PoS(ISFTG)015 http://pos.sissa.it/ can be constructed using small deformations in vacuo ∗ [email protected] [email protected] Speaker. We show how approximate solutions ofproximate the solutions two-body of problem Einstein’s in equations General Relativity, and the ap- of geodesics and of Einstein space-timesdimension. embedded The into method a consists pseudo-Euclidean in flat using(a space expansions circular of of orbit equations higher in around the a case given simple ofEinstein’s solution geodesics, equations) and in Minkowskian a or series Schwarzschild space ofiteration in powers the the corresponding of case linear small of systems deformation of parameter, differential and equations. then solving by ∗ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. c
Salvatore Vitale Laboratoire de Physique Théorique de laUniversité Matière Pierre-et-Marie-Curie Condensée, - CNRS UMR 7600 Tour 22, 4-ème étage, Boîte 121, 4, Place Jussieu, 75005 Paris, France Email: 5th International School on Field TheoryApril and 20 Gravitation, - 24 2009 Cuiabá city, Brazil Richard Kerner New approximation methods in General Relativity Laboratoire de Physique Théorique de laUniversité Matière Pierre-et-Marie-Curie Condensée, - CNRS UMR 7600 Tour 22, 4-ème étage, Boîte 121, 4, Place Jussieu, 75005 Paris, France Email: PoS(ISFTG)015 Richard Kerner ; then, starting from ∞ of the simplified theory, → c , its square also supposedly pro- c / v exact solution Simplified (linearized) theory Exact solutions 2 −→ ...... Both methods are based on successive approximations in vacuo ↓ ↓ of exact theory of approximate theory Full (exact) theory . This is represented on the right side of the diagram below: r , supposed very small in realistic situations, and keeping only the linear term of / r Approximate solutions / MG MG The relativistic two-body problem including gravitational radiation was treated in an oppo- In General Relativity this diagram, which we would like to become commutative, represents Facing a highly non-linear and complex theory like Einstein’s General Relativity one is forced a) when a linearized version of theory exists, to find an In these lectures we present an alternative method of calculus of trajectories and motions of b) when an exact (usually very symmetric) solution of the exact theory is at hand, try to deform Minkowskian space-time and the Newtonianthe solution, trajectories and corrections to were the addedportional metric, to simultaneously with to small parameter being site manner: first, theory, exact which solution can of the be equation considered of as motion a was limit found of in General the Relativity Newtonian when the- the expansion. This approach is represented by the left side of the diagram. two different approaches to thein solution the search literature. in The Generalsists classical Relativity: in way both solving to cases in obtain are quadratures theSchwarzschild represented the perihelion background, geodesic advance, then equation as in treated describing takingpowers an by the of approximate Einstein, motion integral con- of by a expanding test the particle integrand in in a to use approximation techniques in ordertive ways to to find treat realistic the solutions. problem: There are usually two alterna- of solutions which are presented asare infinite series still in far powers from of the some results smallpresent obtained parameter. approach by Although is the we very usual well methods suited of for recurrent post-Newtonian approximations, numerical the calculations. test particles in thesolutions vicinity of of Einstein’s a equations massive body, and a similar method for obtaining approximate it slightly and obtain, byThis successive situation approximations, is be a represented more by general the solution following diagram: of the exact theory. New approximation methods in General Relativity 1. Introduction and then add small perturbationssolution; transforming it into something that is closer to the exact theory PoS(ISFTG)015 (1.2) (1.1) 827), and . 0 = Richard Kerner e 2 4 c c / / 2 4 the greater half-axis v v a ∼ ∼ . ]) ) 5 displayed by the orbit of Mercury, ... e ], [ 4 + will lead to the result that differs only 6 ) Post–post-Newtonian Newtonian theory e 2 ], [ e 3 + ) ]).. + 4 2 9 ], [ e e 1 ∞ 2 ( GM → + − ], [ ), which is below the actual error bar. c −→ 2 π 8 ...... 1 e ( 6 3 1.1 a the mass of the central body, ], [ + 7 1 = M ( is very high, as for the asteroid Icarus ( ], [ φ 6 e ∆ GM a π ↓ ↓ ↓ ↓ 6 = "epicycles" approximations eccentricity φ ∆ ∼ ε (circular orbit, approximations when it can be supposed to be very small. General Relativity r its eccentricity. GM Special exact solution Post-Newtonian e ]) who found that the perihelion advance during one revolution is given by the Successive deformations 1 Schwarzschild background) as an elliptic integral, which is then evaluated after expansion of the integrand in terms is Newton’s gravitational constant, ϕ 2056), but also in the case when . G 0 18% from the result predicted by relation ( This formula is deduced from the exact solution of the General Relativistic problem of motion In what follows, we show how one can obtain the same results without taking integrals, but just It has been successfully confronted with observation, giving excellent fits not only for the or- In what concerns the two-body problem and the description of motion of bodies in the vicinity The second approach has been tested on the two-body problem in General Relativity quite The problem of motion of planets in General Relativity, considered as test particles moving . = e by 0 of planet’s orbit and of powers of the ratio One can note atsecond this term, i.e., point taking that into account even only for the factor the case of planet Mercury, the series truncated at the of a test particle in thevariable field of Schwarzschild metric, which leads to the expression of the angular where by successive approximations around a circular orbit with constant angular velocity, leading to an is bits with small eccentricities (e.g., one of the highest values of of a massive central body, the first approachTh. has been Damour, successfully G. exploited Schaeffer, by L. Ll. Blanchet, Bel, and N. others Deruelle, ([ recently, by one of the authorsHolten, of R. this Colistete paper Jr. (RK), and in C. collaboration Leygnac with ([ with A. Balakin, J.-W. van formula represents one of the besteccentricities it confirmations can of be Einstein’s developed theory into a of power gravitation. series: In the case of small along geodesic lines in the metricway by of Einstein Schwarzschild’s solution, ([ has been solved in an approximate New approximation methods in General Relativity PoS(ISFTG)015 (2.6) (2.3) (2.4) (2.5) (2.1) (2.2) geodesic is a Richard Kerner s . ) 2 ) . Equivalently, the , ν ) x µν 2 ) g δ . ν ( ) 0 x O δ d s = / + (( ρ µ 0 u O , λ d x = . + ) u e s σ σ ( x x = ( . . µ has conserved its geodesic character, λρ µ δ δ 0 0 , Γ x ) µ ) s ν ρ u δ ( , controlling the maximal deviation from ν = = + ds e x µ dx dx ( µ ρ ρ x µ ˜ x ) + λ ds ds ds s dx d µ du λρ ( ds dx dx Γ λ λ ) µ ˜ = x σ λ x with the line element defined by metric tensor ds ds ∂ dx d x µ λρ 4 µ ( 4 Γ V Ds µ µ σ ) = λρ λρ ) + µν Du s ˜ ∂ Γ Γ ν ( g x + µ + + ( ˜ = x ρ µ µ 2 2 2 µ λρ x ⇔ ˜ x x , we arrive at the following condition on the infinitesimal → Γ δ ds 2 2 ds λ d s d s ) d d satisfies the equation: x d s 0 δ ( ) λ ) = x µ ν = d s ˜ x x δ / ( µ µ u µ λρ µ λρ λ Γ Γ d x ∇ 2 λ = u + = ( parametrized with its own length parameter (or proper time) µ µ µ λρ ) x 2 u ˜ s Γ δ ( ds 2 λ d x denote the Christoffel connection coefficients of the metric : , which can take any value, but the excentricity λ r µ ρλ , x / ) Γ δ λ x Suppose now that a small deformation is produced along this geodesic line: In order to check out to what extent the new curve ˜ In order to compare two close geodesics, one needs to see how a parallelly transported vector Given a (pseudo)-Riemannian manifold ( MG µν we should veriify whether its satisfies its own geodesic equation: Applying the Taylor expansion also to the Christoffel symbols, g is modified when transported alongimal the vector one field or is another defined neighborinfinitesimally along geodesic. close a to In given the fact, geodesic first ifa curve, one. an geodesic it infinites- ? may A natural serve The to questiongeodesic answer can produce deviation is equation. be another positive asked curve, if then, the is infinitesimal the deformation new vector curve satisfies also the so-called New approximation methods in General Relativity iterative process of solving ordinarynot linear differential systems. The smallthe parameter initial here circular will orbit. be Itobtained is already amazing at that the with first this order method in the the effect expansion of in the powers of perihelion advance2. is Geodesic deviations a smooth curve if its tangent vector where geodesic equation can be written in its more standard form: and keeping only termsvector linear in PoS(ISFTG)015 (2.7) (2.8) (2.9) into a in the (2.11) (2.10) ) ε we shall s ( which we ) µ µ µ x x x ( Richard Kerner 0 δ λ y , , and is manifestly ν → x . µ . δ ... x ν µ n x σ λ δ ) + close to the geodesic curve n Ds s ν Dn ( ) 0 x µ s λν λ µ ( ∂ y Γ x ρ 2 µ µ 3 , n x x ∂ ) δ λ ∂ transforms as a vector, the second s , after some algebra using the the ) + . δ u ( s 1 ε ρ 3! µ ( + µ µ x x µ )+ n the Jacobi equation µ δ λρσ δ s x ρ x ε ) + ( R σ 2 b 2 s µ 4 ( σ δ ds δ ; in order to make clear the infinitesimal x dx u 0 µ ) = µ + µ = λ λ x s λ b x ν y ( u 2 ν ) = n ds ∂ ∂ µ δ µ dx s ρ ds x ( approximation - one can say that if the deviation dx n 5 ρλσ 1 µ = δ 2! µ σ R 0 λ x λρσ u λ , ds λ + y R dx ) ) + 2 u s s ] µ linear ( 2 ( = δ µ λν b µ µ µ 2 Γ µ u Ds x νσρ ) x 2 D s R δ δ but = ( λ 2 Ds µ µ ∇ x D , ) + 2 ds µ s dx − ( x δ µ δ µ ), then the curve ˜ x 0 µ λρσ λ ) = x 2.7 y s R ( ∂ ν ∂ ) = s µ ∇ ( ). Although it does not look manifestly covariant, one can put forward its b = µ 0 ε ˜ x = [ λ 2.6 y δ . As in the usual differential calculus, we should expand the functions ˜ µ x δ is also a geodesic, but only up to the terms of quadratic and higher order in ) Now, one can become more ambitious and ask that the neighbor curve to the geodesic be a satisfies the equation ( s ( µ x µ Taylor series containing all the higher-orderapproximation: corrections, which we can stop at the desired order of geodesic not only up todeviation the linear approximation, but up to a given order in powers of the small This first-order geodesic deviation equationcovariant. is Note often that called it representsδ only the deliberately neglected. It should be stressed now that although the first deviation then it is easy to check that the vector defined as x and higher-order deviations do not; for example, after a coordinate change New approximation methods in General Relativity In this form of the geodesicof deviation the Coriolis-type equation and one centrifugal-type easily inertial identifiesthird forces, the represented terms relativistic respectively of by generalizations the Eq. second ( and covariant character by replacing thecompensating ordinary terms second containing Christoffel derivative symbols by andcovariant the their form covariant derivatives of to one, the obtain and geodesic the deviation adding well-known equation: including the terms quadratic inquantity the of first the deviations. same However,character it order; of is we this easy shall vectorial tofollowing introduce denote quantity, manner: a let it let covariant now us by introduce the notation with small parameter Developing the geodesic equation upBianchi and to Ricci the identities for second the ordering Riemann of in tensor, the we second get order the terms: following condition for the vanish- have PoS(ISFTG)015 , ], µ n 11 , al- pro- (3.2) (3.1) ) of the (2.12) s , when ], [ ( ) µ s M 10 x ( n µ 0 x δ Richard Kerner ) and their ex- . , ) the gravitational ad infinitum ρ 2 n G 2.11 φ λ d n 1; ), ( ν θ u = 2 σ 2.7 c u sin − + ν ; these vectors are completely 2 n θ σ d n ] and books, e.g. the well-known ρ 2 u 17 r λ when compared to the mass , u + ) ]). The circular orbits and their stability ( 2 m . λ 16 n 13 , dr ρ µ λρ r u ) MG 14 satisfies an inhomogeneous extension of the r Γ 2 σ 1 ( − σ ) B ds ρ s ∂ − db ( 6 n ν 1 + µ λ λ ∂ 2 b u u ( − dt ) = µ λσ ) σ r on the reference geodesic r τ σν ( Γ ds ( ) dn Γ B 2 . As in the case of the first-order deviation, it is sometimes B ) ρ s + − ∂ ,... ( ) ρ µ λν µ 0 s = b µ λτ ( x Γ ) in the equivalent but non-manifest covariant form ]. σ 2 Γ µ 0 u 2 b 18 ν σρ ds λ , Γ ) 2.11 + u − s ( + by solving the geodesic deviation equations ( µ λρ µ λσ = µ 0 s Γ n Γ ν µ σρ ( τ ρ Γ a desired set of geodesics in the neighborhood of the reference ∂ ∂ dx λ µ ∂ + τ σν Γ dx µ 2 4 . b µν 2 ds M + = g d construct ]). << 8 m Let us consider the geodesic deviation equation starting with a circular orbit in the field of The gravitational field is described by the line-element (we have put Let us recall the essential features of the solution of the geodesic equations for a test particle Now we can proceed to the simple application of these formulae to the two-body problem The procedure can be extended to an arbitrarily high order geodesic deviations ]), but only for straight geodesics; a non-geodesic circular motion in the gravitational field of a 12 specified as functions of the congruence of geodesics isset not of given deviation a vectors priori in closed form. Indeed, all that is needed is the a spherically-symmetric massive body, i.e.analysis in in General the Relativity Schwarzschild has metric. been performed The by S.L. geodesic Bazanski deviation and P. Jaranowski ([ central body. 3. Deviations from circular orbits [ monograph by Chandrasekhar [ constant): massive body was also studied byhave Aliev been and Gal’tsov analyzed ([ and studied in several papers [ of mass lowing us to treated as motion of a test particle with negligible mass with convenient to write equation ( New approximation methods in General Relativity Note that the second-orderfirst-order deviation geodesic vector deviation equation, withbut the with the same extra left-hand terms side on thedeviation as and right-hand for side, its the containing first various derivatives. first expressions deviation quadratic in the first tensions to higher order, for given Again, the above formula isducing valid better up approximations to each the time. third-orderfound terms; in The ([ third-order one deviation can vector continue and its equation can be PoS(ISFTG)015 (3.8) (3.6) (3.3) (3.5) (3.7) (3.4) ), and (which 3.4 R is given by ) , Richard Kerner φ 2. ( / , Eq. ( r per unit of mass π 2 ds ) J / = 2 dr θ ` MG ( 12 − . 