Egwald: Topics in Astronomy and Cosmology  Stellar Collapse
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Modelling Stellar Collapse, Neutron Stars, and Black Holes
by Mathew Hay, Elmer Wiens, and Haris Amiri
Department of Physics and Astronomy, University of British Columbia
Research Paper prepared for Kristin Schleich's Physics 407 course: Introduction to General Relativity (2018W).
Abstract
Stars that stop generating energy through nuclear fusion die. Smaller stars transform into white dwarf stars. Larger stars collapse to form neutron stars or black holes.
Much larger stars may blow apart. Various coordinate transformations of the Schwarzschild metric are used to calculate geodesic trajectories of particles and radiation at various
radii about stellar objects collapsing from selfgravitation. Within a black hole, the mechanics of stellar collapse are modelled with the geodesics of the RobertsonWalker metric (FLRW).
I. Introduction
Stars evolve by burning a sequence of lighter to heavier elements through the process of nucleosynthesis. Heavier elements are the result of the consumption of lighter elements.
At each stage, the heaviest existing element accumulates in the core of the star, with lighter elements arranged in layers of decreasing atomic weight moving from the core to
the surface of the star. High mass stars can burn through the entire sequence of hydrogen, helium, carbon, neon, oxygen, silicone, and iron. Because iron has the highest
binding energy of all elements, this sequence of nucleosynthesis stops at iron. Thereafter, the outward pressure of the star’s hot gas cannot withstand the inward pull of gravity.
Some other source of outward pressure is required to stop the star from collapsing through selfgravitation when energy production through fusion of lighter elements into
heavier elements stops.
The type of object that forms after a star collapses depends on its mass. Stars with a mass similar to the Sun's mass (M_{⨀}), and up to 10 M_{⨀}, wind up as white dwarf stars.
After these stars blow off their outer layers as planetary nebulae, further gravitational collapse is prevented by electron degeneracy pressure. Larger stars form neutron stars
or black holes when their iron cores collapse and they expend gravitational potential energy as supernovae. Electron and neutron degeneracy pressure are quantum mechanical,
nonthermal sources of pressure.
White dwarf and neutron stars within a binary system are able to increase their masses by accreting hydrogen from a close companion star. The maximum mass of a stable
whitedwarf is 1.4 M_{⨀}  the Chandrasekhar limit. The maximum mass of a neutron star based on observations is 2.27 M_{⨀}  the TolmanOppenheimerVolkoff limit.
A whitedwarf that exceeds its mass limit will collapse into either a neutron star, a black hole, or shatter completely, producing a supernova. A neutron star that gets
too large will collapse into a black hole and also create a supernova.
The mass of a black hole is contained within a sphere, the event horizon, whose radius equals a parameter called the Schwarzschild radius. During stellar collapse to a black hole,
a star's particles will either be expelled in a supernova or wind up inside the black hole. Once inside, a particle cannot leave a black hole since the escape speed from
within the event horizon hole exceeds the speed of light.
The ratio of the star's radius to its Schwarzschild radius for the neutron star
PSR J0348+0432 in the constellation Taurus equals 2.2. The white dwarf supernova candidate, RX J0648.04418 in Puppis, has a ratio of 1500. For the supernova
candidate Betelgeuse in Orion, it is 1.9 * 10^{7}.
In geometric units with c = 1, and G = 1, the Schwarzschild radius equals twice the mass of the stellar object.
A. The Schwarzschild Metric
Albert Einstein's field equations of general relativity model
the curvature of spacetime in the vicinity of stellar objects [1]. Particularly, they relate the curvature of spacetime and the stressenergy from the presence of
matter or other forms of energy. With certain simplifying assumptions, the Schwarzschild metric is a solution to these equations. In describing the gravitational field around
the stellar object, the Schwarzschild metric and its transformations depend only on the mass of the object (M), the distance (r) to its center, the polar angle (θ),
and the azimuthal angle (φ).
