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Basic Concepts on Black Holes

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Title: Basic Concepts on Black Holes


1
Basic Concepts on Black Holes
  • Cesare Chiosi
  • Department of Astronomy
  • University of Padova, Italy

2
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.
3
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.
4
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
5
In brief
6
Three basic equations
Spherical, symmetric and static distribution of
matter. Assume spherical coordinates r, q,f
7
Singularity Proper time
8
Gravitational redshift
9
Curvature in the 3D space
10
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.
11
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.
12
Eliminate the singularity
13
Light Cones of radial photons
14
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)

15
The motion of a test particle
16
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
17
Final result
with
and integrate
18
The function t(r)
Nothing happens in the proper time t when the
particle arrives at rs. The total proper
time is
19
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
20
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.
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