Bridging	Relativity
Motors and Generators	Space
Quantum Mechanics	Medical Physics
Age of Silicon	Electronics

Relativity 6

Relativity 1	Relativity 4
Relativity 2	Relativity 5
Relativity 3	Relativity 6

The Relativity of Distance and Time

Identify the usefulness of discussing space/time rather than simple space.

Having reached the conclusion that absolute time cannot be defined independently of a particular reference frame, Einstein then goes on to discuss the measurement of space and time intervals.

If the observer, O^/ in the carriage, wants to determine the length of the carriage then this can be easily done by taking a ruler and measuring along a straight line, say along the carriage floor. However the observer, O on the embankment, has a problem in measuring the length of the train; this is because in relativistic terms they cannot be sure that they are seeing both ends of the train simultaneously. What O must do, is arrange two markers, say A and B, that correspond to each end of the train and are seen to be at each end of the passing train by making an observation exactly half way between A and B. This midpoint observation will confirm that A and B were simultaneously opposite the ends of the train. Once this has been checked, then O can also measure AB, which is the length of the train, using a ruler. We can also suppose that before O^/ climbed aboard, both the rulers used by the different inertial observers were compared and calibrated. Despite all these precautions, Einstein points out that we will find that the measurements of distance (as distinct to position) taken by the two observers do not agree. In fact our experience of relativity now tells us that the length of the moving train, as measured from the embankment, will be shorter than the measurement taken aboard the train.

As well as showing that length intervals will differ when compared between inertial observers, Einstein also stated that time intervals will differ between inertial observers.

The Lorentz transformations

Account for the need, when considering space time, to define events using four dimensions.

At this stage Einstein introduces the Lorentz Transformations. Although he derived these in his first 1905 paper on Special Relativity he was aware that Lorentz had already introduced these to explain relativistic phenomena in 1904.

The particular choice of the transformations belongs to the two inertial axis systems shown below.

The Galilean assumptions for this system are:

v Time intervals are independent of the motion of the frame, t₂ – t₁ = t^/₂ – t^/₁.

v The distance between two points is independent of the motion of the frame, x₂ – x₁ = x^/₂ – x^/₁.

v Events at various points are may occur simultaneously for both frames.

v .

v For light travelling through a vacuum in the +x-direction, c^/ = c – v.

ALL THESE ASSUMPTIONS NEED TO BE MODIFIED!

Explain quantitatively the consequence of special relativity in relation to – length contraction and – time dilation.

For physical situations where v ® c it has become clear that the first three assumptions do not hold. The last one is now contradicted by Einstein’s second postulate: The velocity of light in a vacuum is the same for all inertial observers. The theory of relativity uses the Lorentz equations

and using these we can see how the first two assumptions are modified and the last confirmed. These modifications apply for all speeds v < c.

Einstein’s postulate concerning the constancy of the speed of light is easily confirmed from the Lorentz transformations. If a light signal moves through the K frame according to

then we substitute this value of x into the first and last equation to get

and by a simple rearrangement we can show that

x^/ = ct^/.

Thus we conclude that the light signal also moves through the K^/ frame according to a similar equation and with the same speed.

The inverse Lorentz transformations can be found from the above using algebra, or more simply by using physical intuition. If reference frame K^/ moves with a relative speed of v with respect to reference frame K, then we can also say that K moves with a speed of –v with respect to K^/. Hence we have

Einstein demonstrates length contraction in the following manner. Consider a one metre rod in the K^/ reference frame, travelling with speed v in the x^/ direction. The ends are given co-ordinates (x^/₁, y^/₁, z^/₁) = (0, 0, 0) m and (x^/₂, y^/₂, z^/₂) = (1, 0, 0) m, (you should see from here that it is aligned parallel to the relative velocity). Now we measure these ends from the K frame of reference at the time t = 0, ie the measurements are made instantaneously in the K frame. Turning to the first of the Lorentz transforms (not the inverse ones given above) we put in our values to get

and

From these two equations we can see that the length of the metre rule when observed in the K frame is

It follows, very simply, using the inverse transformations that if a metre rule in K, (1, 0, 0) to (0, 0, 0) m is measured from K^/, then

Both observers will record a length contraction when they observe the dimensions in from each inertial frame. The measurement of length of an object that has relative motion, L_v, relates to the length that would be observed if the observer was travelling with the object, L_o, according to the equation

The time dilation is nearly as easy to derive from the Lorentz equations, there is a problem that the clock which is stationary in one frame of reference must be changing position in the other. Suppose a clock was placed at (0, 0, 0) in the K^/ frame. This clock then registers a one second time interval by indicating t^/₁ = 0 and t^/₂ = 1 s. For the first time measurement in the K frame we use the Lorentz transformations

These two equations give conflicting relations between t₁ and x₁, this can be resolved by using both t₁ = 0 and x₁ = 0. For the second time measurement in the K frame we use

From here we find

and

so that

If we place a similar clock at x = 0, in the K frame of reference and apply the inverse Lorentz transformations, then we will find

x^/₂ = -vt^/₂

and

The time interval measured for a system that has relative motion, t_v, relates to the time interval that would be measured if the observer was travelling with the object, t_o, according to the equation

These results will be discussed further under the Twin Paradox.

