- #1
Vagelford
- 7
- 0
I have a question on horizons. Actually I would like to make some
things that I have in mind clear. First of all let's give some
definitions.
Trapped surface: Is a surface on which null vectors point inwards.
That means that for example, if we let a photon on that surface it
could only travel inwards and it could at no time be outside that
surface.
Untrapped surface: Is a surface on which there exist null vectors
pointing outwards. That means that for example, if we let a photon
on that surface, there are null geodesics leading out of that
surface.
Marginally trapped surface: Is a surface on which there exist null
vectors that are tangent to that surface but there are no null
vectors pointing outwards.
A first question could be how exact are those definitions. The
question I would actually like to ask is: What exactly is the
difference between an event horizon and an apparent horizon in
terms of trapped surfaces and null geodesics?
As I understand the subject, an apparent horizon is defined as a
trapped surface that is the border between a region that has
outgoing null geodesics that point outwards and a region that
doesn't have outgoing null geodesics pointing outwards. That means
that the apparent horizon is the lust trapped surface between the
two regions and the next infinitesimally near surface is an
untrapped surface.
On the other hand an event horizon is a marginally trapped surface
an thus it is also a null surface, in the sense that it has null
generators, and it is an asymptotic surface to the outgoing null
geodesics from the inside and the outside.
These things can be seen in the following diagram of the formation
of a spherically symmetric black hole, were the cyan lines are the
null ingoing geodesics, the yellow lines are the null outgoing
geodesics, the green line is the apparent horizon and the event
horizon is defined by the convergence of the yellow lines. We can
see that at late times the two horizons coincide.
http://www.geocities.com/vagelford/Science/black_hole_formation.gif
In the above example, it is easy to identify the horizons and
distinguish between the event horizon and the apparent horizon.
Both horizons are trapped surfaces as it can be seen by the light
cones, but the event horizon is also a null surface while the
apparent horizon isn't at early times. Will these criteria apply
to more general configurations or should I look for a more general
"tool"?
The point is, are these things generic? The case in the diagram is
a special case with a lot of symmetry. Are the above definitions
general or they apply only to especially symmetric settings like
the above? What are the other trademarks that distinguish these
horizons?
Vagelford
PS. The x-axis is distance r and the y-axis is time t.
things that I have in mind clear. First of all let's give some
definitions.
Trapped surface: Is a surface on which null vectors point inwards.
That means that for example, if we let a photon on that surface it
could only travel inwards and it could at no time be outside that
surface.
Untrapped surface: Is a surface on which there exist null vectors
pointing outwards. That means that for example, if we let a photon
on that surface, there are null geodesics leading out of that
surface.
Marginally trapped surface: Is a surface on which there exist null
vectors that are tangent to that surface but there are no null
vectors pointing outwards.
A first question could be how exact are those definitions. The
question I would actually like to ask is: What exactly is the
difference between an event horizon and an apparent horizon in
terms of trapped surfaces and null geodesics?
As I understand the subject, an apparent horizon is defined as a
trapped surface that is the border between a region that has
outgoing null geodesics that point outwards and a region that
doesn't have outgoing null geodesics pointing outwards. That means
that the apparent horizon is the lust trapped surface between the
two regions and the next infinitesimally near surface is an
untrapped surface.
On the other hand an event horizon is a marginally trapped surface
an thus it is also a null surface, in the sense that it has null
generators, and it is an asymptotic surface to the outgoing null
geodesics from the inside and the outside.
These things can be seen in the following diagram of the formation
of a spherically symmetric black hole, were the cyan lines are the
null ingoing geodesics, the yellow lines are the null outgoing
geodesics, the green line is the apparent horizon and the event
horizon is defined by the convergence of the yellow lines. We can
see that at late times the two horizons coincide.
http://www.geocities.com/vagelford/Science/black_hole_formation.gif
In the above example, it is easy to identify the horizons and
distinguish between the event horizon and the apparent horizon.
Both horizons are trapped surfaces as it can be seen by the light
cones, but the event horizon is also a null surface while the
apparent horizon isn't at early times. Will these criteria apply
to more general configurations or should I look for a more general
"tool"?
The point is, are these things generic? The case in the diagram is
a special case with a lot of symmetry. Are the above definitions
general or they apply only to especially symmetric settings like
the above? What are the other trademarks that distinguish these
horizons?
Vagelford
PS. The x-axis is distance r and the y-axis is time t.