Logo: Proper Interval Locality
Visualising proper Interval Locality
A space-time diagram is a graph showing position as a function of time. Conventionally, time runs up the diagram with the past at the bottom and the future at the top. Position is the horizontal (X axis, represented by one dimension). Events can be reasonable visualised using this devise but it has a fundamental weakness in that the interval between events is necessarily distorted. Diagram 1 illustrates this difficulty
Image: Interval on Space-Time Diagram
The space-time diagram allows to visualise position as a function of time, but the price we have to pay is a distortion of the proper interval between events. (A consequence of trying to depict the Minkowski geometrical relationship between space and time on a sheet of paper characterised by Euclidian geometry.) However for any single event we can create a coordinate transformation that allows us to graphically represent the magnitude of the proper interval between the event and any other set of coordinates in space-time.
In diagram 1 let (X2-X1)2 + C2(T2-T1)2 = H2 ; if (X2-X1)2 > C2(T2-T1)2 then
(X2-X1)2 - C2(T2-T1)2 = S2 and if (X2-X1)2
The change in equation representing the change from space-like to time-like intervals.
Now let event 1 be positioned at the origin of the space-time diagram. So that X1 = 0 and T1 = 0.
We can use the coordinate transformation x = (S/H)X and t = (S/H)T to visualise the interval from the origin to an event at (X, T).
Image: Shows interval to origin for given set of rectilinear coordinates
In diagram 2; point A represents the coordinates of position and time relative to a given frame of reference. The distance from the origin to Point B now represents the proper interval from the origin to event 1. The proper interval will be fore-shortened unless for space-like intervals T is 0 or for time-like intervals T is infinite.
Now let us extend the coordinate transformation to the gridline (1, T)
Now let us extend the coordinate transformation to the gridline (1, T)
Image: Gridline transformation (Lily Diagram)
Diagram 3
Diagram 3 shows the proper interval transformation for the gridline (1, T). Note here C is set at 1.
We see the plot of the gridline transformation touches the origin twice, when T = 1 and when T = -1. That is for these two events the proper interval joining them to the origin has collapsed to zero. According to the principle of proper interval locality the world states at (1, 1) and (1, -1) are therefore not independent of the world state at the origin. More specifically a quantum system experiencing event (0, 0) can receive momentum from a quantum system at (1, -1) and donate momentum to a quantum system at (1, 1).
Image: Quantum Interaction and Gridlines Transformed (Rose Diagram)
Diagram 4
In diagram 4 the coordinate transformation is shown for the gridlines (-1, T), (-2, T), (-3, T), (1, T), (2, T), (3, T), (X, -1), (X, -2), (X, -3), ), (X, 1), (X, 2) and (X, 3). It is important to remember that this gridline transformation is unique to the event (0, 0). Superimposed on the diagram are the world lines of two quantum objects Q1 andQ2. Q1 is initially at rest relative to our inertial frame of reference and placed at X = 0 Whilst Q2 is in relative motion and its path passes through the event (2, 2). The Proper interval between events (0, 0) and (2, 2) has zero magnitude, the two quantum objects are touching in space-time and therefore the can exchange energy. An interaction is indicated by the change in direction of the objects at (0, 0) and (3, 3). The curved white plot is the proper interval between event (0, 0) and the world line of quantum object Q2. Note that this object passes through event (3, 0) the distance and the proper interval between this event and the origin have the same magnitude.
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