"Falling photon paradox"

The "falling photon paradox" counter-argument.

This first counter argument shows that the conventional physical interpretation gives rise to a "falling photon paradox". This argument utilises the following diagram. In the diagram, increasing time is upwards and increasing radius is to the right in Schwarzschild coordinates and in the new coordinates. In the Kruskal-Szekeres chart, time and distance follows the same convention in the right hand quadrant and are reversed in the upper quadrant. The central singularity is identified by the thick black line bordering the dark grey area which represents negative radius. (Only positive radii are considered here.) The thick red line represents the event horizon at r=2m in the Schwarzchild and new coordinates but in the K-S coordinates it is not so clear what the red line bordering the large light red triangle represents. (Some claim it is the boundary to another universe and to a white hole by symmetry arguments.)

In the conventional interpretation, the blue and green curves shown in the Schwarzschild graph of the above diagram are considered to be one continuous curve representing the motion of a single 'infalling' photon. In the Kruskal-Szekeres chart, the events a, b and c appear to be a nice progression of sequential events on a continuous path. Unfortunately this is an illusion because it relies on the flawed and unjustified interpretation that the green curve in the Schwarzschild chart is the path of a photon going from right to left and backwards in time, rather than a photon going from left to right and going forward in time.

This gives rise to the 'falling photon paradox'. At time (a) in Schwarzschild time, the single photon following the blue and green curves is in two locations simultaneously, one location below the event horizon (b) and one location above the event horizon (a). How can a photon be both sides of the event horizon (separated by an arbitrary distance) at the same time?

Even worse, it can be seen that a photon arrives at location (c) before it was emitted at time (a). This is a serious violation of causality.

The correct interpretation is shown in the following diagram:

Here, all photon paths consistently move forwards in time (in all 3 coordinate systems). No one claims that the brown and light blue paths are the paths of a single photon, even though they share the equation of an 'outgoing photon'. It is understood that if an 'outgoing photon' is slightly below r=2m then it follows a path towards r=0 and if the 'outgoing photon' is slightly above r=2m then it follows a path outwards to infinity. A consistent and logical argument by symmetry is that the green and dark blue paths are not the path of a single photon either. Immediately the 'falling photon paradox' is eliminated. There is no longer a requirement for a single photon to be in different locations at the same time and no photon arrives at its destination before it is emitted so causality is no longer violated.

Now, it can be clearly seen in the Kruskal-Szekeres chart that no photon passes through the event horizon at r=2m. The problem is not in the mathematics of the Kruskal-Szekeres transformation of Schwarzschild coordinates, but in the initial (flawed) assumptions of the motion of a photon in Schwarzschild coordinates.

In the two interactive java applets above the magenta and green curves are light paths while the black and blue curves is the path of a free falling particle. The verticle red line marks the event horizon at r=2m. In the first applet the equations contain absolute functions and paths that contain complex values with non zero imaginary parts are plotted. When the erronous absolute functions are removed and only purely real results are plotted te paths in the second applet are obtained. The point marked (a) is the apogee of the particle trajectory and can be dragged left and right as desired. If the apogeee is dragged to a point between r=0 and r=m it can be seen that the trajectory always returns to the event horizon. In the second applet containing the correct equations it can be seen that the particles and photons never cross the event horizon. The point labelled c can be dragged left and right or up and down and the light curves effectivey mark the future and past light cones at that point. Here, future light cone is defined as the Northern quadrant of intersecting light paths in the direction of increasing coordinate time and this definition consistently applies above and below the event horizon. If point c is placed on the particle path on the opposite side of the event horizon to the part that contains the apogee it can be seen that these imaginary paths are not within the future or past lightcones. This makes it clear that these imainary paths are not the path of a real particle. These imaginary paths are the ones plotted in text books to 'prove' that the path of a particle or photon passes through the event horizon. If the reader clicks on the details button of the applets, the equations for the plotted paths can be seen and altered as desired. Nothing is hidden. The 'realonly' function in the applets tells the applet software to only plot those points that are purely real and contain no imaginary part.

More anomalies in the conventional interpretation.

In the conventional physical interpretation of light rays in Schwarzschild coordinates, light rays (green and magenta curves) are interpreted as only being able to travel towards the central singularity as indicated by the arrows on these two diagrams. The trajectory of a free falling particle is depicted by the blue and black curves.

In the first diagram it can be seen in the lower left corner that the particles is travelling outwards from the central singularity while the light rays are unable to do so. Does that strike you as odd? It should.

By reversing the arrow on the section of the magenta curve to the left of the event horizon, symmetry is restored and we are no longer required to accept the concept of photon going backwards in time, below the event horizon.

In the second diagram, the top right hand corner shows the conventional alleged descending path of a real particle in Schwarzschild coordinate time. Imagine that at point c the particle emits some photons. It can be seen that the particle arrives at the central singularity in a time interval (dt) that is less than the time taken by the photons. In other words in the conventional interpretation, the particle is falling faster than the local speed of light measured using the same coordinate time. One of the fundamental principles of Special and General Relativity is that a physically real particle never exceeds the local speed of light and a particle with mass never overtakes a photon. Clearly the conventional interpretation that the blue curve below the event horizon represents the path a real falling particle is in error. If the ad-hoc insertion of an absolute function in the in the equation of the falling particle's motion is removed, the path that extends the motion of the particle through the event horizon and down to the central singularity is shown to be imaginary. What is left in the new interpretation, when these considerations are taken into account, is shown in the last applet above where the particle only ever returns to the event horizon after passing apogee.