which gives a property of the rotation or transformation matrix MTM = 1. The transpose of a matrix interchanges the rows and columns by Mj ^ Mji. It is then possible to use this information to find the entries of the matrix M.

The unchanged length of a vector under this transformation, called its invariance, means that the components of V and V' are ct, x, ,y, z and ct', x', y', z' respectively with

-(ct')2 + (x')2 + (y')2 + (z')2 = -(ct)2 + x2 + y2 + z2. (6.5)

This problem is simplified by considering the transformation as a velocity in the x direction. This means that y' = y and z' = z. As a trial solution consider x' = y(x — vt) and x = y'(x' + vt'). The factors 7, y' are to be determined. By reversing the roles of x and x' it is evident that y = y'. For a particle or photon moving with x = ct then x' = y(c — v)t. Further, since the speed of light is regarded as an invariant x = ct implies x' = ct'. This leaves the two equations, ct' = Yt(c — v), ct = Yt'(c + v).

At this point it is easy to see that by multiplying these two together and dividing out the tt' term, we obtain

which is the famous Lorentz-FitzGerald contraction factor. This was derived by Lorentz and FitzGerald to contract the length of a body moving in the aether to cancel out the influence of the aether. This leaves the following Lorentz transformations t' = y(t — vx/c), x' = y(x — vt), y' The transformation matrix then has the form

( |
y |
—yv/c |
0 |
0 |

yv/c |
y |
0 |
0 | |

0 |
0 |
1 |
0 | |

\ |
0 |
0 |
0 |
which defines the Lorentz transformation. The y factor is what is responsible for the time dilation and length contraction effect. An observer on one frame will observe a clock on a frame moving with velocity v tick away with time intervals slowed or increased by At' = yAt. Similarly a meter stick pointing in the direction of motion will appear shrunk by a factor L' = l/y. There is a caveat that must be made about this. To observe another reference frame photons must "bounce off" items on that frame and reach the observer who's frame is designated as at rest. This adds complicating factors, which implies that a moving cube will appear rotated. It is also commonly said that the mass of a body is changed by 7m. It is my sense that this statement is best not used, at least extensively. Vector quantities, such as momentum, transform, but scalar quantities like mass do not. So after this bit of analysis it is good to step back and look at the big picture. This forces a modification of Newton's laws. This modification will be examined in more detail later. However, it modifies the first law of motion by saying that a body in a state of motion will remain in that state of motion if not acted on by a force as measured by an observer in an inertial reference frame, where the speed of light is invariant on all such reference frames. So not only is the motion of some massive body a constant if free of any acting force, but now on all possible reference frames the speed of light is the same. As a technical matter, the velocity of a particle is now considered in spacetime. It includes the standard spatial velocity v, but in addition there is a time-like velocity. The spacetime velocity is written as Ua so that U1 = Ut and the components for a = 2, 3, 4 are the components of the spatial velocity. This modifies Newton's first law of motion by including an invariance principle. The second law must be extended to spacetime, or four dimensions, where momentum, force and acceleration are extended to four-vectors. There is further a subtle issue with what time, "t" is used in the dP/dt term, where this equation must make sense to all observers. Hence transformations between frames changes the meaning of time. The second law of motion must be reformulated accordingly. However, given these modification the relativistic second law of motion is essentially the same. The third law is similarly modified. Newton's third law indicates that space has a symmetry structure, where the physics is the same no matter where one's frame is or how it is oriented. Special relativity demands that this symmetry principle be extended to one that includes the Lorentz transformations. The symmetry of spacetime has an additional set of "rotations" that can inter-change spatial coordinates with time. These modifications are required in order to understand the physics of a relativistic rocket and photon sail. It is clear that an understanding of dynamical principles in the framework of special relativity must be reformulated. To start this process consider the length invariance of the vector V above. In three dimensions the length of a vector form the origin is |V|2 = x2 + y2 + z2. In spacetime this vector includes a ct component in four dimensions. The length of this vector is then This length is often labelled by s, or t, and is called the invariant interval, and defines the proper time. The time variable t is then the coordinate time. The negative sign on the (ct)2 term is why flat spacetime of special relativity is often called pseudo-Euclidean. There are a number properties for this interval. To consider a particle travelling in the x direction let x = vt and set y = z = 0. It is then clear that s2 = (v2 — c2)t2 < 0, (6.11) which is called a time-like interval. Similarly for x = ct we have s2 = (c2 — c2)t2 =0 , (6.12) and is designated a null interval. It is then apparent for a particle travelling faster than light, called a tachyon, that s2 > 0, which defines a space-like interval. The tachyon is a sort of fiction, or a particle that vanishes in superstring theories. It is unwise to start seeing the tachyon as some way of travelling faster than light. The invariant interval defines the proper time of the particle which travels from this origin to the tip of this vector. It is also the length that defines the path of a particle, called a worldline, in spacetime. For a particle or photon travelling the speed of light this interval is zero. A photon has no internal clock. This structure defines a light cone. For a point defined as the origin there is a set of null lines that pass through this point. This gives the past and future light cones through this point. This defines points in the past of the origin that may causally interact with this origin, as well as points in the future of the origin that may be causally influenced from the origin. The past and future events or points may interact with the point of origin must propagate on either time-like curves from points within the cone or are propagated along the cone as null lines. This is shown in Figure 6.2, where one spatial dimension has been suppressed. A property of special relativity is that there is no universal frame from which time is measured. For instance, two points separated by a distance or ruler may have their clocks synchronized according to a reference frame at rest with respect to them. If there are flash bulbs at these points that pulse simultaneously in a frame at rest with respect to these bulbs the light pulses will meet at the center. However, for a frame moving with some large velocity with respect to this spatial distance since the speed of light is an invariant the wave fronts no longer meet at the center. This means that to this observer one bulb flashes before the other. Figure 6.3 illustrates this effect. This means that clocks can only be synchronized according to a chosen inertial reference frame. Fig. 6.3. Lack of simultaneity on spacelike intervals. Fig. 6.3. Lack of simultaneity on spacelike intervals. As a formalistic sideline, some books on relativity put the negative sign in the interval on the spatial terms so that s2 = (ct)2 - x2 - y2 - z2 , (6.13) which reverses the sign convention for time-like and space-like intervals. This is a matter of convention, where some people prefer to have positive intervals for time-like intervals and others prefer that the spatial part of the interval remains positive to fit with our usual three dimensional sense of things. It does not matter which convention is used, but one must stick with one through the calculation of a problem. A similar interval exists for momentum and energy. A particle with a momentum p and an energy E has a spacetime momentum of with PT defined accordingly. The momentum-energy interval is then What is defined on the left hand side of equation 6.14? To answer this, we first compute the Lorentz transformation of the momentum vector in spacetime P' = MP. It is left as an exercise to the reader to perform this, where the energy and momentum terms in the four vector are found to transform as Since y ^^ as v ^ c it is clear that light speed is an infinite barrier that imposes a fundamental speed limit in the universe. For the momentum term use the fact that kinetic energy of a particle is defined in Newtonian mechanics as the displacement of a force through a distance For simplicity, we consider this motion only in one direction so that the vector notation may be suppressed. We further use the fact that F = dp/dt so that |

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