(More on Twin Paradox: see this web site)

The transformation of velocities.

The significance of . . . the constancy of the speed of light.

For this section we continue with our K (observer on the embankment) and K^/ (observer in the carriage) inertial frames, the relative motion still being the speed v parallel to the x-axis. Suppose the passenger in the carriage walks forward with a speed w. Then in the K^/ frame we have

If the speed observed from the K frame is W, then we might expect

using a Galilean calculation. However, we should guess this won’t work as the Galilean transformations should be replaced by the Lorentz transformations. In this case we take the first equation above and transform them so that

thus we find

Hence we have

This gives us the relativistic velocity transformation for velocities in the same direction as v. This equation helps us to understand why the Lorentz transformations confirm Einstein’s hypothesis that the speed of light is the same for all inertial observers. For instance, if w = c, then you find that W = c!

Another case, not discussed in Einstein’s text, is that where the velocity is perpendicular to the x-axis. This time we will use w_y and w_x = 0, the corresponding components in the K frame will be W_x and W_y. Although we have w_x = 0, there is an x-component W_x because of the relative motion between the frames. In fact W_x = v, this should be expected and confirmed by the above equation. For the perpendicular motion, that we are discussing, we find that

These equations have been produced so that we can again consider the special case w_y = c and w_x = 0, for light moving perpendicular to the axis of relative motion. For this case

and the relative speed according to the K observer is

Again this confirms Einstein’s postulate about the constancy of the speed of light for both inertial observers.

Einstein on inertial mass

Explain qualitatively and quantitatively the consequence of special relativity to the equivalence between mass and energy.

We are nearly at the end of our commentary on Einstein’s explanation of special relativity. We are only halfway through his text and the rest is worth following if you also want a reliable introduction to the theory of General relativity. We should not finish without some comments on one of the most famous equations of the twentieth century, that is

Einstein starts with the claim that the kinetic energy of a body of mass m is

where m is now called the rest mass and v is the motion of the body. In the inertial frame of reference that is moving with the body v = 0, but the kinetic energy is not zero, rather the inertial mass at rest still has an energy of

For any other inertial observer moving at speed v relative to the rest frame of the object, the kinetic energy will be greater than . Using one of the typical mathematical approximations that were so necessary before the invention of computers, Einstein was able to write

For the Galilean approximation it is necessary that v << c, in which case we have

where all the terms to the right of become infinitesimally small. Einstein explained that the kinetic energy of a body had two parts, the rest part, and the familiar Galilean-Newton part . We are usually unaware of the massive c part of the kinetic energy when we calculate kinetic energies in our familiar inertial frames of reference. The reason for this is because we actually measure and calculate changes in kinetic energy for a particular inertial frame thus we get

where the large constant terms, , subtract out of the expression.

Einstein was well aware that energy came in different forms: heat, electromagnetic, gravitational and mechanical; he also followed the developments in understanding nuclear energy. For relativity, he usually considered inertial systems in outer space (away from appreciable gravitational fields). Even for these, he realised there would be exchanges of energy through electromagnetic radiation, say E0. Thus for the total energy of a body he wrote

From here he reasoned if a system was self contained that is then changes to the system could occur according to

Here we disregard a plausible minus sign because a reduction in mass would correspond to a large amount of energy given out from the system This is one of the most famous equations of the twentieth century, as late as 1920 Einstein wrote “A direct comparison of this relation with experiment is not possible at the present time …”.

This mass to energy transformation is the basis of nuclear power, and in the case of our sun quite a natural and beneficial power source. Even though Einstein’s theory anticipated and explained nuclear energy it would be quite wrong to attribute the discovery of nuclear energy (or weapons) to Einstein.

In his famous undergraduate textbook (The Feynman Lectures on Physics vol.1), Richard Feynman takes Newton's second law (F = d(mv)/dt) and states that Einstein showed the mass changed with speed as

where we now use as the rest mass of the moving object.
Feynman then claims: "For those who want to learn just enough about it [relativity] so that they can solve problems, that is all there is to the theory of relativity - it just changes Newton's laws by introducing a correction factor to the mass." High school students who wish to learn more would be well advised to read chapters 15 to 17 of this textbook, unfortunately not yet available from the web.