1 . , , 0 0 s 2 2 r ` . + = = . + 1 r θ MG -axis. There is another first integral 1 2 ds dr 2 ds 3 z d r ` R MG 2 = − ε = + 2 r 1 − ` θ MG u 1 of special simple orbits, namely, the circular u 1 2 r MG , , 2 ) = = 2 3 2 ` , we have r ` R − 7 R R ε MG dt ds R ) gives the well-known result 1 2 MG constant, the expressions for ε = 2 + 0 − 3.5 = 2 ` ⇒ − r − 1 ω can be integrated owing to the conservation of the world-line 2 ( R 1 2 r r = MG perturbations ε = = = 0 ϕ for stable circular orbits to exist. − r φ = = ds d ds dt ds d 2 2 = ) = ε = = MG r 2 is then directed along the 2 6 t φ ds MG dr ds u d J u 3 ≥ , showing that the circular orbits are characterized completely by spec- ) R − ]. However, to get explicitly an approximate parametric solution to the ,` R ε ( 20 , 2 , ) can in principle be integrated; indeed, the orbital function R ` ( 19 close to a circular orbit. We recall that on the circular orbit of radius − 3.4 2 ) ), must both vanish at all times. This produces two relations between the three s ( µ 3.5 n MGR , and the energy per unit of mass by m / , Eq. ( J 2 = ds Let us write down the four differential equations that must be satisfied by the geodesic devia- Observe that for circular orbits In particular, the equation for null radial velocity gives As the spherical symmetry guarantees conservation of angular momentum, the particleThe orbits angular momentum The equation for the radial coordinate The equation ( From this a simplified expression for the radial acceleration is easily derived: / ` r 2 an elliptic integral [ d Then the null radial acceleration condition ( tion 4-vector dynamical quantities corresponding to the conserved world-lineby energy. Denoting the magnitude of equations of motion one can alsoones. study leading to the requirement is a geodesic in the background Schwarzschild metric) we have: ifying either the radial coordinate, or the energy, or the angular momentum of the planet. are always confined to an equatorial plane, which we choose to be the plane New approximation methods in General Relativity Hamiltonian, i.e. the conservation of the absolute four-velocity: PoS(ISFTG)015 φ n , (3.9) θ (3.13) (3.10) (3.11) (3.12) (3.14) (3.16) (3.15) n -axis slightly z Richard Kerner . . . The harmonic 0 . 0 ϕ n = = 0 0 0 r ϕ 0, too, the only non- 0. n 0. ): and ds φ , dn = = r u = r = , n 2.7 θ φ θ . 0 θ , ds n u t dn n . n = t r 3 ϕ ` R 0 r φφ γ MG n n n 2 R Γ 2 + r 2 = − ) ∂ s − σ 2 2 ` . n 1 = + of the four-velocity on the circular R R MG ε r ρ r 2 φ , using Eq. ( φ u n 2 d ds r 1 and cos ds t n ` u dn λ n − R 2 ) u 2 ds by choosing MG u t 2 = d 1 4 cos u ) ( θ R − MG θ r λρ 4 and r tt n θ 0 2 t 2 R Γ 2 R ε n t Γ MG ), we get 0 , d r 2 u ds σ ds − dn ∂ θ − ∂ 1 3.8 n − ε MG ) = ( + + 8 2 4 γ 1 2 ` 2 R ( φ 2 ` ρ R R 2 + MG ds R ds − R s MG − dn 2 dn 0 2 ) φ − λ = + ω u u − r ( R θ = MG n 1 d n 2 ds r φφ t r λρ 2 2 const., so that sin cos 2 Γ n Γ ) ) − 2 4 2 2 θ 0 φ ds 2 d = ds d 1 ε ` R n R u MG ( R + 2 3 ( ` 2 MG + 2 2 4 R t / 2 − ` + MG r R − 1 3 2 2 displays the frequency which is equal to the frequency of the circular 2 π n ) = ( ds dn 2 s − 2 θ ) and the definitions ( λ = ds ( − R t d = n 1 u 2 θ 2 θ 4 2 θ 3.7 n d r tt n ds λ 2 2 Γ 4 ds ` ). We get without effort the first three equations, for the components ) for d 2 R is independent of the remaining three variables d 3 ds 2 2 ε θ 0 + 2.7 d 3.9 2 ds − n M R r 2 r 2 n const., 2 n 2 2 ds d ds = d R 0. In this case the deviation from the plane will be described by the above solution, i.e. = r = : z t Using the identities ( The deviation Taking into account that only the components The four equations are much easier to arrive at if we use the explicit form of the first-order It takes a little more time to establish the equation for The characteristic equation of the above matrix is n vanishing terms in the above equation are: orbit are different from zero, and recalling that we have chosen to set motion of the planet itself: This can be interpreted as the result of a change of the coordinate system, with a new The system of three remaining equations can be expressed in a matrix form: deviation equation ( New approximation methods in General Relativity because and oscillator equation ( inclined with respect to theplane original one, so that thea trigonometric plane function of with the the orbit periodcoordinate does equal effect, not it to allows coincide the us period with to of the eliminate the the planetary variable motion. Being a pure PoS(ISFTG)015 to R (3.22) (3.24) (3.20) (3.23) (3.17) (3.18) (3.21) (3.19) or con- ) in the 1 in the s = 1.1 ν added to the u Richard Kerner µ ϕ u which takes into u . ; however, before ) ) . ∆ µν ) s s 0 s ( g ω µ ω = , ( ( and n r 0 ϕ t n sin u u 0 t M ∆ R ∆ 6 n is exactly what produces the . − 0 = t 1 ω R n MG , R q , MG 2 . 0 ) 6 r 0 and s − n = MG − ω R 1 , ω MG √ ( ) 1 2 R 2 s MG : 6 ` r sin − ) ω . R 2 ( 0 0 r t 1 − ϕ 0 2 ϕ R MG ω n ∆ ∆ 1 − ∆ > of the unperturbed circular orbital motion. 6 , because we have t R cos r 0 = r 0 2 + + = u q + r 0 √ − n n s s R s 9 R ∆ ϕ n ) ) 1 ) n t r `/ ε = − R ϕ u = MG u , ) reduces to r 0 0: we get both the correct value and the correct sign. 2 u ) 6 4 r 0 = ∆ 2 ∆ n s R ( ` n ∆ ( R MG 0 ) = 3.7 ( − → ω ω s 0 2 ), the solution is the following: ω ( 2 ( 1 2 + ω ω ) r e 2 R = 2 n r ε cos λ r , with the subsequent variations R 3.15 MG 2 r = r 0 ) and ( ` 2 ∆ R 4 n ∆ r 0 MG − λ n − 3.6 2 = + ), the general solution may also contain the following vector: 1 ( ω ` R ω R 2 3 2 MG ) = R R 2 3.15 s ( r − = = n 1 , are arbitrary; 0 t ϕ 0 0 n n 2 4 ϕ = ` R 3 ∆ θ n 0, which means that the radial velocity remains null; and = and r u t ∆ ∆ The only independent amplitude is given by Let us display the complete solution for the first-order deviation vector Obviously enough, the general solution contains oscillating terms cos The discrepancy between the two circular frequencies Let us choose the initial phase to have (with When inserted into the system ( in the matrix operator ( s account only the non-trivial degrees of freedom: we analyse in detail thisstants: part as of a solution, matter of let fact, usto because consider of the the presence terms of linear first and in second-order the derivatives with variable respect This condition coincides witha the neighbor transformation one, of with the radius initial circular geodesic of radius limit of quasi-circular orbits, i.e. when so that the characteristic circular frequency is which corresponds to the perihelion position. Whatwith remains the to fundamental be circular done is frequency to compare this frequency New approximation methods in General Relativity which after using the identities ( corresponding components of the 4-velocity in order to satisfy the condition perihelion advance, and its value coincides with the value obtained in the usual way ( linear approximation. PoS(ISFTG)015 , , 1 ϕ 0 − n ) 2 (3.30) (3.31) (3.25) (3.27) (3.29) (3.26) (3.28) e and − 0 t 1 n ( Richard Kerner and the greater r 0 R . This shows that n , R 0 corresponds to the ) MG s , . = ) with ω s s ( e ω ...... ( sin . , + + 0 t )] R sin n 3 t 3 MG ) ) 0 ϕ 0 3 + n ω R s ( MG MG 3 + ( ( , R s R MG cos π 3 e 2 becomes the dimensionless infinitesimal R 1 MG 135 − ) only through its square. In contrast, we 3 − r 0 135 r 0 R n 1 R R n MG 1 + [ − 6 + 2 q 1 a MG ) 2 , − ) 2 √ q ' 1 R R 2 ) = MG MG ) / 10 s 2 t 3 ( MG from which we obtain the perihelion advance after function was chosen so that R ( 0 4 R ω 2 ) ( 0 ` 2 π T 2 R ω 27 T ∆ ( e sin 27 s ) = + ) if we identify the eccentricity − = s 0 t cos , 1 cosine + = n ) ( e ϕ π ω s 2 R ( ∆ a 3.25 MG ω + + ω 3 s ( MG 1 sin R ) π + ϕ 0 cos 6 1 n R MG ) = r 0 2 t ε n = + ( 0 r s − T ϕ − 0 1 ∆ (exactly like in the Ptolemean system, except for the fact that the Sun is ( ω R ' , which also fixes the values of the two remaining amplitudes, to the aphelion. r 0 T = = = n π ω t r . ; but there is also the additional difference, that the circular frequency of the ϕ r 0 R n = R ], s epicycle 21 with a Then obviously one must have Note that at this order of approximation we could not keep track of the factor But if the circular frequency is lower, the period is slightly longer: in a linear approximation, What we see here is the approximation to an elliptic orbital movement as described by the The trajectory and the law of motion are given by The coefficient considered as a small parameter, and truncating all the terms except the linear one, leads to the our method can be ofof interest very when massive one and compact has bodies, to having consider a the non-negligible ratio low-eccentricity orbits in the vicinity obtain without effort the coefficients in front of quadratic or cubic terms in half-axis containing the eccentricity (here replaced by the ratio Kepler result [ we have placed in the center instead ofa the circle). Earth, and As that a thee matter epicycle of happens to fact, be the an development ellipse into rather power than series with respect to the eccentricity epicycle is now slightly lower than the circular frequency of the unperturbed circular motion. one revolution presence of an where the phase in the argumentperihelion, of and the hence keeping the terms up to the third order in parameter controlling the approximation series withutives powers consecutive terms of proportional to the consec- New approximation methods in General Relativity defines the size of the actual deviation, so that the ratio which looks almost as our formula ( PoS(ISFTG)015 , 1 is − ) ω 2 (3.34) (3.35) (3.37) (3.38) (3.33) (3.36) (3.32) e − ) into the 1 ( , we must go 3.24 ) Richard Kerner , 2 . , , e # , )–( ) # # . This shows that + s i ) ) with second-order t r ) s s ϕ 1 ω s R ( µ C 3.22 C MG ω C ω 2 x ω ( 2 2 2 ( ( ( 2 ) cos sin cos r 0 cos . n : ) R ϕ MG 2 R 8 , 2 MG M ω : = ( R R . 2 MG ) ) δ 7 2 R s ) MG s − 5 6 s 1 − ω ω t ω r ϕ 4 6 + ω 2 b 2 b 1 and + b R ( 2 ( + R 2 r ) ( 2 s 2 ) ) MG ω R . But the particular solution includes the sin − δ R 2 . The expressions on the right-hand side 5 ) only through its square. In contrast, we ) MG sin + ) 27 ε r 0 R 3 s 6 r 0 MG R 0 R d ) ) ( n ds n 3 2 2 ) M 2 ω − R µ ) MG R ) and cos − ) ( ) 72 R n − 6 MG R ) 1 M R 2 MG ( 7 18 s 6 R 1 R 3 MG R MG − )( R MG ( MG 2 ω − 2 )( 2 − + 6 2 has the same characteristic equation of the matrix − 11 − 18 0 2 ) d 2 ds R − M 2 − ( 1 MG ) R 2 1 R MG ( − R ) 3 s 1 MG R ( + 6 1 90 MG 2 )( ( 3 ( 1 ( 5 2 ) µ − ` M MG R − R + , sin R R ( b 2 − 2 MG 1 − ε 1 5 ( MG 1 M 18 , MG 2 − R : ( − 2 6 in order to obtain the geodesic curve ( ( ) 33 − ω ) 1 + s 6 3 µ − ( , ( 2 − x 0 ( − − µ R 2 MG R = 3 5 MG b ω d " 2 ds δ 5 = , , constant ones, and the terms oscillating with angular frequency " " ϕ ) 2 1 2 s t ) shows the explicit quadratic dependence of the second-order devi- R − ) ) C − : 0 , C d ds R 1 2 ε ) R MG M 2 R MG ( ) s 2 ω R 6 6 2 ( 3 ` MG M r 0 ( ) 2 2 2 2 6 4 R h − ` r 0 MG n ) ϕ ) R − − 1 3 2 ( r 0 n r 0 ( b − ( 1 1 2 n n ( ( − R ( ( 1 MG 2 ( R 2 3 2 and R R d ds 2 ) = = s = r r ( d ds = 2 r on the first-order deviation amplitude ϕ , and the general solution containing oscillating terms with angular frequency C 2 ε b r 2 ) δ b are functions 0 µ d s 2 b ds ) and a tedious calculation, we find the following set of linear equations satisfied by ( b R ϕ MG µ 2 C n 2.11 and ) for r After inserting the complete solution for the first-order deviation vector ( The solution of the above matrix for Now we need to calculate In order to include this effect, at least in its approximate form as the factor It is obvious that at this order of approximation we could not keep track of the factor The common factor . Being interested in the quantities directly accessible to observation, we give here the eplicit C , t ω 3.15 beyond the first-order deviation equations andthe investigate quadratic the effects. solutions of the equations describing of no interest because it isterms already linear accounted in for the by proper2 time expressions of geodesic deviation; again, we show only the components ( our method can be ofof interest very when massive one and compact has bodies, to having consider a the non-negligible ratio low-eccentricity orbits in the vicinity system ( obtain without effort the coefficients in front of terms quadratic or cubic in the second-order deviation vector ation vector New approximation methods in General Relativity containing the eccentricity (here replaced by the ratio C PoS(ISFTG)015 . , µ µ Z x x , so : ∈ r 0 (4.1) R n µ (3.39) (3.40) k x : = . For ω e ) e enables us to Richard