Radial motion of a free falling particle
Here
the radial motion of a particle in terms of velocity and acceleration relative
to radial displacement from the centre of the black hole is analysed. The coordinate
velocity of a free falling particle is given by:
where r is the radial displacement and R is the apogee, which
can be thought of as the radius that an initially stationary particle is
released from. More accurately R is the maximum (or minimum) height that a
particle gets to before turning around. (At the apogee the velocity is
momentarily zero.)
The
coordinate acceleration of the free falling particle is given by:
These derivation of these equations by Kevin Brown can be seen in his excellent mathpages website.
The
easiest and most intuitive way to analyse these type
of equations is to plot them on a graph. Initially there are some paradoxical
and apparently contradictory aspects when the measurements of an observer at
infinity (Schwarzschild coordinate observer) and the measurements of a local
observer are compared, but it soon becomes apparent how these contradictions
can be logically resolved.
This chart illustrates some of the paradoxical nature
of time and motion in strongly curved spacetime above and below the event
horizon of a Schwarzschild black hole. The convention used here is that
positive velocity represents motion from right to left or reducing radius and
positive acceleration is acceleration towards the centre. The thick red line is
the acceleration of a free falling particle whose velocity is represented by
the thick green line. All the curves are velocity plotted against radius,
except for the red line which is acceleration against radius. It is only the
direction of the acceleration, rather than the scale of the magnitude of the
acceleration that is important to the discussion here. The light blue shaded
area is envelope of sub luminal motion. The thin blue line that outlines the
shaded area is the speed of light in Schwarzschild coordinate time (which is
the point of view of an observer at infinity.) It can be seen that from this
observers point of view, the speed of light is zero at the event horizon
(r=2m). The thick blue line is the speed of light according to a local observer
which is always c, but note that it appears to reverse direction across the
event horizon.
Assuming
the particle has initial motion (the thick light green curve) away from the
event horizon it can be seen it initially accelerates away but at a certain
radius the acceleration reverses direction and starts slowing the particle
down. It eventually reaches apogee at point R before falling back towards the
event horizon. On the return trip it is initially accelerated towards the event
horizon, but as it approaches the event horizon the combined effect of the
gravitational time dilation and time dilation due to its motion slows its
coordinate motion down bringing the particle to a halt at the event horizon
(from the point of view of a coordinate observer at infinity.) The coordinate
acceleration at the event horizon is zero. From the point of view of local
observers the particle is always accelerating and increasing in speed,
appearing to approach the speed of light as it approaches the event horizon. (Note
that the y coordinates have been scaled by a factor of 5 for clarity.)
One
obvious paradox is how can a single particle appear to be moving at the speed
of light according to one observer and stationary according to another
observer? Another obvious paradox is if the particle actually crosses the event
horizon it must reach and exceed the speed of light (as represented by the
brown line.) The local velocity of the particle and the coordinate velocity of
the particle below the event horizon also appear to be going in opposite directions which appears to be a contradiction. Below the
event horizon the coordinate motion is from left to right away from the centre
towards the event horizon. The positive acceleration is acceleration towards
the centre slowing the particles rise until it comes to a stop at the event
horizon where the coordinate acceleration is also zero. The coordinate velocity
of the particle below the event horizon is greater than the coordinate speed of
light and a local observer would also agree that the local speed of the
particle is greater than the local speed of light. This indicates that the
curves below the event horizon in this diagram do not represent the motion of a
real particle and so only the curves above r=2m are taken to be physically valid
for a particle with an apogee of greater than 2m.
In this chart the same equations are used as in the
previous chart but the apogee is set to a value that is less than r=2m. The
green curve representing coordinate motion has a particularly interesting
feature to anyone who knows anything about black holes. The particle curves
back to the event horizon instead of falling inevitably to the central
singularity. This contradicts just about everything that has ever been
published in text books about black holes but that is the difficult truth that
comes out of the maths. The motion is confirmed by the equation for coordinate
acceleration (the red line) At the apogee the
coordinate acceleration is extreme, but directed away from the centre of the
black hole. As the particle rises the acceleration very rapidly switches
direction and starts slowing the particle down bring the particle to rest at
the event horizon where the coordinate acceleration is zero. You may have read
in a text book that the acceleration at the event horizon is infinite and
nothing can remain stationary there. That however is the
acceleration measured by a local stationary observer and so the argument
is self destructive, because if the acceleration is infinite at the event
horizon there can not be a local stationary observer at the event horizon. Once
again it can be noted that on the side of the event horizon opposite to the
apogee local and coordinate observers disagree on the direction of the free
falling particle. One says it is rising while the other says it is falling.
Once again, the velocity of the particle on the side of the event horizon
opposite to the apogee, is greater than the local
speed of light according to the measurements of local observers relative to the
local speed of light and according to an Schwarzschild observer at infinity
relative to the coordinate speed of light. Once again we can dismiss as
unphysical the curves that are not on the same side as the apogee. In both
scenarios, motion only occurs on the same side as the apogee and the particle
never crosses the event horizon.
A quick guide to using and understanding these interactive graphing applets.
The two applets above compare the velocity of a free falling particle (green and magenta curves) and the light envelope defined by the black and blue curves/lines. As usual the event horizon is marked by the vertical red line. The first applet shows coordinate measurements while the second shows local measurements made by an observers placed all along the trajectory of the particle. The coordinate velocites are converted to local velocities by dividing by the factor of (1-2m/r) and it can be seen that the local speed of light is always constant. By dragging point a left and right it can be seen that the velocity of the particle on the opposite side to the apogee is always greater than the speed of light in both coordinate and local terms and it safe to assume that those sections of the curves are not representative of a physically real path. The conclusion is that only the parts of the curve on the apogee side of the event horizon are physically real and that the trajectory starts and ends at the event horizon and does not pass through. It can also be clearly seen that the text book description that a falling particle inevitably falls into the central singularity of a black hole is incorrect. In fact it is impossible for a real particle with mass travelling at less than the speed of light to arrive at the centre of a black hole and the acceleration acting on a particle becomes infinite as the particle aproaches the centre and the acceleration of the particle is directed away from the centre.
(c) 2008 KevPegrume