The Electromagnetic Christodoulou Memory Effect in Neutron Star Binary Mergers
TThe Electromagnetic Christodoulou Memory Effectin Neutron Star Binary Mergers Lydia Bieri, PoNing Chen, Shing-Tung Yau
Abstract
Gravitational waves are predicted by the general theory of relativity. In [6] D.Christodoulou showed that gravitational waves have a nonlinear memory. We provedin [3] that the electromagnetic field contributes at highest order to the nonlinear mem-ory effect of gravitational waves. In the present paper, we study this electromagneticChristodoulou memory effect and compute it for binary neutron star mergers. Theseare typical sources of gravitational radiation. During these processes, not only mass andmomenta are radiated away in form of gravitational waves, but also very strong mag-netic fields are produced and radiated away. Thus the observed effect on test masses ofa laser interferometer gravitational wave detector will be enlarged by the contribution ofthe electromagnetic field. Therefore, the present results are important for the plannedexperiments. Looking at the null asymptotics of spacetimes, which are solutions of theEinstein-Maxwell (EM) equations, we derived in [3] the electromagnetic Christodouloumemory effect. Moreover, our results allow to answer astrophysical questions, as theknowledge about the amount of energy radiated away in a neutron star binary mergerenables us to gain information about the source of the gravitational waves.
The main goal of this paper is to discuss the electromagnetic Christodoulou memory effectof gravitational waves and to compute this effect for typical sources. In [6] D. Christodouloushowed that gravitational waves have a nonlinear memory. In our paper [3] we proved thatfor spacetimes solving the Einstein-Maxwell (EM) equations, the electromagnetic field con-tributes at highest order to the nonlinear memory effect of gravitational waves. In the presentpaper, we also calculate it for neutron star binary mergers. We find that for typical constel-lations, very strong magnetic fields enlarge this effect considerably. Fields which are strongenough have so far only been known to be produced during mergers of neutron star binaries.The latter are well known to be frequent events. There is a vast astrophysical literature aboutthis.Moreover, our results in [3] and in the present paper, are also important from a purelyastrophysical point of view. Namely, the knowledge about the amount of energy radiatedaway in a neutron star binary merger allows to tell in the experiment what type of source thegravitational waves are coming from. Thus, our findings in the gravitational wave experimentwill contribute to astrophysical results.A major goal of general relativity (GR) and astrophysics is to precisely describe and fi-nally observe gravitational radiation, one of the predictions of GR. We know from the work[6] of Christodoulou that also these waves radiate. That is, in a laser interferometer grav-itational wave detector, this will show in a permanent displacement of test masses after awave train passed. The latter is known as the Christodoulou nonlinear memory effect. In L. Bieri is supported by NSF grant DMS-0904583 and S.-T. Yau is supported by NSF grant PHY-0937443and DMS-0904583.Lydia Bieri, University of Michigan, Department of Mathematics, Ann Arbor MI. [email protected] Chen, Harvard University, Department of Mathematics, Cambridge MA. [email protected] Yau, Harvard University, Department of Mathematics, Cambridge MA. [email protected] a r X i v : . [ m a t h . DG ] J un