Kerner + and , where can be written ω 1 i π ( µ 2 ) . a ) h k # r 0 and the eccentricity , 2 R ) n a s ( + and 2 ) ω ) . ) 1 s s 2 containing the functions , represented by exactly ) s = + ( e ω µ ω ω 1 ( 3 ( h = ( ( h ) − 3 sin t r r 0 s ϕ R R 1 h h sin cos n h M ( ω cos ( R r ϕ a = t 32 ω a 2 sD sD − sD O 5 s d ds s + ) + ) + of a matching ellipse compared to the ) + ) ω s + r 0 s s ω R , and another with frequency 3 s 2 a n M ω R resonance terms ω ω 2 ) 3 ω 3 2 2 3 2 ( = cos ( 0 2 ( d − M ds R e s 1 18 cos ( sin turns with angular frequency 2 sin ω M ` R r t ϕ 4 R + 6 µ 2 ) 3 C C C b 12 r 0 sin and cos 3 M R − − R n 5 ( 1 ) ) + ) + ) + , . s , − s and the aphelion when s when second-order deviation is used. The perihelion s M r 0 R s R 2 n 2 ω ω ω d π ( ds R O ω ( ω ( ( k 3 / − 2 = 3 2 r 0 ) + 1 sin d e sin − cos ds n M ( R 2 2 t ε = " r ) sin 3 3 cos ` 2 ϕ 4 B r 0 ` M 2 − 6= R R ) B s R B 3 2 1 n and ( e 0 ( M ω R 2 6 − R R ω 2 3 2 2 2 ) − ) = d + and r 0 ds r 0 1 n a R n ( 0, there is no perihelion advance and a ( 2 d ds 2 = R 2 6= = ( ε → 0 d 2 ds a M R R = 2 R MG , whose circular frequency is the same as the eigenvalue of the matrix-operator ϕ 2 s on the right-hand side. It is easy to see that after reducing the expressions on the δ ω and cos s With the third-order approximation we are facing a new problem, arising from the presence of As a matter of fact, the equation for the covariant third-order deviation ω of an ellipse that has the same perihelion and aphelion distances of the orbit described by in matrix form, with principalthe part same linear differential in operator theseparated as into third-order in two deviation parts, the one lower-order oscillating deviation with frequency equations. The right-hand side is resonance terms acting on the left-hand side. The fact that the second-order deviation vector Matching the radius for perihelion and aphelion, wee obtain the semimajor axis 4. Third-order terms and gravitational radiation right-hand side, which contain the terms of the form and the like, we shall get not only the terms containing sin New approximation methods in General Relativity get a better approximation of the ellipticincluding shape second-order of the deviations resulting is orbit. not Thedistances an trajectory ellipse, described to by but see we that canand match aphelion the perihelion distances and of aphelion the the perihelion Keplerian, is i.e., obtained elliptical when orbit are In the limit case of the second-order deviation increases thefirst-order semimajor deviation, when axis which do not create any particular problem, but also the PoS(ISFTG)015 . r 0 = R n , j ; the (4.3) (4.2) , etc, , i s ω ω 2 where ]). scos , Richard Kerner ? ) ω i . 3 x ( . sin µ R , MG s on the right-hand side , and so on, so that the 4 ω s ), from the origin of the 5 ω P , we can not only recover e cos , j are replaced by theirs values. α is different from 0 t ..., x will be also equal to 0. Next secular 2 ds i dh + 3 α ω 3 , but get also some algebraic relations x ω µ ω α and h 3 m , e ϕ µ α ds ∑ + dh b 2 , = µ ω of a given mass distribution n 2 13 e are complicated functions of i j when dV j Q + r contains only even powers of the small parameter ϕ x , h 1 i r ω x , ω will give rise to the unique non-singular solution of the t e µ ω = + , which is in contradiction with the bounded character of the Z k s ω , etc. k = , 3 D → i j ω ; as a result, the correction Q , ω ω to vanish, while the next term 2 and 1 ω k , ω C 1 , and 4 : ω k r 0 R n B ω = ]: by an infinite series in powers of the infinitesimal parameter, which in our case e ] was first to understand that in order to solve this apparent contradiction, one 38 ω 22 be the vector pointing at the observer (placed at the point are point masses. −→ OP α m Let Similarly, as there are no resonant terms in the equations determining the fourth-order devia- The equations resulting from the requirement that all resonant terms on the right-hand side be The decomposition of the elliptic trajectory turning slowly around its focal point into a series It is well known that gravitational waves are emitted when the quadrupole moment of a mass Of course, it is only a linear approximation, but it takes the main features of the gravitational More precisely, let us denote the tensor The proper frequency of the matrix operator acting on the left-hand side is equal to Poincaré [ 3, see Ref. [ , 2 , 1 where tion, because all four-power combinations ofing sine with and frequencies cosine 2 functions will produce terms oscillat- terms appear at the fifth-orderproduce approximation, resonant as terms products again, of which the willresulting type enable series us representing to the find frequency the correction canceled by similar terms on thethem left-hand here. side are However, rather one complicated.deviation easily We equations observes do forces not that attempt the to absence solve of resonant terms in the second-order of epicycles around a circular orbitsition can of also gravitational serve waves for emitted obtaining ancenter. by approximate a spectral celestial decompo- body moving around a very massive attracting distribution is different from zero, and thetive amplitude of of the the quadrupole wave moment is with proportionalmass respect to coincides the to with third time the deriva- (in origin the of reference the system Cartesian in basis which in the three centerradiation dimensions, of see emitted Ref. by [ thefields system are well not relativistic into and account, themensions wavelength provided of of the the gravitational source velocities radiation (quadrupole approximation). is and large the compared gravitational to the di- same frequency, but the resonancerise terms to secular of terms, the proportional basic to deviation frequency we have on supposed the from right-hand theis beginning. side eliminated The will in term the give proportional differential to equation for is the eccentricity defining the corrections has to take into accountreplacement possible of perturbation of the basic frequency itself, which amounts to the New approximation methods in General Relativity where the coefficients terms containing the triple frequency 3 Then, developing both sides intothe a former series differential of equations powers for of the the vectors parameter PoS(ISFTG)015 . ) ), s to ( can R e P 4.5 ϕ (4.6) (4.7) (4.4) (4.5) ratio, , dt ds = ) R M / s r 0 ( n r ds d f = . Richard Kerner and dt d f ϕ a ) ) 2 2 2 e e − + sin 1 1 , which is not shown ( ( 2 , i.e., s s e that appears in the first are calculated to obtain by means of successive , = + ) r 0 ) ω calculated from Eq. ( 2 2 s R n ) ( e . ) t t ϕ − ( . and 1 j j ϕ 3 ( , with orbit equation and angular 0, so cos allows the two total powers 2 Q R a 2 e dt 3 ) 1 ) using second-order geodesic devia- m 2 2 → d r 0 2 + as function of m R µ r n m ii R M 1 x ( 3 P ( = and Q + 3 dt 1 1 12 . We have the explicit solutions d M O m m emitted by the system over all directions is obtained from R ( M 1 3 4 + ) ) P G t − ϕ during one orbital period (between perihelions), ( s p R i j GM P ; the values of 3 6 14 1 ]): Q cos = 2 3 dt m e − / ) in the limit . 38 d 3 , ϕ e 1 dt + e M R d R i j 2 3 , 1 q π Q ( a ) with our solution 2 calculated with geodesic deviations depend on the 3 − (third Kepler’s law), but an increased one, dt , ) . So we finally get . The choice of d 2 1 P 2 1 i j GM / a 4.4 ϕ 3 ) 5 3 m Q 2 ) a q 3 dt √ Gm 2 e = π d t 5 ε + e √ 2 cos c G 1 = − R 0. Figure 1 shows this comparison for a small eccentricity and a 5 e = − 1 m = T s ( ( 1 we need only the derivatives with respect to in Eq. ( + = → 2 2 ( a T 1 5 i j P P m R a M dt 2 1 dQ = m r 4 5 , we have c G e 0 and ratio. and fix the values of 8 now reads = 15 2 R P e MG m = P . 1 R m , so to calculate ) of the geodesic deviation case has to use s ( | As we want to compare the two total powers We shall calculate the Because the emitted total powers There are many possible ways to compare a Keplerian orbit with a relativistic one. Here we Then the total power of gravitational radiation When applied to Keplerian motion of two masses P t in the Kepler case is obtained from the numerical solution for −→ OP P here because it is a verycalculus large computer expression program. that nevertheless can be easily obtained using a symbolic approximations, starting with Up to first orderbe in equal when tion, to inspect the non-negligible effects of the ratio and we see that the period is not This effect is the directand consequence higher-order of geodesic deviations. the form of angular frequency non-negligible velocity given by the total power and be applied successively to obtain assume New approximation methods in General Relativity coordinate system coinciding withis the approximately center described of by mass ourlength of of solution this the in vector two a is much orbiting Fourier| greater series bodies than form. the whose characteristic motion dimensions It of is the also radiating system, supposed i.e. thatgiven the by the following expression (see Ref. [ an exact ellipse (up to the second order in PoS(ISFTG)015 . The upper 1 m Richard Kerner = M can be considered of the central body ) M ]. M using geodesic devia- / , with 41 P T m , ( 40 . , ) 39 M ]), and once such an embedding / . In both approaches the emission m 27 1 6 ( ], [ < is not a negligible quantity, its presence 29 R MG m ) is kept small, the ], [ r 0 R n . Other applications can be found in problems 28 during one orbital period m M t 15 (i.e., e and arbitrary eccentricity, whereas our scheme is best ] using geodesic deviations can only be calculated if addi- 23 R MG P 1. When the mass negligible when compared with the mass m << ) M / in three cases as function of m ( P The total power contributions to the gravitational radiation formula are included. This approach, but using R M 1 . The curves are taken from the reference [ The next challenge is to include finite-size and radiation back-reaction effects. In the post- Now we shall describe the departure from the initial Minkowskian or Schwarzschild metrics in The shortcoming of this method, which was entirely focused on the trajectories, was the total The two approaches are complementary in the following sense: the post-Newtonian scheme Another expected feature of Figure 1: as Caution is required as the use of quadrupole approximation is not allowed for high values of . 0 , so the exact amplitude and shape of = R M is given, all intrinsic geometric quantities of the embedded manifold can be expressed in terms Newtonian scheme some progress in thismay direction be has regarded already as the been first made. term In in this an expansion aspect in our result terms of the embedding functions. EmbeddingsEuclidean of flat the spaces exterior Schwarzschild are geometry known in since pseudo- a long time ([ of gravitational radiation is estimated usingmation. the quadrupole formula, based on a flat-space approxi- appearing in the Schwarzschild background metric.of More planet’s precisely, trajectory the remain successive valid approximations as long as the dimensionless parameter lack of any variation of theinvariable gravitational for field, all i.e. orders the of Schwarzschild geodesic metricthe deviation. which case This of was test fact maintained particles reduced with the mass validity of the method only to tional adapted for small eccentricities, but arbitrary values of of gravitational lensing and perturbations by gravitational waves. gives better results for small values of (blue) curve represent the result(black) obtained curves by correspond post-post-Newtonian to the approximations, approximations servingsecond-order obtained approximation as via (middle) a the and reference. epicycle the method,e third-order The with approximation lower first (right). order only The (left), excentricity with has the a common value the post-Newtonian expansion scheme, is well developed in Refs. [ as negligibly small, i.e. Figure 1: must inevitably alter thetherefore geometry represented of by a the power initial series in Schwarzschild the metric, small parameter and its influence can be New approximation methods in General Relativity tions converge very fast in respect of the orders of geodesic deviation. PoS(ISFTG)015 . N (5.1) (5.2) which ]; nev- µν 25 o g 2. Here we / ) Richard Kerner 1 , including the of dimension ε , etc., which can embedding func- + 3 N n − ( E r n . , 3 2 = , − 2 r , N . 