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Special Relativity: Kinematics

Lorentz Transformations and Minkowski Diagrams

Problems on time dilation and length contraction

Problems on Lorentz Transformations and Minkowski Diagrams

The Lorentz Transformations

Michelson and Morley's experiments (see the Introduction to this topic) showed that there was no difference in the speed of light when the earth was moving through the ether in different directions, suggesting that there was no such thing as an ether. However, the properties of the ether underpinned much of physics and, understandably, physicists were not willing to give it up easily. In the 1890s, G.F. Fitzgerald and H.A. Lorentz independently proposed that any length (including Michelson and Morley's experimental apparatus) must shrink in the direction of motion through the ether by a factor = . In fact, Fitzgerald and Lorentz saw that for the laws of physics to be preserved in all inertial reference frames, the Galilean transformations of Newtonian physics had to be replaced. However, no rationale or theory was provided for these particular transformations; Fitzgerald and Lorentz deduced their transformations from the mathematics of electromagnetism and not from any understanding of the relativistic nature of motion. It was not until 1905 that Einstein's theory showed the rationale behind the Lorentz transformations (sometimes called the Lorentz-Fitzgerald transformations).

It is possible to derive the lorentz transformations from the postulates of Special Relativity). However, the derivation is long and not particularly enlightening because there are several assumptions which are hard to justify without delving deeply into the mathematics of spacetime. The result of the derivation is:


Δx = γ(Δx' + vΔt)      
Δt = γ(Δt' + vΔx/c 2)      

where:

γâÉá    

What does all this mean? The primed variables ( x' and t' ) refer to a coordinate system, call it F' , that is moving with speed v with respect to another frame F (the unprimed variables, x and t , refer to F ). Further, F and F' have their x -axes pointing in the same direction and the velocity of F' is entirely in the x -direction. makes this clearer:
Figure %: Relative motion of F and F' .
The Δ s refer to differences in space or time between events. Δx , for example, is the distance, measured in F , between two events; likewise Δt' is the time interval between two events measured in the frame F' . Thus, the lorentz transformations relate allow us to translate distances and times measured in the reference frame F' into those measured in frame F . The inverse Lorentz transformations allow us to transform from F to F' :


Δx' = γ(Δx - vΔt)Δt' = γ(t - vx/c 2)      

Also, the Lorentz transformation in the y and z -directions are just Δy = Δy' and Δz = Δz' .

Note that in the limit v < < c (that is, when the velocity involved is nowhere near the speed of light), γ 1 and the transformations reduce to x = x' + vt' and t = t' . As we would expect (from the correspondence principle), these are the familiar Galilean transformations. We will now see how the lorentz transformations can be easily applied to show the results we have already derived.

Lorentz and Simultaneity

If two events are simultaneous in F' , then Δx' = x' and Δt' = 0 . Plugging into the equation for Δt we find: Δt = , which is non-zero unless x' = 0 or v = 0 . Thus the events do not occur simultaneously in frame F ( Deltat 0 implies that there is a time difference between the events).

Lorentz and Time Dilation

If two events occur at the same place in F' then Δx' = 0 and Δt' = t' . Using the second equation, the separation in time between the events in F is: Δt = γΔt' (for Δx' = 0 ). Similarly if events occur in the same place in F , Δx = 0 and Δt = t . Then the second inverse transformation tells us: Δt' = γΔt (for Δx = 0 ). Thus we again have arrived at the seeming-contradiction we saw in Section 2. However, here it is clear that one equation applies when Δx = 0 and one when Δx' = 0 ; the nature of the Lorentz transformations themselves assure us that these cannot both the satisfied for any two events.