6] Christodoulou showed how the nonlinear memory effect can be measured as a permanentdisplacement of test masses in such a detector. He derived a precise formula for this per-manent displacement in the Einstein vacuum (EV) case. The present authors proved in [3]that when electromagnetic fields are present, they will contribute to this nonlinear effect athighest order. In fact, we showed that for the EM equations this permanent displacementexhibits a term coming from the electromagnetic field, which is at the same highest order asthe purely gravitational term that governs the EV situation. Moreover, we showed that theinstantaneous displacement of the test masses is not changed at leading order by the electro-magnetic field. To see this, we investigated spacetimes of solutions of the Einstein-Maxwell(EM) equations at null infinity.Typical sources for gravitational waves are binary neutron star mergers and binary blackhole mergers. As the former are known to be much more frequent, it is likely that grav-itational waves as well as the nonlinear memory effect will first be measured from binaryneutron star mergers. During such processes mass and momenta are radiated away. More-over, large magnetic fields are produced and radiated away. The radiation travels at the speedof light. That means, it moves along null hypersurfaces of corresponding spacetimes. There-fore, in order to fully understand all the different situations, one has to investigate spacetimeswhich are solutions of the Einstein equations. Taking into account the strong magnetic fieldswhich are generated during binary neutron star mergers, we consider spacetimes solving theEinstein-Maxwell equations. As the sources are very far away, we can think of us as doingthe experiment at null infinity. Therefore it is very important to understand the geometry ofspacetimes especially at null infinity, that is when we let t → ∞ along null hypersurfaces inthe corresponding spacetimes.In this paper, we discuss the electromagnetic Christodoulou memory effect and computeconcrete examples for binary neutron star mergers. In [3], we derived this effect in theregime of the EM equations. First, we recall the Bondi mass loss formula obtained in [15] forspacetimes solving the EM equations. ∂∂u M ( u ) = 18 π (cid:90) S (cid:18) | Ξ | + 12 | A F | (cid:19) dµ ◦ γ (1)Compared to the formula obtained in [8] for spacetimes solving the EV equations, we havean additional term, | A F | , from the electromagnetic field. (See [3].)As shown in the work of Christodoulou [6], Σ + − Σ − is the term which governs the permanentdisplacement of test particles. Using this fact, Christodoulou shows that the gravitationalfield has a non-linear “memory” which can be detected by a gravitational-wave experimentin a spacetime solving the EV equations. Here, Σ denotes the asymptotic shear of outgoingnull hypersurfaces C u that are level sets of a foliation by an optical function u , which we willdiscuss below. Σ + and Σ − are the limits of Σ as u tends to + ∞ respectively −∞ .In our paper [3], we study the permanent displacement formula for uncharged test parti-cles of the same gravitational-wave experiment in a spacetime solving the EM equations. Wederive Σ + − Σ − in the EM case, and we find that the electromagnetic field changes the lead-ing order term of the permanent displacement of test particles. Moreover, investigating theexperiment for our setting in [3], we prove that the electromagnetic field does not enter theleading order term of the Jacobi equation. As a result, to leading order, it does not change theinstantaneous displacement of test particles. But the electromagnetic field does contribute2t highest order to the nonlinear effect of the permanent displacement of test masses.To study the effect of gravitational waves, we follow the method introduced by Christodoulouin [6]. The analysis is based on the asymptotic behavior of the gravitational field obtainedat null and spatial infinity. These rigorous asymptotics allow us to study the structure of thespacetimes at null infinity. To foliate the spacetime, we use a time function t and an opticalfunction u . We denote the corresponding lapse functions by φ respectively a . Whereas eachlevel set of t , H t is a maximal spacelike hypersurface, each level set of u , C u , is an outgoingnull hypersurface. Along the null hypersurface C u , we pick a suitable pair of normal vectors.The flow along these vector fields generates a family of diffeomorphisms φ u of S . Using φ u we pull back tensor fields in our spacetime. In this manner, we can study their limit atnull infinity along the null hypersurface C u . Building on these, we then take the limit as u goes to ±∞ , which allows us to investigate the effect of gravitational waves. For a detailedexplanation of the structure