1 ) , 0 = ν , given by the set of µ 4 − − − −− , V B z N (+ ν ]. Locally, any n-dimensional Riemannian ∂ ,... or A 27 2 z ) , µ 1 ]). defined as ∂ = 16 24 AB η ,... − − −− = B . In the Minkowskian case such a wave can appear even , 3 A µν ε o g (+ + 3 as usual) in a pseudo-Euclidean space , 2 cannot be obtained directly from the embedding functions, but , induced metric with 1 , , µν ) 0 o g µ x = is the ( , and exterior or interior Schwarzschild solutions can be embedded in ) 4 A ν z , V embeddings, which may require a relatively low dimension of the “host” µ = , with signatures A µ z x N E − − − − , yet unspecified, depends on the topology of the Riemannian space under consid- global N (+ : A ; then we consider the effects of short-range behavior of the type 1 A general analysis of deformations of the embedded Einstein spaces was given in [ Here we shall explore not only the first linear approximation, but also the effects of second In the case of spherical gravitational waves the exact solution cannot be found at once; instead, Consider the embedding of a four-dimensional Riemannian space parametrized by local coor- Consider a global embedding of a Riemannian space The metric tensor of The inverse metric tensor − r a six-dimensional are interested in space if the Riemannian spaceple, to the be de embedded Sitter possesses spacewith some signature can particular be symmetry. embedded globally For in exam- a five-dimensional pseudo-Euclidean space manifold can be embedded in a (pseudo)-Euclidean space of dimension The dimension tions z eration, and may be quite high, as acknowledged in [ and third order in the expansion of deformations in powers of small parameter we shall find successive approximationsas considering first the dominant terms at infinity, behaving dinates (denoted by New approximation methods in General Relativity of derivatives of the embeddingthe functions space-time which coordinates. depend Embedding on techniques four werecosmology, like “internal” also the used parameters change in which of certain are signature, models as of in primordial ([ ertheless, only the theoreticalRicci-flat deformations setup of was known considered,Schwarzschild. exact without Einstein any spaces, concrete and in solution the describing first place, Minkowskian or corrections describing gravitational waves. Wetions start of with flat the Minkowskian space-time investigation embedded ofEuclidean in wave-like space. five deforma- or, It if will necessary, appear moreonly that dimensional in the pseudo- third wave-like behavior order of terms the proportonalif Riemann to the tensor embedding can is be in obtained dimensions five are dimensions; necessary but in in order the to case accomodate of it. spherical gravitational wave, more extra give more information about the source. 5. Isometric embeddings and their properties should be computed from the covariantthe components superscript as notation their in order inverse to matrix. make From difference between now the on “basic” we induced use metric PoS(ISFTG)015 . ) q , p N ( (5.5) (5.3) (5.4) (5.6) is the E 4 ˜ V . The last 4 V Richard Kerner ; sometimes a deforma- 4 V . ... . B ) + z manifold. µν µ induced by a vector field in 4 x o 2 g λν and their partial derivatives. Let us ( ... 4 V ρ ∂ V A A ∂ A + z z w and an infinitesimal deformation producing − 2 µ µν 4 ε ∂ 2 g V µρ 2 o + g ε ) + ν B µ z ∂ + x ν ( + ∂ µν 17 A A 1 g v and can be implemented as a local coordinate trans- νρ z ε ε o g 4 2 µ V λ µ + ∂ ∂ ) + µν µ o x g ( λρ AB A o = g η z 2 1 = µν → g = ) we have the expression for its partial derivatives: ) µν µ o g µν λ x 5.2 ( λ ( o Γ ∂ A as follows: z µν ε o g than its part orthogonal to the embedded ) 4 ˜ q V , p N ( E The initial embedding of a Riemannian manifold When seen from the ambient pseudo-Euclidean space, the new embedded manifold Our first aim is to express all important geometrical quantities e.g. the connection coefficients From the definition of induced by the following deformation of the initial embedding functions: Figure 2: and the curvature tensor, in terms of embedding functions the new embedding start with Christoffel connection New approximation methods in General Relativity will be considered asinfinitesimal a parameter background, and its infinitesimal deformations expanded in terms of an result of an infinitesimal deformation ofIt is the obvious initial that manifold such ainto field, its which normal is part defined (in on thepart the sense induces embedded of an submanifold, the internal can pseudo-Euclidean diffeomorphism be metric) of decomposed andformation. a part Such tangent deformations to doto not consider have exclusively any the physical deformationstion meaning, orthogonal having but non-vanishing to it parallel the and is embedded orthogonalambient not parts space always can necessary have less non-zero components in the PoS(ISFTG)015 . (5.7) (5.8) (5.9) 0 (5.10) (5.11) (5.12) (5.13) = # B z Richard Kerner σ ∇ µ ∇ ν . ∇ . . Nevertheless, the most B 0 − z A z B ρ = z . µ ∇ 0 σ ) µ ∇ , B ∇ B = ∇ z z ν leads to the identity A σ λ z ∇ B B ∇ µ ∇ λ z z . . µ ν ν ∇ B ∇ 0 z ∇ ν ∇ ∇ λ we get the identity for arbitrary indices ) = A λ A ∇ 2 µν ρ µν z A , z ∇ z ∂ 0 o ρ µ − R , B A ρ A z B 0 ∇ z ∇ z = = z ν ( ∇ µ ρ ν ρ = B AB ∇ ∂ ∇ z µλ ∇ ∇ η A g ρ z 18 ν µν + ) + ( λρ − µ ν = g B ∇ B ∇ o g ) z z λ ∇ ∇ A B B . The components of the contravariant metric tensor ν z σ z λ z µ AB ∇ ν − λ ∇ ∇ µν σ ∇ η ∇ ∂ A µ o ) it gives ∇ g ν ∇ A z λν µ = ∇ z ν µ g ∇ AB µ A 5.5 ∇ ∇ µ ∇ z η ∂ µν λ − A λ ∇ ρ o z ν Γ AB ∇ AB ∇ ρ ∇ + η η ν ∇ µ ( ∇ − µν = AB ∇ µ g η − = ∇ λ µν B o z ∇ g σ ) looks compact, but is in fact highly non-linear and complicated. This is so AB µν λρ ∇ η ν o R ∇ 5.13 A z ρ ∇ µ ∇ " : λ AB The expressions derived in this section will be very useful in the development of a power series Using this result, let us form the following combination of covariant derivatives which vanishes The definition of the Riemann tensor by means of derivatives of the embedding functions , η ν , µ because it contains many Christoffel symbolscontain involved the in contravariant the metric second covariant tensor derivatives, which expansion of infinitesimally deformed embedding. are obtained as rational expressionsimportant in point third here and is fourthembedding that powers functions, the so of that Riemann the tensor Einsteinwill equations depends lead expressed only in to terms on second-order of partial first the differential embedding and equations. functions second derivatives of New approximation methods in General Relativity When substituted into the definition ( Now, an alternative (although implicit) definitiontion of that Christoffel states the symbols vanishing is of contained covariant in derivatives of the the equa- metric: which after substitution Taking the combination identically: Applying the derivation and using the Leibniz rule we get: Recalling that so that we can write given by formula ( which is the well known Gauss-Codazzi equation. PoS(ISFTG)015 . . ) q B , z p N ( ν (6.1) (6.4) (6.2) (6.3) (6.5) ∂ E A there is z µ ) ∂ µ x ( AB A Richard Kerner η v keep the internal ε = ) q , µν p ) + N ( o g µ in a pseudo-Euclidean E x ( 4 + ... : o A ε V . z . # B 3 ) + . The deformed embedding z , µ ε ν 2 x B , ∂ ( z 1 A ν A , v ∂ 0 w µ A satisfy the identity ) 2 ∂ w µ = ε B A x µ + , , ( ν ∂ Λ 0 B B , A ) + z v , z µ + µ ν = B A x B ∂ µν = A ( C A Λ h w ... z A A , ν Λ ε : ε z µ v ∂ + N ∂ N + ε A CB + , z η A µν 19 q ,... = µ , z ε µν 2 g 2 N p ∂ ) + + q , 2 o g µ E + = 1 C ε B + x + B A = ( B Λ z ˜ p z + = v A ν as dynamical fields. We know however that most of them z ν → µν ∂ AB µν → ∂ g A ,... η 1 µν g z A A B 4 h v ε µ ) = z , , with o µ ∂ µ V ) A + ∂ x " which will not alter the intrinsic geometry of the embedded mani- ( − ) q ; A µν AB 2 ˜ , z 4 µ o η ε g ˜ x V + ( with → p Const. obviously do not change the internal metric = + = A can also be developed in a series of powers of ( ) v = 4 µ ˜ µν V x A ˜ ( g v components of A z ten with signature q constant matrix, , N p B A E Λ Let us consider an isometric embedding of an Einsteinian manifold then the first-order correction vanishes if the matrices The geometric character of our approach enables us to eliminate unphysical degrees of free- New approximation methods in General Relativity 6. Infinitesimal deformations of embeddings space Consider now an infinitesimalterms deformation proportional of to the the embedding consecutivedefines defined an powers Einsteinian by of space a a small converging series parameter of The induced metric on Among possible infinitesimal deformations of the embedding functions a large class of functions fold. The translations metric unchanged. Indeed, if we set Also the generalized Lorentz transformations of the pseudo-Euclidean space with defining infinitesimal rigid rotations (Lorentz transformations) of the pseudo-Euclidean space dom using simple geometrical arguments.linearized equations Remember for that gravitational in fieldsmetric the the tensor: starting traditional point approach is leading the to following development of the thus introducing do not represent real dynamical degrees of freedom due to the gauge invariance. The metric tensor PoS(ISFTG)015 a or- (6.7) (6.9) (6.6) (6.8) to the . To take ) 4: λ ]), although x − span a basis ( satisfying the tangent 37 Richard Kerner k ) N v , in other words B A 4 v z ], [ V µ ,..., 36 ∂ 2 that is . On the other hand, , ],[ 4 1 = N V 34 A E = z can be decomposed along k µ 4 functions 4 V ∇ , − ) k A ( N ) represents four degrees of free- X µ tangent to . ). ξ . 6.6 0 N ) N ν describe the same gravitational field. λ E E = ∇ must satisfy the following obvious x ; therefore any vector ( B µν 4 4 + ) z g k V V A ν ( µ ξ 4. This means that general non-redundant X ∇ . µ 0 A and − )) ∇ v ν = N x µν + (as seen in ( AB 20 k g 4 η to µν v µν V h 4 1 g A 4 independent fields = − µ = v ∑ k N B = − ∇ v N A ; therefore its global deformations can be described by a µν z ) ) = ˜ g µ µ ) imposes four independent equations, which reduce the num- x ∂ ( → A 6.8 AB v describes nothing else but a diffeomorphism of ; this is why in the linearized Einstein equations one may impose η µν 4 g µν − − − − ]). V g generating gauge transformation ( ) is transversal to tangent to the submanifold 25 µ (+ can be chosen at will provided they induce a non-singular global vector N ξ to the embedded hypersurface 6.8 ) E λ : x (see [ the four partial derivatives (let us remind that ( ) ) k λ A A ( x if we note that any vector field in the embedding space ( X ) v µ x transversal ( with signature A , while the relevant degrees of freedom are contained in v 4 4 , 5 V 1 E One may ask the following question: what if the deformation destroys the initial symmetry of Vector fields tangent to the four-dimensional embedded manifold The orthogonality condition ( The unphysical degrees of freedom can be easily eliminated from the embedding deformation -dimensional pseudo-Euclidean space but needs a flat embedding space of higher dimension ? It deformations can be decomposed along single function the embedded manifold so that the deformedN manifold cannot be embedded in the initiallyis sufficient known that global embeddingsfor of the the embedding of Kerr the metric exterior need (or interior) more Schwarzschild than solution (see six [ flat dimensions sufficient an example, the de Sitter spacespace can be globally embedded in a five-dimensional pseudo-Euclidean Schwarzschild’s metric can be obtained(the from angular Kerr’s momentum) metric tends as to zero. a limit when the Kerr parameter ber of independent deformation functions dom which are redundant in four gauge conditions e.g. a coordinate change, which has no influence on any physical or intrinsicfour geometrical arbitrarily quantities. chosen independent smoothvector vector fields fields in thogonality conditions field on New approximation methods in General Relativity itself does not correspondchanged to by a any gauge directly transformation without measurable changingis quantity. the the components source of of In the measurable Riemann fact, gravitational tensor effects. its which In components particular, the may gauge be transformation embedded Riemannian space The basic fields functions does not alter the Riemann tensor so that both ˜ The arbitrary vector field For any value of of four vector fields in orthogonality condition ( PoS(ISFTG)015 = α z (6.14) (6.11) (6.10) (6.12) (6.13) , N E , : # ... # Richard Kerner µν σκ g 1 g j v ρλ ν 1 g ∂ i v λκ µ o g ∂ νσ o i j g Accordingly, any deformation η . µρ B m + z o g ν + " ∂ ... 2 B : N A z ε m v + ν µ E + ∂ , , it would contain the initial embedding , ] ∂ , A # µν m ,...., k m 2 g w 2 + v + ! . ρσ . To this end we must calculate the approxi- E 2 , µ . If the covariant metric is decomposed as in ] N ε i j B 4 2 g ε 1 ∂ 0 ˜ v , m ⊕ E V η µν ν v A + = + νσ N g v ∂ , o B g AB A E A 0 ε µν 21 z v w η ,... 1 g µ = µρ ν + β
