**Problems
with Relativity (Page 1)**

Name and Address:

Tom Hollings. E-Mail carmam "at" tiscali.co.uk

All references are to A. Einstein - " Relativity, The Special And The General Theory" Methuen & Co. 1920

The theory of relativity as expounded by Einstein gives a very clear insight into what happens in the real world, especially at extremes of velocity and/or mass. There are, however, some questions which need answering.

I include this link to professor Dingle's book, because anybody who has the slightest doubt (and in fact anybody who does not) about the truth of the Theory of Special Relativity should read it. I read it long ago, and re-read it recently. I was astounded by it - both times. It is even more relevant now than when professor Dingle wrote it in 1971. I urge you to read it with an open mind.

Two other people deserve a mention here. They are Bernard Burchell, who wrote the web page Alternative Physics and rewrote my original version of "7. The Twin Paradox" (which I first put on a science forum) to make it more realistic and more readable. Also Hans Zweig, who wrote Relativity Unraveled They have both helped me tremendously.

Contents (and links).

1. The Lorentz transformations

5. The equality of inertial and gravitational mass

6. The constancy of the velocity of light

8. Experimental evidence on the constancy of the velocity of light

1. THE LORENTZ TRANSFORMATIONS.

When Hendrik A. Lorentz devised his transformation formulae in 1890 he thought that they applied only to electrically charged bodies, but Einstein incorporated them into his special theory of relativity assuming that they applied to all bodies. This is the equation for mass increase.

The theory tells us that mass increases with velocity, becoming infinite at the speed of light.

where m = the mass of the body

m0 = the rest mass (proper mass)

v = the velocity of the body

c = the velocity of light

Lorentz was however, nearer the mark. Any body which has been accelerated to an appreciable velocity for the increase in mass to be tested (and proved) has been accelerated by an external force which is itself electro-magnetic and therefore constrained to the speed of light. These formulae therefore apply only to bodies which receive an acceleration from an external force, and the increase in mass (and length contraction) is with respect to the reference frame from where the force originated. The increase in mass (and the length contraction) is an illusion. If an electro magnetic force force is used to accelerate a body, the electro
magnetic field is itself constrained to the speed of light, so it cannot accelerate the body past that speed. The observed effect is as though the body has increased in mass. A simple (probably oversimple - so please don't take it too literally) analogy may help. A tow truck (all tow trucks used in this example have a top speed of 20mph) goes out to rescue a broken down lorry. It starts the tow, but finds that it cannot go faster than 20mph. The driver calls for assistance, and another tow truck arrives to help. Now there are two tow trucks pulling together, and therefore twice the force. The broken down lorry can be accelerated more but still cannot be moved faster than 20mph. They then try to measure the mass of the lorry by hitting it sideways with yet another tow truck traveling alohgside. When they are moving slowly, the tow truck can push it sideways for some distance, but as they approach 20mph, the acceleration from the sideways push gets less and less, and at 20mph the sideways acceleration is zero. The drivers are puzzled at first, then use the Lorentz equations to find out what is going on. They use 20mph in place of c, and conclude that the mass of the lorry increases with speed, becoming infinite at 20mph. This conclusion fits relativity theory. The more power they apply, the more the mass increases and the less the speed increases, until at 20mph, all the power goes to increase the mass, and none to increase the speed. It does not matter how many tow trucks are used, the result is the same, 20mph is the limiting velocity.

Imagine now a space rocket, which is propelled by ejecting a small amount of matter (the rocket exhaust) at high speed from the rear, so imparting a thrust in the opposite direction. We will assume that the exhaust velocity is 3,000 m/s and the mass of the rocket is 30,000 Kg (very similar to NASA's Mercury-Redstone rockets). Now we can use the Lorentz transformation to find the new mass. The velocity between exhaust and rocket is 3000 m/s, so :-

m = m0 / sqrt( 1 - ( v / c )^2)

m = mass of rocket at velocity v as measured by the essential observer

(Remember that Einstein's observer, properly called the essential observer, is always at rest relative to the motive force. In this example therefore, the essential observer is in the same frame as the rocket exhaust).

