GRAVITATIONAL COLLAPSE TO BLACK HOLES & NEUTRON STARS Burkhard Zink Theoretical Astrophysics
OVERVIEW Both black holes and neutron stars are so-called compact objects, i.e. they are particularly dense High densities can result from gravitational collapse, when larger, less dense objects lose part of their pressure support and collapse to a smaller, denser phase General relativity is central in the formation of black holes, and still very important for neutron stars I will first discuss both object classes, and afterwards how they are believed to being produced
NEUTRON STARS Neutron stars are ultradense objects of ca. 1.4 to 2 solar masses, with radii of ca. 10-15 km A large part of the star's interior is above the nuclear saturation density (ca. 10^14 g cm^-3) For this reason, they form unique laboratories of the state of matter at supernuclear densities Many neutron stars are endowed with strong magnetic fields, and are observationally visible as pulsars Since they are so compact, they may also be effective emitters of gravitational radiation, which in turn may provide clues to properties of matter at supernuclear densities
INTERNAL STRUCTURE OF A NEUTRON STAR
THE EQUATION OF STATE The equation of state (EOS) relates thermodynamics potentials to local primitive properties of the star (e.g. density and temperature) It can be built from models based on nuclear physics In the core, the EOS is mostly unknown: it can be a standard baryonic phase rich in neutrons, with a smaller component of protons, electrons and hyperons. Or it may be a quark phase of some description, yielding strange or hybrid stars. The core will also exhibit superfluid and superconducting components Going to the surface, and the direction of decreasing density, matter will have a number of different (maybe very complicated) transitional phases until we finally reach the surface, which will likely be solid iron with degenerate electrons
EQUILIBRIUM MODELS OF NEUTRON STARS Neutron stars are very compact, and therefore also probes of strong (general relativistic) gravity A model of a neutron star can be constructed by assuming a relativistic equilibrium state, that is a spacetime which has a timelike Killing vector. This kind of model is of course idealized (after all, the neutron star is produced by collapse!) but useful in practice. Another very common assumption is spherical symmetry, which is reasonable for most neutron stars apart from very rapidly rotating ones. If we further assume an ideal fluid with so called barotropic structure (the pressure depends only on the density), we can use the ideal fluid energy momentum tensor and solve for the remaining components of Einstein's field equations The resulting system are the TOV (Tolman - Oppenheimer - Volkoff) equations which are central to neutron star physics
THE TOV EQUATIONS
ROTATING NEUTRON STARS Neutron stars are probably born with a substantial amount of angular momentum, so even idealized models of young neutron stars benefit from having rotation taken into account However, whereas the TOV equations are a system of ODEs, rotating models result in a system of PDEs and need to be solved for using sophisticated numerical techniques The resulting models are the foundation of a number of present studies in neutron star physics
BLACK HOLES Black holes are even more compact than neutron stars: here, several solar masses are compressed within only a few kilometres "radius" They do not possess a distinct surface as such, but are delimited by a so-called event horizon which is the limit beyond which light cannot escape to infinity In many respects, black holes are objects which are very unique to general relativity, since they exhibit the geometric nature and causal structure of spacetime in a peculiar way
THE SCHWARZSCHILD SOLUTION Similar to the TOV equations before, we can construct a simple model of a black hole by assuming stationarity and spherical symmetry, but now assume vacuum but still require the spacetime to have a non-zero mass at infinity The field equations can be solved analytically and result in the Schwarzschild metric, which is singular at the surfaces r = 2GM and r = 0. The angular part however corresponds to a normal sphere. Interpreting r as a radius (which is not obvious, and in particular incorrect when r < 2GM) we can see that the part r > 2GM is a natural description of a spacetime outside the singular surface In fact, it can be shown that M has the physical meaning of a mass in that any orbit around the center will be equivalent to having a star of mass M replace the black hole