Lorentz and Length Contraction

In the section on length contraction we noted that any measurement of length requires that the coordinates of the ends of the object be recorded simultaneously. To measure the length of a moving train, for example when might place two time bombs, primed to go off simultaneously, at opposite ends of the train. The length of the train is the distance between the explosions. Note that if the explosions were not simultaneous (say the explosion at the rear occurred first), the train would move between the explosions and you would measure an incorrect length (too long, in this case). Thus if we have a pole of length l' in frame F' and it is lying along the x' -axis, what is the length in F ? In F we make our simultaneous measurements and we have Δx = x and Δt = 0 . From the first Lorentz transformation we have: Δx' = γΔx (for Δt = 0) . Δx is by definition the length in F , and since the pole is not moving in F' , Δx' is its length in F' . Thus l = l'/γ , just as we discovered in Section 2. We could also analyze a situation when a pole is at rest in F , and find the seeming-contradictory result l' = l /γ . As we have seen the former equation applies only to situations where Δt = 0 and the latter to those where Δt' = 0 . Everything depends on which frame the simultaneous measurements are made. (See Section 2.)

Velocity Addition

Consider a truck (just for a change) moving with velocity v 1 in the x -direction with respect to the ground. Inside the truck a ball is thrown with velocity v 2 with respect to the truck, also in the x - direction. Call the frame of the truck F 1 and the frame of the ground F 2 . The question is this: what is the velocity of the ball with respect to the ground? Under the Galilean transformations the answer is intuitive and obvious: the ball is moving with velocity v = v 1 + v 2 with respect to the ground. Things are quite different in relativity. We know that v , the velocity of the ball with respect to the ground is given by v = , where the subscripts refer to the frame F 2 . Since F 1 is moving with respect to F 2 , we can use the lorentz transformations to write:


Δx 2 = //Δt 2 =      

Thus:

v = =    

However, we know that the velocity of the ball inside the truck is v 2 = . Using this we can simplify our expression for v :

v = =    

This is the velocity additional formula, and it is the true (so far as we know) equation for determining the relative velocities of moving objects. Note that when v 1 < < c and v 2 < < c , the equation reduces to the familiar v 1 + v 2 (as the correspondence principle would anticipate -- we hope that the Galilean form would continue to work for 'normal' speeds). This equation only applies when the speeds being considered are being measured in different frames. Here, the speed of the ball is being measured in the frame of the truck and the speed of the truck is being measured in the frame of the ground. When the speeds are both measured in the same frame, the usual v 1 + v 2 formula still applies.

Minkowski Diagrams

A Minkowski diagram or spacetime diagram is a convenient way of graphically representing the lorentz transformations between frames as a transformation of coordinates. They are especially useful for gaining a qualitative understanding of relativistic problems. We make a spacetime diagram by representing frame F as the coordinate axes x (horizontal) and ct (vertical). We are ignoring the y and z directions, since they are uninteresting. The plot of an object's x - position versus time on the Minkowski diagram is called its worldline. Notice that light, traveling one unit of ct for every unit of x will follow the line x = ct , inclined at a 45 o angle.

Figure %: Minkowski or spacetime diagram.
What do the axes of F' , moving with velocity v along the x -axis of F look like? Take the point (x', ct') = (0, 1) . From the lorentz transformations we can find that this point transforms to (x, ct) = (γv/c, γ) . As shown in the angle between the ct' and ct axes is given by: tanθ 1 = x/ct = v/c . Actually, the ct' axis is just the worldline of the origin of F' . The point (x, ct) = (γv/c, γ) is a distance = γ from the origin, so the ratio of units on the ct' axis to those on the ct axis is just this value, namely:

=    

This approaches infinity as vc and is one if v = 0 . Similar analysis shows that the x' axis is an equal angle from the x -axis and that the ratio of units is also equal (see ). Thus, the faster F' relative to F , the more its coordinates are squished towards the x = ct line.

The advantage of a Minkowski diagram is that the same worldline applies to both sets of coordinate axes (that is, to x and ct , as well as to x' and ct' ). The Lorentz transformation is made by changing the coordinate system underneath the worldline rather than the worldline itself. In many situations this allows us to visualize the perspectives of the different observers more easily. If we had a very detailed and accurate Minkowski diagram we could use it to read off the values for Δx , Δct , Δx' , and Δct' . To find the spacetime coordinates of an event in F , one can read the value off the x and ct axes; to find the coordinates in a moving frame the x' and ct' axes corresponding to the appropriate velocity can be constructed (using the angle formulas explained above), and the value read off using the units derived for x' and ct' , above.

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