m0 = 30,000 Kg (proper mass of rocket or rest mass when v = 0)

v = 3,000 m/s - rocket's velocity relative to the exhaust

c = 300,000,000 m/s

m = 30000 / sqrt( 1 - (3000 / 3e8)^2) = 30000.0000015000000001125 Kg

The mass increase is therefore 0.0000015 Kg or 0.0015 gram which is simply not measurable compared to 30,000 kilograms. For all intents and purposes the mass increase is zero. A further point to note here is that the mass increase is measured against the exhaust which is providing the motive force, and no matter what the velocity of the rocket when measured against its starting point (or anything else for that matter), the velocity between rocket and exhaust never changes, so the rocket mass is always 30,000.0000015 Kg (disregarding the loss of mass due to fuel used). In other words, the mass is fixed at 30,000.0000015 Kg for the values used above between rocket and xhaust, and the extra 0.0000015 Kg is an insignificant amount. As there is no significant mass increase with velocity, and certainly no accumulative mass increase, there is no theoretical upper limit to the velocity of the rocket.

It therefore follows that as the mass increase is virtually zero, m aproximates very closely to m0. If the acceleration is regulated to 1g for the comfort of the crew, the space ship can reach an enormous velocity, and time on this space ship will pass at exactly the same rate as back at home on earth. "The effects of gravity are indistinguishable from the effects of acceleration" [AE] (with the qualification in section 5).

I know that relativists would say that the mass increase has to be measured relative to the starting point of the rocket, but why is that? Einstein used the (essential) observer against which to measure the mass increase, with the tacit assumption that the starting point was where the propulsion unit was located, as in a particle accelerator. With that assumption, it is reasonable to refer the mass increase to the starting point. If we assume the propulsion unit (rocket motor) is remote from the rocket, then it is perfectly true that the rocket cannot exceed the speed of the rocket exhaust, as a particle in a particle accelerator cannot exceed c.

This is analogous to a space vehicle which uses light sails for propulsion. The sails are deployed in the vicinity of a star (the sun), and the light hitting the sails imparts a tiny acceleration away from the sun. This acceleration will propel the vehicle away from the sun, and the velocity will gradually but steadily increase. As the vehicle approaches light speed however, the energy from the light striking the sails gets less and less, and the acceleration gets less and less.

Quote from "Begin The Adventure" by Homer Tilton and Florentin Smarandache. Begin the Adventure

"A sailing vehicle which depends on light from the sun to accelerate it remains in that way connnected to the sun, its reference is the sun, and its speed is limited to less than the speed of light c, relative to the sun. Propulsive energy cannot reach a vehicle traveling away from the sun faster than that. It is limited to the speed of light for the same reason that a cablecar is limited to the speed of the cable pulling it."

2. DEFINITION OF SIMULTANEITY. Chapter IX.

Einstein uses lightning strikes at two places on the railway track, and says that although they can be said to be simultaneous as judged from the embankment at a vantage point exactly equidistant from them, when judged from the moving train, they are not. He uses this as an argument for there being a different

I have difficulty in accepting this argument for the following reasons. In judging whether the lightning strikes are simultaneous, he uses light itself as a medium for carrying the information to the observer, without making any correction for the known velocity of light. The two observers must be positioned as to be equidistant from the lightning strikes, but without prior knowledge of the timing (simultaneity) of them - how? Even allowing for the considerations of the able meteorologist (chapter VIII) who can so position them, any number of other observers on the embankment, who are positioned so as not to be equidistant from the two lightning strikes, see the same two lightning strikes, but do not observe them to be simultaneous, and indeed, observe the timing difference between them to be larger or smaller depending on where on the frame they are. This leads us to the conclusion that there can be an infinite number of time scales within one frame of reference - a conclusion which is not in accord with reality.

Also, we could, with equal validity, have visualised two workmen with hammers, and used sound to convey the information to the observer. The results then achieved would be markedly different from those using light, but nonetheless would be perfectly valid. Of course you point out that we should use the fastest medium that we can - which is light. Yes - use it by all means, but acknowledge the fact that it has a finite velocity and compensate for it. To get accurate results we should be using as a medium something which carries the information instantaneously - but we know of no such medium. If we postulate the existence of such a medium, and use it in a thought experiment, two occurrences judged to be simultaneous from the embankment (wherever the observer is positioned) will also be judged to be simultaneous from the moving train.

Chapter IX, paragraph 2 states "Are two events ... which are simultaneous with reference to the railway embankment also simultaneous relative to the train? We shall show directly that the answer must be in the negative."

"Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity)."

