Bondi-Sachs Formulation of General Relativity (GR) and the Vertices of the Null Cones

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1 Bondi-Sachs Formulation of General Relativity (GR) and the Vertices of the Null Cones Thomas Mädler Observatoire de Paris/Meudon, Max Planck Institut for Astrophysics Sept 10, IAP seminar

2 Astrophysical Motivation c The Australian Observatory

3 Astrophysical Motivation c The Australian Observatory

4 Astrophysical Motivation Supernova 1987A was a core-collapse supernova, which is the explosion of a star undergoing a gravitational collapse... involves time varying multi-poles of the stellar mass-energy distributions... generate gravitational waves...

5 Gravitational Waves -Motivation of Bondi s Ideametric = Minkowski + perturbation g µ = µ + h µ choose coordinates equation of motions j 1 c 2 y h µ =2appleT µ

6 Gravitational Waves -Motivation of Bondi s Ideametric = Minkowski + perturbation g µ = µ + h µ choose coordinates equation of motions j 1 c 2 y h µ =2appleT µ PROBLEM of THIS DESCRIPTION linearised metric theory is not gauge invariant other coordinates also yield : metric = Minkowski + perturbation Eddington (1922): Gravitational waves travel at the speed of thought

7 Gravitational Waves Pirani showed in Motivation of Bondi s Ideagravitational radiation is characterised by the Riemann tensor fundamental speed is the speed of light

8 Light-cone approach of GR The Bondi-Sachs Formulation stargazer gravitational waves star

9 Light-cone approach of GR The Bondi-Sachs Formulation stargazer gravitational waves star What are the properties of the Bondi-Sachs Formulation of GR and which role plays the vertex?

10 Out-look Simplifying Assumptions The Bondi-Sachs Metric The Field equations Mass-loss and Gravitational Waves Regularity the Vertex Summary

11 Simplifying Assumptions

12 Simplifying Assumptions Axial Symmetry

13 Simplifying Assumptions Axial Symmetry + Vacuum Space-Times

14 The Origin of the Coordinate System in Axial Symmetry Axis of symmetry in a four-dimensional space-time is a totally-geodesic, time-like 2-surface A contains time-like geodesic curves given by the symmetry choose in A A a time-like geodesic (world-line of an observer) which traces the vertices of null cones VERTEX = ORIGIN

15 a Light-Cone axis of symmetry r x A out-going light ray u = const x A = const r varies u u = const axial observer u c( ) = r c( ) =0 O c( ) null ray u = const x A = const r varies out-going light cone u = const vertex

16 The Bondi-Sachs Metric x 0 := u is null coordinate k := g r u out-going null rays g r ur u =0 null vector of the x A := (x 2,x 3 ) are constant along null rays k r x A =0 x 1 := r is a luminosity distance det(g AB ) r 4 = f(x A )

17 The Bondi-Sachs Metric x 0 := u is null coordinate k := g r u out-going null rays g r ur u =0 null vector of the g 00 = g 11 =0 x A := (x 2,x 3 ) are constant along null rays k r x A =0 x 1 := r is a luminosity distance det(g AB ) r 4 = f(x A )

18 The Bondi-Sachs Metric x 0 := u is null coordinate k := g r u out-going null rays g r ur u =0 null vector of the g 00 = g 11 =0 x A := (x 2,x 3 ) are constant along null rays k r x A =0 g 0A = g 1A =0 x 1 := r is a luminosity distance det(g AB ) r 4 = f(x A )

19 The Bondi-Sachs Metric x 0 := u is null coordinate k := g r u out-going null rays g r ur u =0 null vector of the g 00 = g 11 =0 x A := (x 2,x 3 ) x 1 := r are constant along null rays k r x A =0 is a luminosity distance det(g AB ) r 4 = f(x A ) g 0A = g 1A =0 g 01 6= 1 g AB = r 2 h AB

20 The Bondi-Sachs Metric x 0 := u is null coordinate k := g r u out-going null rays g r ur u =0 null vector of the g 00 = g 11 =0 x A := (x 2,x 3 ) x 1 := r are constant along null rays k r x A =0 is a luminosity distance det(g AB ) r 4 = f(x A ) g 0A = g 1A =0 g 01 6= 1 g AB = r 2 h AB ds 2 = e 2 +4 du 2 2e 2 dudr + r 2 h AB (dx A U A du)(dx B U B du) h AB dx A dx B = e 2 cosh(2 )d 2 +2sin sinh(2 )d d + e 2 sin 2 cosh(2 )d 2 Bondi [1960], Sachs [1962], van der Burg [1966]