at null infinity, see [6] by Christodoulou.Understanding gravitational radiation and therefore null infinity heavily relies on the rig-orous understanding of the corresponding spacetimes. The methods introduced in [8], usedin [14], [15] and [1], [2], reveal the structure of the null asymptotics of our spacetimes. Inthese works, stability results were proven. The authors showed that under a smallness con-dition on asymptotically flat initial data for the EV respectively EM equations, this canbe extended uniquely to a smooth, globally hyperbolic and geodesically complete spacetimesolving the EV respectively EM equations. The spacetime obtained is globally asymptoticallyflat. The main achievements are generally two-fold: First, existence and uniqueness theo-rems were proven. To ensure these, one has to impose smallness conditions. Second, precisedescriptions of the asymptotic behavior of the spacetimes were derived. We stress the fact,that the results about null infinity are largely independent of the smallness. An elaboratedgeometric-analytic procedure led to these results. And many mathematical theorems wereproven on the way. However, the outcome exhibits a physical result in point two from whichChristodoulou in [6] derived the Christodoulou memory effect of gravitational waves in theEV case and the present authors in [3] in the EM case. In what follows, let us, discuss thenew physical results and compute the effects for different binary neutron star constellations.First we recall the Einstein-Maxwell equation. The electromagnetic field is represented by askew-symmetric 2-tensor F µν . The stress-energy tensor corresponding to F µν is T µν = 14 π (cid:0) F ρµ F νρ − g µν F ρσ F ρσ (cid:1) The Einstein-Maxwell equations read: R µν = 8 πT µν D α F αβ =0 D α ∗ F αβ =0 . (2)Let S t,u be the intersection of the hypersurface H t and the null cone C u . Let N be the space-like unit normal vector of S t,u in H t and T be the timelike unit normal vector of H t in thespacetime. Let { e a } a =1 , be an orthonormal frame on S t,u . We have the following orthogonalframe ( T, N, e , e ). This also gives us a pair of null normal vectors to S t,u , namely L = T + N and L = T − N . Together with { e a } a =1 , , they form a null frame. The following is a pictureof the null cone C u together with the null frame ( L, L, e , e ).3 _C CLLH S__ We can decompose the Weyl curvature tensor and the electromagnetic field with respect tothe null frame or the orthogonal frame. The asymptotics of these components are studiedin [14] and [15]. These asymptotics are important for the understanding of the geometry ofnull infinity. For simplicity, we will only list the components of the spacetime curvature andelectromagnetic field that are used in our discussion. Please see [14] and [15] for more detailson the asymptotics.Let
X, Y be arbitrary tangent vectors to S at a point in S . Given the null frame e = L , e = L and { e a } a =1 , , let χ ( X, Y ) = g ( ∇ X L, Y ) and χ ( X, Y ) = g ( ∇ X L, Y ) be the secondfundamental forms with respect to L and L , respectively. Let (cid:98) χ and (cid:98) χ be their tracelessparts. We also need the following null components of the Weyl curvature α W ( X, Y ) = R ( X, L, Y, L )and the electromagnetic field F A = α ( F ) A F A = α ( F ) A F = 2 ρ ( F ) F = σ ( F ) (3)We have the following limit of the above quantities at null infinitylim C u ,t →∞ r (cid:98) χ = Σ , lim C u ,t →∞ r (cid:98) χ = 2Ξlim C u ,t →∞ rα W = A W , lim C u ,t →∞ rα F = A F As shown in [6], the permanent displacement of the test masses of a laser interferometergravitational-wave detector is governed by Σ + − Σ − wherelim u →±∞ Σ = Σ ± Theorem 1 [14], [15] We have the following equations for Σ , Ξ and A W ∂ Σ ∂u = − Ξ and ∂ Ξ ∂u = − A W
4n our paper [3], we prove that in a spacetime solving the Einstein–Maxwell equations,Σ + − Σ − is governed by the following relation. Theorem 2 [3] Let F ( · ) = (cid:90) ∞−∞ (cid:0) | Ξ( u, · ) | + 12 | A F ( u, · ) | (cid:1) du . (4) Then Σ + − Σ − is given by the following equation on S : ◦ div/ (Σ + − Σ − ) = ◦ ∇ / Φ . (5) where Φ is the solution with ¯Φ = 0 on S of the equation ◦ (cid:52) / Φ = F − ¯ F .
