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Space time Theory


Why was the Special Theory of Relativity needed?

The velocity addition problem

The observed constancy of the speed at which light travels tells us that the Newtonian model of space and time is flawed. But the flaws don't become noticeable until we start trying to describe things moving near the speed of light.
Reference frame 1 One reason why Einstein's Special Theory of Relativity was needed was because of the special problems that cropped up when scientists noticed that the speed of light is a constant everywhere in every direction. This caused problems with the Newtonian model for measuring time. One of these problems is called the velocity addition problem.
The velocity addition problem is illustrated in the figures above and below. In the top figure we see the red driver's frame of reference or rest frame. To the driver of the red car in her rest frame, the blue car is travelling forward at velocity U and some purple object is flying out of the blue car at velocity V.
In the figure below the point of view of the blue driver is illustrated. To the driver of the blue car, the purple object is flying forwards out of her window at velocity V', and the red car is driving backwards. Reference frame 2
The velocity addition problem asks the question:
Given U and V', what is V?
If we use Newton's model for time as being experienced exactly the same for all observers, we wind up with the answer: V = U + V'. How?
Suppose, as Newton believed, the red and blue drivers measure time and distance precisely the same. According to the blue driver, in time T the distance of the purple ball from the blue car is Xball = V' T. In the same time T in the reference frame of the red car, the blue car has travelled the distance Xcar = U T. According to the red driver, the total distance the purple ball travelled is the distance the blue car travelled from the red car plus the distance the purple ball travelled from the blue car, or X = Xcar + Xball = U T + V' T = (U + V') T.
But X = V T, and that gives us the velocity addition formula:
V = U + V'
So, for example, if the blue car is going 30 mph and the driver of the blue car measures the purple ball going 60 mph, then the driver of the red car should measure the purple ball going 90 mph, because U = 30 mph, V' = 60 mph and hence V = U + V' = 90 mph.
Sounds reasonable, right?
Okay, now suppose that U = half the speed of light, and V' = the speed of light (maybe the purple object is a laser pulse). Now what is V? The above formula tells us V = one and a half times the speed of light.
This is contrary to observed behavior of Nature. Therefore the Newtonian model of time as being experienced equally by all observers must not be a good model for Nature.

Time must be relative

If the speed of light is the same even if the light source itself is moving at some speed relative to the person doing the measuring, then how does this affect the way different observers measure time and space and combine them into spacetime?
Let's take the red and blue cars from the previous example, and let's mount a laser on top of the blue car. We'll put a mirror on the blue car as shown in the figures, and aim the laser at the mirror perpendicular to the direction the blue car is travelling.
Blue frame of reference

Here is the laser pulse viewed in the frame of reference of the blue car.

Within the time interval T' measured by the driver of the blue car operating the laser, the laser light travels a distance 2 L from the laser to the mirror and back.
Meanwhile, the driver of the red car sees the blue car go whizzing by. According to the red driver, the laser took a total time T to hit the mirror and return. (She's willing to agree with the driver of the blue car that the distance between the laser and the mirror is L, since neither driver has any velocity in that direction.) She measures the blue car to have travelled a distance X = U T and the laser pulse to have travelled a total distance of 2 D = c T in time T.

Here is the laser pulse viewed in the frame of reference of the red car.

Red frame of reference
Here is where the problem arises. Since we just deduced above that c T' = 2 L and c T = 2 D, the only way we could have T' = T would be if D = L.
Therefore, the red and blue drivers do not measure time equally..
Now how can we sort this out and find out how their measurements of time should differ in order to account for the observed constancy of the speed of light?
What both drivers agree on is the distance L perpendicular to the motion of the cars. Using the Pythagorean Rule on the laser pulse's path in space, we get L2 + (X/2)2 = (c T/2)2 . From the blue driver's point of view L2 = (c T'/2)2 , and putting these together and using X = U T we get: (c T')2 + (X)2 = (c T)2 or T = T' / (1 - (U/c)2)1/2 This is stupendous! The drivers of the blue and red cars don't measure the same time for the laser to hit the mirror and come back! We are forced to conclude this if we want to be consistent with the observed constancy of the speed of light (so far) in Nature.
This has profound implications for the mathematical modelling of space and time as observed in Nature. This means we have to expand the idea of Euclidean analytic geometry to include the observer-dependent relativity of measurements of time and space. This opens up a gigantic can of mathematical worms, eventually bringing us to black holes, wormholes and at least the abstract mathematical possibility of time travel.

What we've learned here is called Relativistic time dilation. The process that occurred in the blue driver's rest frame with in time T' was perceived by the red driver to have occurred in time T = T' / (1 - (U/c)2)1/2, which can be much much greater than T' if U is close to the speed of light c.

Space must be relative, too

In the previous frame we learned that if we want to use geometry to model space and time together, in order to account for the observed constancy of the speed of light, observers moving at constant velocity relative to one another perceive the passage of time differently.
Now we will figure out whether these observers measure space differently as well.
Let's suppose the driver of the blue car below has measured the length of her car with a ruler to be length L'. She sees the red car driving by her at velocity -U. The blue driver sees that it takes a time T' for the front of the red car to pass from the front end to the back end, and she calculates that therefore L' = U T'.
Length contraction 1 L' is the blue car length, and T' is the time for the red car to pass, as measured by the blue driver.
The driver of the red car, on the other hand, sees the blue car rushing past her. She measures the time T it takes for the blue car to pass her front bumper. She then calculates the length L of the red car to be L = U T.
So we have L/L' = T/T'. Now we just need to know the relationship between T and T'. But we calculated that already in the previous frame.
Length contraction 2 L is the blue car length, and T is the time for the blue car to pass, as measured by the red driver.
Recall that in the previous section, the blue driver was timing a laser pulse and its reflection from a mirror. These were events that happened at the same place according to the blue driver, and their time separation was measured to be T', with only a single clock needed for the measurement.
The red driver saw the laser pulse and return happening in different places (because she saw the laser as moving with the blue car) and so the red driver's time measurement T could only be made with a minimum of two (synchronized) clocks.
Constraining the speed of light to be constant gave us this relationship between the time T' measured by the blue driver and the time T measured by the red driver:
T' = T (1 - (U/c)2)1/2.
The blue driver with the laser is like the red driver in this example, trying to measure a time interval between two events happening at the same place. The blue driver in this example is like the red driver from the previous section, trying to measure a time interval between two events in two different places.
Therefore, the relationship T = T' (1 - (U/c)2)1/2 must hold between T and T' in this example. (The roles of T and T' are switched compared with the last section because the roles of the red and blue drivers with respect to time measurement has switched, as explained above.)
But this means that the relationsip between the lengths L and L' must be
L = L' (1 - (U/c)2)1/2
Therefore, if we want to make a mathematical model for spacetime that is consistent with the observed constancy of the speed of light, we have to conclude in this model that measurement of space and time is not the same for all observers.

What we've learned here is called Relativistic length contraction. The blue car measured to have length L' in the blue driver's rest frame was measured by the red driver to have have the length L = L' (1 - (U/c)2)1/2, which can be much much smaller than L' if U is close to the speed of light c.

Is there anything that is the same for all observers? That is what we'll look at next.