Lorentz Transformation

The combination of the three phenomena is going to be used to formulate the general coordinate transformation between two inertial frames of reference. We shall consider two frames of reference, and . The origin of is in motion at a constant velocity of in the frame of reference, in the direction. Let’s say there exists an arbitrary event . We are going to find that, for a given time and location in one frame of reference, when and where does event occur in the other frame. To formalize Lorentz Transformation, we will define an event which will serve as our reference point. Event occurs at the origin when both frames of reference overlap, meaning and both refer to the same point. We shall let the moment occurs be . We can write ‘s coordinate as in both frames.

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In a similar fashion, we shall write the coordinates of event as and for the and frame of reference, respectively. Note the usage of instead of , this is for the sake of unit consistency.

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Firstly, acknowledge that time dilation is used to find the time delay between two events which occur at a different time, but at the same position in the primed frame of reference. On the contrary, simultaneity is used to find the time delay between two events which occur at the same point in time but at a different position in the primed frame of reference. So by introducing an intermediate event, we can find the time delay in the lab frame between two events which occur at different time and positions. Let be the intermediate event. It occurs at the origin of , and at the same time as in the frame. This means the coordinate of in is . The coordinate of in shall be .

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We can immediately evaluate the value of using time dilation on the interval between event to :

Because occurs at the origin of frame, and is moving at in the direction, we easily find that:

So the event’s coordinate in is .

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In , and occurs simultaneously with the spatial difference on the axis of . From simultaneity, two events which occur simultaneously with spatial difference in will have a time delay between them in . In , the time delay between and due to simultaneity is

Where . Therefore, the time at which occurs in is

As for the position at which occurs, it is the sum of the distance has moved, plus the distance under length contraction. This means:

Let’s evaluate :

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As for and , since there is no length contraction nor motion in those axis, it is simply

So ultimately, for a given event which occurs at the coordinate in the frame, it occurs at in the frame of reference.

Note the symmetrical relationship between time and displacement in the Lorentz transformation equations.