This statement is seriously in error. Just because you cannot tell whether two events are simultaneous does not mean that they are not simultaneous. Let us put our trust in the able meteorologist, and position ourselves on the embankment exactly equidistant (at point M) from the two lightning strokes A and B. An observer in the speeding train at position M* is exactly at position M on the embankment when the lightning strokes occur, but as he is speeding towards B, and away from A, he sees the flash from B before he sees the flash from A, and assumes them to be not simultaneous. If I now tell him the velocity of light, the velocity of his train, and the distance M to A (which is the same as M to B), he can easily work out the distance that he has travelled and the distance that the light has travelled. This will tell him that the lightning strokes were in fact simultaneous.

The inference from this discussion is that when the velocity of light is taken into account and compensated for, an occurrence judged to be simultaneous in one frame is also simultaneous in another. Einstein uses this definition of simultaneity to determine that one reference frame has a different time scale to another which is in (non accelerated) motion relative to it, but when the velocity of light is properly compensated for, there is no need for different time scales, and absolute time can be used throughout all reference frames, whatever their (non accelerated) motion. This however, comes into conflict with the assumption that the velocity of light is the same in all reference frames, which is discussed in section 8.

3. CLOCKS IN MOTION.

Most physics books use a light clock in their proof of the time dilation effect, one "tick" of the clock being the time it takes a pulse of light to travel from the source to a mirror and back. When the pulse arrives back at the source, the electronics ensures that another pulse is initiated, ad infinitum. The observer O also sees the pulses of light as they are initiated.

In A the clock is at rest relative to the observer (both are in reference frame F1), and the light pulse travels the dotted path to the mirror and back in T = 2L/c. In B to D, the clock is moving (now in reference frame F2) relative the observer who is still in F1, and he sees that the pulse has further to travel - hence the time dilation. When the clock is in motion relative to the observer in F1, the observer sees the clock running slow according to the Lorentz equation :-

equation 1.

As seen from F1 :-

T = time in F1 : T* = time in F2

v = the velocity of the body

c = the velocity of light

This time dilation depends on the velocity of light being the same for all observers. Bear in mind that this not observable in any real sense. It is inferred, and because of this inference it is also called a "perceived" velocity. Note that an inferred or perceived velocity is not an actual velocity, and Einstein himself used the word "judged" when refering to this time dilation. On page 87, we can read this :-

"As judged from K, the clock is moving with velocity v; as judged from this reference body, the time which elapses between two strokes of the clock is not one second, but 1 / sqrt( 1 - (v^2 / c^2)) seconds, ie a somewhat larger time. As a consequence of its motion the clock goes more slowly than when at rest. Here also the velocity c plays the part of an unattainable limiting velocity."

Note that very misleading penultimate sentence. The impression is given that the clock is actually running slow, not just judged to be running slow. The last sentence is equally misleading. The unattainable limiting velocity is just as judged from K, and not physically unattainable.

Remember that the above discussion depends on the velocity of light being the same for all observers. This has not been proved, only inferred. For a detailed discussion on light, go to Alternativephysics/light
This is an excellent web page by Bernard Burchell.

Let us put the clock far out in space, so there is no reference point as to its velocity, or equivalently, imagine the clock alone in the universe. Let us further imagine that it is in a perfect vacuum. In line with the discussion in section 8 of this paper about what constitutes a vacuum, I shall define a perfect vacuum as there being no atoms whatever in the path of the light beam, no matter what the velocity of the clock or its position in space. Now we have absolutely no way of knowing whether it is in motion or not. We are in F1, measure the clock’s rate, and find one tick to be T = 2L/c. The light pulse has travelled the path as in A. Now a force of 1G is applied for 4,252 hours, so the clock is accelerated, and then the force removed. The clock must now be at half the speed of light relative to its velocity during the first measurement of its rate, but of course there are no reference points, so there is no way of knowing this. The rate of the clock is measured, and found to be T = 2L/c. In a vacuum, in its own reference frame, the rate of the clock does not alter.

This paragraph agrees with Einstein and Alternativephysics, but the next paragraph is written bearing in mind what Alternativephysics has just shown us, and is in conflict with Einstein.

Now the same experiment is done in a medium, which could be air. We will neglect the fact that air resistance will stop us reaching high velocities. The observer in the clock’s reference frame notes that he is stationary with respect to the air, and checks the clock’s rate, which he finds to be T = 2L/(c/n). The air then starts to move (in other words the wind starts to blow, and it blows at half the speed of light), the observer again measures the rate of the clock (he is stationary with respect to the ground, but he is standing in a rather strong wind). He finds that the clock is running slow, as shown in Fig.2 B to D, according to the equation :-

Equation 2.

The index of refraction of the medium is n.