Modeling spacetime within the framework of the Schwarzschild metric requires some fundamental assumptions. Schwarzschild spacetime is spherically symmetric in that there is
no change after rotation around a chosen origin. Additionally, the metric is a vacuum solution to Einstein's field equations, with Ricci scalar R = 0. Furthermore, the
Schwarszchild spacetime is static, which differs from the dynamic nature of the FriedmannLemaitreRobertsonWalker (FLRW) metric considered in Section C. For the
purposes of this paper, Schwarzschild spacetime provides an approximate model of the collapse of a shell of dust that is infalling to a nonrotating stellar object.
In solving for the geodesic equations of the Schwarzschild metric, the dθ/dτ component will be assumed to be zero constraining motion of particles to null
geodesics, or particle freefall radial geodesics.
The Schwarzchild Metric
ds^{2} = (1  2M/r)dt^{2} + (1  2M/r)^{1}dr^{2} + r^{2}dΩ^{2} (1)
dΩ^{2} = (dθ^{2} + sin^{2}(θ)dφ^{2})
Using the Schwarzschild metric given in Eqn. 1, the geodesic equations for a particle travelling in Schwarzschild geometry can be obtained. The derivation is omitted and
the resulting first order terms for the general fourvelocity of the metric are given below.
dt/dτ = ε / (1  2M/r) (2)
dr/dτ = √(ε^{2}  1 + 2M/r  l^{2} / r^{2} + 2M l^{2} / r^{3}) (3)
dφ/dτ = l / r^{2}
(4)
Here, the value of ε is set by the initial radial velocity at some time τ_{0} and the value of l is related to the angular momentum.
With ε = 1 and l = 0 for radial infall of a particle from infinity, the following relationships between proper time τ, and r and coordinate
time t are obtained [2]:
r(τ) = (3/2)^{2/3} (2M)^{1/3}(τ  τ) (5)
where τ is the particle's final proper time at r = 0, and
t(r) = t + 2M [(2/3)(r/2M)^{3/2}  2(r/2M)^{1/2}
+ ln ( (r/2M)^{1/2} + 1 ) / ( (r/2M)^{1/2}  1 ) ] (6)
where the value of t selects a particular radial geodesic (see FIGS 2 and 3).
The Schwarzschild metric has a coordinate singularity at r = 2M. This singularity can be removed by transforming the metric to more convenient coordinates.
Let v = t + r^{*} with dr^{*} = dr / (1  2M/r). Integrating yields
r^{*} = r + 2M ln  (r/2M 1)  (7)
Substituting dt = dv  dr^{*} into the Schwarzschild metric results in the following version of the metric with EddingtonFinkelstein coordinates.
1. EddingtonFinkelstein Coordinates
ds^{2} = (1  2M/r) dv^{2} + 2dvdr + r^{2} (dθ^{2} + sin^{2}(θ)dφ^{2})
(8)
The null geodesics are v = C1, and v = 2 * r* + C2, where C1 and C2 are different constants for each null geodesic trajectory.
FIG. 1: EddingtonFinkelstein Coordinates
Null Cones in grey;
Null Geodesics: blue; Radial Geodesics: red
Variable v is timelike for r > 2M, spacelike r < 2M
Matlab Program to generate Figure 1: Schwarzschild spacetime in EddingtonFinkelstein Coordinates.
2. KruskalSzekeres Coordinates
A coordinate transformation into KruskalSzekeres coordinates extends the visible regions of spacetime allowing an observer to examine the behaviour
of spacetime within a black hole, white hole, and theoretical alternate universes. Transforming Schwarzschild spacetime into KruskalSzekeres coordinates
gives the resulting metric.
ds^{2} = (32M^{3} / r) e^{r/2M}(dV^{2} + dU^{2}) + r^{2}dΩ^{2}
(9)
subject to the constraints with which one can derive the Schwarzschild radius implicitly
V^{2}  U^{2} = (1  r/2M) e^{r/2M} (10)
where the transformation is given by
V = √(r/2M  1) e^{r/4M} sinh(t/4M)
U = √(r/2M  1) e^{r/2M} cosh(t/4M) (11)
V = √(1  r/2M) e^{r/4M} cosh(t/4M)
U = √(1  r/2M) e^{r/4M} sinh(t/4M) (12)
where Eqn. (11) is subject to r > 2M and Eqn. (12) subject to r < 2M.