ε ∂ ∂ o g A , m = [ . A z " + + = α z ε α ... 2 ... N µ , v AB ε µν ∂ = [ E + αβ m + o g − α η + ε η ˜ z ,.. µν µν = B 2 2 g ρσ + 2 v g , 1 2 g ν 2 1 µν ε ∂ ε g B A νσ = z + v + o g ν µ ∂ µν ∂ ,... µν A µρ j 1 z g 1 g o , g µ i ε ε ε ∂ AB , + η + − N " µν AB 2 µν µν ,... o η ε g o g g 2 , = + = 1 = = = µν µν g g and another one in the complementary subspace as its linear subspace, so that ,... N N B , , therefore the deformed embedding functions can be written as E E ] A The answer is that as long as we investigate only the first-order corrections to geometry, we From this one can see that the deformations towards extra dimensions do not contribute to Our principal aim now is to establish the explicit form of connection and curvature components 0 ), , A z 5.3 then we have the following formulae defining the corresponding decomposition of But the initial embedding functions had their components entirely in the first subspace should not worry about this issuegiven, its for infinitesimal the deformations following cannot two lead reasons: toif a first, global the when modification bigger of a the embedding global embedding; space embedding second, is was introduced, say New approximation methods in General Relativity of the initial embedding canspace be decomposed in two parts, one contained in the initial embedding [ space and its pseudo-Euclidean metric could be represented as a blockwise reducible matrix with so that the induced metric of the deformed embedding will be the first-order corrections offunctions. any This geometrical is quantities why we obtained shallterms from not linear consider the in such the deformed deformations infinitesimal embedding while parameter investigating at first only the induced on the infinitesimally deformed embedding mate expression of the contravariant metric( tensor PoS(ISFTG)015 . (6.18) (6.20) (6.16) (6.19) (6.15) (6.21) (6.17) . µν o g B z ρ , ∂ σ Richard Kerner ∂ . − B z ε µν σ µν µρ o o 2 g µν Γ o . g ρ ∂ ) A ν ∂ 2 v A ∂ ε v ρ − ( ρ + ∂ . B ˜ ∂ z O µρ + . νρ ρ B + o + g ε B z ˜ o g ∇ B ν v # ν µ µ v ∂ A σ ∂ ˜ ∇ z ∇ ∂ + 2 µ µν A λ ˜ ∂ z ∇ ..., µν σ ∇ νρ σκ A ) λ A o z Γ o 1 g 2 g + v ˜ ∇ µν λ ρ µ A ε ρ ν ∂ z ( ρκ 1 ∂ Γ µν λ ∇ ˜ ρ o ∇ g O 2 Γ ∂ − + 2 − + λρ A λρ λσ B ε B v A o g 1 o v g g ˜ z v λρ ν + ν ρ 2 2 1 1 o ν g AB ∇ ˜ ∇ ˜ ∇ η µν ∇ µ λ 22 + − µ ν AB µ 1 Γ ∇ ˜ ∇ = η ∇ ˜ ∇ ε A ) we find the final expression A − ε ε z ˜ µν µν z + ρ 1 1 g g λ = yields: + + µν ρ ρ ∇ ˜ 6.18 µν λ ∇ A A 1 A g ∂ ∂ z z z µ o ρ Γ ν ν ˜ ∂ − − ∇ µν λρ ˜ = ∇ ∇ o g − o g µ µ µρ µρ ρ ) and ( ˜ AB 1 1 µν ∂ λ g g ∇ ∇ AB µρ ν ν η Γ 1 η g − ∂ ∂ = = 6.17 ν = = ∂ + + µρ A will be useful for the derivation of geodesic equations in the deformed ˜ z ), ( o g + νρ νρ µν ν λ ν 1 1 µν λ ˜ g g ∂ 1 νµ λρ ∇ Γ νρ ): µ µ 1 ˜ 6.17 Γ µ 1 g R + ∂ ∂ ˜ µ ∇ ∂ νρ 5.13 o g λρ λρ , transform as connection coefficients, therefore their difference must transform as µ o o g g λρ ∂ µν λ 1 1 2 2 o g ˜ Γ 1 2 = = λρ 1 g µν λ 2 1 and 1 This expression has a tensorial character as it should be, because by definition both quantities The coefficients Let us start by computing the first (linear) correction to the components of the Christoffel In what follows we shall keep only the first order terms linear in Γ µν λ Combining together ( The Riemann tensor induced on theprevious deformed section embedding ( is defined by the same formula as in the a tensor, and this is true for any term of the development into series of powers of Γ space-time, but they areRiemann not tensor, necessary which for can be the determinedof computation as the of follows. deformed embedding the Let functions us first-order ˜ develop deformation second covariant of derivatives the One easily checks that connection, which develops in Taylor series as then by definition we have: New approximation methods in General Relativity which shows that also theof contravariant the metric embedding tensor functions, depends although in exclusively a on quite the complicated first manner. derivatives while the second term after some algebra gives PoS(ISFTG)015 (6.22) (6.25) (6.26) (6.27) (6.23) (6.24) Richard Kerner ) to the Einstein ε . . B z o R λ ) into the definition of B z µν ∇ ρ 1 g λ νρ 6.20 ∇ 1 2 B ν 1 Γ , but also the “deformed" co- z ); therefore the first-order cor- A ∇ − A ρ z z A , just two types of terms linear in ∇ λ v λρ 2 µ , λ 5.10 ∇ o 0 (6.28) ε R ∇ µ ∇ = A λρ µ ∇ . v 1 µνλρ g ∇ λ µν λρ AB o R λρ µν + o 1 ∇ R R 1 ν R o g B ε η νλ = v ! µν ∇ 1 g 1 2 1 λρ R ρ 1 g o g − λρ + − ∇ µν o + g B ν µν v o 23 g and λρ ∇ o µν ρ g µν 1 2 1 A R µνλρ o 1 g ∇ 1 2 z R 1 R B − µ 2 1 λρ λ v − µν o o R g ∇ ∇ ρ νλ − o g A µ o µν g z ∇ ρ µν µν ν ∇ = ν 1 R 1 o λ δ g g = 1 ∇ R λ ∇ µ 2 2 1 1 ), we shall encounter, besides the zeroth-order initial Riemann A ν δ AB µρ z − − η ∇
1 λ R 6.21 µν µν ∇ = − µ 1 1 R R ∇ = = AB νµλρ . Now, when we insert the expressions like ( 1 R µν µ ε η 1 ˜ G ∇ and the second-order corrections proportional to we use not only the deformed embedding functions ˜ 4 ˜ V µνλρ o R In what follows, we shall always suppose that the initial Riemannian manifold is a solution of Note that in order to calculate the components of the Riemann tensor induced on the deformed : ε The terms of the second type vanish by virtue of the identity ( Einstein’s equations, i.e. an Einstein space whichvature, is too. Ricci-flat and Therefore consequently the has linear zero correction scalar cur- (of the first order in small parameter Riemann tensor components ( tensor rection to the components of Riemann tensor can be written as follows: Finally, The first-order correction to thetions is: Einstein tensor, i.e. the left-hand side of Einstein’s equa- tensor will reduce to: In the absence of any extra gravitatingthe matter central (besides spherical the matter body generating forform the basic of Schwarzschild’s solution, a solution) e.g. matrix the acting equations on the to first-order solve correction can to the be Ricci written tensor: in manifold New approximation methods in General Relativity variant derivations To establish the form of linearof correction the to first-order Einstein’s correction equations to we the needcomputed Ricci to as tensor know and follows: the the components Riemann scalar. These quantities are readily Consequently, the first-order correction to the Riemann scalar is: PoS(ISFTG)015 0 ε = (6.35) (6.34) (6.31) (6.32) (6.33) (6.29) (6.30) should λρ o R Richard Kerner ). in vacuo − , reduces then to , B ε w ), and the first order µν 6.19 ν T 6.3 ∇ ρ G ν µ ... 4 δ ) π c ∇ µ 8 λ µ A x δ z ( − ρ A = = h ∇ . λρ 3 , one readily gets: T ! + ε 2 ) µν # B B ε 1 κσ R z v . o g λρ ) + ν # 0 σ o µ g ∇ ∇ x µν λρ = µ ( A o g µν o g z A ∇ o g 1 2 ρ B A w ˜ µν z ), can be written equivalently as 1 2 ∇ 2 ν w . o − g ε ρ ˜ 0 ∇ − + σ 1 2 ν ∇ A B ρ δ 6.29 ν = ˜ z 24 z ) + − δ κ µ + µ µ σ ρ δ λ ν µ λρ B ˜ x ∇ δ ∇ v ( δ 1 R λ A λ ν A " µ ˜ v ∇ ! ! v δ ∇ ρ G ε 4 " µ λρ π c ∇ o ∇ AB ( g 8 = ) + A η − v µ µν σ κσ λρ ρ x 1 o g Γ ( = 1 ) which is valid also on the deformed manifold: R ∇ 2 1 A − z λρ µν − o g 1 5.10 R λρ ρ ) is non-singular; in fact, it is its own inverse: σ ) = o g δ µν µ o λ g x κ AB ( 6.28 δ of the deformed embedding is given by the formula ( 2 1 η A ˜
z − = µν g µν µν λ 1 R 2 Γ of the formula ( ε 0: = From the expansion, identifying the terms proportional to Let us expand the deformed embedding functions into the power series of small parameter The second order correction to the Christoffel connection can be found from the expansion in In the case when the energy-momentum tensor is present (supposing however that it describes o R as before, this time keeping the terms up to the third order: correction to the Christoffel connection coefficients was already given in eq. ( powers of and this in turn, due to the idempotent property ( which may prove to be more practical fortensor further has calculations a especially when particularly the simple energy-momentum form. The metric tensor ˜ New approximation methods in General Relativity But this amounts to the Riccion flatness the up Ricci to tensor in the ( first order, because the operator acting on the right and the influence of matter weak enoughthe in full order Einstein’s to tensor keep on the the basic solution right-hand unchanged), side. one must The use first correction, linear in From this we infersatisfy that the in equation an Einsteinian background the first-order correction only two terms due to the fact that the initial solution is an Einstein space in vacuo so that PoS(ISFTG)015 (6.38) (6.37) (6.39) (6.40) (6.36) , µρ λ 1 Γ Richard Kerner , κ νσ ) 1 Γ ν ) ν − + ↔ + . ↔ λ νρ µ + 1 ) ( Γ B B µ ˜ z z ν ( − κ µσ ρ κ B z − B ˜ 1 ∇ ↔ ∇ Γ z κ + µ ρ λ νρ µ µρ κ ∇ ˜ B ∇ ( ∇ 1 µνσρ Γ 2 z Γ κλ µ A µρ κ o µσ o R ρ g ˜ − z 1 o − µσ κ ∇ Γ R B ∇ λ + µσ 1 B − v Γ µ ˜ − ∇ v µσ 2 g ρ A B ν ∇ κ B B 2 z g z κλ ∇ A ˜ v v + . ∇ κ 1 ∇ ρ g ν + h ρ ρ + ), because they contain the products of first ∇ ∇ σ ∇ − µσ + µρ κ ∇ ∇ B µ A 2 B ∇ ν µ g z µσ 1 κ νσ Γ ˜ z µνσρ w ν ∇ ρ 1 R ∇ ∇ 2 Γ 1 5.10 A ρ R σ − ∇ A λ νρ ∇ v w ∇ µσ B ∇ ν − 2 Γ µσ 25 + ν σ σ µ 1 g w A ∇ 1 g ˜ B and in A . This gives the following result: v ∇ ρ ∇ ∇ A ∇ µσ κ 0. z h + ε κ A ν µ + ∇ w 1 ˜ κ ρ Γ z + = ∇ ∇ µ ∇ σ ∇ λ µσ ∇ functions. To write down the second-order correction to B + ∇ ∇ µνσρ ˜ 2 µ + R κ νσ − ∇ w v µσ κ νσ µ 2 R µ B 1 µνσρ ∇ B ρ λ νρ 2 Γ 1 µσ Γ ∇ v ˜ 2 A A R w 1 R ∇ ∇ Γ z z ρ o − g ρ ν and − − σ σ A µσ ∇ µσ κ ∇ ∇ z A B = ∇ ∇ µ o AB g w µ , and the correction to Einstein’s equations reduces again to the 2 v A Γ ν ν w 2 η v σ R ∇ σ ∇ ρ = ∇ ∇ A σ µσ ∇ A − ∇ v ∇ 2 z 0. With a little algebra we find: ν ∇ 0, the second-order correction to the scalar curvature contains only are zero according to ( ν ν σ νρ κλ R σ = µ ∇ ∇ 2 AB o 2 g Γ AB = R ∇ ∇ = ∇ ∇ η µσ A η ν ν B z o z ∇ ∇ νρ σ g ν = + + + = + µν λρ 1 ∇ ∇ ˜ R R µ A AB AB z ∇ η η λ µνσρ µνσρ ∇ 3 2 R terms contain only R + − = µ 1 Γ ∇ 1 Γ -functions contained only in AB w µνσρ η 2 R Finally, the third order correction to the Riemann tensor can be obtained by expanding the Obviously, if the first-order deviation was Einsteinian, i.e. not only the initial space-time was with the Gauss-Codazzi formula applied to the deformed manifold: where the Einstein equations we need also the explicit form of the correction to the Ricci tensor: and the scalar curvature vanishing second-order Ricci tensor, and expanding all the quantites in powers of the first contribution, Ricci-flat, but also and second covariant derivatives of the originalmetric, embedding functions contracted with the Euclidean New approximation methods in General Relativity As we shall see, theorder third expansion term order of corrections the to RiemannThe the tensor, second therefore order Christoffel terms we symbols in do do the not expansion not write of them appear the Riemann down in tensor explicitly the are here. given third by the following formula: The terms proportional to PoS(ISFTG)015 all hy- ] (7.1) z 4 is the , (6.41) (6.42) y M A , v x , ct Richard Kerner = [ 0 µ x = , B : v ε ρ orthogonal to the 0 (7.2) µνσρ ∇ , 4 o R , = µ . Let us choose the simplest 5 1 ) µσ ∇ E µσ o v R − ... , A 3 g ) v ρ 0 1 µσ λ + ∇ ) + = 3 g µ − ∇ µ 5 ν x ∇ z + N ( v ∇ ( µνσρ h , , λ µσ 1 3 − R z + 1 ε B R ∇ 1 v ν = µσ ( 4 µσ ρ , 2 4 g ∇ 0. 5 ) + 1 z 2 g ∇ µ , E + − ν = x y + v ( ∇ 26 → ρ = w µν A µσ 4 2 v ∇ 3 3 R µνσρ 2 R z ε ν λ 2 M R ) can be replaced by partial derivatives, and all second , ∇ ∇ x µσ v µσ µ ) + 1 g = 1 λ parameterized by cartesian coordinates µ g ∇ 6.21 ; therefore the only non vanishing component of x + 2 ∇ 4 4 ( + z µ v V , M AB µσ ∇ ε 3 ct η R = µνσρ λν = µσ λν 3 5 R o g 1 o z o g g z µσ = ⇒ = o g = 3 R µρ = 0 2 R = νρ 3 R µρ 2 R . This leads to the following equation resulting from the requirement of vanishing of the ε In a Minkowskian space-time Again, in order to write down the third correction to Einstein’s equations, we should get the with the first four components denoting a Minkowskian space-time vector in cartesian coordinates: case of embedding in five dimensions: the last cartesian coordinateperplane. considered All as covariant an derivatives extra in ( dimension of remaining fifth coordinate deformation, expanded in a series of powers of In order to keep the Einsteinmust equations have satisfied after deformation up to the second order terms, we derivatives of linear embedding functions are identicallytrivial zero. deformations Therefore of in order the to Minkowskian investigate spaceorder non in embedded as a hyperplaneRicci we tensor: must go the second Again, because the third-order correction to the scalar curvature is given by the formula and if the previous two orders were Ricci-flat,reduce the to third order the correction Ricc-flatness to of Einstein’s the equations third will order, 7. Plane waves in a flat background space-time connection coefficients identically vanish,tensors. as well The as flat the Minkowskian spacespace components with can of more be the than embedded Riemann four as and dimensions a and Ricci hyperplane signature in any pseudo-Euclidean We shall not consider infinitesimal deformations ofwith the coordinate first four transformations coordinates in because they coincide explicit expression for the third order correction to the Ricci tensor. New approximation methods in General Relativity PoS(ISFTG)015 ). 7.2 (7.3) (7.6) (7.4) depends on w Richard Kerner , 0 = µ k ρ k -axis: z − 0 (7.5) ρ k = ) µ y k , ρ x . . For the sake of simplicity, let us start k 2 ( coordinate are identically zero. ) 0 k µ 5 w v kz k z 2 ), this deformation does not contribute to ν ) λ ε k − k µ ν t only: 7.5 x ν k ω y k ) + µ ( k = kz − ( f ρ and 27 2 cos − 0 k t x v ν A ) results in the following simple equation : ρ k ω ) = ( k λ µ µ k x ) = 7.2 k ( µ µ cos v λ x k k ( A ν v ε into ( k = ) = − µ 5 2 x 0 z v µ as a plane wave propagating along the µνλρ k ρ 2 ( 4 R k v ν M k λ k µ k λν o g ) in which the covariant derivatives can be replaced by partial derivatives given that ) given in the previous section. 