Simultaneity in Special Relativity
According to Galileo's principle of relativity (which predates Einstein's theory by several centuries) it is impossible to assign an unambiguous or absolute value to the velocity of any object in motion, for the velocity is always a relative value (relative to the frame of the observer). So therefore if an observer was on board a moving vessel, and tossed a ball up and down in the air, the ball would remain 'at rest' relative to the observer (it would be tossed up and down in the same manner as it would if it was to be tossed into the air by someone standing stationary on the ground outside the moving vessel (of course this is not a stationary frame because the earth is rotating and the earth is also moving in orbit).
If a ball was tossed with a velocity of ten miles an hour on board a train moving at 50 miles per hour, to an observer 'at rest' with respect to the train (moving with the train at the same velocity) the measured velocity of the tossed ball would be 'ten miles per hour'. To an observer outside the moving train (standing 'stationary' on the ground beside the train tracks) the velocity of the moving ball would be measured as 60 miles per hour (here we assume that the ball was tossed going with the motion of the train). The velocity of ball (10) and moving train (50) were additive. So to the person 'at rest' with respect to the train, the ball had a velocity of 10 miles per hour and to the person for whom the train was in motion with respect to the observer's frame, the ball had velocity of 60 miles per hour, and both measurements are correct, for all measurements of velocity are 'relative' with respect to the observer's reference frame (therefore we can never assign an 'absolute velocity' to the moving ball, and can only measure 'relative velocity').
According to Einstein's theory of relativity, the speed of light is a fixed velocity in the reference frame of all observers. This then leads to some bizarre and unexpected 'side effects'.
An observer on board a space vehicle moving at just under the speed of light, measures the velocity of a beam of light and finds that the velocity is constant ('c', the speed of light).
A second observer watches the exact same event, and sees the rocket moving at just under the speed of light, while the speed of light remains at a constant value ('c', the speed of light). The velocity of the light wave remains constant, and this is quite different from the behaviour of a tossed ball on a train, where the velocity was additive.
Therefore since the speed of light is observed to be a fixed constant in all reference frames, two observers have different interpretations of the same event.
A clock is designed which measures the passage of time by bouncing bursts of light between mirrors. Because the speed of light is a fixed constant a precise measure of time can be made by bouncing light pulses through a fixed distance in space, and the clock 'ticks' and advances in time with each measured bounce of the light beam. In the illustration above, the clock and the observer are both 'at rest' (there is no relative difference in velocity between clock and observer).
If the clock is in motion relative to the observer then it can be seen that path followed by the light beam now appears to be enlongated. Given that Einstein's theory of Special Relativity requires the speed of light to be a fixed constant for every observer in every frame, this then implies that the passage of time must have slowed down, as measured by the moving clock, relative to an observer 'at rest' (for whom it is the clock which is in motion). The length of the path (the diagonal side of the triangle) is dependent upon the velocity of the clock, and using the Pythagorean theorem to determine the length of one side of a triangle, when you know the length of the other two sides, this then leads to the formulation of the equation for velocity induced 'time dilation':
where 't' is the time as measured on the object in motion, and v is the velocity with which the object is moving relative to the observer (these observers consider themselves to be 'at rest'), and t' is the time which has passed in the reference frame of the observer at rest. This equation is the 'Lorentz transformation', which was not originated by Einstein, but what was unique was the imaginative approach employed by Einstein to derive exactly the same equation, after which time Einstein would then go on to justify some of the bizarre consequences of the fixed value of the speed of light as it relates to our normal perceptions of the passage of time.
The equation is employed as follows. If the time which has passed as measured by the moving clock was one second (t), and the calculated square root of the 'time dilation ratio' was '0.4' then the time as measured by the observer at rest (t') would be t' = t / 0.4 = 1 / 0.4 = '2.5 seconds' (which means that time is passing faster for the observer at rest, since for every one second that passes on the moving clock 2.5 seconds has passed for the observer 'at rest').
It is for this reason that two observers can watch the same event and see two different outcomes. For the observer on the moving craft, the passage of time has 'slowed down' and therefore this observer sees the speed of light as a 'fixed constant' (this observer appears to be 'at rest' with respect to the light beam, since for every observer in every frame in motion, the local frame appears to be a 'rest frame'). This is just one of the bizarre consequences of the fact that the speed of light is always measured as being a fixed constant value by all observers no matter what their relative velocity with respect to each other.
Because the speed of light is always measured by all observers as being a fixed constant (no matter what their relative velocity might be) it is possible to use the speed of light to define the length of a meter stick (even though the earth is in constant motion through space). When the meter stick is 'at rest' relative to the observer, it is possible to measure the 'proper length' of one meter, given that the speed of light is a fixed and constant value in all 'inertial reference frames' (both meter stick and observer are 'at rest' in that both move with the same constant velocity through space). However when the meter stick is in relative motion to the observer, given that the speed of light remains constant, this then introduces ambiguity into the definition of 'length' (given that for science 'length' can only be properly defined as a property of the method of measurement).
The relativity of both time and distance that emerge in relativity theory due to the constancy of the speed of light, then leads to one of the most bizarre results of Einstein's theory of relativity. Here we see two beams of light being sent to two opposite walls of a measuring box, which is 'at rest' relative to an observer. Given that the distances to both walls are equal and given that the speed of light is an invariant constant, both beams of light strike the opposite walls of the box at precisely the same time.
A second observer watches the exact same event unfold. This observer is not 'at rest' relative to the box, but instead the box appears to be in motion with a certain velocity with respect to the reference frame of the observer. The light, moving with a constant velocity, reaches one wall first, and then there is a delay and the light, moving once again with the same constant velocity, reaches the second wall. Later the two observers compare their observations, and they find that they are unable to agree that two events they both observed occurred 'simultaneously'. It is typical to assume that when some event is observed to occur at '4 o'clock' that means that it happened at '4 o'clock' for every observer everywhere in the universe. You would only need to mention that an event occurred at '4 o'clock' and everyone could consult their local clock and everyone would then come into agreement on the time at which some event occurred. This is not the way the universe really works, and two events which seem 'simultaneous' to an observer in one reference frame, might be observed to have occurred at different times to an observer in another reference frame. This bizarre result is one of the strangest consequences of the theory of relativity, as it violates our normal 'common sense' understanding of the meaning of the 'passage of time'.
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