In Einstein's new spacetime, the light cone is the same for all inertial observers. There is no longer a discrepancy in speed between past and future or left and right. But, to keep the uniformity of space and time we have to give up something else - our notion that everyone can agree on the simultaneity of events. Time, as well as space, is now a 'relative' concept.
Einstein offered the following "thought experiment" to show that simultaneity is relative:
A Trip on Einstein's Train
Lightning strikes the front and back ends of a moving train. The lightning strikes leave marks on the train and on the tracks.
Consider two observers:
Stan receives ("sees") the light emitted from events F and B simultaneously, and concludes that: "since I am midway between the marks on the tracks, and since the speed of light is a constant, then events F and B are simultaneous."
Mo receives ("sees") the light emitted from event F, then from B, and concludes that: "since I am midway between the marks on the train, and since the speed of light is a constant, then events F and B are NOT simultaneous: F occurred before B."
Or, in terms of the spacetime diagram:
where the green worldline belongs to Stan, and the pink worldline belongs to Mo. Even though the light cones originating at events F and B arrive simultaneously at Stan, the light cone originating at F definitely reaches Mo before the one from B.
In Einstein's spacetime, the Galilean transformations that we had before to translate the information between different inertial observers (having a relative velocity, v, between them) are now replaced by the Lorentz transformations.
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Galilean
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Lorentz
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x' = x - vt
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x' = g(x
- vt), g = 1/[1-v2/c2]1/2
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t' = t
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t' = g(t
- vx/c2), g
= 1/[1-v2/c2]1/2
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Inertial observers in Einstein's spacetime do not necessarily agree about where or when events take place. One observer may see two lights flash at the same time, while another observer may see that one definitely flashes before the other. Similarly, one observer may say that two events occur at one location, although at different times. Another observer may see the same events occuring at different places as well as at different times.
However, all inertial observers will agree about causally-linked events in spacetime. If two events occur at the same place and the same time according to one inertial observer, then these causally-linked events will be seen the same way by all inertial observers.
Under the Lorentz transformations, the speed of light (in empty space) is the fastest any object can go. While Einstein's theory makes simultaneity a relative notion, it does not change the idea of causality. So, two causally linked events in the view of one inertial observer will be known as causally linked to every other inertial observer that sees them. If speeds faster than light were possible, then causality could not be maintained.
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