21 Einstein Equations 6 main equations 3 conservation equations 1 trivial equation 4 hypersurface equations 2 evolution equations R uu =0 R ua =0 R ur =0 R rr =0 R ra =0 R AB 1 2 g AB g CD R CD =0 g AB R AB =0 Tamburino et al. [1966]

22 Structure of the Field Equations r R 1 2 imply... R µ µ Bondi-Sachs Lemma: If the main equations hold on one null cone and the optical expansion rate of the null rays does not vanish, i.e. 6= 1, on this cone, then the trivial equation is fulfilled algebraically and the supplementary equations hold everywhere on this null cone provided they are fulfilled at one radius r=r>0. =0

23 Hierarchy of Main Equations Hyper-surface equations R rr =0:,r = J (0) (, ) R ra =0: U,rr A = J A (,, ) g AB R AB =0:,r = J (1) (U A,,, ) evolution equations,ur = J ( ) (,, U A,, ),ur = J ( ) (,, U A,, )

24 Initial values to Integrate the Einstein Equations (1) axial geodesic (x ) free data (x ) free data r = R>0 specify, U A, and U A,r for all values of xa

25 Initial values to Integrate the Einstein Equations (2) axial geodesic (x ) free data (x ) free data r = R>0 specify, U A, and U A,r for all values of xa

26 Condition for Out-Going Gravitational Waves... is the Sommerfeld radiation condition:... ansatz for the asymptotic solution (u, r, x D )= c(u, xd ) r lim r = const, lim r = const r!1 r!1 x A = const u = const x A = const u = const + O(r 3 ), (u, r, x D )= d(u, xd ) r + O(r 3 ) Bondi et. al.[1962],van der Burg [1966]

27 Asymptotic Solution of the Main equations... consequence of the integration of the main equations (x )= 1 (u, x A )+O(r 2 ) U A (x )=U A 1(u, x A )+O(r 2 ) e 2 (x ) =1 2M(u, x A ) r + O(r 2 ) Bondi s choise of the functions of integration 1(u, x A )=0 U1(u, A x A )=0

28 Asymptotic Solution of the Main equations... consequence of the integration of the main equations (x )= 1 (u, x A )+O(r 2 ) U A (x )=U A 1(u, x A )+O(r 2 ) e 2 (x ) =1 2M(u, x A ) r + O(r 2 ) Bondi s choise of the functions of integration 1(u, x A )=0 U1(u, A x A )=0 Bondi chooses an asymptotic Minkowskian observer

29 The Bondi-Mass & Mass-Loss the function of integration aspect of the isolated system M(u, x A ) is the Mass the Bondi Mass m(u) = 1 4 m(u) I of an isolated system is M(u, )sin 2 d d the time variation of the Bondi mass as measured by the asymptotic observer is d du m(u) = 1 4 I h i c 2,u(u, )+d 2,u(u, ) sin 2 d d

30 The Bondi-Mass & Mass-Loss the function of integration aspect of the isolated system M(u, x A ) is the Mass the Bondi Mass m(u) = 1 4 m(u) I of an isolated system is M(u, )sin 2 d d the time variation of the Bondi mass as measured by the asymptotic observer is d du m(u) = 1 4 I h i c 2,u(u, )+d 2,u(u, ) sin 2 d d -an isolated system can only lose mass - -via gravitational radiation-

31 Infinity vs. Origin stargazer gravitational waves star

32 Infinity vs. Origin gravitational waves stargazer Bondi s Observer at infinity star

33 Infinity vs. Origin gravitational waves stargazer Bondi s Observer at infinity passive star

34 Infinity vs. Origin gravitational waves stargazer Bondi s Observer at infinity passive active star

35 Infinity vs. Origin gravitational waves stargazer Bondi s Observer at infinity passive active star to study a gravitational active source need inertial observer at the origin

36 The Vertex - Problems (1) a null cone is not differentiable at its vertex (2) Bondi-Sachs metric is not defined at r=0 (3) require additional regular structure at the vertex with a Taylor expansion in regular coordinates (4) order of approximation of the metric near the vertex determines how the light rays leave the axial observer

37 The Vertex - Problems (4) order of approximation of the metric near the vertex determines how the light rays leave the axial observer axial observer spherical emission of the light rays & linear approximation non-spherical emission of the light rays & quadratic approximation

38 The Vertex - Problems (5) How does one move the origin of the coordinate system along the time-like curve defining it?