The conclusion is that in a medium, where that medium can flow unimpeded through the path of the light pulse, moving clocks can go slow, even within their own frame of reference. If the clock’s physical frame impedes the flow of the medium, and slows it down to a value which is less than its value away from the physical frame, the clock will not run as slow. If the clock is completely enclosed, e.g. in an enclosed vehicle which is travelling at half the speed of light, the enclosed medium is also travelling at that speed, and the clock’s rate is T = 2L/(c/n). Popular teaching states that all clocks run slow when in motion relative to the observer. What we have just discovered is that when in a medium, a clock does run slow at velocity, even in its own reference frame.

4. CLOCKS AND GRAVITY.

Let us expand on the discussion about clocks. Let us imagine a pendulum clock which is constructed to the maximum possible accuracy, so that placed alongside an atomic clock on the surface of the Earth, the two keep exactly the same time (this is a thought experiment remember, so such a clock is feasible, and its timekeeping will not be affected by such variations as temperature, pressure and humidity). Now the two clocks are placed on the surface of the Moon. The atomic clock speeds up while the pendulum clock slows down. Which clock is correct? The answer is that neither clock is correct, both are subject to errors caused by gravitation (or acceleration), but we are led to believe that the atomic clock is an absolutely accurate clock - which it is not. We are told that time itself speeds up as gravity (or acceleration) decreases, and slows
down as gravity (or acceleration) increases, but this is simply not so, it is the effect on the clock itself which is being observed, not an actual variation in time, as proven by the opposite effect on the pendulum clock. If my only reference was the pendulum clock, I would have to conclude that time speeds up as gravity (or acceleration) increases, and slows down as gravity (or acceleration) decreases, coming to a stop in zero gravity (or acceleration). If we wish to remain unbiased, this is a perfectly valid conclusion. Gravity affects both clocks, but in opposite senses, and of course in one far more than the other.

5. THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASS.

In chapter XIX, Einstein makes the following statement.

"Bodies which are moving under the sole influence of a gravitational field receive an acceleration, which does not in the least depend on the material or the physical state of the body. For instance, a piece of lead and a piece of wood fall in exactly the same manner in a gravitational field (in vacuo) when they start off from rest or with the same initial velocity."

When watching a piece of lead and a piece of wood fall, they appear to fall in exactly the same manner. They do not. The lead actually falls faster, but the difference in acceleration is so minute that it cannot be measured, and can be completely ignored under all but very extreme circumstances.

The Equivalence Principle (chapter XX) states "It is not possible by experiment to distinguish between an accelerating frame and an inertial frame in a suitably chosen gravitational potential, provided that the observations take place in a small region of space and time".

Einstein states (chapter XX) that all objects when dropped, will fall to the floor with equal acceleration, whether the chest is in a gravitational field or is being accelerated by an outside force. We are now in a position to show that this is not the case. Appearances can be deceptive. We will assume that we are standing on the surface of the Earth. If you picture a mass the equivalent of the Earth, but compressed to a size similar to that of the wood or lead under discussion (it is immaterial what this mass is, but it might be convenient to picture a miniature black hole), and hold it suspended by some means, when that mass is dropped, the observed acceleration will not be 9.8m/s^2, but 19.6m/s^2 . As the Earth’s gravity is the former value, and as is that of the miniature black hole, we can immediately see that the gravitational attraction is a result of the attraction of the two bodies’ gravitational fields. This applies whatever the mass of the bodies, and explains why the wood and the lead appear to behave the same - their mass is so tiny compared to that of the Earth, that for all practical purposes when dealing with the Earth, they are identical in mass.

Referring to Einstein’s book again, if the man in the chest is being accelerated at 1G by an outside force (the hypothetical being pulling on the rope, or a reaction motor etc) and drops a piece of lead or a miniature black hole, they will both fall with an acceleration of exactly 9.8m/s^2 - not a hair under or over. The objects are quite simply left behind as the chest accelerates away, and will hit the floor of the chest after 2 seconds if they are at a height of 20 meters to start with. [experiment 1].