FIG. 2: KruskalSzekeres Coordinates
Radial freefall of particles from rest at infinity
Intersect observers at coordinate time t = 0
Observers at radial distances r = 2.2M, 2.5M, 3M, 3.5M
FIG. 3: Radial freefall of a particle
Particle's Radial Distance r (red) and Coordinate
Time t (blue) Versus Proper Time τ
t = infinity at r = 2M, t = 0 at r = 0
Matlab Program to generate Figures 2 and 3: Schwarzschild spacetime in KruskalSzekeres Coordinates.
B. Neutron Stars
The TolmanOppenheimerVolkoff limit describes the theoretical maximum stable mass of a neutron star before it will collapse into a black hole under its
own gravity. Discussed in [3], an approximation was made by modeling the neutron star as a degenerate cold fermi gas. By assuming that
the main source of energy within a star comes from thermonuclear reactions, one can model the neutron star as a collection of noninteracting fermions
(neutrons). Above some critical mass, no stable configurations can be found for the fermi gas. This indicates that above some critical mass (the TOV limit),
degeneracy is no longer strong enough to maintain the star's structure and the gas will collapse to a singular point  a black hole.
The results from [3] resulted in a theoretical maximum of 0.7 M_{⨀}, less than that of the Chandrasekhar limit which models the critical mass of
a white dwarf. Through modelling the evolution of a neutron star from formation to a cold neutrinofree star, a different maximum value is obtained. Accounting
for strong nuclear repulsion, [6] obtained a value closer to 2.27 M_{⨀}. As this is the largest observed mass of a neutron star, this is presently
set as the TOV limit. If a neutron star is under 2.27 M_{⨀}, it can gain mass either through accretion of hydrogen from nearby companion stars, or the merger
of two binary neutron stars can o
ccur in which the sum of their masses exceed the TOV limit and the gravitational force dominates collapsing the star into a black hole.
C. The Robertson Walker Metric
The FriedmannLemaitreRobertsonWalker (FLRW) metric is an exact solution of Einstein's field equations. A universe characterized by the FLRW metric is based on three
assumptions. The first assumes that the space inside a FLRW universe is homogeneous. The universe is the same everywhere, but it is able to change as a function of
time unlike a Schwarzschild universe. Additionally, the FLRW universe is spatially isotropic. There is no preferred direction of expansion or distribution of matter
within the FLRW universe. Finally, the matter that is contained within the FLRW universe can be characterized solely by its pressure and density allowing one to solve
for the equation of state and subsequent field equations from these two variables alone. The sign of the dimensionless, timedependent a(t) factor is positive. While
the universe is presently seen to be expanding, corresponding to an increasing value for a(t), regions of contracting spacetime, such as the collapse of a star into a
black hole, correspond to a negative value for da/dt. The value of a(t) is determined by the Friedmann equation, a solution to Einstein's field equations [4],
((da(t)/dt)/a(t))^{2} = 8 π G ε(t) / (3c^{2})  κ c^{2} / (R_{0}^{2} a(t)^{2})
(13)
where ε(t) is the energy density, R_{0} is the radius of curvature, and possible values of κ are as below, along with the fluid equation and equation of state.
The RobertsonWalker Metric
ds^{2} = dt^{2} + a(t)^{2}(dr^{2} + S_{κ}^{2}(r) (dθ^{2} + sin^{2}(θ)dφ^{2})
(14)
S_{κ}(r) =  R sin(r/R)  (κ = +1)  
r  (κ = 0)  (15) 
R sinh(r/R)  (κ = 1)  
The value of S_{κ}(r) describes the curvature of spacetime, with radius of curvature of spacetime R, corresponding to positive, zero, and negative curvature respectively.