39 The Vertex - Problems (5) How does one move the origin of the coordinate system along the time-like curve defining it?

40 Solving the Vertex - Problem... define a regular coordinate system along the axial geodesic... define a null cone in the regular coordinate system... transform the regular metric to a Bondi-Sachs metric

41 The the Vertex The radial expansion of the Bondi-Sachs metric functions......starts at different positive powers...shows a strict angular behaviour in terms of associated Legendre polynomials in the coefficients...contains at higher order coefficients time derivatives of the lower order ones...contains strict numerical factors in the expansion coefficients Example : (u, r, ) = apple apple 2(u)P2 2 ( ) r 2 + 3(u)P 2 3 ( )+ 5 6 d 2 du (u)p 2 2 ( ) r 3 + O(r 4 )

42 The the Vertex The radial expansion of the Bondi-Sachs metric functions......starts at different positive powers...shows a strict angular behaviour in terms of associated Legendre polynomials in the coefficients...contains at higher order coefficients time derivatives of the lower order ones...contains strict numerical factors in the expansion coefficients Example : (u, r, ) = apple apple 2(u)P2 2 ( ) r 2 + 3(u)P 2 3 ( )+ 5 6 d 2 du (u)p 2 2 ( ) r 3 + O(r 4 ) The metric near the vertex is very rigidly fixed

43 Initial values to Integrate the Einstein Equations (1) axial geodesic (x ) free data (x ) free data r = R>0 specify, U A, and U A,r for all values of xa

44 Implications from the Regularity Conditions at the Vertex for Initial Data on a Light Cone axial geodesic r =0 TM, E. Müller, gr-qc/

45 Implications from the Regularity Conditions at the Vertex for Initial Data on a Light Cone axial geodesic r =0 set = U A = =0 and determine U,r A (, ) r=0 TM, E. Müller, gr-qc/

46 Implications from the Regularity Conditions at the Vertex for Initial Data on a Light Cone axial geodesic ll r=0 (u) free data l l r=0 (u) free data set r =0 = U A = =0 and determine U,r A (, ) r=0 (x )= (x )= 1X nx n=2 l=2 1X nx n=2 l=2 r n ln (u)pl 2 (cos ) r=0 r n l n (u)pl 2 (cos ) r=0 TM, E. Müller, gr-qc/

47 Regularity at the Vertex and asymptotical flatness Regularity conditions allow timedependent initial data that are TM, gr-qc/

48 Regularity at the Vertex and asymptotical flatness Regularity conditions allow timedependent initial data that are asymptotically flat TM, gr-qc/

49 Regularity at the Vertex and asymptotical flatness Regularity conditions allow timedependent initial data that are asymptotically flat not-asymptotically flat TM, gr-qc/

50 Regularity at the Vertex and asymptotical flatness Regularity conditions allow timedependent initial data that are asymptotically flat not-asymptotically flat This can be demonstrated by solutions derived from a quasi-spherical approximation of the Bondi-Sachs metric TM, gr-qc/

51 Example: the Scalar Wave Equation in Flat- Space Null Coordinates... the wave equation 0= = 1 r h 2(r ),ur +(r ),rr + 1 i r 2 ra r B (r )... a solution that is asymptotical flat 1X lx = l=0 m= l apple r A l u(u +2r) l+1 Y lm (x A )... a solution that is not asymptotical flat = 1X lx l=0 m= l B l e u+r I l+ 1 2 (r) p r Y lm (x A )

52 Summary In the Bondi-Sachs formulation it can be shown that an isolated system can only loose mass via gravitational radiation vacuum initial data on a light cone are fixed by the regularity conditions at the vertex - data are given by free functions along the curve tracing the vertex regularity conditions do not restrict whether the initial data are asymptotically flat or not