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In other words, we shall derive the Lorentz transformations—which are just the equations giving the four coordinates of an event in one inertial frame in terms of the coordinates of the same event in another inertial frame.
We take ) O′ flashes a bright light, which she observes to create an expanding spherical shell of light, centered on herself (imagine it’ s a slightly foggy day, so she can see how the ripple of light travels outwards).
However, these transformations presuppose that time is a well-defined universal concept, that is to say, it’s the same time everywhere, and all observers can agree on what time it is.
Once we accept the basic postulate of special relativity, however, that the laws of physics, including Maxwell’s equations, are the same in all inertial frames of reference, and consequently the speed of light has the same value in all inertial frames, then as we have seen, observers in different frames do not agree on whether clocks some distance apart are synchronized.
At time Question: how do O and his observers stationed throughout the frame S see this light as rippling outwards?
To answer this question, notice that the above equation for where the light is in frame S′ at a particular time O and his observers in frame S will say the light has reached a spherical surface centered on O.Please report examples to be edited or not to be displayed.Rude or colloquial translations are usually marked in red or orange.(This is just the old story of synchronizing the two clocks at the front and back of the train one more time.) That is why O does not see O′’s sphere: the arrival of the light at the sphere of radius is said to be a Lorentz invariant: it doesn’t vary on going from one frame to another.A simple two-dimensional analogy to this invariant is given by considering two sets of axes, Oxy and Ox′ y′ having the same origin O, but the axis Ox′ is at an angle to Ox, so one set of axes is the same as the other set but rotated.Now imagine viewing this from a faster train overtaking the clock train—from this view, the front clock will be the first to start.The important point is that although these events appear to occur in a different order in a different frame, neither of them could be the cause of the other, so cause and effect are not switched around. Let us try to visualize the surface in four-dimensional space described by the outgoing shell of light from a single flash, This means the surface is a cone with its point at the origin.If they are synchronized in S′ by both being started by a flash of light from a bulb half way between them, it is clear that as viewed from S the light has to go the same distance to each of the clocks, so they will still be synchronized (although they will start later by the time dilation factor).Let us now suppose that O′ and her crew observe a small bomb to explode in S′ at -axis).How can O′ and O, as they move further apart, possibly both be right in maintaining that at any given instant the outward moving light pulse has a spherical shape, each saying it is centered on herself or himself?Imagine the light shell as O′ sees it—at the instant !