Scourge Contradiction

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There seems to be a contradiction in Fry’s heuristic models.

The first model which describes time compression is the three-spacecraft example:

We will now accelerate all three ships to the velocity C with respect to their starting point D. At this velocity the three ships cease to exist materially insofar as the observer at D is concerned, since they have entered the plane of energy, and are also at the zero point of the curve of time with respect to him.”

To be perfectly clear here: Fry has stated that a ship which achieves a velocity of C with respect to the inertial frame of reference will cease to exist materially insofar as the observer on Earth is concerned.

However, in the next example, in the voyage to Alpha/Proxima Centauri model, the ship never ceases to exist with respect to the observer on Earth:

“Since we postulated at the beginning of this analogy that our craft was capable of unlimited acceleration, and since the postulated force continues to act, our velocity will continue to increase and we will have between ourselves and the earth, a rate of increase in the degree of separation which is greater than that specified by the quantity C. We can do this from our point of reference although, as will be explained later, we cannot do it from the point of view of an observer upon the earth.”

And…

“We can see that, even if our energy level bad been so close to infinite that the outward trip had required only one second, if during the one second trip we had emitted enough light to make observation possible, the astronomer upon the earth would note that the trip required four years and one second, and so would have undeniable proof of the mathematics which postulate that only with infinite energy may the velocity C be achieved.”

Fry is describing what looks to be two different results under the same conditions. In the first case, the ship being observed from the inertial reference frame on Earth appears to disappear as it reaches the speed of light. In the second case, a ship moving away from the Earth at the speed of light and greater only appears to be moving at 1/2C with respect to the observer on the Earth for the duration of the trip.

Analysis

The contradiction invokes a much deeper issue which is relevant to the entire theoretical structure of Curvity: if we go with the Curvity interpretation of GR observations, then there's a difference between the observations of the accelerated and the unaccelerated observers. In this case, the observer onboard a ship moving at C wrt to inertial frame cannot see the Earth at all, but the observer on the Earth can still see the ship, and it appears to be moving at a velocity of 1/2C.

The result is a disturbing asymmetry: how can one observer see the other, but not the other way around? Curvity says that there's a 'preferred' reference frame. The problem with a preferred reference frame is that, by definition, Relativity forbids a preferred reference frame - motion is a totally relative phenomenon.

It may not be a deal-breaker because by measuring the Doppler shift of the CMB radiation, an observer should be able to determine his/her state of motion wrt to the universe.

The idea of motion inducing a condition of one-way observation though, is genuinely troubling.

Possible Solution

Let us ignore ships C and B and focus on the visibility of the ship A from the reference point D (the observer). The visibility goes through three phases. For further illumination, let us suppose that Observer D and Ship A both have a very intense light source pointed at each other, which are pulsed at one second intervals. Each light pulse is also encoded, such that they start at one and count up, and that before starting the experiment, a number of light pulses have passed both Observer D and Ship A.

The first phase is when Ship A is at a velocity just below that of the speed of light with respect to Observer D. As suggested by Relativity, and shown by experiments with fast moving particles with short half lives, the time on Ship A slows down the closer it gets to the speed of light. From Ship A’s perspective, the pulses from D are also taking longer and longer to arrive because but they can still be viewed from the back of Ship A and are still counting up. As the velocity of Ship A is increased closer and closer to the speed of light, the pulses headed at Observer D take longer and longer to arrive, as they do at Ship A. Time is slowing down between them.

For phase two, Ship A has reached the speed of light and Ship A can no longer see the one second pulses from D because A is forever between the light pulses, neither catching up to, nor having them pass. They are no longer visible nor are they counting up or down. Time between A and D has stopped and the same thing happens to D, where the one second pulses from A never reach D. They cannot see each other.

If Ship A stayed at the speed of light forever, then Ship A would disappear in the sense that no new light pulses would ever arrive at Observer D. As Ship A approached light speed, Observer D would count pulses up until they stopped arriving.

Phase three is when ship A exceeds the speed of light. Ship A can now see the one second pulses from D, but there are three notable changes:

  • The first is that the pulses are arriving at the front of Ship A because they are pulses that had already passed and Ship A is now catching up to those light pulses, coming at them from behind.
  • Secondly, the pulses are now counting down.
  • Ship A will also observe an identical copy of itself, let us call it Ship AB, as in backward in time, separate ahead of itself and leave, heading back toward Observer D, also in front. Everything on Ship AB is happening in reverse. Ship A, of course, is still leaving behind one second pulses of light that are counting up.

From observer D’s perspective, Ship A is visible “again”, and this is where Fry’s first model needs clarification. I put “again” in quotations because Observer D cannot tell the difference between the first and third phase, only that Ship A’s light pulses take longer and longer to arrive, or more accurately, time on Ship A seems to be going slower and slower.

