terminology.
Coordinate
time.
Coordinate
time is the elapsed time between two events as measured or calculated by an
observer located at infinity. The clock of such an observer is considered to be
unaffected by gravity or motion.
Proper time.
Proper
time is the elapsed time of a clock as measured by an observer co-located and
co-moving with the clock. This observer always measures the local speed of
light to be c and always perceives the proper time to advance at a rate of one
second per second. More often it is more useful to compare the proper time rate
relative to the time rate of the clock located at infinity and it should be
clear from the context when this is done in the text and is usually denoted as d(tau)/dt.
Local time.
Local
time here means the time interval between two nearby events as measured by an a stationary observer in close proximity to those events.
It can be thought of as a special case of proper time. Stationary here means
spatially stationary with respect to the Schwarzschild coordinate system where dr=0. In the extreme curvature of spacetime in the vicinity
of a black hole, nearby may have to be taken as meaning an infinitesimally
small region. Below the event horizon infinite acceleration would be required
for an observer to remain stationary so we have to use the concept of a
momentarily stationary observer by considering a
infinitesimally small time interval. This is similar to the concept using to
using an array of momentarily co-moving inertial observers in Special
Relativity when analysing acceleration in flat Minkowski space. An inertial
observer can not be truly stationary with respect to an accelerating particle,
not even for a moment, but by considering an infinitesimally small time
interval a useful mathematical approximation is achieved and the same concept
is used here to justify the use of stationary observers below the event
horizon. To reject this notion you would have to reject other ideas such as Rindler spacetime that are based on the same principle of
momentarily commoving observers. It is also shown in the text of this work that
real particles have apogees below the event horizon (in coordinate and proper
measurements) and that at the apogee a reasonable approximation of stationary
is achieved if an infinitesimal time interval is considered. It is a bit like
using calculus to find the velocity of an accelerating object. As the time
interval is made smaller and smaller the velocity is approximated more
accurately but if we try to take the time interval to exactly zero to obtain
the exact velocity at the ‘instant’ then the result becomes infinite. The
notion of infinitesimals and taking the limit is important to all calculus and
nearly all physics and that is the notion being used here in this work.
Background
time
This
is not a definition but an interesting idea that worth exploring further and
some might find it a helpful concept. The proper time (dtau)
of the Schwarzschild solution can be thought of as the time indicated by a
clock moving distance (dr) in coordinate time (dt). This does not have to be a real
physical clock, as is made clear by considering the proper time of a photon. We
can say the same about the local time and extend the idea to the notion of
‘background time’. It does not have to be a real physical clock. For each
infinitesimal point in Schwarzschild spacetime we can imagine an intrinsic
‘background time’ for that location. This background time controls all physical
process at that location. The speed of light at that point is determined by the
background time at that point. The speed of all particles passing through that
point is controlled by the background time of that location. All physical
processes such as the speed of thought or the speed of electricity or any
chemical processes is controlled by the background
time at that location. The proper time of a particle passing through that
location is jointly determined by the relativistic velocity of the particle
relative to that location and the by the background time of the location. The
end result is that if an observer was at the location, the speed of light and
all other physical processes appears perfectly normal. Just as we do not have to have a real
physical clock travelling at the speed of light to be able to say that the
proper time of a photon is zero, we do not have to have a real physical
stationary observer at a given location to be able to say what the intrinsic
background time is that location. The background time
is determined by the distribution of matter which is effectively the
gravitational potential at that point, but in turn the background time
determines how gravity acts on particles at that point. In short we can say:
Gravity
tells time how to tick and time tells gravity which way is up.
Imaginary
and real proper time are explained in section 3.5
©
2008 KevPegrume