D. Black Holes
In order to study what would happen to observers as the process of entering a black hole from the surface or edge is `reversed', thereby beginning with the observer in the center of the black
hole and moving outwards, requires a set of radial geodesics obtained from the FLRW metric.
d^{2}t / dτ^{2} = (a(t) da(t)/dt) / (1  κ r^{2}) * (dr / dτ)^{2}
(16)
d^{2}r / dτ^{2} = 2 (da(t)/dt / a(t)) * (dt/dτ) * (dr/dτ)  κ r / (1  κ r^{2}) * (dr/dτ) ^{2}
(17)
As an observer moves from the center of the black hole to the edge, which is impossible, but since the process is being `played back' it may be a valid thought experiment,
time t becomes 'zero', and no longer spacelike. In the FLRW metric, time and proper time are the same. When radiation dominates over matter, as is the case inside a
black hole, the expansion of the universe takes the form a(t) ∝ t^{1/2} [4].
FIG. 4: a(t) in blue; da/dt in red;
(da(t)/dt / a(t)) in black; r(t) in green
Another way of examining this process considers the scale factor a(t) to be proportional to t^{1/2}. Starting at time t = 0 and r = 2M, a photon will travel a path that increases
rapidly, before slowing down. The scale factor is the inverse of the previous case, only this time modelled after a contracting universe. The result is the same, as the photon
eventually approaches r = 0, with dr/dt = 0. The speed of light is a relationship between space and time, and at the center of a black hole, time becomes spacelike.
Time 'blows' up to infinity, and if time is spacelike, the relationship between time and space, dr/dt, becomes 0. Fig. 4 shows a possible path of a photon
inside a black hole, starting at t = 0 and r = 2M.
Modelling the black hole similarly to blackbody photons in the early universe, and the processes inside the star as analogous to the big bang, the timedependent relationship of temperature
is given in Eqn. (18).
T(t) ≈ 10^{10} K (t/1s)^{1/2} (18)
At t = 0, or at the edge of the black hole, the temperature diverges. In simpler terms, the temperature at the edge of the edge of the black hole is incredibly hot. At the center
of the black hole,
when t = ∞, the temperature is 0. This is comparable to the observation that supermassive black holes have a predicted temperature of 1.4 x 10^{14} Kelvin,
which is almost absolute zero. When an observer is drawn into a black hole, they are accelerated to nearly the speed of light. Their molecules collide much more rapidly, heating
up to hundreds of millions of degrees Kelvin [5]. This also explains why when astronomers observe and try to measure the temperature of black holes, they record very
high temperatures. They are actually observing the radiation from the observers or the material drawn in, masking the radiation escaping from the black hole itself.
E. Discussion
Modelling stellar collapse within the framework of general relativity is one way of predicting some of the properties of neutron stars and black holes. Schwarzschild spacetime is one of
the simplest metrics to model black holes, but assumes no Hawking radiation, charge, or rotation. The FLRW metric assumes isotropy and homogeneity in the universe where matter is
equally distributed. By deriving the geodesics of the FLRW metric, the path a photon travels in a black hole, assuming radiation dominance, can be investigated. The scale factor
in the equation is dependent on the equation of state. A more accurate understanding of the equations of state for neutron stars could cement an experimental value for the TOV limit,
using more suitable spacetime metrics.
F. References
[1] Einstein A. Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik. 1916; 354(7):769822.
[2] Hartle, JB. Gravity: An Introduction to Einstein's General Relativity. Pearson, 2003.
[3] Oppenheimer JR, Volkoff GM. On Massive Neutron Cores. Physical Review. 1939 Feb 15; 55(4):374–81.
[4] Ryden, BS. Introduction to Cosmology. Cambridge UP, 2003.
[5] Fisher L. Are black holes hot or cold? Available from: https://www.sciencefocus.com/space/areblackholeshotorcold/ [accessed Dec 6th, 2018].
[6] Bombaci I. The maximum mass of a neutron star. Astronomy and Astrophysics. 1995; 305:871.
