Portal Paradox Part 3: The Portal Paradox Theory

The Theory

OK so, given the past however many thousands of words I've written on what portals might actually look like and how they might operate we can consider the original reason for this text: the paradox. In order to investigate it I decided to take all the axioms we discussed and talked through and compile those into a simulation and simulate the system using a bit of Python and some simple constant-velocity iterative mechanics. To keep things simple I have stuck to 2D circular portals hanging in a pure void in 2D space and they are operating on a single point I have called the "ball".

Let's consider options A and B and how they might come to be.

Option A: "Plop"

The most common argument for this option is "speedy thing goes in, speedy thing comes out". In other words, the ball is stationary to start with, so it must come out of the portal stationary. In other words, the velocities of the portals relative to the ball are irrelevant.

Option B: "Whoosh"

For the ball to come out at speed, velocity must come from somewhere. Suppose you were to drop the ball into a portal at velocity v, but the counterpart portal was moving at velocity -v, the velocities should cancel out and leave the ball stationary. If you're having trouble visualising this, Mythbusters did a fantastic related experiment firing a ball out the back of a truck at 30 mph, whilst the truck was moving at 30 mph.. The ball ends up with zero velocity as the two cancel each other out. Option B assumes that the portal velocity is added to the ball's velocity. So, in the paradox scenario, the orange portal is moving at velocity v toward the ball which has 0 velocity, as it passes through, the ball has the velocity of the portal added to its velocity and it leaves the blue portal with velocity v.

Summary

Ultimately the result of the paradox, option A "plop" or B "whoosh" comes down to one simple question.

Do portals transfer their velocity to whatever passes through them?

Option A - no

Option B - yes

The code I wrote (below) summarises the issue.

position_relative_to_gate = entity.centroid - input_.centroid
velocity_relative_to_gate = entity.velocity - input_.velocity
# corrections for rotation
new_position = output_.centroid + position_corrected_rotation
new_velocity = output_.velocity + velocity_corrected_rotation
if alternative_method:  # no velocity transfer
    new_velocity = entity.velocity

If we add the velocities on we get option B: the ball flies out of the blue portal (whoosh) because the (negative) velocity of the orange portal gets added to the ball and so it comes out of the blue portal moving.

But, if we instead do not add them the ball enters with 0 velocity and leaves the blue portal with 0 velocity, remaining stuck indefinitely in the blue portal's space (plop).

Which is more physical? To understand this I think it is important to look at frames of reference.

Inertial Frames of Reference

All motion must be measured relative to something. If you were to run 100 m, you are implicitly running from a fixed point where you originally started. This coordinate system is known as a "Reference Frame" or "Frame of Reference". "Inertial Frames of Reference" are frames of reference moving at constant velocity (including stationary). For example, if you were to watch a train go by, that train would be moving in your inertial frame of reference. The origin point would be you, and the train appears to be moving. If, however, you were on the train then from your frame of reference the train would appear stationary because your frame is now also moving. However, it is generally believed that there is no universal fixed reference frame that everything can be measured against. There is no "Aether". So when we look at the portal paradox, we actually need to consider the frames of reference.

Frames of reference are the first thing you learn about when tackling relativity and are one of the most important parts of it. Whilst our portals are not moving at relativistic speeds (thank heavens), there is another rule that underpins relativity we should consider.

Equivalent Inertial Frames of Reference: "All inertial frames are totally equivalent for the performance of all physical experiments."

In other words, it is impossible to distinguish between frames of reference that are moving at a constant velocity. For example if you were in the void of space and a football appeared to be slowly coming toward you, it would be impossible to tell if the football is moving toward you, or you are moving toward it, or some combination of the two, and the laws of physics must remain true in both!

Inertial Frames of Reference and Portals

So, returning to our portal paradox, we encounter the crux of the issue. The gateway is only one frame of reference, except it is also simultaneously two: the blue portal and the orange one. The gateway is one singular frame of reference where everything must operate as per the laws of physics (this is the equivalent of the "continuous space" axiom). So, from the gateway's point of view it is stationary itself, and a ball comes along and enters it. Then, if option B is true, the ball leaves the gateway with the same momentum it entered (whoosh), but if option A is true then the ball appears to stop dead in its tracks for no apparent reason (plop). With this perspective in mind it seems clear that option B has to be true to maintain the laws of physics.

Another way to look at it, if that answer doesn't seem satisfying, is that from the perspective of the gateway this interaction would necessarily be the same as if the blue portal wasn't displaced from the orange one and instead sat on top of it like a door. From the orange portal's perspective the ball comes toward it and enters the gateway, then leaves via the blue portal's frame of reference. The two are, in fact, the same frame of reference so the ball must carry on through it unperturbed.

Problems with this

Option B, whilst being the clear answer with relativistic laws in mind still has issues, like the fact that the portals' frames of reference are not identical and just because I say the ball "leaves" the orange frame and goes over to the blue frame... does it leave? Why is it not still in the orange frame but at new coordinates? Should momentum not be conserved in that frame too? I am not sure the answer to these questions but I suspect that the real conclusion is that the frames of reference of the ends of the gateway can not differ in velocity or orientation, which would make sense. They could be separate, and move, but would always be "in sync" with each other. This would agree with Jeremy Thille's video concluding the paradox can not happen. I think this is the most likely outcome.