Fidos and Frefos and the Equivalence Principle

      In considering the description of events near the horizon of a static black hole from the viewpoint of an external observer[1], it is useful to imagine space to be filled with static observers, each located at a fixed (r, θ, φ). Such observers are called fiducial observers, or by the whimsical abbreviation, FIDOS. Each Fido carries a clock which may be adjusted to record Schwarzschild time t. This means that Fidos at different r values see their own clocks running at different proper rates. Alter natively, they could carry standard clocks which always record proper time τ. At a given r the relation between Schwarzschild time t and the Fidos proper time τ is given by

 (1.1)

Thus, to the Fido near r = 2MG, the Schwarzschild clock appears to run at a very rapid rate. Another possible choice of clocks would record the dimensionless hyperbolic angle ω defined.

        The spatial location of the Fidos can be labeled by the angular coordinates (θ, φ) and any one of the radial variables r, r^∗, or ρ. Classically the Fidos can be thought of as mathematical fictions or real but arbitrarily light systems suspended by arbitrarily light threads from some sort of suspension system built around the black hole at a great distance. The acceleration of a Fido at proper distance ρ is given by 1/ρ for ρ <<MG. Quantum mechanically we have a dilemma if we try to imagine the Fidos as real. If they are extrememly light their locations will necessarily suffer large quantum fluctuations, and they will not be useful as fixed anchors labeling spacetime points. If they are massive they will influence the gravitational field that we wish to describe. Quantum mechanically, physical Fidos must be replaced by a more abstract concept called gauge fixing. The concept of gauge fixing in gravitation theory implies a mathematical restriction on the choice of coordinates. However all real observables are required to be gauge invariant.

     Now let us consider a classical particle falling radially into a black hole. There are two viewpoints we can adopt toward the description of the particle's motion. The first is the viewpoint of the Fidos who are permanently stationed outside the black hole. It is a viewpoint which is also useful to a distant observer, since any observation performed by a Fido can be communicated to distant observers. According to this viewpoint, the particle never crosses the horizon but asymptotically approaches it. The second viewpoint involves freely falling observers (FREFOS) who follow the particle as it falls. According to the Frefos, they and the particle cross the horizon after a finite time. However, once the horizon is crossed, their observations cannot be communicated to any Fido or to a distant observer.

       Once the infalling particle is near the horizon its motion can be described by the coordinates (T,Z,X, Y ) defined. Since the particle is freely falling, in the Minkowski coordinates its motion is a straight line

 (1.2)

where p_Z and p_T are the Z and T components of momentum, and m is the mass of the particle. As the particle freely falls past the horizon, the components p_Z and p_T may be regarded as constant or slowly varying. They are the components seen by Frefos. The components of momentum seen by Fidos are the components p_ρ and p_τ which, are given by

 (1.3)

For large times we find

 (1.4)

Thus we find the momentum of an infalling particle as seen by a Fido grows exponentially with time! It is also easily seen that ρ, the proper spatial distance of the particle from the horizon, exponentially decreases with time

 (1.5)

       Locally the relation between the coordinates of the Frefos and Fidos is a time dependent boost along the radial direction. The hyperbolic boost angle is the dimensionless time ω. Eventually, during the lifetime of the black hole this boost becomes so large that the momentum of an infalling particle (as seen by a Fido) quickly exceeds the entire mass of the universe.

          As a consequence of the boost, the Fidos see all matter undergoing Lorentz contraction into a system of arbitrarily thin “pancakes” as it approaches the horizon. According to classical physics, the infalling matter is stored in “sedimentary” layers of diminishing thickness as it eternally sinks toward the horizon (see Figure 1.1). Quantum mechanically we must expect this picture to break down by the time the infalling particle has been squeezed to within a Planck distance from the horizon. The Frefos of course see the matter behaving in a totally unexceptional way.

Fig. 1.1. Sedimentary layers of infalling matter on horizon.