SPECIAL RELATIVITY

BASIC IDEA.

**An attempt to explain the theory in an accessible way without Bowdlerization.**

It is easy to believe that "everything is relative," that the famous man on the train is at perfect liberty to say: "I am still, it is the man beside the track who's moving." (As long, that is, as the train is moving at *constant velocity*. If it is accelerating - it could be imagined jerking violently about, like a plane in clear air turbulence - then it would be obvious it was the man on the train who was moving. This problem is dealt with in General Relativity by the Principle of Equivalence.) Equally the man standing beside the track may claim to be still.

There must be no way we may, in a sense, tell the two men apart. They must be **equivalent**. (Hence Einstein's phrase: "All inertial [i.e. not accelerating] observers are equivalent.")

How could this idea be cast so as to be used in a scientific theory?

The method is to say that it follows - from the idea that you cannot tell the two apart - that they must be able to write down their laws of physics in the same way. (Their formulae are the same, are "invariant", hence "form invariance".) Thus if the man on the train has a law of physics: x = 2p + y, (where x, p and y are measurements) then the man beside the track may write down the formula as: x' = 2p' + y' (where x', p' and y' are the corresponding measurements taken with measuring apparatus at rest beside the track).

To spell the idea out: both men look at something exemplifying the law of physics "x = 2p + y", say an owl swooping down and carrying of a rabbit in its talons, or anything at all. The man on the train measures the x, the p and the y with measuring apparatus on the train and notes with satisfaction that, when he puts the measured figures into the formula, the number to the left of the equals sign is equal to the number on the right. The man beside the track using identical apparatus, but stationary relative to the tracks, finds that, with his measurements plugged into the formula, the formula also works. (The two men's measurements may be different - for example they will probably not agree about the velocity of the owl: if the owl, at some point, is at rest relative to the train it is nevertheless moving relative to the tracks.) This simply follows common sense-ically from the "*everything is relative/they're equivalent/you mustn't be able to tell them apart*" idea.

(Form invariance only applies to *general *laws of physics. The sort of big, important laws. It does not apply to many everyday laws, which are often approximations and the like. Relativity provides a heuristic method for checking general laws of physics: if they are not form invariant they must be wrong.)

If the simple formula, v_{light} = c, is a general law of physics it must, by hypothesis, be form invariant. (v_{light} is the velocity of light in a vacuum and c is the number 300,000,000 i.e. 3x10^{8} meters per second. The figure is approximate.)

These are the two

keynote axioms of special relativity:

- Form invariance.
- v
_{light} = c is form invariant.

# Discussion of the Constancy of the Velocity of Light.

The second axiom is very odd indeed. It is not surprising a theory with such an axiom gives counter intuitive results.

It is important to slow c down to make things easier to visualize. (Einstein did this. It is interesting how reluctant one is to take this necessary step.) Imagine c is 10 miles per hour.

One way for v_{light} = 10 m.p.h. to be form invariant is for v = 10 m.p.h. (where v is the velocity of *anything*) to be form invariant. This is the case used in Special Relativity.

Take the example of three people standing still with a bicyclist cycling towards them at 10 m.p.h.. One of them runs towards the cyclist, expecting thereby to increase their closing velocity. If the person runs at 5 m.p.h., the closing velocity might be expected to be 15 m.p.h.. But then v_{cyclist} = 15, for the person, so that v_{cyclist} = 10 would not be form invariant.

Another of the three runs away from the cycle at 7 m.p.h.. Again the closing velocity between her and the bicycle cannot change to 3 m.p.h. but must remain at 10 m.p.h.. The one remaining at rest also **measures** v_{bicyle} to be 10 m.p.h.. All three **measure** the cyclist to be closing on *them *at 10 m.p.h..

# How Is the Claim "v_{light} = c is form invariant" Rationalized?

### Its An Axiom.

You could simply say it is an axiom of the theory, stated without proof. Its validity standing or falling depending only upon whether or not the theory works.

This famous experiment failed to disclose any difference in the velocity of light between when it was moving upstream, against the current, or downstream, with the current, of the hypothetical "luminiferous ether" - the medium through which light was supposed to propagate, as sound waves propagate through the air.

In the 19th. Century Maxwell had devised his electrodynamics. (Unlike Newton's dynamics his electrodynamics was consistent with relativity.) The theory included the derivation of a formula for the velocity of light.

v_{light} = 1/sqrt(e_{o}u_{o}) (where e_{o} & u_{o} are the permittivity and permeability of free space, respectively.)

The permittivity is the constant of proportionality in the formula for the force acting between two charged objects. The permeability is the constant of proportionality in the formula for the force acting between two electric currents - moving charges. Both may therefore be easily measured on a laboratory bench.

There is something strange here: Imagine two laboratory benches, moving relative to each other. Both measure the permittivity and permeability and plug the values into the formula for v_{light}. They then compare the values they get with the same lump of moving light - both of them measure the velocity of the same piece of light and compare this figure with the number they have calculated. How can they both be correct?

One way to resolve this problem is for v_{light} = 1/sqrt(e_{o}u_{o}) to be form invariant.

# Reference Frames.

Let two people stand side by side looking along a long, flat, straight road. Out of each their chests projects a long ruler. The two rulers project for miles, side by side, along the road. (Ignoring the drooping, of the rulers, produced by gravity.) Each ruler has a mark on it for each mile - each person is the zero mark, one mile down the road are the 1 mile marks and so on. From each distance mark dangles a clock.