Which brings us to the final answer about what observer D would see once Ship A has a velocity faster than the speed of light. The boring answer is more of the same, as in the pulses would still be counting up and still take longer and longer to arrive. However, to satisfy the requirements that nothing can be observed to travel faster than light from Observer D’s perspective, we must understand what light and, more importantly, time, is actually doing and Boyd’s experiment provides the answer.

Below is a diagram where the vertical axis is time, increasing downward, the horizontal axis is distance and the origin is Observer D where the encoded light pulses arrive. Ship A is traveling across the diagram from left to right as distance increases between them. Each yellow line represents an encoded light pulse from Ship A to Observer D and the slope represents the speed of light. If Ship A is at a low speed with respect to D, the pulses would arrive at Observer D at one second intervals. As Ship A gets closer and closer to the speed of light (the vertical line in the middle) the pulses from D’s perspective get farther and farther apart.

Figure 1: Diagram of Observer D and Ship A

As demonstrated by the Boyd Experiment, once Ship A goes faster than light, it will be going back in time and thus a single encoded pulse from Ship A will exist in two places at once for a duration. It may seem strange for two photons to suddenly pop into existence between Ship A and Observer D and then hurry off in opposite directions, but this is exactly what happens in the Boyd Experiment. Observer D only ends up seeing one of the photons as does Ship A, which it should be pointed out, at speeds above the speed of light, Ship A is emitting not receiving. It is a third observer, who can see both photons at the same time. The “two” photons are a mirage created by a photon going both forward and backward in time, as one is headed forward in time to Observer D and the other “seems” to be going backward to Ship A.

Let us continue our trip with Ship A, which, after a time at a velocity faster than the speed of light, decelerate back to a zero speed with respect to Observer D, but now a great distance away. As Ship A decelerates and once again attains a speed exactly the speed of light with respect to Observer D, Ship A will no longer observe itself in the forward view finder, nor will it observe the light pulses from D.

As Ship A slows to a speed lower than the speed of light, it will see some interesting artifacts:

  • It will see Ship AB once again, but in the rear view, headed backward, with everything going in reverse, including the encoded photons arriving which Ship A transmitted at Observer D.
  • There will also be a third ship, Ship AC, now headed forward in time to meet Ship AB, at which point they will merge and cancel each other, along with any encoded pulses that Ship AB transmitted.

From Observer D’s perspective, nothing will seem amiss.

Here are a few clarifications:

If Boyd can see the wave in negative time coming back to cancel the inbound wave, then why can’t Observer D also see Ship A in negative time?

It has to do with distance and the true sequence of events for Ship A’s photons to get to Observer D. Because Ship A is flying directly away from Observer D, any photons from Ship A that are given off while at a speed faster than light (FTL), will sort themselves out before getting to Observer D. By “sort themselves out” I mean that the photons that exist in both positive and negative time will have cancelled each other out. In other words, remember from Observer D’s perspective, that the light impulses emitted from Ship A while Ship A is traveling faster than light, seem to pop into existence at a point between Ship A and Observer D, one photon going toward Observer D and the other going back to meet Ship A (going backward in time). And ALL photons from Ship A at FTL speeds would seem to be in two places at once. Even placing a mirror to try and bounce the negative time light impulse back toward Observer D wouldn’t work, because, from Ship A’s perspective, the light is coming from it at the back of the mirror, not the other way around. With the mirror, the light pulse would seem to emerge from the back of the mirror and head toward Ship A.

If Observer D could suddenly move to a position where Ship A passed it at FTL from left to right, then it should be possible for Observer D to see Ship A in all three phases. Just like Ship A can slow down and see itself in all three phases.

Which brings up a possible test – If things are zipping around our universe at the speed of light or faster, it might be possible for an astronomer on Earth to see them in all three phases, but they would have to know what to look for.

Above, I mentioned:

It may seem strange for two photons to suddenly pop into existence between Ship A and Observer D and then hurry off in opposite directions

It isn’t strange from the perspective of the photon because it sees linear time from the moment it leaves Ship A and arrives at Observer D. It is only a third observer who, at zero velocity w/r/t Observer D but at a half way point along Ship A’s journey, who would see the photon going back toward the Ship A while it’s other half travelled towards Observer D.

The strangest bit about all this is what an observer would see if they were observing the point at which Ship A dropped below light speed. There would be nothing but empty space and then suddenly boom, two ships would appear, one going forward, Ship A, and an identical one going backward, Ship AB. Just like you can’t hear a sonic boom from a jet flying faster than the speed of sound until it has past. If our observer had a good enough telescope, they could also look back and see Ship AC heading toward Ship AB.