Download gravitational collapse to black holes

GRAVITATIONAL COLLAPSE
TO BLACK HOLES
& NEUTRON STARS
Burkhard Zink
Theoretical Astrophysics
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
INTERNAL STRUCTURE OF A
NEUTRON STAR
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
THE TOV EQUATIONS
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
THE KRUSKAL DIAGRAM
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
PENROSE DIAGRAM OF A
SCHWARZSCHILD BLACK
HOLE
Dienstag, 8. Mai 12
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)
Dienstag, 8. Mai 12
PENROSE DIAGRAM OF A
ROTATING BLACK HOLE
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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.
Dienstag, 8. Mai 12
ONION SHELL STRUCTURE
BEFORE CORE COLLAPSE
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
SPACETIME DIAGRAMS OF
GRAVITATIONAL COLLAPSE
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
ADVANCED SUBJECT:
THE CAUCHY HORIZON
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12
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
Dienstag, 8. Mai 12