The simple relationship of real and imaginary time

The simple relationship of real and imaginary time.

Now we come to another paradox, that is the crux of the whole matter. The contradiction of the apparent paths followed by a falling clock as measured in proper and coordinate time. (The falling clock paradox.)

In these two spacetime diagrams only the real part of the time component of the paths is plotted.

In the first image with the apogee above the event horizon the motion in coordinate time (black curve) never extends below the event horizon, while the motion in real proper time (green curve) extends all the way to the central singularity.

In the second image, the apogee is below the event horizon. The motion with a real coordinate time component (black) contradicts the motion of the same particle plotted using the real component of proper time (green). This is the paradoxical aspect and is only resolved by studying the motion in full complex time as discussed in the section after the applet below.

Derivation of the equation of free falling motion here.

In the following interactive java applet the points discussed above are demonstrated. Dragging point 'a' either side of the event horizon indicated by the vertical red at r=2m it can be seen that the path in proper time contradicts the path in coordinate time. Clicking on the 'details' button reveals the equations used to plot the curves.

A quick guide to using and understanding these interactive graphing applets.

The reader may have noticed something odd about the the above diagrams. It apparently shows the path of the particle in proper time as rising from the central singularity and then passing outwards through the event horizon before turning around at apogee and falling back again. Are we not told that nothing can pass outwards through the event horizon? The text book answer is that the outward part of the the curve (magenta part) represents the particle leaving a 'white hole in the infinite past'. The nice 'get out clause' for this argument is that it is difficult to prove what happened in the infinite past. In the infinite past anything could have happened, even possibly white holes. This argument can be shown to be false by the following counter arguments. The equation for the curve is obtained by integration so it contains a constant of integration. That means we a free to move the curve upwards in the positive coordinate time direction. By drawing the curves of two particles, we can make the infinite past of one particle be in the same time and space of the 'infinite future' of another particle that is returning to the central singularity of the black hole. Clearly the same location in time time and space can not be a white hole and a black hole simultaneously. White holes have never been observed in our current universe,so the text book argument is not empirically supported. It can also be observed that the proper time for a particle to fall to the central singularity is finite and can be very brief, of the order of minutes or seconds. In proper time the infinite past argument does not apply. It has to shown that the white hole existed in the last few minutes if we are attach any physical significance to the arguments that trajectories in black holes happen in brief finite intervals and not in the infite periods indicated by the Schwarzschild equations. Since text books attach more physical significance to the finite proper time of a falling particle than the infinite coordinate time they have show that white holes existed in the finite past. The arguments presented in this website show that white holes are not required to explain the motion af a falling particle.

Complex time and motion

In these next two spacetime diagrams showing the motion of a free falling particle in complex time, only the part of the trajectory on the falling side of the apogee is shown to reduce clutter. The rising part of the trajectory is merely a mirror image across the horizontal axis. The black curve represents the motion the falling particle in coordinate time when the complex coordinate time has a none zero real component. The red curve is imaginary component of the same falling particle is complex coordinate time. The green curve is the motion in complex proper time when the real component of the proper time is non zero. The magenta curve is the corresponding motion in complex proper time when the imaginary component of the proper time is non zero. Note that the motion in complex proper time is never 'mixed complex' where 'mixed complex' is defined here as a complex number with both a non zero real component and a none zero imaginary component. The proper time is always pure real or pure imaginary. The coordinate motion on the other hand does have some mixed complex time components.

The first diagram shows the apogee above the event horizon. It can be seen than the coordinate motion below the apogee (point a) and above the event horizon has no imaginary component and the corresponding motion in proper time is purely real.

The second diagram shows the apogee below the event horizon. It can also be seen that the coordinate motion between the apogee and the event horizon has no imaginary component and the corresponding motion in proper time is purely imaginary. This is the key to resolving the paradoxical relationship between coordinate time and proper time. If the regions where the coordinate time has a non zero imaginary component (mixed complex) are regarded as being non physical and discarded then there is no conflict between coordinate motion and proper motion.

It should be noted that proper time can sometimes be purely imaginary (below the event horizon) but once this is accepted, the relationship between coordinate time and proper time is completely self consistent.

A quick guide to using and understanding these interactive graphing applets.

The two applets above illustrate the points disussed above. In the first applet, the real and imaginary components or coordinate and proper motion of a free falling particle have been plotted. By demanding tha time component of the physically real motion of a particle is non-complex in all coordinate systems, the logical and consistent behavior demonstrated in the second applet is obtained. This can be alternatively stated as 'by demanding that the physics of a falling particle be logicical and consistent, then we have to discard as unphysical the motion of a particle that has real and imaginary components that are both simultaneously none zero in any coordinate system. Not only does this solve the paradoxical behavoir of a black hole but it also reveals something fundemental about the nature of time itself. Proper time for a physically real particle can be purely real (positive or negative) or purely imaginary relative to the time of a coordinate observer at infinity but never partly real and partly imaginary at the same time. Coordinate time of a physically real particle is always purely real and only ever advances.

To coin a Wheeleresque phrase, it can be said that gravity tells time how to tick and time tells gravity which way is up.

What does it mean for a falling clock to have imaginary time?

Although imaginary time would normally be rejected as being unphysical there is a logical explanation. The imaginary time is proper time of a falling clock as measured by a Schwarschild observer at infinity. The observer is outside the event horizon and is measuring the time of a clock below the event horizon. Since the regions above and below the event horizon are causally cut off and timing signals can not be sent in either direction across the event horizon, it is not surprising that the proper time of a clock below the event horizon is imaginary and can not be defined with absolute certainty relative to the time of a clock above the event horizon. In Schwarzschild coordinates the local time varies relative to the time of a clock at infinity. At any radial distance all physical processes are governed by the local coordinate time and all physical process slow down as you descend towards the event horizon. To a local observer, everything appears completely normal. In effect even the brain of the observer, (or the cpu of some artificial programmable device) slows down. At the event horizon coordinate time stops and the brain or cpu stops too and it is not surprising that no sensible measurements can be made exactly at the event horizon. Below the event horizon, local time is imaginary and the proper time of a falling clock that is ticking forward one imaginary unit time every real coordinate unit time appears to be advancing normally at the rate of one second per second according to a local observer. Here the brain or cpu of the observer is operating in imaginary time as are all other physical process. This can be clarified by thinking of the twins paradox in special relativity. The twin that accelerates away and then turns and returns is unable to make any measurements to determine that he is ageing slower than his non accelerated twin until they are once again alongside each other and directly compare ages and clocks. The travelling twin is completely unaware that he is ageing slower and that his brain is operating in slow motion and everything appears normal to hi. It is not too large a step to imagine that if the brain of a local observer below the event horizon is operating in imaginary time then clocks advancing in imaginary time will appear completely normal to her and if the clock is advancing in real proper time it will be unphysical to her just as a clock above the event horizon advancing in imaginary time is unphysical (non existent) to a local observer whose brain is functioning in real time. So now the paradox of motion across the event horizon is solved. Proofs showing that particles fall to the central singularity in real finite proper time are flawed because real proper time is unphysical below the event horizon. When we look at the imaginary proper time of a falling clock below the event horizon, we get complete agreement with the all the measurements made in coordinate time (velocity, acceleration and spacetime location) as shown in section 3.2 of this website. In fact, the event horizon at the Schwarzschild radius of r=2m, is final destination of any free falling particle with an apogee in the range r=0 to r = +infinity.

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