6.40 6.40 According to our general analysis, by virtue of ( The linear contribution coming from the expressions containing third-order deviation linearly A wave-like behavior of the Riemann tensor can be produced if we assume that The fact that there are no wave-like solutions at the first order of deformation of Minkowskian This means that the only hope to produce contributions to the Riemann tensor behaving like inserting the derivatives of The only contribution to the thirdformula order ( correction to the Riemann tensor has the form given by the does vanish because the derivatives of the corresponding Indeed, if we set: variables orthogonal to the worldlines parallel towith the the vector first order deformation inMinkowskian the hyperplane direction of fifth coordinate, i.e. orthogonal to the embedded the Riemann tensor, which remainsorder zero deformation even depending at on the the variables second order. Now let us add up the second New approximation methods in General Relativity Any function of linear combination of cartesian coordinates is an obvious solution of Eq. ( spacetime suggests that the same situationmanifolds will embedded prevail when in we shall acontrary investigate other pseudo-Euclidan was Einsteinian flat true, one space,which could e.g. would keep contradict the the the wave-like Schwarzschild absence propagatingMinkowskian space-time. of deformations solution. such also solutions in If among the the the flat first-order limit, deformations ofa propagating the gravitational field, i.e.order the deformations gravitational of embedded waves, Einsteinian is manifolds. totensor in The consider the third the case order third of variation (and deformations for offormula higher) the all ( Riemann orders orthogonal to the embedded manifold reduces to the But in fact, thiswhich deformation is does the not only observable have quantity, any identically vanishes: physical meaning, because the Riemann tensor, The vanishing of the Riemannlike tensor a is deformation not of surprising, a because plane into the a deformation cylinder, considered which looks does not alter its intrinsic flat geometry. PoS(ISFTG)015 B (7.7) (7.8) (7.10) (7.11) ) is the ) is sat- , with 7.7 7.9 Bxy Richard Kerner ) = y , x ( must have some non v w ) . y 2 2 µρ , v k x ∂ ρ 2 ( ∂ c w w µ 2 ∂ xy = : let us put ∂ 2 w y , , because the Minkowskian metric v v − ω 2 v . Taking into account the form of xx v ρ ∂ B w can be easily generalized. As a first 0 (7.9) 2 2 µρ xy µρ 2 C and µρ 2 µρ µ ∂ ) ∂ 2 − ∂ 2 = x = µρ ∂ y 3 R v 2 ) , A ∂ C ): v = w B x 2 2 xy C v 2 ( µρ 2 ρ o xy g 2 C y 2 − − − µρ 7.8 2 ∂ , etc. w xy ∂ yy ∂ y , the only non vanishing term in ( + ∂ + µ = = = w y 3 R A µρ v v v 2 µρ x − xx 2 v o 3 ∂ ( g R Bxy 2 2 2 µρ µρ µρ ρ 2 28 − − B and ∂ x ∂ ∂ + ∂ , 2 − and on w w w = = A w , two other components of Riemann tensor will appear 2 x 2 2 xy 2 = xx yy component ( y Ax ρ ρ ∂ 2 ∂ ∂ µ ρ ) xy yy = ∂ = y − − − µ µ xy , ρ 3 3 µ R R x + w = of the Ricci tensor vanish if the same condition ( xx 3 ( = = R yy µ w . ) ρ o g ρ ρ 1 3 R ρ xy µρ 2 x yy − yy xx µ + o ( g ∂ µ µ 3 R ρ 3 3 v , imposing the dispersion relation cm R R y v xx 2 µ and µ 0; therefore, to make the Ricci tensor vanish up to the third order means ∂ 3 R ) ; so that we have = xx B xy o g xy = does not depend on ( o g = , ρ v ) w xy y µ xx ∂ ( 3 R x ∂ ), the only non vanishing components are: Now, given that Then the only non vanishing second derivative is This particular form of the “modulating" function 6.40 and all other components obtained fromknown this symmetries one of Riemann’s by tensor, permutations like of e.g. indexes allowed by the well that the following equation must be satisfied: one containing There is no contribution to the Ricci tensortensor coming is from diagonal and The components isfied; but now we shallwhich also will make be sure true that if the all following other trace components is of zero: the Ricci tensor vanish, too, New approximation methods in General Relativity all Christoffel symbols vanish in cartesian coordinates. The function vanishing second order derivatives; let us make the simplest choice and set ( This is the wave equation for = Const having the dimension step, let us consider an arbitrary quadratic form in variables Besides the non vanishing component now: which have the same structure as the PoS(ISFTG)015 and , the v (7.12) w , which , typical φ i ixy 2 2 e 0. The only + Richard Kerner 2 = y w − 2 satisfying the two- 2 , provided it satisfies ∇ ) , the vanishing of the embedding functions x y y x is linear both in , , thus leaving only two x C ( and − w x linear µνλρ character of the gravitational = 3 R ) 3 A 3 ε ε ( as the modulating function will take on the factor O y , , becoming infinite at spatial infinity, iy x -plane) Riemann tensor, multiplied by y represent a physically acceptable case. y , ) + + , y y x , namely, x , x quadrupolar ( x C 0. Finally, we can generalize our result by ( φ , i and w e = x 2 and txxz ε w 3 → R A 2 yy 29 , make vanish not only the Ricci tensor, but also Rie- iy ∂ ) + should depend on the angular variables. We should w + txyt kz + A 3 R x w − w t 2 xx ∂ ω ( . This suggests the plane, w cos y , which is acceptable for an infinite plane wave. , one can form their complex combination ) ε 0 (or a constant, which would not have any physical meaning at − xy lead to a constant (on the kz x = = w depended only on one transversal variable, say 5 satisfies the two-dimensional Laplace equation − w z t if it has the form and far from the source should contain a factor propagating in radial direc- w w ω 2 ε A ( y v − and 2 2 x k 2 c = 2 ω However, although from a purely mathematical point of view an The quadratic functions The particular form of plane wave solution suggests also the form of a spherical wave. The But if we take a cubic or higher degree polynomial in Taking into account that the corresponding Riemann tensor The same is true for any homogeneous polynomial of two variables ). , we can compose by superposition a transversally polarized plane wave of arbitrary shape and A not expect total vanishing of the second-order correction to the Riemann tensor like it happened mann’s tensor as well (due to the second derivatives of the zero-order first-order deformation tion, while the second-order deformation w spectrum, propagating with the phase velocity equal to the speed of light. dimensional Laplace equation becomes ations, solution we to claim the that third-order only correction theThis to quadratic is Einstein’s because functions equa- constant of and linear functions z the oscillating factor cos for the spin 2fields. 8. Spherical waves in a flat background New approximation methods in General Relativity leading to the extra condition ondegrees of the freedom coefficients for the function wave, which deforms the space simultaneouslypropagation; notice in that two if directions perpendicular to the direction of the two-dimensional Laplace equation stating that the deformationdimensional of Euclidean Minkowskian ambient space-time space embeddedorder leads in as to small a parameter the hyperplane vanishing in of an the five- Ricci tensor up to the third Riemann tensor will become linear (or higher degree) in under the rigid rotation in the all). Ricci tensor would impose provided non-vanishing components of the Riemann tensor are then which is physically unacceptable. Therefore onlythe the second plane degree polynomial gravitational fully wave, characterizes polynomials conveying being it only two degrees of freedom. The two independent PoS(ISFTG)015 t (8.3) (8.1) (8.2) and r in vacuo Richard Kerner 0 and the only = v µνλρ r 1 R θ ∂ can be neglected and in fact 2 3 , ... − sin θ r r 2 ) + or . θ φ φ θ = µ v sin 2 θ θ 2 x 1 R v 2 . r tt − ( − ) ∂ ϕ r 5 sin cos − r , keeping only the radiative part. Let us h r ∇ kr r = 3 ϕ / = = = ε v − θ θ ∇ embedding functions do vanish identically. t r t ) φ , we need to insert the expressions for second r 2 ϕ r , R r ∇ φ θϕ r ϕϕ ω as seen from the host space, this deformation ) + t v , Γ ( Γ Γ r t the non-vanishing combinations are: µ ∇ 4 ( ∂ ), it is easy to see that x r r 5 ( r µνλρ M linear cos v 30 5 , 2 R θ θ 8.3 θ v = w t A = 2 and tr 2 2 R v 5 t ∂ . We suppose that is depends on the variables ε cos θ v 4 ) = t = θ ∇ M µ θ ) + v x θ θ µ r 1 ∇ ( sin t r x 5 − ∇ 2 ( R v t − − r 5 , v v ∇ = = = given above ( 2 ε rr ∂ ) θ t r rt = r θ r ). Because we are looking for radiative solutions, we shall neglect all r θθ θ ϕϕ 2 , R = 5 t Γ Γ Γ z ( v 7.2 5 r v . In a flat space parameterized by spherical coordinates the non-vanishing ∇ v r is a function only of that we need to calculate are: ∇ v µ ν λ ρ 2 R With the choice for Let us start with the first-order deformation of Minkowskian space-time embedded as a hyper- In order to make vanish also the the second order correction to the Einstein equations We consider the same deformation of the fifth coordinate as before: The first order deformation of the Ricci tensor is still zero, because the deformation is per- terms which decay at spatial infinity more rapidly than 1 and in the case when components of New approximation methods in General Relativity in the case of planeapproximation; waves. static It terms is vanishing at important spatial that infinity there like will be no propagating terms at that order of set does not contribute to thesecond-order correction first-order to correction the Riemann to tensor, the Riemann tensor. In order to evaluate the only: Being perpendicular to the embedded manifold covariant derivatives of Christoffel symbols are: plane in some pseudo-Euclidean space;dicular it to has the one Minkowskian component hyperplane along one extra dimension perpen- describe the approximation to the staticthe part source of of the spherical space-time gravitational deformation waves. inevitably produced by This will remain true in any curvilinear coordinate system, too. we have to satisfy the eq. ( formed orthogonally to the embedded manifold.the first-order In correction the to case the Riemann of tensor thecoordinates, is Minkowskian automatically when zero. background all even This second is easy derivatives to of see in the cartesian PoS(ISFTG)015 Ω (8.4) there Ω 2 Richard Kerner Ω cos Ω + Ω Ω 2 sin 2 ) 3 sin ω Ω sin ( 2 2 2 cr 2 A Ω r A k 2 cos 2 2 r − ) r + 2 ) A r Ω cos 2 Ω Ω 2 ( Ω r Ω Ω 2 ( c Ω 2 2 − ω 2 . whose frequency is the double of the sin cos 2 sin cos : Ω , with the substitution 4 Ω r A A cos cos ) r θ 2 Ω cos / r 2 2 2 2 , − 4 ≡ c + t A r sin . The same is also true for the components cos → − ( ω ) 2 Ω sin Ω 5 2 2 Ω 2 v Ω A k kr 2 r − Ak 2 2 r 2 cos and cos2 sin − − A Ω ω sin r t Ω 2 + 31 Ω 4 − k + , ω r ω 2 cos 2 ( Ω 2 ω 2 sin Ω Ω c 2 c 2 cr Ω 2 A k . Nevertheless, although the presence of terms behaving sin A k 2 2 r sin instead, so that their sum will produce a constant. − cos cos c 2 sin A ω + 2 2 c 2 Ω Ω ∼ k = k → k 2 Ω A 2 2 3 cancel each other, and for each term containing sin r A ) = = Ω A θθ sin − r 4 t r rt 2 2 cos R 3 , Ω 2 R t ): k = + θ ). The contributions to the second-order corrections to Riemann’s ω Ω cos ( r θ 2 t 2 6 2 kr Ω A cos v sin θ θ − sin − Ω t sin 2 t cos 2 R k = = ω = 1 ω cr r − 2 = ϕϕ A θ θ 2 R Ar r Ω could be interpreted as a gravitational wave, it should not be there at spatial θ ϕ ϕ θ 2 = R 2 , and there is no reason to observe such frequencies in a gravitational wave whose R r 1 ) = − r Ω r , θ θ t t are the same as the ones coming from ( 2 R 5 v , which will give rise to terms like sin2 µ ν λ ρ Ω 2 R The very form of these undesirable terms suggests how they can be suppressed. It is quite clear . v and cos basic frequency tensor infinity, i.e. in the planecorrections of wave the limit, third order; where what the is wave-like worse, we behavior get could everywhere the be quadratic observed expressions like only sin in the at infinity like source was supposed to oscillate with the frequency and they do vanish provided that (recall that of the second-order correction toof the metric tensor, which is also quadratic in thethat first derivatives in order to cancelan double additional frequencies (here sixth) (quadratic dimension, terms) of we the should form: add a deformation toward Now all the cross termswill sin be a corresponding term with cos New approximation methods in General Relativity The explicit calculus yields theshorter, we following have result put (in what follows, in order to make the formulae Among the second-order corrections tonents the of Ricci radiative tensor character, there i.e. are behaving only at two infinity non-vanishing like 1 compo- PoS(ISFTG)015 ) = r (8.5) (8.7) (8.6) , t ( 5 v ), from which Richard Kerner . Under these with the third- . µν 6.40 o g 5 or 6 2 , . 2 2 6 = A r A r 4 w . A 2 4 2 ϕ + r , A + , 2 2 A ) ∇ r 2 k 2 ϕ + ϕ 2 k k w , 4 2 2 ∇ A 2 r ϕ θ 2 A = r 2 ω , ∇ ϕϕ 2 r 2 r 2 ω ( rr c ϕ 2 o 2 g θ A A 0. We shall suppose that the second- c 2 . g + ∇ A 2 r . + w 2 0 = , 4 ϕϕ = w θ θ sin ω θ r = 2 r o θ t g 2 = 2 2 µν 2 2 A R r c A ∇ = 1 kA g + θ θ 3 cr w θ sin t θ ω w ϕ ϕ , 2 2 ∇ R − t t θ 2 = R = θθ ∇ = sin θ ϕ ϕ θ 2 32 , o θ g rt 2 R 4 0 r r ϕ ϕ 2 ω ∇ g r = + 2 2 2 R R 2 r = c A θθ = 2 w r 4 r 3 o g A , but each term of this expansion can be represented as a r tr ϕ ϕ ∇ , , 2 can be dealt with and eventually cancelled via introducing r ε 2 − θ θ g r 2 2 v t 2 4 ), considering all more rapidly decaying terms as negligible R 4 4 r − r ∇ 2 A 1 r R = ω − , 2 2 2 ∇ − r − , 2 t , there are no more radiative terms in the second-order Riemann c r t k + 2 A 2 A 2 r ∇ Ω t r rt 2 or r v 2 2 2 A θ θ 2 t R = c 2 r t ω 2 = ω 2 2 ∇ sin − 2 R t t t c r tr − 1 A . Therefore, our strategy is based on the analysis of the radiative terms 2 = R 3 ∇ r θ R − do not depend on time 2 = tt = = 2 r g Ar A sin tt w θ θ θ θ 3 R 2 r R = ) = 2 r t R , t ( ϕ ϕ 6 t v 2 R and Ω cos The terms behaving like The third order correction to the Riemann tensor is given by the equation ( As in the plane wave case, we expect the radiative behavior to appear in the third order of With two simultaneous deformations in two extra dimensions, opposite in phase, 1 − and of course, extra dimensions. Bytime-dependent the functions, same being token, now reduced the to the correction following to form: the metric tensor does not contain any at this stage. we can easily calculate the third-orderthe correction Ricci tensor to contains the only Ricci the tensor. contractionorder Note of Riemann that the tensor, unperturbed in because metric the in tensor third this order, particular case series of negative powers of first (behaving at spatial infinity as approximation. But now Einstein’spowers of equations small in deformation vacuo parameter can be expanded not only in a series of A short calculus gives the non vanishing components of the second-order Ricci tensor: order deformations represent as beforedeformation a functions monochromatic spherical wave, whereasassumptions the the third-order ten independent componentsas of follows the third-order correction to the Ricci tensor are New approximation methods in General Relativity Ar tensor, whose only non-vanishing components are the following: PoS(ISFTG)015 2 − r (8.8) (8.9) , (8.11) (8.10) behave v v ϕ θ rr , ∇ 3 R ∇ 5 or 6, and ϕ w θ , ϕ ∇ = ∇ Richard Kerner w and ∇ w w A ϕϕ r ϕ . r tr o g ∇ ∇ ∇ 2 ; by an appropriate 3 Ω ϕϕ R r ϕ v ∂ + , v θ ∇ 3 ∇ tt v at least. θ r θ cos 3 ∇ θ R − . 