Basic Concepts on Black Holes

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Basic Concepts on Black Holes

• Cesare Chiosi
• Department of Astronomy
• University of Padova, Italy

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Generalities 1
General Relativity set the maximum mass for a
neutron stars to a value that depends on the EOS
in use. If a collapsing nucleus has a mass in
excess of this value the collpase cannot halted
and a Black Hole (BH) is formed.
A BH is a region of space-time enclosed by the
event-horizon, a region whose gravitational
field is so strong that no matter no radiation
can escape from this surface. A BH can be
detected only by its gravitational effects on
nearby objects.
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Generalities 2
The existence of BH has been predicted long ago
by Laplace with very simple considerations
suppose a test partcle with mass m in the
gravitational field of an object with mass M, and
assume that the velocity of the test particle is
zero at infinite distance.
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Generalities 3
Rotation and charge remain to a BH. The charge
can be later easily neutralized by accretion of
matter. Any other properties of the material
collapsing to a BH is definitly lost. To
describe a BH General Relativity is
required. Let be a non rotationg, highly
concentrated object with mass M, the
gravitational field around is governed by the
Einstein solutions. Each line-element ds
(distance Between two events in the
four-dimensional space) is given by
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In brief
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Three basic equations
Spherical, symmetric and static distribution of
matter. Assume spherical coordinates r, q,f
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Singularity Proper time
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Gravitational redshift
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Curvature in the 3D space
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Motion of a test particle 1
If a particle locally moves with velocity v over
the spatial distance ds ? the interval of proper
time dt decreases at increasing velocity. One has
dt ds 0 for v c. E.G. photons traveling
along geodesic of length ds 0.
For particles with mass gt 0, v lt c and dt 2 gt
0 ? ds2 lt 0, the sparation is said Time-Like.
The Universe line of material particles are
always time-like
Separations with ds2 lt0, dt2 lt0 would require
vgt c and are named Space-Like. E.G. The
distance between two simultaneous events.
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Motion of a test particle 2
Null geodesic (ds20) yield the propagation of
photons and describe the so-called Light-Cones
in the space-time.
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Eliminate the singularity
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Light Cones of radial photons
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The two solutions meaning
The first solution represents photons moving
inward with speed c.

• The second solution changes sign at r rs
• It is gt 0 for r gt rs the photons can be
  emitted outward (dr gt0)
• For r ? rs the cone rotates inwards
• For r rs no photons are emitted ouwards
• For r lt rs the solution becomes negative,
  all photons are emitted
• inwards and no photon can be emitted
  outward (leave the star)

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The motion of a test particle
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Eulero-Lagrange
Consider for simplicity only radial motion and
assume some initial distance ro with v 0 and t
0. Performing some substitutions and
algebric manipulations we get
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Final result
with
and integrate
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The function t(r)
Nothing happens in the proper time t when the
particle arrives at rs. The total proper
time is
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The observer at roo
An observer at roo sees a different story. In
fact the relation between t and t is
The fact that such an observer sees the t-clock
to slow down as r ? rs means that t(r) will
reach r rs only at too.
Events inside rs are fully masked for such an
observer
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Consequences

• For an observer at rs the collapse proceeds
  quickly but smoothly through
• the surface.
• Once the surface of the stars has fallen inside
  rs no static solution exists
• and the collapse towards the central
  singularity cannot be opposed.
• The singularity at r0 is real but the physical
  conditions are not known.

• For a distant observer the scene is very
  different. At his t-clock
• the collapse of the star surface slows down
  as r ? rs which indeed
• can be reached only for too
• The surface of the star will appear as at rest
  and the emitted light more
• and more shifted toward the red (z ? 00 )
• For this observer the stars will disappear from
  his view and will be
• detectable only via the gravitational
  interaction with nearby objects.