Let us now assume that he is in the gravitational field of the Earth with the floor of the chest standing on the surface of the Earth. The objects are again at a height of 20 meters, so that at 9.8m/s^2 acceleration, they would take 2s to hit the floor. The piece of lead does indeed fall at that rate, but the miniature black hole falls at 19.6 m/s^2 (actually the miniature black hole and the Earth will each fall towards each other with an acceleration of 9.8m/s^2 each, giving a total acceleration of 19.6 m/s^2), and hits the floor after 1.4s . [experiment 2]

To summarise then, if the experiment is done in the accelerated chest, the objects will hit the floor after 2s. If the experiment is done on the Earth, the black hole will hit the floor after 1.4s, while the lead will hit the floor after 2s. He can immediately decide from this experiment whether he is in a gravitational field or is being accelerated by an outside force. If a black hole with a mass the same as that of the Earth falls faster than a piece of lead, then so does a mass of half the Earth, as does a mass of one hundredth, or a thousandth etc. In principle, if the man’s instruments are sensitive enough, he can detect whether he is in a gravitational field or being accelerated, whatever the mass of the objects which he drops.

When watching a piece of lead and a piece of wood fall on Earth, they appear to fall in exactly the same manner. They do not. The lead actually falls faster, but the difference in acceleration is so minute that it cannot easily be measured, and can be ignored for all practical purposes.

Is it possible that Einstein did not know this? When Johannes Kepler wrote his equations for planetary orbital motion in the early part of the 17th century, he used the masses of both the primary and secondary bodies, so he knew they had to be additive.

5a. Objections.

Some are rather specious, like saying that a miniature black hole would have gravity gradient effects. Yes it would. That in itself proves that gravity and acceleration are different. The objection I liked most however was that if I was correct, then a heavy satellite would orbit faster than a lighter one (in the same orbit) . Yes, absolutely correct. But, as above, the effect is far too small to be noticed. This got me to wondering just how large (massive) a satellite would have to be for this effect to be noticed, which in turn led to a rather unexpected conclusion.

Here is the scenario, and although a satellite has not been put into orbit at the stated distance, there is no reason why it cannot be, so in that respect, it is real. I am going to put a satellite into a specific orbit and calculate its orbital period and velocity. I will then calculate the orbital period and velocity of a heavier satellite in the same orbit.

The orbits are assumed to be circular.

The units used are Kilograms, meters, and seconds. The orbital distance is 384,900,000m from the centre of the earth. The formula to use to determine the satellite’s period (Ps) is :-

Ps = 2 * pi * sqrt( R^3 / G * ( Me + Ms ))

Where R = distance to satellite from the centre of the earth (or to be more precise from the centre of mass of the earth satellite system) ie orbit radius = 384,900,000m

G = the gravitational constant = 6.67e-11

Me = the mass of the earth = 5.97219e24 Kg

Mm = the mass of the moon = 7.34767e22 Kg

Ms = the mass of the satellite (for a man made satellite not normally taken into account, here it is assumed to be 1,000 Kg)

I used Fortran to create a flexible program to calculate orbital velocities from various orbits and masses. The results from using a calculator may not be an exact match but will be close enough. The program is available here for you to check and experiment with, but it treats the masses as point sources, so will not be accurate with a low radius orbit around a large mass :-

http://myweb.tiscali.co.uk/carmam/sat11.exe

The source code is here :-

http://myweb.tiscali.co.uk/carmam/sat11.f95

Ps = 6.2831853 * sqrt( 384,900,000^3 / 6.67e-11 * (5.97219e24 + 1,000))

= 27.51428811520548 days.

For a satellite of 1,000,000,000 Kg the period (in seconds) is the same to 9 decimal places. It can now be seen why the mass of a man made satellite is not normally taken into account when calculating orbital velocity, as increasing the mass a millionfold will result in an orbital period difference of 2e-10 seconds in 27.5 days.

The circumference of the orbit is :-

C = 2 pi R = 6.2831853 * 384,900,000 = 2,418,398,092.031479 m

Therefore the velocity of the satellite is :-

Vs = C / Ps = 2,419,371,395.58 / 2,378,191.224559946

= 1,017.31573486328125 m/s

The orbital radius used above is the radius of the moon's orbit so now its period is calculated :-

Pm = period of orbit of the moon.

Mm = mass of the moon = 7.347673e22 Kg

Pm = 2 * pi sqrt( R^3 / G * ( Me + Mm ))

= 6.2831853 * sqrt(384,900,000^3 / 6.67e-11 * (5.97219e24 + 7.34767e22))

= 2,362,744.336301510490677977736859628 seconds

= 27.346577966452667 days

The circumference of the moon's orbit is the same as the satellite’s (but not concentric with it) C = 2,418,398,092.031479 m

The velocity is :-

V = C / Pm = 2,418,398,092.031479 / 2,362,7744.36301510490677977736859628

= 1023.5547 m/s

The moon is faster than the man made satellite by 6.23895263671875 m/s, and if the satellite were launched to be on the opposite side of the earth from the moon when it went into orbit, the moon would gradually catch up with it until they collided. This would take about 6 years.

