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The Light Cone
--------------
In special and general relativity, a light cone is the path
that a flash of light, emanating from a single event (a single
point in space and a single moment in time) and traveling in
all directions, would take through spacetime. Light cones
are always oriented in the time direction and light rays always
travel at 45° angles.
Consider the (1,-1,-1,-1) form of the metric.
For the (1,-1,-1,-1) form of the metric:
Lightlike (LL): c^{2}dt^{2} - dx^{2} = 0 v = c
Timelike (TL): dτ^{2} = c^{2}dt^{2} - dx^{2} > 0 v < c
Spacelike (SL): ds^{2} = -c^{2}dt^{2} + dx^{2} < 0 v > c
For the (-1,1,1,1) form of the metric:
Lightlike (LL): -c^{2}dt^{2} + dx^{2} = 0 v = c
Timelike (TL): dτ^{2} = -c^{2}dt^{2} + dx^{2} < 0 v < c
Spacelike (SL): ds^{2} = c^{2}dt^{2} - dx^{2} > 0 v > c
Therefore,
ds^{2} = -dτ^{2}
or,
ds = idτ
In the timelike case, time is the bigger factor. The
distance light travels is greater than the spatial
separation. 2 events can take place at the same position
but not at the same time. Therefore, a causal relation
between the two events is possible, i.e. one event can
cause or influence the other. Timelike events are inside
the light cone.
In the spacelike case, space is the bigger factor. The
spatial separation is greater than the distance light
travels. 2 events can take place at the same time but
not at the same position. Therefore, any causal relation
between the two events is impossible, i.e. one cannot affect
the other. Spacelike events are outside the light cone.
Simply put, they show how far something is apart compared
to ct. In the timelike case, if you are fast enough, you
can be at event A and at event B, it is only a matter of
time until you see the second event. In the spacelike
case, the events C and D are too far apart in space. You
cannot see both of them together, no matter how fast you
travel. As soon as event C happened and you go as fast
as possible, event D will have happened before you arrive
there.
Rindler Wedge
-------------
In region I the worldlines are timelike. At first sight
this would seem to be at odds with the definition of the
light cone where region I is spacelike and region II is
timelike. However, this is misleading. The light cone
requires that the worldlines pass through the origin in
the Minkowski frame (X = 0,T = 0), which they clearly do
not. If an observer did pass through the origin, then
indeed region I would be spacelike and not reachable.
What is really happening is that every point (event) along
the hyperbolae will have their own past and future light
cones. All events from the perspective of the observer at
a particular point are then either outside (spacelike),
inside (timelike), or on (light like) the associated light
cone. Since observers traveling along the hyperbolae can
never exceed the speed of light, these worldlines stay within
the light cone and are, therefore, always timelike.
In region II, space and time trade places. The spatial
coordinate in region I becomes timelike, and the time
coordinate becomes spacelike, i.e.
Schwarzchild metric:
dτ^{2} = (1 - r_{S}/r)dt^{2} - (1/(1 - r_{S}/r))dr^{2} - (1/c^{2})r^{2}Ω_{2}^{2}
For r < r_{S} the metric becomes:
ds^{2} = -(1 - r_{S}/r)dt^{2} + (1/(1 - r_{S}/r))dr^{2} + (1/c^{2})r^{2}Ω_{2}^{2}
Rindler metric:
dτ^{2} = (Rα/c)^{2}dω^{2} - dR^{2}
ds^{2} = -(Rα/c)^{2}dω^{2} + dR^{2}
Consequently, the hyperbolas in region II are spacelike.
In these diagrams, light rays are constrained to travel at 45°
and are shown in pink. Understanding this fact allows any
number of different scenarios to be investigated. For example,
consider Alice and Bob in a rocket.
Alice is ejected at point, P, and follows the blue trajectory.
At Q Bob shines a light ray at Alice which she receives. At
R Bob sends out another light ray which Alice receives just
as she hits the singularity. At point S Bob shines a third
light ray at Alice but this doesn't reach her because she has
already hit the singularity and no longer exists!
From Bob's perspective, he see's Alice falling towards the
event horizon, E, at a slower and slower rate, seemingly
taking forever (ω = ∞) to get to the point where she gets
'stuck' on the horizon.
Once Alice is behind the event horizon she cannot escape. In
order to do so she would have to exceed the speed of light
which is not possible. Her fate is sealed and she is torn
apart by the extreme gravitational tidal forces as she heads
towards the singularity.