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Space Time Theory


How does the Special Theory of Relativity work?

The Lorentz transformation

Recall that in the previous section, we tried to model space and time in a way that would be consistent with the observed constancy of the speed of light in Nature. The resulting model for space and time measurements has observers measuring time and space differently when they are moving relatively to one another. The two basic differences between what relatively-moving observers measure can be summarized as:
Relativistic time dilation:
The process that occurred in the blue driver's rest frame with in time Tb was perceived by the red driver to have occurred in time
Tr = Tb / (1 - (U/c)2)1/2.
This means that a clock appears to tick more slowly to an observer who perceives that the clock is moving than it does to an observer who is in the rest frame of the clock.
Relativistic length contraction:
The blue car measured to have length Lb in the blue driver's rest frame was measured by the red driver to have have the length
Lr = Lb (1 - (U/c)2)1/2.
This means that the length of some object appears to have a shorter length to an observer who perceives that the object is moving than it does to an observer who is in the rest frame of the object. (This applies to the length parallel to the direction of motion only.)

We will now state (without going through the grunge work of proving) that the mathematical model that best encompasses the observed constancy of the speed of light in Nature (pretending for now that gravity doesn't exist) is the one where the spacetime coordinates of two such observers as described above are related through what is now called a Lorentz transformation:

c T = (1 - U2/c2)-1/2 c T' + (U/c)(1 - U2/c2)-1/2 X'


X = (U/c)(1 - U2/c2)-1/2 c T' + (1 - U2/c2)-1/2 X'
This transformation tells us how some observer O, who sees another observer O' moving at velocity U, translates measurements made by O' into measurements valid in the O frame of reference. In our specific case, we'd say this is how the red driver sees things that are happening in the frame of reference of the blue driver.
For example, in time dilation: in the blue car's frame of reference, the laser pulse did not travel in the Xb-direction at all, so X' = Xb = 0, leaving T = T' /(1 - U2/c2)1/2.
In length contraction: in the red car's frame of reference, the red driver must measure the length of the blue car at a single moment of the red driver's time. This means we have Tr = T = 0 and so c T' = - (U/c) X', leading to X = X' (1 - U2/c2)1/2.
The Lorentz transformation is the foundation of relativistic geometry, which we will examine next.

The geometry of relativity

Red and blue cars In order to talk about relativistic geometry, we need some coordinate systems to compare. The red and blue cars in the figure to the right each have a (schematic) coordinate system attached to their individual rest frame.
The Lorentz transformation
c T = (1 - U2/c2)-1/2 c T' + (U/c)(1 - U2/c2)-1/2 X'


X = (U/c)(1 - U2/c2)-1/2 c T' + (1 - U2/c2)-1/2 X'
gives a mathematical prescription for comparing how the red and blue drivers measure space and time.
The path of a light ray is given by X = c T or X = - c T. You should verify for yourself that if X = c T then X' = c T', so the paths of light rays do not change under Lorentz transformation. The purple line in the figure represents the path of a light ray.
If we draw the time axis in units of c T instead of just plain T, then the paths of light will always be lines at 45° angles.
To see relativististic effects in action, it is helpful to draw one coordinate system in the other coordinate system.
The coordinate axes of the red driver's system are formed from the line X=0 (the T axis) and the line T=0 (the X axis), as shown by the red set of axes on the figure below.
The coordinate axes of the blue driver's system are formed from the line X'=0 (the T' axis) and the line T'=0 (the X' axis).
Euclidean rotations If we want to draw the T' and X' axes in the (T,X) coordinate system, we have to use the Lorentz transformation. Plugging into the Lorentz transformation, the line X'=0 (the T' axis) reduces to the formula cT = (c/U) X. The line T'=0 (the X' axis) reduces to cT = (U/c) X. These lines form the blue set of axes shown on the figure to the left.
Now we can see from the spacetime geometry how relativistic length contraction happens. The blue bar in the figure represents the blue car. The first animation sequence shows how the blue car moves in the blue driver's spacetime coordinate system. The length in space for the blue driver is defined at a specific moment of the blue driver's time T'.
How does the red driver measure the length of the blue car? She defines the length in space of the moving blue car as distance in her X-coordinate that the blue car occupies at a specific moment of the red driver's time T.
To the red driver, the blue car looks shorter than the blue driver says it is.
It is also true that to the blue driver, the red car looks shorter than what the red driver says it is. Neither observer has a privileged or more true version of reality. That is an important part of the mathematical model for spacetime called Einstein's Special Theory of Relativity.

The velocity addition problem, solved

Recall the Pythagorean Rule (which the Mesopotamians knew at least a millenium before the Pythagoreans). By taking differentials the Pythagorean Rule can be written as what is called the Euclidean metric dL2 = dX2 + dY2 (Note: this is for two space dimensions. In higher space dimensions you just add more terms to the formula. In three space dimensions it's dL2 = dX2 + dY2 + dZ2.)
Now what to do about spacetime?
It turns out that finally about 4000 years after the Mesopotamians experimentally (but not theoretically) figured out the Euclidean metric, Albert Einstein hit on the spacetime analog, which we now call the Minkowski metric: Lorentz transformation dS2 = c2 dT2 - dL2 where dL2 is the Euclidean metric on the space part of the spacetime. For example, in two space and one time dimensions, the Minkowski metric becomes: dS2 = c2 dT2 - dX2 - dY2 In the cases we've been looking at, the relevant Minkowski metric is simply dS2 = c2 dT2 - dX2 Recall the Lorentz transformation (using b = (U/c) for shorthand): c T = c T' / (1 - b2)1/2 + X' b/ (1 - b2)1/2

X = c T' b/ (1 - b2)1/2 + X' / (1 - b2)1/2
The Lorentz transformation has the amazing property that it leaves the Minkowski metric unchanged so that c2 dT2 - dX2 = c2 dT' 2 - dX' 2, which is demonstrated in the above figure. The two purple lines represent different values of the Lorentz-invariant interval dS2.
In particular, the bright purple line representing dS2 = 0 is important because it represents the path of light (which is described by both X = cT and X' = cT'). This shows that the Minkowski metric invariance under Lorentz transformations really does encode the observed constancy of the speed of light in Nature. So we're on the right track here in trying to model the natural world using the language of mathematics.
How does the Minkowski metric help us solve the velocity addition problem and come up with a new rule for adding velocities that will encode the observed behavior of the constant speed of light as a maximum observable speed in nature? dX = dX' / (1 - b2)1/2 + c dT' b / (1 - b2)1/2 = dT' (V' + U)/(1 - b2)1/2

c dT = c dT' / (1 - b2)1/2 + dX' b /(1 - b2)1/2 = c dT' (1 + U V'/c2)/(1 - b2)1/2
The velocity we're looking for is V = dX/dT. Dividng the top line dX above by the bottom c dT, we get for V:
Relativistic velocity addition rule
dX/dT = V = (U + V')/(1 + U V'/c2)
as the velocity addition rule consistent with the observed behavior of nature.
Let's check: if the blue car is rolling at U = c/2 and a laser on the blue car is travelling at V' = c then the red driver sees the laser travelling at V = (c/2 + c)/(1 + 1/2) = c so this Minkowski metric models very well the observed behavior of Nature that one can't seem to make light travel faster than c, not by putting a laser on a moving car, or a rocket or any other type of moving vehicle. If either U=c or V'=c, the velocity addition formula reduces to V=c.
Isn't that clever? How could nature be that smart?