Comparing this with the EV case studied in the last chapter of [8] and used in [6], where thecorresponding formula was F ( · ) = (cid:82) ∞−∞ | Ξ( u, · ) | du , we find that new the electromagneticpart | A F ( u, · ) | appears in the integral. In fact, in our proof, we derive the limitingformulas and obtain the said electromagnetic contribution in Σ + − Σ − . (See [3].) Gravitational Wave Experiment
How will our findings relate to experiment? In what follows, we are going to show howthe electromagnetic field enters the experiment. In particular, we will discuss the instan-taneous and the permanent displacement of test masses. For a detailed explanation of theexperiment we refer to [6] and for a detailed derivation in the EM case we refer to [3].Consider a laser interferometer gravitational-wave detector with three test masses. We denotethe reference mass by m , this is also the location of the beam splitter. The masses m , m , m are suspended by equal length pendulums of length d . The motion of the masses in thehorizontal plane can be considered free for timelike scales much shorter than the period ofthe pendulums. Now one measures the distance of the masses m and m from the referencetest mass m by laser interferometry. We observe a difference of phase of the laser light at m whenever the light travel times between m and m , m , respectively, differ.The motion of the masses m , m , m is described by geodesics γ , γ , γ in spacetime.Denote by T the unit future-directed tangent vectorfield of γ and by t the arch length along γ . Let then H t be for each t the spacelike, geodesic hyperplane through γ ( t ) orthogonalto T . At γ (0) pick an orthonormal frame ( E , E , E ) for H . By parallelly propagatingit along γ , we obtain the orthonormal frame field ( T, E , E , E ) along γ , where at each t the ( E , E , E ) is an orthonormal frame for H t at γ ( t ). Then we can assign to a point p inspacetime close to γ and lying in H t the cylindrical normal coordinates ( t, x , x , x ).Supoose that the distance d is much smaller than the time scale in which the curvatureof the spacetime varies significantly. Then the geodesic deviation from γ , namely the Jacobiequation (6), replaces the geodesic equation for γ and γ . Let R k l = R ( E k , T, E l , T ), thenwe write d x k dt = − R k l x l (6)5e can decompose R k l into the Weyl curvature and the Ricci curvature R k l = W k l + 12 ( g kl R + g R kl − g l R k − g k R l ) . From the EM equations (2) we find R = 12 ( | α ( F ) | + | α ( F ) | ) + ρ ( F ) + σ ( F ) (7)The component R observes the term | α ( F ) | , where α ( F ) is the electromagnetic fieldcomponent with worst decay behavior, but entering R as a quadratic. Hence, R is of theorder O ( r − ). Whereas the leading order component of the Weyl curvature is of the order O ( r − ). We give a detailed proof in our paper [3]. Thus, the electromagnetic field does notcontribute at highest order to the deviation measured by the Jacobi equation. As a conse-quence, it does only change at lower order the instantaneous displacement of the test masses.However, we are going to see that it does change the nonlinear memory effect.Using the relations from theorem 1 and our theorem 2 as well as the fact that Ξ → u → ∞ and taking the limit t → ∞ , we conclude that the test masses experience permanentdisplacements after the passage of a wave train. In particular, this overall displacement ofthe test masses is described by Σ + − Σ − ∆ x A ( B ) = − d r (Σ + AB − Σ − AB ) (8)where from our theorem 2 one sees that the right hand side of (8) includes the electromagneticfield terms at highest order.Let us now derive formula (8). We will use L = T − E and L = T + E . Then we writethe leading components of the curvature α AB ( W ) and of the electromagnetic field α A ( F ) asfollows: α AB ( W ) = R ( E A , L, E B , L ) = A AB ( W ) r + o ( r − ) α A ( F ) = F ( E A , L ) = A A ( F ) r + o ( r − )Let x k ( A ) with A = 1 , k th Cartesian coordinate of the mass m A . From [6] and[3] one sees that there is no acceleration to leading order in the vertical direction. One startswith m , m being at rest at equal distance d from m at right angles from m . Thus toleading order it is ¨ x A ( B ) = − r − d A AB (9)In particular, the initial conditions are as t → −∞ : x B ( A ) = d δ BA , ˙ x B ( A ) = 0 , x A ) = 0 , ˙ x A ) = 0.Integrating gives ˙ x A ( B ) ( t ) = − d r − (cid:90) t −∞ A AB ( u ) du . (10)From theorem 1 equation ∂ Ξ ∂u = − A W and lim | u |→∞ Ξ = 0, one substitutes and concludes˙ x A ( B ) ( t ) = d r Ξ AB ( t ) . (11)6s Ξ → u → ∞ , the test masses return to rest after the passage of the gravitationalwaves. Now, we use theorem 1 equation ∂ Σ ∂u = − Ξ and integrate again to obtain x A ( B ) ( t ) = − ( d r ) (Σ AB ( t ) − Σ − ) . (12)Finally, by taking the limit t → ∞ one derives that the test masses obey permanent displace-ments. This means that Σ + − Σ − is equivalent to an overall displacement of the test massesgiven by (8): (cid:52) x A ( B ) = − ( d r ) (Σ + AB − Σ − AB ) . The right hand side of (8) includes terms from the electromagnetic field at highest order asgiven in our theorem 2.In the next subsection, we are going to apply our results to astrophysical data for binaryneutron star mergers.