2 θ 3 θ ∇ 1 A ∇ 2 w − θ 1 2 ∇ θ r sin ϕ w , ∇ w ∇ sin ∂ is satisfied, which we shall − ϕ r w , which also behaves like , contain, besides the factors + r θθ 1 v ∇ 2 θθ ∇ ϕ ϕ + are left. Ω r o w r r g ϕ o g ∇ ω ∂ − 3 2 w θ R ∇ r ∇ 1 + − ∂ + θ 2 w sin w ∇ = − r − v r θ v ∇ v ϕ r t θ ϕ t θ 2 2 r 2 r and θ 1 c ∇ kA ∂ ∇ ∇ t 2 t ∇ ∇ θ 2 θ sin t cos sin r k . Explicitly, these expressions are given ∇ = ∇ ) ∇ 3 ∇ R w − v + ] + φ , w r r ϕ v , w = ϕ = w Ω w ϕ t ∂ t r θ ∇ ϕ 3 θ ( ϕ r ∇ R ∇ t , contain the common factor 2 ∇ r θ r t θθ r , r 3 ∇ ∇ R 3 2 cos ∂ R ∇ θ ∇ θ ϕϕ t ∇ , contain the common factor corresponding to the r Q A 3 3 or 33 R ∇ R , sin − rr , 2 + 1 r 3 w v R w w w + 2 r ) = contain the factor θ v θ − θ θ k r and w , which makes the components µ r ∂ w ∇ ∇ 1 ∂ x r ∇ ∇ ϕ r and r − = ϕ ϕϕ 1 r ( r − ∇ θϕ ∇ 3 r R 2 ∇ + ∇ w ∇ tr ∇ 2 3 ϕ − R c ϕ 3 v i R w ω . Let us do it systematically, by groups of terms displaying w , = [ ∇ v r w , ϕ ∇ θ ∞ and tt ϕ θ ] ∇ θθ 2 ∇ r ϕϕ t ∇ 3 v θϕ R 3 R ∇ ϕ ∂ 1. It is quite easy to determine the behavior of each of these t o θ g 3 θθ → ∇ R ϕ ] = we shall ensure that it also behaves like ∇ 3 ∇ − ∇ R r ) v = t + ∇ = ] t ϕ ϕϕ v ∇ w w θ , ∇ , the factor t by ϕϕ t o g t θ θ v θ − 3 o R g ∇ , ∇ ∇ t ∇ + v r r r r θ ( r + ∇ o g − ; besides, the factors containing the derivatives of the first-order correction w ∇ ∇ v w ∇ 2 v θ t − r r − ∇ θθ v ∇ r ∇ ∇ t θ r o g r ∇ ∇ ∇ , = [ ∇ r h v [ r . r θθ ∇ 3 = o ϕϕ by 1 and g ∇ − r 3 r θ R = [ r tt ∇ 3 o R g -like term is cancelled if the dispersion relation θθ = 1 3 R − rr r - Finally, the remaining four components, - The last three components, Therefore, there are no radiative terms at infinity in- The the first last three components, three components of the Ricci Other two terms present in 3 R all start with the term proportional to by the formulae with second derivatives of at least like tensor In these formulae we displayedreplaced only the generic form, commoncomponents for at both spatial indices infinity, similar behavior: choice of the function angular part of the three-dimensional Laplace operator, v The supposed true from now on. Then only the terms behaving like at least. The last remaining expression that may cause trouble is which behaves like Now, there is andependence obvious on solution to the above equations which consists in choosing the linear New approximation methods in General Relativity PoS(ISFTG)015 1 − re- r , but ϕ ϕ (8.13) (8.12) r 3 R and , we observe r Richard Kerner θ r in other compo- , 3 ) R ’s also start with 0 2 , v ϕ − , ϕ = t r , the terms behaving θ 3 ) R ( can be developed into , 0 ) of the Riemann tensor. kr A θ − w ϕ t t , w 3 r R ω 2 θθ ( θ ( θϕθϕ 0 w 3 R θ ∂ Acos θ w ) along with the spherical wave 2 ∇ 2 ) decay at spatial infinity like θθ ∼ θ sin , B 8.12 ) ∇ v θ ∂ v , which will result in the following ϕ 8.10 2 ) + , ϕ ϕ , ϕ θ ∇ im ) sin , ( ; still, certain components present in the e ϕ p A 2 ϕ θ , ( ∇ − w 0 ) + r r θ p given by ( ( + w ϕ − A 1 ) , θ r w 6 0 w θ , ϕ at spatial infinity), the Riemann tensor has at least ( = ∞ 5 ∑ 0 θ ∂ 34 p 1 ∇ w ) + − ϕ = r θ ϕ cos ∇ B ) = , v , θ θ ∂ θ ϕ θ A ( , ( ∇ A 0 even at spatial infinity: , but this corresponds to a pure gauge and does not change θ sin θ , cos w r r , namely − ∇ ( θ , which can be interpreted as a spherical gravitational wave. A ) , so that there is a spherical gravitational wave at spatial infinity. 1 A w 1 = ϕ − w sin − , r r θ − ( Constant ) , starting with the zero-power term: 0 θϕθϕ r ϕ w 3 = R , 0 2 θ ϕϕ ( w ∂ 0 ), and keeping only the leading terms in negative powers of w ) inserted in the last three components will produce the leading factor behaving 2 , we are sure that all the components of the third-order correction to the Ricci ) Ω ) deformations 1 m ( 2 . But there is one component, ϕ 6.40 − 1 ε , − r − sin θ r ∼ ( k Q , which we wanted to avoid. 1 − However, there is no reason to require from these four components of the third-order correction What we should prove now is that although we have the approximate Ricci-flatness satisfied r whose leading term is independent of The obvious solution is This is in agreement with our result for the plane gravitational wave, for which the component nents. In fact, there isseries an of alternative negative choice, powers admitting of that the functions found in the second-orderRiemann approximation tensor into ( the formula for the third-order correction to the the induced metric. Otherspecting solutions should periodicity, contain i.e. the dependencecondition they on must the azimuthal be angle proportional to that will ensure the vanishing of the radiative part of the component to the Ricci tensor to vanish identically, while keeping the terms behaving like like the function This form of the second-order deformation makes the terms ( (see the full list ofsecond-order terms ( in the Appendix 1) that only the first two terms in the expansion of the can produce, when combined with theat first-order infinity deformation like This ansatz cancels identically the four components of the Ricci tensor, New approximation methods in General Relativity Inserting the second-order deformations tensor have their leading term at infinity behaving as asymptotically (up to the terms behaving like or faster, and taking intothe account dependence that the terms with the second derivatives of some components behaving like Riemann tensor will behave like PoS(ISFTG)015 . A w , we ) (8.14) 1 + , l φ ( l im e Richard Kerner ) θ ...... 0 0 0 0 0 0 cos ( = 6= 6= 6= 6= 6= 0, the solution is the m l P 6= µν µν µν µν µν µν 3 3 3 3 3 3 R R R R R R ) 1 m 1 in form of portent waves of + 3 and the subsequent terms of 2 , , , , , , 2 ε m A + ) 0 0 0 0 0 0 φ v √ l ; if -axis; in the case of a spherical ϕ ( im 6= 6= 6= 6= 6= 6= z , θ − l e θ ) − ( 1 1 θ cos A 1 ρ ρ ρ ρ ρ ρ µ νλ µ νλ µ νλ µ νλ µ νλ µ νλ θ − + 2 w 2 K 3 3 3 3 3 3 R R R R R R θ θ cos m ( = sin m l . cos cos 0 0 P 0 Q 2 1 r A lm 6= 6= 6= 2 C = C l µν µν µν φ 35 − 2 2 + l R R 2 R = ∑ im 1 . m 2 e + , , 0 0 0 , 2 0 ε ) 2 0 0 m 0 ∞ = θ ∑ l √ 6= 6= θϕθϕ 6= R cos 1 1 ) = ( ρ ρ ϕ m + − , which is the angular part of the Laplace operator in spherical ρ µ νλ µ νλ l µ νλ , P 2 2 θ θ R R 2 R θ w ( ϕ ϕ A cos cos ∇ 1 ∇ (as well as the functions and w 0 0 0 0 0 0 ϕ ε ϕ ) 1 ∇ ∇ 0, the obvious solution is ϕ C The non-vanishing components of Riemann and Ricci tensors , 0 ϕϕ ϕϕ 0 0 0 0 0 0 = θ ε o g o g ( ) can be decomposed into the sum of the Legendre polynomials as follows: m A 0 + + w 2 3 4 5 6 θ r 1 1 1 1 1 1 w r r r r r 8.12 ∇ θ Table I: θ for the approximate spherical wave solution in Minkowskian background ∇ ∇ θ ∇ θθ o g θθ o g was zero when the direction of propagation was along the The knowledge of the asymptotic angular behavior should enable us to reconstruct theLet angular us summarize the result of our approximation at this stage in the following table: The functions They appear in the expressions for the third-order correction to the Ricci tensor via the combi- Now, if we set xyxy 3 the development ( distribution of matter in the source of the gravitational wave. so that these terms becomeThese the in sum turn of can linear be combinationsdifferent superposed frequencies of with in the order the second-order to first-order deformations produce deformations the wave packets with an arbitrary frequency spectrum. nation coordinates. Therefore, as eachLaplace of equation the with associated the corresponding Legendreshall radial polynomials get part is included, a which solution gives of the the factor full New approximation methods in General Relativity R wave the propagation is alongthe the Riemann radial tensor becomes direction, precisely so that the "purely transversal" componentcombination of of two functions, PoS(ISFTG)015 ) . 2 θθ and in all − R r 1 θ − Richard Kerner 2 r sin . This leads us to can never descend, = M ) t ϕϕ ( R m at infinity and in the first 2 6 6 1 A 0 0 2 r r proportional to 4 − r the space-time containing the θθ 5 2 1 0 0 0 1 R r − r 2 c can never become negative or even hit 2 k ) 2 t . A 4 ( 2 ) 2 r 4 t 2 2 r 4 4 ω − 1 M ( c 2 r r 2 A 2 c A ω M 2 A usually much smaller than 3 ) − t 36 ( are displayed in the following Table II: m 3 1 0 0 0 r ε 2 2 , with r 2 ω ) 1 0 0 2 r 2 t c ( A m + r 1 0 0 0 M rr The non-vanishing components of second-order Ricci tensor tt θθ ) = 2 2 R 2 R R t ( M Table II: is beyond our possibilities at the moment; still, we proceeding step by step, we shall ε for the approximate spherical wave solution in Minkowskian background ( The table above shows that in the asymptotic radiative region the dominant terms satisfy Ein- Our strategy, in what follows, will consist in creating other deformations of the embedding Now, contrary to what may happen in electrodynamics, where the radiation may come from a Before going into details, let us draw attention to the physical nature of the problem. The This is why we should combine the embedding of our approximate spherical wave with a stein’s equations, i.e. up to the terms behaving at infinity like New approximation methods in General Relativity spherical wave contribution is Ricci-flat. Thesors, remaining are terms, similar be to it the thethe Coulomb-like Riemann source, terms or where the in they Ricci the dominate. ten- electromagnetic(and However, radiation although should the case be) contributions in to not the the equal zonethe Riemann maximal near to tensor number zero, may of orders The for Riccicomponents our of tensor approximation the to should Ricci be vanish tensor, considered starting at asthree from valid. all lowest the powers The terms orders, of non-vanishing decaying or the like small at parameter least up to functions, perhaps in extra dimensions,tensor in displayed order in to the cancel Table the II. non-zero To components make of the the Ricci Ricci tensor vanish at all orders of the zero value. This means that there is some minimal value below which charge distribution with total average zero, the function spherical gravitational wave wasground, produced and is by valid the only asymptotically, deformationscoming at from spatial of the infinity. center the of But coordinates the flat cannotmust presence be Minkowskian be of justified some if spherical back- there time-dependent radiation is mass no distribution, matterradiate. because near we the Whatever can center. the be There distribution, sure ifseen that as a it an static is almost mass bounded point-like will in mass not varying space, with from time, a very great distance it will be Schwarzschild background. The embeddingthan six of flat the dimensions; combined letsolution. solution us will recall the certainly properties need of more embeddings of the exterior Schwarzschild powers of cancel at least the dominant terms, of which the most important is so that we can write the conclusion that the natural backgroundmetric, for with time-dependent a perturbations spherical which gravitational are wave the is real the source Schwarzschild of radiation. PoS(ISFTG)015 (9.1) , with . µ 6 x ] proposed ,... , 2 , 29 1 Richard Kerner in order to give ct = MG 2 z ) , using hyperbolic 2 ) ,... φ − B d and 5 , , cosh θ A 1 = θ z 2 1 2 + ν 1 ! ( sin cos dx µ r − r 2 with denotes Newton’s gravitational MG dx = θ 2 ) , 6 d ]. An embedding of the exterior G r λ z ( − , x d 2 ) 27 ( r 1 1 2 ,
# φ − µν 1 2 g − sin MG dr = 4 1 θ ν . = − is parameterized by the coordinates φ 2 − − − −− r dx 4 z r MG sin µ 2 MG V = r 37 (+ , 3 r dx − MG x = can be now identified with the measure of the ratio B 2 1 − in front of the definitions of 5 in the non-vanishing components of the Riemann and z , z ε ν 1 diag 2 − ct θ ∂ MG " 1 ω , = A MG = z φ Z , using trigonometric functions of time in order to parametrize 2 µ AB ) x − ∂ η = 2 , cos sinh r 3 AB z 1 2 dt θ η 2 = ! c 1 = sin x −−−− 2 r , r MG ds ct r = 2 MG 4 (++ 2 = z − 0 1 ], who also proved that the embedding of Schwarzschild’s solution in a five- − x 1
28 . The rapidity of the time variation will also play an important role, as can be = M MG / is the mass of the central gravitating body and = | 3 so that 1 M , z )) with signature t 2 ( , 6 1 m E , ( 0 Isometric embeddings of Einstein spaces in pseudo-Euclidean flat spaces of various dimen- The yet unspecified small parameter Here The embedded four-dimensional manifold In what follows, we use Fronsdal’s embedding of the exteriorIt Schwarzschild is space-time worthwhile to into look at the non-vanishing components of the Riemann tensor in the Schwarzschild Let us define the flat metric by Then it is easy to check that the induced metric on the embedded manifold has indeed the = max µ Schwarzschild solution which isfound of by Kasner particular [ interest to us , cited in Rosen’s paper, has been a similar embedding into pseudo-Euclideanfunctions instead space the with trigonometric the ones. signature Fronsdal’s embedding is defined as follows: sions and signatures can be found in J. Rosen’s paper in [ Ricci tensors generated by the wave. dimensional pseudo-Euclidean space is impossible.space Kasner’s embedding uses a pseudo-Euclidean New approximation methods in General Relativity 9. Spherical wave and the Schwarzschild background | the two timelike dimensions, and admits closed time-like curves. In 1959 C. Fronsdal [ constant. (Note the dimensional factor inferred from the presence of the factor a pseudo-Euclidean flat space with one time-like and five space-like dimensions. space-time. They