If they release a rabbit at time zero and it runs off, down the road, at 5 m.p.h., it will pass the 1 mile mark when the clock hanging therefrom reads 1/5th. hour; pass the 2 mile mark when the dependent clock reads 2/5th. hour, etc.. They may calculate the velocity of the rabbit by forming the quotients 1/(1/5), 2/(2/5), etc..

Each ruler with its hanging clocks is called a** reference frame**. The term comes from "reference framework". That is, in three dimensions, you have a *framework*or* scaffolding* of forward going, x-direction, rulers (the ones described above); of sideways, y-axis, rulers and of upwards, z-axis, rulers. Hanging all over this scaffolding are clocks. The 4 **coordinates** of an **event**, e.g. someone snapping their fingers, are given by noting the 3 numbers, x, y & z, on the nearest rulers and the time on the nearest clock. (This coordinate is an example of a "**four vector**".)

# The Physical Content of the Theory, Clocks and Rulers.

Going back to the two on the road with the clock/rulers, reference frames, sticking out of their chests:

Extend their reference frameworks in the negative direction. That is give them long rulers sticking out of their backs, again with clocks hanging therefrom.

At time zero - this is for simplicity - measured on both sets of clocks, let one of them run off down the road with velocity v, relative to the other.

(They both agree about their relative velocity. Next to the origin of the moving frame, 0', the marks, x, on the resting ruler and the times, t, from the hanging clocks give x/t = v. Next to the origin of the resting frame, 0, the marks, x', on the resting ruler and the times, t', from the resting, hanging clocks give x'/t' = v.)

At time zero two stars are shot, by one of the observers, from a Roman Candle firework. One is shot up the road and the other down the road in a backwards direction. The former has a velocity, v_{star} = c and the latter v_{star} = -c. On the basis of v = c being form invariant these two formulae are form invariant. Slow c down to 10 m.p.h., say, for ease of visualization.

Thus the running observer chases off after the +c star. If v = 5 you might expect, for the running observer,

v'_{star} = 10-5 = 5. That is, x'/t' = 5 (where x' is the number on the moving ruler next to the star and t' is the time on the moving clock underneath the x' mark). But this would violate v_{star} = 10 being form invariant.

There are only three ways out of this problem:

- Shrink the moving ruler.
- Make the moving clocks go slowly - a longer time between ticks.
- A bit of both.

To get the theory to work properly the third case is used. Hence **Fitzgerald-Lorentz Contraction** and "**Moving clocks run slow**". (If you do not use the third case you can tell the two frames apart.)

This makes v_{star} = c form invariant for the "front" frame, the one coming out of the chest, but the problems of the "back", the negative going, frame have been compounded. v_{backwards going star} = -c is not remotely form invariant.

We have fiddled the rulers and fiddled the clocks; there seems nothing left to fiddle.

There is in fact one thing left: we may put the moving clocks out of synchronicity, as measured by the resting clocks. When this is done, just so, v_{star} = +c or -c becomes form invariant.

# Transformation Equations

To put what has been said into mathematical form we need equations which transform the measurements of one frame to the corresponding measurements made by the other. For example, if a cow drops dead next to the 2 mile mark on the running observer's ruler, x'=2, and the dangling clock next to the cow reads t'=5, what is the reading on the still ruler and on the still clock right next to the same event? The transformation equations answer this question by connecting x', t', x, & t. (For completeness they also connect y' & y, z' & z.)

They are:

x' = k(x - vt)

y' = y

z' = z

t' = k(t - (v/c^{2})x)

(Where k = 1/sqrt(1 - v^{2}/c^{2}))

The reader is urged to draw the two reference frames, with accommodating values for v and c, one below the other, on a sheet of paper, for the relative position they are in an accommodating time after time zero.

# Velocity of Light Is the Upper Limit to Relative Velocity

Since the running observer can obviously never catch up with the star he must be going slower than the star.

In the light of the previous paragraph, consider attempting to push a tennis ball violently away. Slow c down to 10 m.p.h.. As soon as the tennis ball gets anywhere near 10 m.p.h., relative to one's chest, it must become reluctant to accelerate further. How would this feel? As though its mass had gone up - which it has, mass being the proportionality factor connecting force and acceleration.

As the tennis ball goes faster its kinetic energy rises. As shown in the previous paragraph its mass rises also. If its E_{k} rises by a small amount, dE, then dE = c^{2}dm.

This equation is derived by integrating F=rate of change of (mv), with respect to distance, in the normal way. Unlike Newton however the mass is allowed to vary according to the formula from special relativity:

m = km_{o}

(Where m_{o} is a constant equal to the mass of the tennis ball when it is at rest, the "**rest mass**".)

(As far as Sporus is aware E=mc^{2} is *proved *only for kinetic energy. That it applies to potential energy is merely a truth of experience - is not proved from theory. This seems an odd situation.)

(To emphasize that the theory has nothing to do with the **position** of the observer it would have been better to disappear the persons above and just leave the two frames moving relatively.)

### Setting the Clocks.

Either of the two persons can, ex hypothesi, synchronize the clocks by sending a piece of light down the ruler. When it passes a clock the clock sets itself to x/c (where x is the figure on the ruler above the clock). This seems perfectly common sense-ical.

(

**Any help? Why? Message me?**)(

Twin paradox)