Binary neutron star mergers
We compute the electromagnetic Christodoulou memory effect for typical sources, that isfor different constellations of binary neutron star (BNS) mergers.In a binary neutron star or binary black hole system, the two objects are orbiting eachother. In Newtonian physics, they would stay like that forever. However, according to thetheory of general relativity such a system must radiate away energy. Therefore, the radius ofthe orbits must shrink and finally the objects will merge.As binary neutron star systems are much more frequent than binary black hole systems,it is very likely that gravitational waves as well as the nonlinear memory effect of gravita-tional waves will be detected first from the former systems. The magnetic fields producedand radiated away during the merger of two neutron stars are among the largest magneticfields known in astrophysics. In fact, in the electromagnetic Christodoulou memory effectthat we derived, the magnetic field enlarges the nonlinear displacement of (non-charged) testmasses significantly. As we are going to show in this subsection, the contribution from themagnetic field is very important, as it is very big for a large part of the known constellations.Astrophysical data gives for typical neutron star binaries a range of possible constellationswhich allow the mass and the magnetic field to vary within given boundaries. Typically, themass of a neutron star is around slightly more than 1 M ◦· and the radius of a neutron staris 3 - 30 km. Thus, the typical mass for a BNS system ranges between 2 . . M ◦· .In such a system, as the neutron stars are spiraling around each other, they are radiatingaway gravitational and magnetic energy. The inspiral goes with increasing speed and theBNS system emits an increasing amount of electromagnetic and gravitational energy, whichbecomes extremely large when the orbit radius is about 10 - 100km. For the detection of theelectromagnetic Christodoulou effect, the largest contribution will come from the last phaseof the inspiral, starting when the orbit radius is about 10 times the neutron star radius.In the literature, we find that the merger times range from a few milli-seconds up to 1000ms. We would like to compare the amount of gravitational energy radiated away during the M ◦· = 1 solar mass ≈ . · g erg. On the other hand, the amount ofmagnetic energy radiated away could vary drastically depending on different constellations.Typically, the rate of change for the magnetic field is dBdt ≈ − G(ms) − and themagnetic field produced in the merger is about 10 − G . Comparing the energy from the radiated mass, i.e. purely gravitational, and from the mag-netic field, we observe that during the merger of BNS very large magnetic fields are producedand radiated away in certain scenarios.Consider the following data. Assume: Total mass of BNS is initially 2 M ◦· , 1% of the totalmass will be radiated away during the (whole) merger, radius of each neutron star is 10 km.Under the assumption, the gravitational energy radiated away is about 3 . × erg.In the physics literature, one finds many linearized models. However, the Einstein equationsbeing nonlinear, the main information usually gets lost in linearized models. As we do inves-tigate the nonlinear problem here, and as the results of [6] and [3] show the Christodouloumemory effect of gravitational waves to be a nonlinear phenomenon, we consider a corre-sponding nonlinear model for the neutron star binary mergers. Thus, we use the results ofZipser’s global stability work [14] and [15] for the initial value problem in spacetimes satisfy-ing the Einstein-Maxwell equations. We assume that outside the neutron star, the magneticfield decays like r − / . Such decay at spatial infinity is suggested by the decay obtained in[14], [15]. One might want to consider situations with a slightly different decay of the mag-netic field. This would not affect the main picture, as one finds during the computations thatthe decay of the magnetic field does not play a role here. Thus, we work with the nonlinearmodel explained in the following paragraph.Now, consider such a BNS system with the magnetic field B initially being B = 10 Gand dBdt = 10 G(ms) − on the surface of the neutron star. Assume that the merger time is1000 ms. We estimate the total magnetic energy radiated away using the following model.We assume that through the merger, the matter of the neutron star stays in a ball of radius10 km. We compute the contribution from the magnetic field outside the support of thematter of the neutron star. As a result, we simply use the vacuum magnetic constant whencomputing the magnetic energy density. Moreover, we assume that outside the neutron star,the magnetic field decays at the rate of r − / . Using this model, the energy radiated awayfrom the magnetic field is about 4 . · erg. In this case, the addition of a magnetic fieldhas a small contribution to the memory effect.Next, consider a BNS system with the above data, but where the magnetic field B is initially B = 10 G and dBdt = 10 G(ms) − on the surface of the neutron star. Assume that themerger time is 1000 ms. We compute that the total magnetic energy radiated away is about4 . · erg. This will be one order of magnitude higher than the gravitational energyradiated away. This situation is consistent with astrophysical data. Also, in the numericsimulation in [9], [10] and [11], it is observed that the magnetic field could increase by twoorders of magnitude during merger when one starts with magnetic fields around 10 to 10 G. When we start with a stronger magnetic field, the merger would take longer and allow · cm s − and 1 M ◦· ≈ . · erg G = 10 − kg · C − s − = 10 − g · C − s − Conclusions:
We find that among the variety of different constellations of BNS systemsthere is a large part for which the magnetic field contributes to the Christodoulou effect atthe same highest order as the purely gravitational term.