are displayed in the following Table III: these coordinates the dimension of length). usual Schwarzschild form: PoS(ISFTG)015 MG Richard Kerner δ 3 4 4 1 0 0 0 r r G 3 4 M G 4 4 4 1 32 0 0 0 r r 2 M 4 2 8 ω correspond to the same r 4 4 1 2 A 0 0 2 r r 2 c A 3 MG 3 δ G 2 µνλρ 3 3 3 3 3 3 3 1 1 0 0 MG r r r 0 0 r r r MG G 3 1 R δ 0 0 0 0 M 2 r 2 2 4 M 12 θ . The resulting first-order correction 2 2 2 2 2 2 G G r 2 ω M k 2 2 2 . The resulting embedding induces the 2 2 2 r 1 2 0 0 2 2 sin r r r 2 1 c 0 r r δ A 2 M M A M 2 2 MG MG 2 A δ δ 2 2 2 + δ − 1 0 0 r r r − + M MG MG 4 4 r M r r 1 r MG 1 0 0 0 0 0 0 MG − → 38 r θ MG r r 1 M 0 0 δ MG 2 1 0 0 0 0 δ − sin 1 0 0 0 2 θ k 1 0 0 0 0 2 2 A sin ) are displayed in the following Table IV: − θ r 0 0 0 M 2 First order Riemann tensor due to a small mass δ is quite different, as shown in the Table V below: r 0 0 0 0 MGr r sin 2 The Riemann tensor for the Schwarzschild background r 0 0 0 − θ θ deformation around the Schwarzschild background r t MGr θ θ trtr δ r t θφθφ θ θ 2 2 r 2 R 2 Table IV: R t 2 R R θ θ trtr − r t θφθφ o o R o R o R Table III: R Second order Riemann tensor for a spherical wave in Minkowskian background θ θ r t θ θ trtr r t θφθφ 1 1 R 1 R 1 R R Table V: set of indices as the contributionsamong of the a negative powers spherical of wave in Minkowskian background, the repartition One can note that although the non-vanishing components of the New approximation methods in General Relativity The Ricci tensor is null, of course.be Now, produced a that small alters deformation the of the componentsconsists Schwarzschild of in embedding the a may Riemann small tensor variation keeping ofSchwarzschild the the metric space-time total corresponding mass, Ricci to flat. the It totalto mass the Riemann tensor (linear in PoS(ISFTG)015 is → λ r (9.2) 2 ) 0 3 is the only v ) ( 00 3 v r 0 3 − Richard Kerner − v 0 2 00 3 at infinity, found (+ v other deformation − 0 3 2 A − rv v r ) − r ( 3 2 v B 2 = signature) for the embedding λ 2 4 3 ) C v − 2 2 6 6 + r r C 6 1 0 6 r BD , + 6 ) ... r BD ( − + f 14 4 −−−−− E r − λ ct + (+ ). Supposing that in the radiative zone 3 2 D r 5 5 cosh BC BC ε r r 5 ), and the remaining two space-like dimensions, 1 0 2 8 r + λ 39 − − 2 9.1 = C r 2 2 v + C r 2 B + 2 , 4 B ) r 4 4 BD 2 1 r 0 r r ) = 4 ( r + f − ( 2 f λ λ 2 ct B equal to zero. sinh A v 3 3 1 0 0 λ BC r r 3 = 1 2 v 2 2 1 0 0 r r B are positive-definite, and cannot cancel similar terms generated by the wave-like The contributions to the second-order Ricci tensor, coming from a , but keep the embedding using Fronsdal’s hyperbolic functions. Producing extra r 2 r 1 0 0 0 / − r MG tt rr θθ , will be supposed to be flat and the corresponding deformations of first and second order 2 2 R 8 R 2 R z Table VI: will represent the two degrees of freedom of the spherical wave, producing the contributions Expanding the function f in a series of negative powers of r as follows: Let us produce a first-order deformation of the first three embedding functions, deforming the Our aim now is to improve the approximation via cancelling more terms in the Ricci tensor. We saw already in the previous section that six pseudo-Euclidean dimensions were needed to We shall choose the first six dimensions (with the As one can see, all the contributions to the second-order correction to the Ricci tensor behaving Schwarzschild metric is asymptotically Minkowskian, we neglect the terms coming from the ε and 7 yet unspecified constant with dimension of length: with all other components embedding of Minkowskian space-time embedded in six pseudo-Euclidean dimensions. Here way to produce negative contribution todisplayed the in the Ricci Table tensor IV. able to cancel the positive-definite terms deformations in the first two pseudo-Euclidean dimensions carrying the signature correction in the second-order Ricci∞ tensor (proportional to we obtain the following corrections to the Ricci tensor in four dimensions: The most important terms after the radiative ones are the terms behaving like z in to Riemann and Ricci tensors displayed in the Tables II and IV. embed the spherical gravitational wave;for the global embedding. exterior Schwarzschild Most requiressame probably, also it six-dimensional six is flat dimensions impossible space toThis without accomodate is their the why two metrics embeddings we being inSchwarzschild shall modified background the look in even an at for eight-dimensional pseudo-Euclidean the the spaceseven lowest common with space-like one orders. directions. embedding time-like and of spherical gravitational waves and the New approximation methods in General Relativity of the Schwarzschild space-time defined as in ( at infinity as PoS(ISFTG)015 (9.3) (9.4) satisfy . This ) Q − 7 , and equal to zero. + Richard Kerner 1 f P ( deformation A w ) r ( . 3 3 w − r R 6 6 T R = r r 6 6 6 1 12 r r 3 20 − and − w 2 ... . − 2 r S 5 5 Q , Q r r + 5 5 5 k 6 ) 1 r r , r 6 12 0 − ( T − r g = + λ ct 4 P 5 Q 4 R r S 4 4 2 P r r 1 4 r r 6 ; This leads to the conclusion that the leading - 2 − + k cosh , 2 4 0 R r λ A 3 Q 40 is also a series of negative powers of r: r 1 2 3 3 = 1 0 0 + = r g P 3 2 − = Q r 2 w P − + ]). 2 P 2 r 2 2 2 1 0 0 r , k P behaving at infinity like 32 r ) 2 − r A ( µλ g ) = 2 R r 1 0 0 0 r λ ( ct g tt rr θθ sinh 2 2 2 R R R at spatial infinity, to the solutions found by Boardman and Bergmann λ 4 = − 1 r w The contributions at the second-order Ricci tensor, coming from a Table VII: But there is no reason to neglect the second-order deformation of the first two embedding Combining the results in the Tables II, VI and VII, we see that all the components of the In this article we have set forth a new formalism that makes it easier to consider small perturba- At this stage of approximation we get The combined embedding of the Schwarzschild metric in six pseudo-Euclidean dimensions ]) or by Robinson and Trautman ([ at infinity, which was avoided up to now. This is why we shall set the function 1 31 − and the other components zero. The function New approximation methods in General Relativity deformation. Moreover, they produce ar first-order contribution to the Ricci tensor, behaving like Inserting this deformation alonetensor into we obtain the the formulae following result, for displayed in the the second-order Table VII correction below: to the Ricci the following two relations: which cancels the components of functions. Let us choose the second-order correction of the same form as the first-order one: second-order correction to the Ricci tensor can be cancelled provided the constants tions of a given Einsteinian background withoutcomputations unphysical based degrees on of the freedom deformations that mar ofgeometric traditional the criterion metric. selecting Here physical the degrees of embedding freedom provides us and eliminating with the clear unphysical ones. 10. Discussion and conclusions and the wave-like deformationsembedded of in two a extra pseudo-Euclidean dimensions space produce of eight a dimensions, four with dimensional the manifold signature term in the second-order correction to thebackground embedding functions is of proportional Schwarzschild to (or the Minkowskian) square of the wave number, manifold represents an approximateterms radiative behaving solution like of Einstein’s([ equations similar, up to the PoS(ISFTG)015 (10.1) . ) ) φ 2 , Richard Kerner , (1995) p θ φ , r t d ( θ , 1986 W 2 ) and up to the terms ibid 2 sin ε − . 2 4 , pp.5009-5024, (2000); θ − d r 17 ( 2 r , of the form 3 (A), (1985); − ε 2 Physical Review Letters dr 1 − 2 2 r (3), p. 189-203 (1971) where the non-radiative terms prevail. These Q r + 14 D, (1998) 41 r (A), MG Class. and Quant. Gravity 2 − 1 by a term proportional to 2 − r 2 769 / 2 Physical Review dt or faster, we can try to go farther and cancel the third-order (pro- 49 Q 2 Annales de l’institut Henri Poincaré c 4 − r 2 2 r bring its total charge. The Maxwell tensor of such a distribution behaves Q Q + Ann. Physik r MG 2 , and the energy-momentum tensor is proportional to 2 − ) terms of the Ricci tensor in the vicinity of the central mass, i.e. in the non-radiative − 3 1 r Annales de l’institut Henri Poincaré ε = . The exact solution is then given by the following modification of Schwarzschild’s metric: 2 Q ds e-Print: gr-qc/0009016 We have succeeded the construction of wave solutions in flat space, or in an asymptotically flat The treatment of this problem isNow the that subject the of Einstein the work equations in are progress. satisfied up to the second order ( In order to produce a contribution to the Ricci tensor that would cancel the third-order terms In our case, the yet non-vanishingBecause components we of are the looking for Ricci the tensor third-order are effect, exactly we of shall this modify Schwarzschild’s form. embedding [1] Einstein A 1916 [2] Ll. Bel, [3] T. Damour and N. Deruelle, [6] A. Balakin, J.W. van Holten and R. Kerner, [4] L. Blanchet, T Damour, B.R. Iyer,[5] C.M. Will, A.G. P. Jaranowski Wiseman, and G. Schäffer, References This metric can be inducedof by the an first embedding two identical embedding with functions.the Fronsdal’s energy-mementum It one, tensor is with of not the new Ricci-flat, electromagneetci coefficients because fieldelectric of its charge a Ricci density, sperically tensor symmetric is distribution proportional of at to infinity as Schwarzschild manifold at spatial infinity. These solutionspossibility can display of any linear imposed form superposition due of towe the Legendre can polynomials. extrapolate it Once towards a the smaller radiative values solution of is chosen, generated as the side-effectsReissner-Nordstroem of metric, which the is wave-like not deformation,tions Ricci-flat of because we the of energy-momentum may the tensor presence follow ofcharge in the the the electromagnetic example Einstein field equa- generated of by the the central electric functions by replacing the term New approximation methods in General Relativity can be seen as the corrections tofor Schwarzschild the metric emission close of to gravitational the central waves detected body that at are spatial responsible infinity. decaying asymptotically as portional to region. Note thatthe up wave, to which the is secondbackground a order, with the there third outcoming was order gravitational wave: no effect.perimposed the without information corresponding provoking Let Riemann about any tensors modification us any can in be consider the characteristicfrom su- lowest the the of orders Tables of IV combination the and of expansion, VI. as the it follows Schwarzschild PoS(ISFTG)015 Richard Kerner The mathematical ( pp. 4725-4742, (2001); 1018 39 18 30 , pp. 4573-4590, (2002); (Paris) 19 263 (2), pp. 265-281 (1967) υνταξις A 8 Σ (10), pp. 899-912 (1981) D63, 027502 (2001); e-Print: , No. 11, p. 1185 (1993) , (New York: Oxford University Press) 25 13 , (San Francisco: Freeman) , p.463 (1962) (3), pp. 365-376 (1996) J. Math. Phys. , Vol. , 10, pp.2111-2122, (1993). Phys.Rev. (5), pp. 1318-1324 (1959) 1794 28 , pp.339-351, (1988). A 265 , 5 30 αθεµαθικη (2), pp. 193-203 (1984) Proc. Roy. Soc. 115 , 16 Gravitation HM 115 145; 1989 Class. and Quant. Gravity 42 , (1978) 27 27 Class. and Quant. Gravity (Amsterdam: North-Holland), (1960); (New York: Academic Press), (1972) 180; 1961 GRG-journal , p. 130 (1921) J. Math. Phys. , (1965) 249 659 ibid (3), p. 778 (1959) A Physical Review 10 ; original title ‘ Journal of Mathematical Physics , pp. 1559-1571 (1967) 116 Proc. Roy. Soc. London , , University of Paris-VI, (2002) (The New Astronomy), Pragae (1609) General Relativity and Gravitation 174 Class and Quant. Gravity , p.126, Classical and Quantum Gravity 43 Almagest Les méthodes nouvelles de la mécanique céleste General Relativity and Gravitation , The Mathematical Theory of Black Holes (Leiden) (1916) Ann. Inst. H. Poincaré A Ann. Inst. H. Poincaré A Proc. Roy. Soc. Gen. Rel. Grav. Ph.D. Thesis Physical Review Gravitation and Cosmology Relativity: the General Theory Am. J. Math General Relativity and Gravitation Physical Review Reviews of Modern Physics Astronomia Nova PhD. Thesis ) ´ ´ ´ nski S L 1977 nski S L 1977 nski S L and Jaranowski P 1989 ˙ ˙ ˙ za za za e-Print: gr-qc/0205019 gr-qc/0010098 e-Print: gr-qc/0102099 Misner C W, Thorne K S and Wheeler J A 1970 syntax [8] R. Kerner, J.W. van Holten and R. Colistete Jr. [9] R. Colistete Jr., C. Leygnac and R. Kerner, [7] R. Kerner, J. Martin, S. Mignemi and J.-W. van Holten. [10] Ba [11] Ba New approximation methods in General Relativity [13] V Aliev and D.V. Gal’tsov, [27] J. Rosen, [28] E. Kasner, [29] C. Fronsdal, [30] P.S. Wesson, General Relativity and Gravitation, [32] I. Robinson and A. Trautman, [33] G. Qi and B.F. Schutz, [24] R. Kerner and J. Martin, [25] R. Kerner, [26] B. Giorgini and R. Kerner, [12] Ba [14] J. Droste [15] Ptolemaios ca. 145 AD [19] J. L. Synge [20] S. Weinberg [21] J. Kepler [22] Poincaré H 1892-1899 [23] R. Colistete Jr., [31] J. Boardman and P.G. Bergmann, [34] B. Carter, [18] Chandrasekhar S 1983 [35] R.H. Boyer and R.W. Lindquist, [16] Darwin C G 1958 [17] Sharp N A 1979 [36] C. Romero, R. Tavakol, R. Zalaletdinov, PoS(ISFTG)015 65 Richard Kerner 90 , pp. 865-879 14 , Prog. Theor. Phys. 1087 1860 96 48 Classical and Quantum Gravity 435 43 Phys. Rev. D 131 Prog. Theor. Phys. 1497; 1993 Phys. Rev. 47 Phys. Rev. D (1997) New approximation methods in General Relativity [37] J.E. Lidsey, C. Romero, R. Tavakol and S. Rippl, [39] Tanaka T, Shibata M, Sasaki M, Tagoshi H and Nakamura T 1993 [38] Peters P C and Mathews J 1963 [40] Tanaka T, Tagoshi H and Sasaki M[41] 1996 Poisson E 1993