaystyle m q as solutions to eqs 7 are given by 4 note we treat the extremal solution separately because it is much more than the degenerate state of the nonextremal counterpart ψ e r n 1 2 ln l 2 l m 2 γ e r n 0 with l ρ 2 z 2 displaystyle 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 22 thus the extremal reissner nordström metric reads d s 2 l 2 l m 2 d t 2 l m 2 l 2 d ρ 2 d z 2 ρ 2 d ϕ 2 displaystyle ds 2 frac l 2 l m 2 dt 2 frac l m 2 l 2 d rho 2 dz 2 rho 2 d phi 2 23 and by substituting l m r z l cos θ ρ l sin θ displaystyle l m r quad z l cos theta quad rho l sin theta 24 we obtain the extremal reissner nordström metric in the usual t r θ ϕ displaystyle t r theta phi coordinates d s 2 1 m r 2 d t 2 1 m r 2 d r 2 r 2 d θ 2 r 2 sin 2 θ d ϕ 2 displaystyle ds 2 left 1 frac m r right 2 dt 2 left 1 frac m r right 2 dr 2 r 2 d theta 2 r 2 sin 2 theta d phi 2 25 mathematically the extremal reissner nordström can be obtained by taking the limit q m displaystyle q to m of the corresponding nonextremal equation and in the meantime we need to use the l hospital rule sometimes remarks weyl s metrics eq 1 with the vanishing potential γ ρ z displaystyle gamma rho z like the extremal reissner nordström metric constitute a special subclass which have only one metric potential ψ ρ z displaystyle psi rho z 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 6 7 d s 2 e 2 λ ρ z ϕ d t 2 e 2 λ ρ z ϕ d ρ 2 d z 2 ρ 2 d ϕ 2 displaystyle 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 26 where we use λ displaystyle lambda in eq 22 as the single metric function in place of ψ displaystyle psi in eq 1 to emphasize that they are different by axial symmetry ϕ displaystyle phi dependence weyl vacuum solutions in spherical coordinates edit weyl s metric can also be expressed in spherical coordinates that d s 2 e 2 ψ r θ d t 2 e 2 γ r θ 2 ψ r θ d r 2 r 2 d θ 2 e 2 ψ r θ ρ 2 d ϕ 2 displaystyle ds 2 e 2 psi r theta dt 2 e 2 gamma r theta 2 psi r theta dr 2 r 2 d theta 2 e 2 psi r theta rho 2 d phi 2 27 which equals eq 1 via the coordinate transformation t ρ z ϕ t r sin θ r cos θ ϕ displaystyle t rho z phi mapsto t r sin theta r cos theta phi note as shown by eqs 15 21 24 this transformation is not always applicable in the vacuum case eq 8 b for ψ r θ displaystyle psi r theta becomes r 2 ψ r r 2 r ψ r ψ θ θ cot θ ψ θ 0 displaystyle r 2 psi _ rr 2r psi _ r psi _ theta theta cot theta cdot psi _ theta 0 28 the asymptotically flat solutions to eq 28 is 2 ψ r θ n...