THE SCHWARZSCHILD SOLUTION It was long believed that the singular surface at r = 2GM invalidates the solutions (i.e. it is not a physically valid model), but it can be shown that a simple change of coordinates removes the singularity at this event horizon The singularity at r = 0 remains instead, it is an actual physical feature of the solution but is hidden by the event horizon as we will see It is important to remember that r = 0 is not at the center of the black hole, but into its future. The conformal diagrams I will show in a second will illustrate this Generally, the Schwarzschild metric only describes part of the spacetime. To describe the full solution, several coordinate maps need to be employed, exactly in the same way as several maps are needed to fully cover a sphere The most well-known representation of the full Schwarzschild black hole is the Kruskal diagram
THE KRUSKAL DIAGRAM
PENROSE DIAGRAMS Closely related to the Kruskal diagram are conformal or Penrose diagrams These result from applying conformal transformations (i.e. those which keep angles intact) to the spacetime, and from compactification, i.e. mapping an infinite coordinate range to a finite interval A simple example of a compactification is the map y = atan(x) which map R^+ into [0:Pi/2] For general relativity, conformal maps are important since they leave the light-cone structure invariant - they therefore represent the overall causal structure of space-time while removing less important details They are also useful to understand gravitational collapse to black holes, as we will see
PENROSE DIAGRAM OF A SCHWARZSCHILD BLACK HOLE
ROTATING BLACK HOLES Again similar to neutron stars, rotating models can be constructed when neglecting the assumption of spherical symmetry The resulting class of black holes is called Kerr black holes, and they are described by the Kerr metric above In constrast to neutron stars, black holes are "simpler" in that they can be uniquely determined by the parameters mass M and angular momentum J (and also charge Q if applicable) Kerr black holes also exhibit an ergosphere, inside which all lightcones are tilted with respect to observers at infinity (so all local matter is "forced" to rotate with the black hole)
PENROSE DIAGRAM OF A ROTATING BLACK HOLE
BIRTH OF NEUTRON STARS AND BLACK HOLES All models of neutron star and black hole discussed so far show many interesting features, but they also have a serious flaw: they are stationary Unless we wish to describe a neutron star which is eternal (and therefore even predates the universe itself) or black holes which are eternal and always connected with white holes, we have to consider how these objects are actually produced in the first place In both cases, the standard model is gravitational collapse In particular, neutron stars are typically produced by the collapse of iron cores (which will be discussed in a minute), and black holes by the collapse of neutron stars We will begin with neutron stars
SUPERNOVAE: BIRTH SITES OF NEUTRON STARS Supernovae are distinctly luminous transients which are believed to be caused by an explosion connected with the death of massive stars They are entirely distinct from novae, which are caused by hydrogen nuclear burning on the surface of a white dwarf A supernova is massive explosion, but it also often leaves a remnant: a new-born neutron star We will now discuss some aspects of the standard model of supernova explosions
DEATH OF MASSIVE STARS Stars on the main sequence are generally powered by the nuclear conversion of hydrogen into helium, and the resulting thermal pressure opposes the gravitational forces acting on the star If the hydrogen fuel is exhausted, the star leaves its dynamical equilibrium (and also the main sequence) and takes an evolutionary path with depends mainly on its mass Lower mass stars (below ca 10 M_sun) will generally compress into white dwarfs, which are supported by degenerate electron pressure We will focus on the higher mass stars which can give birth to black holes If the central pressure inside the star is high enough (due to higher mass), the star will start burning helium at its core. If that fuel is exhausted, it will continue with carbon and so on The burning sequence will end in the last exogenic link, which results in iron.