Acknowledgment:
We thank Demetrios Christodoulou for fruitful discussions and his in-terest in this work.
References [1] L. Bieri.
An Extension of the Stability Theorem of the Minkowski Space in GeneralRelativity.
ETH Zurich, Ph.D. thesis. . Zurich. (2007).[2] L. Bieri.
Extensions of the Stability Theorem of the Minkowski Space in General Rel-ativity. Solutions of the Einstein Vacuum Equations.
AMS-IP. Studies in AdvancedMathematics. Cambridge. MA. (2009).[3] L. Bieri, P. Chen, S.-T. Yau.
Null Asymptotics of Solutions of the Einstein-MaxwellEquations in General Relativity and Gravitational Radiation.
Submitted. (2010).http://arxiv.org/abs/1011.2267[4] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner.
Gravitational Waves in GeneralRelativity. VII. Waves from Axi-Symmetric Isolated Systems.
Proc. Roy. Soc. A. (1962). 21-52[5] M. G. J. van der Burg.
Gravitational Waves in General Relativity X. Asymptotic Ex-pansions for the Einstein-Maxwell Field
Proc. Roy. Soc. A. (1969). 221-230[6] D. Christodoulou.
Nonlinear Nature of Gravitation and Gravitational-Wave Experi-ments.
Phys.Rev.Letters. . (1991). no.12. 1486-1489.[7] D. Christodoulou. Mathematical problems of general relativity theory I and II.
Volume1: EMS publishing house ETH Z¨urich. (2008). Volume 2 to apppear: EMS publishinghouse ETH Z¨urich.[8] D. Christodoulou, S. Klainerman.
The global nonlinear stability of the Minkowski space.
Princeton Math.Series . Princeton University Press. Princeton. NJ. (1993).99] Bruno Giacomazzo, Luciano Rezzolla, Luca Baiotti Can magnetic fields be detectedduring the inspiral of binary neutron stars?, arXiv:0901.2722 [gr-qc][10] Bruno Giacomazzo, Luciano Rezzolla, Luca Baiotti
Accurate evolutions of inspirallingand magnetized neutron-stars: equal-mass binaries, arXiv:1009.2468v2 [gr-qc][11] Luciano Rezzolla, Bruno Giacomazzo, Luca Baiotti, Jonathan Granot, Chryssa Kou-veliotou, Miguel A. Aloy
The missing link: Merging neutron stars naturally producejet-like structures and can power short Gamma-Ray Bursts, arXiv:1101.4298 [gr-qc][12] Yuk Tung Liu, Stuart L. Shapiro, Zachariah B. Etienne, Keisuke Taniguchi
Generalrelativistic simulations of magnetized binary neutron star mergers, arXiv:0803.4193v2[astro-ph][13] Shapiro, S. L. and Teukolsky, S. A.
Black Holes, White Dwarfs and Neutron Stars: ThePhysics of Compact Objects
Wiley-Vch New York, NY (1983)[14] N. Zipser.
The Global Nonlinear Stability of the Trivial Solution of the Einstein-MaxwellEquations.
Ph.D. thesis. Harvard Univ. Cambridge MA. (2000).[15] N. Zipser.
Extensions of the Stability Theorem of the Minkowski Space in General Rel-ativity. - Solutions of the Einstein-Maxwell Equations.
AMS-IP. Studies in AdvancedMathematics. Cambridge. MA. (2009).
Lydia BieriDepartment of MathematicsUniversity of MichiganAnn Arbor, MI 48109, USA [email protected]
PoNing ChenDepartment of MathematicsHarvard UniversityCambridge, MA 02138, USA [email protected]