They represented the equality of apparent horizon
and event horizon for de Sitter universe.
If [[theta].sub.+] = 0 or [[theta].sub.-] = 0, then S is referred to as a future or past apparent horizon
. These surfaces naturally arise as boundaries of future or past trapped regions.
Part I describes numerical techniques for different types of problems, including 3+1 or N+1 formulations, finite-differencing methods, matter field equations, initial-value problems, deriving gravitational waves, and locating the apparent horizon
of a black hole.
Hawking smashes the idea of a black hole by saying that instead of there being an inescapable event horizon, they should think of a far less total "apparent horizon
Saha and Mondal  have studied thermodynamics on apparent horizon
with the help of gravitationally induced particle scenario with constant specific entropy and particle creation rate [GAMMA].
For a first approximation of the Moon's altitude, the differences in these elevations allowed us to calculate that the photograph's apparent horizon
, formed by the distant Clark Range roughly nine miles from Glacier Point, was raised about 5[degrees]above the ideal geometric horizon.
The surface gravity on the apparent horizon
is given by
"What was really amazing," says Suen, "is that you could see the apparent horizon
pop out faster than the shock waves." The apparent horizon
is the surface that defines the edge of the newly forming black hole.
Akbar and Cai  found that the Friedmann equations at the apparent horizon
can be written in the form dE = TdS + WdV (E, V, and W are the energy, volume inside the horizon, and work density, resp.) in general relativity (GR), GB gravity, and the general Lovelock theory of gravity.
You might improve your chances by observing from extreme elevations, where you can lower your apparent horizon
by at least a half degree.
One such boundary is the apparent horizon
with the radius (for the spatially flat universe),
It is notable that starting from the Clausius relation to the apparent horizon
along with the modified dispersion relation, one can easily get the modified Friedmann equation with bounce solution to the FRW universe [14,15], but the answer is not direct in Padmanabhans conjecture in the emergent universe.