ONION SHELL STRUCTURE BEFORE CORE COLLAPSE
THE IRON CORE The central 1.5 solar masses of material in the onion shell structure consist of iron ash This core is supported by degenerate electron pressure, similar to the ultrarelativistic limit of the white dwarf sequence Subsequent burning will tend to accrete more iron to the core, but at around 1.5 M_sun mass the limit reaches its Chandrasekhar mass, which we will discuss in a moment Even without the mass limit, the iron core loses pressure support by electron capture, in which electrons are captured by protons to form neutrons (which are at this stage an energetically favorable state due to the high density and therefore high Fermi energy of the electrons) For both reasons, it is clear that the iron core cannot exist forever, and will collapse under its own weight
THE CHANDRASEKHAR MASS At around 1.5 M_sun mass, the iron core will reach a limit where it cannot support itself anymore by relativistic electron pressure The Chandrasekhar limit at 1.4 M_sun (the rest is thermal pressure) is the maximal mass which can be supported by a relativistic Fermi gas, and therefore both the maximum mass for white dwarfs and also iron cores For a gas of this type, we have a Gamma (= ratio of specific heats) of almost 4/3 In Newtonian physics, it is known that Gamma = 4/3 polytropes are marginally stable to radial perturbations, i.e. they will statically deform under compression without offering resistance In general relativity, all Gamma = 4/3 polytropes are unstable, i.e. the fundamental radial mode does not have a real frequency and any radial perturbation will grow exponentially This is the foundation of stellar collapse
THE TURNING POINT THEOREM The turning point theorem states that for every location dm/drho = 0 in a sequence of equilibria we have a change of stability in radial mode, starting with the fundamental mode Change of stability means: the frequency of the mode is neutral, i.e. omega^2 = 0, at this point If we have a stellar model and constantly add mass to it, any point dm/drho = 0 will in practice imply that the sequence ends there, that is, the star will either collapse or expand exponentially when adding further mass; typically, the inward ram pressure of material (and loss of internal pressure) cause a collapse For iron cores, this turning point is at 1.4 M_sun (the Chandrasekhar mass) for typical parameters associated with a degenerate electron gas
CORE COLLAPSE When the central iron core becomes unstable, it collapses in a mostly radial fashion. Initial angular momentum is conserved during the collapse, and therefore the spin rate of the core will increase during collapse The binding energy which become available during this phase (10^53 erg) is the primary power source of a supernova Due to the high densities, the nature of matter (of the equation of state) changes during collapse
CORE COLLAPSE (II) The initial iron core has a central density of ca. 10^10 g cm^-3 During collapse, at densities beyond 10^11 g cm^-3 neutron drip occurs, where it is favorably to have a free neutron fluid in addition to the nuclei This further softens the EOS => collapse accelerates At somewhat higher densities the material becomes opaque to the neutrinos which are generated by weak processes producing neutrons This introduces a neutrino pressure and slows collapse again
CORE COLLAPSE (III) When densities beyond ca 2 10^14 g cm^-3 are reached in the center of the collapsing core, we have reached the nuclear saturation density, i.e. the strong interaction becomes important to stabilize the material Gravitation and inertia still compress the material to higher densities, but the transition to a very stiff equation of state essentially halts the collapse => core bounce In the center of the iron core, a proto-neutron star has been born At this stage, the material will be rich in neutrons, and at higher densities transitions to hyperonic matter may also be important
CORE COLLAPSE (IV) The core will now have roughly 0.7 M_sun The sudden bounce will produce a mostly radial shock wave which travels outwards from the core through the stellar material In early days, it was believed that this shock may travel through the rest of the stellar envelope and heat / eject the material so that a supernova occurs However, numerical models have cast doubts on this simple model, since nuclear disintegration removes too much energy from the shock Numerical models nowadays paint a much more complex picture
SUPERNOVA LAUNCH MECHANISM When the outgoing shock stalls, it may be revived in a number of different ways Processes in the proto-neutron star core generate a large amount of neutrinos which could inject energy into the shock In practice, the transfer will rely on strong convection processes between the neutrino sphere and the gain radius Another option are magnetic fields which are amplified during the collapse In all cases though, a substantial amount of material needs to be ejected to produce a visible supernova
