Sterling Relativity and Sterling Transformations in Portal Math Sterling, Mhd Abstract: In the world where space is flat and unbroken, Newton’s laws of motion hold for all inertial frames (related by a Galilean Transformation). In Portal Math, Newton’s laws of motion hold in all inertial frames, but to change from one inertial frame to another you have to use a Sterling Transformation. (All of these experiments are going to be in zero gravity by the way.) According to the Galilean transformation, we can shift our frame of reference any direction we want, and everything will still obey Newton’s laws. But this is not true in portals, even when both of the portals are standing still! Consider the situation where you have two portals on the wall, and you throw a cube at the portal. From reference frame A, the cube is moving at +v velocity. Now if we do a Galilean transformation and shift down, and look ahead in time a bit, we see this new reference frame B. But in this reference frame B, the cube is now traveling at –v. This by itself it not a problem, velocities can of course change over time. But in this example, there is no force acting on the cube. There is no force, but the velocity changes, this violates Newtons Laws of Motion. But we said that Newton’s laws of motion must hold in all inertial frames, and all we did was shift one of the coordinates down. The solution? You cannot shift between reference frames in any direction you please. You have to shift along the path that the object in question is taking. So actually, to shift from A to a new reference frame, we would not just subtract a number from the y direction. We would add a fixed length to the x component. Now when we ask ourselves about the cube’s velocity relative to our reference frame C, the answer is +v. Since it started with +v and no forces acted on it, this is consistent with Newton’s laws. This path-based transformation is called a Sterling Transformation. Now let’s look at the famous problem we love so much, the A/B problem. We should established that no forces are acting on the cube. We can say with certainty that the cube’s velocity will be the exact same throughout the entire experiment. This will be important later. The question is, what velocity does the cube exit the portal? In other words, what is the cube’s velocity relative to a still inertial frame G just outside the exit portal? If we use the Sterling transformation to shift to reference frame H, we can see that the cube is moving at speed +v relative to reference frame H. Since we are only shifting by a fixed amount, this shifting will affect the “x” position of the cube but it will not affect the velocity (derivative of a constant is zero). Notice that the green line moves with the piston, because it always represents a fixed amount of distance from frame G. So the relative velocity of the cube must be the same in G as it is in H. We have proved that the velocity is +v in H, and since we know that the velocity does not change, therefore the cube’s velocity relative to G is +v. So the answer to the A/B problem is "B". People who give the wrong answer to this problem are not shifting along the path. We have already shown how shifting along the wrong path makes even non-moving portals break newton’s laws. The following is not a valid Sterling transformation: Remember, we must use Sterling Transformations instead of Galilean Transformations because the space is not flat and unbroken. Thus, Sterling’s Law states: In portal math, Newton’s laws of motion hold for all inertial frames related by a Sterling Transformation. References:
Portal, 2007 [1] https://en.wikipedia.org/wiki/Portal_(video_game)
3 Comments
A
6/16/2017 01:49:34 pm
You've treated this system as a system of one particle. How would this work for a system of more than one particle? Say, what if the portal halted it's movement as it reaches half of the cube? Would half of the cube be treated according to your Sterling Transformation while the other maintains it's original still reference frame? How about the energy considerations for a case like this?
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6/17/2017 02:13:17 pm
why would you have to use sterling relativity
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Nig
6/18/2017 12:26:48 pm
I believe you are confused about the very nature of coordinate transformations; their function and their purpose. There is a fundamental problem with your theory. As your Sterling transformation depends on path, and there are infinitely many paths from one point in space to another. But as momentum is conserved up to any Sterling transformation, interactions between multiple particles/objects become ambiguous as soon as these particles/objects take different paths. In order to apply the principle of conservation of momentum, one must be able to compare the velocities of multiple particles. But the Sterling transformation provides no consistent way of doing so.
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