THE REMNANT: A NEWBORN NEUTRON STAR In case the shock wave removes most of the remaining stellar material (many solar masses), only a small fraction accretes back to the proto-neutron star in the core The PNS will cool, condense further, and after a while settle down to a colder (on nuclear scales) equilibrium star => a neutron star If magnetic fields are substantial, a strong dipolar component will cause pulsar activity to begin, and will also inject energy in the remaining and cooling nebula of leftover stellar material (pulsar wind nebula) The crab nebula is an example
ANOTHER MECHANISM: ACCRETION-INDUCED COLLAPSE Besides the standard core collapse scenario, neutron stars may also be born in accretioninduced collapse of a white dwarf When a white dwarfs accretes matter from a companion star (cataclysmic variables) it may reach and exceed its Chandrasekhar limit in the process As soon as this happens, the first radial mode will go unstable and the star collapses to a proto-neutron star, similar to the iron core collapse scenario In contrast to core collapse, white dwarfs in CVs are usually rapidly rotating, so the newborn neutron star should have a very high spin Also, no extra stellar shells have to be removed
COLLAPSE TO BLACK HOLES The production of a black hole is quite different from core collapse or AIC Typically, the progenitor of a stellar mass black hole will be a neutron star which goes unstable The cause of instability is the same as in iron core collapse or AIC: the fundamental radial mode of the star goes unstable Similar to the Chandrasekhar mass, neutron stars have maximal masses (now primarily determined by strong forces) in the order of 2-3 solar masses, and if a neutron star accretes material beyond its limit it will collapse A very typical scenario would be a failed supernova, or the merger of two neutron stars after an inspiral
THE FINAL PRODUCT We know how a black hole is structured when in equilibrium: it has an event horizon, a singularity inside that horizon, and also an ergosphere outside But these features are eternal - how does a collapsing neutron star connect with an "eternal" horizon which extends through all of time? It doesn't! The horizon must be dynamically produced during the collapse. The same is true for any singularities We will look at this now, using a spherically symmetric collapse for simplicity
SPACETIME DIAGRAMS OF GRAVITATIONAL COLLAPSE
TRAPPED SURFACES AND APPARENT HORIZONS (I) The structure of spacetime can lead to a focussing of light rays In particular, a bundle of light rays evolves in terms of local expansion, vorticity and shear (consequence of the Raychaudhuri equation) Given a 2-surface, we can decompose the null cone on the surface into an outgoing and an ingoing null vector Normally, the ingoing congruence will converge, whereas the outgoing one will diverge In a trapped surface, also the outgoing bundle converges
TRAPPED SURFACES AND APPARENT HORIZONS (II) The outermost surface where this condition marginally holds is an apparent horizon If an apparent horizon exists, it will always be contained within an event horizon Apparent horizons are surface ("slicing") dependent objects, whereas event horizons are global objects In gravitational collapse, an event horizon will first appear at the center of the star, then expand outwards with the speed of light. Shortly after, an apparent horizon will "suddenly" appear for a set of observers, and will quickly approach the event horizon in typical gauge choices
ADVANCED SUBJECT: THE CAUCHY HORIZON
THE BUCHDAHL LIMIT AND COSMIC CENSORSHIP For static, spherically symmetric stars the maximal mass is given by the Buchdahl limit: the ratio 2M/R must always be less than 8/9 (although typical neutron stars are quite far from this extreme case) A more interesting question concerns rotation: Since Kerr spacetimes have a naked singularity for a/m > 1, could it be possible to dynamically produce a naked singularity (one that is not covered by an event horizon) when the progenitor has J/M^2 > 1? The cosmic censorship conjecture says: no! But it is just a conjecture, and not shown in general. Naked singularities can be pathological and either cause causality issues or restrict the space of possible field solutions quite a bit, so it sounds reasonable, but it is not proven There has been some (weak) numerical evidence opposing the conjecture
SUMMARY Gravitational collapse is responsible for producing two of the most compact (and arguably most enigmatic) types of stars: neutron stars and black holes Neutron stars are typically produced in the iron core collapse after the final hours of the life of a massive star (Stellar mass) black holes are usually produced when neutron stars accrete material beyond their mass limit In both cases the fundamental radial mode becomes unstable, initiating collapse