Using the programme, put the Earth into the same orbit as Jupiter, you will see that they collide in about 12,000 years

Here is the unexpected conclusion which has emerged: No trinary star systems will be found in the universe. I define a trinary system as 1) a system in which the central more massive body has two other bodies in orbit around it in the same plane and which are nearly equal in orbit radius, or 2) where there are three bodies orbiting around their common centre of mass.

If a star system such as 1) formed in the first place, the two stars which were similar in mass but less massive than the primary would collide to form a binary system: or if 2) if the triangle formed by the three stars was equilateral (possible but not probable), due to the differing velocities this triangle would shift to be non equilateral (this would seem to be a more probable starting point, and is similar to system 1), and then the two closest stars would approach and collide. As they did so, a binary system would form. A trinary system can only exist for a very short time relative to the age of the universe, and could only be found in very young star systems.

The calculations above show that a trinary star system is not stable. As can be seen from the above, because satellites of differing masses in the same orbit move at different speeds, there will not be any trinary systems in the universe, except perhaps in very young star systems, which will not last long before they collapse.

5b. An alternative thought experiment.

This thought experiment stresses the distinction between the force of gravity and other forces with which we are familiar, such as the force of an engine pulling and accelerating a train, or the powder in a gun which accelerates a bullet to produce its muzzle velocity.

To distinguish the force of gravity from such other forces consider an idealized experiment in which a train is moving along an embankment on a planet on which the force of gravity is negligible. In one case we let an engine accelerate the train. In a second case we imagine a large body ahead of the train which attracts the train due to its gravitational pull. We can also imagine this second case as a train falling, or racing, to earth.

If the train were in uniform motion then it would be valid to compare a walk forward on the train with a laser firing a pulse of light, or a gun shooting a bullet from the rear of the train in the direction of the train's motion. The velocity of the walker, the bullet or the photon remains constant relative to the velocity of the train.

But if the train is accelerating because of the engine pulling it this is no longer true. In that case the walker, at each step, is in touch with the instantaneous velocity of the train, so that his walk can remain essentially constant with respect to the instantaneous velocity of the accelerating train. But the bullet or the laser beam do not remain in contact with the train so their velocity will decrease relative to that of the accelerating train as time passes.

On the other hand, if the train were falling towards earth, or pulled forward by a large gravitational mass, the acceleration would be due to gravity and the bullet fired from the gun (and possibly the laser light) would also be subject to the continuing force of gravity so the velocity relative to that of the train would be constant as is the case for the walker. This differentiates the case of gravitational acceleration from the force producing acceleration which acts only on the train.

This thought experiment was devised by Hans Zweig, and is in his book, which can be found at aquestionoftime.com

6. THE CONSTANCY OF THE VELOCITY OF LIGHT

Because of the principle of relativity, light ought to have the same velocity, no matter which frame of reference we are in. Various measurements have been made by eminent physicists who have come to the conclusion that the velocity of light is the same in all frames of reference.

This statement appears to hold true, as the velocity of light has been measured in various frames of reference which are in uniform translation with respect to each other. To put this succinctly, if the velocity of light is measured from a certain star which is at rest relative to us, it is found to be 300,000 Kilometers per second (Km/s). If the same experiment is done with a star which is approaching us at 1,000 Km/s, it might be thought that the velocity of light would be measured at 301,000 Km/s. This is not the case, the velocity of light is still measured at 300,000 Km/s. This difficulty led Einstein to his theory of time dilation. Chapters VIII & IX.

There are however, two snags. The first is that whatever the speed of approach or recession, when the star’s light reaches our atmosphere, it slows down (or speeds up), and assumes the velocity for the atmosphere’s index of refraction. The velocity of light in our atmosphere is not c, but a smaller value which is c/n, where n is the index of refraction of the "standard" atmosphere. c/n = c/1.00029 = 299,900,000m/s approximately. The second is the rather arbitrary use of the word vacuum. Einstein made great importance of being precise in his terminology, so there could be no mis-understanding, and yet a vacuum is not defined. The very terms solid, liquid, and gas are themselves rather arbitrary, and depend on temperature and pressure, so what exactly is a vacuum? Is it one atom per cubic millimeter on average? Is it one atom per cubic centimeter? Is it one atom per cubic meter? Space is not a vacuum, it is full of dust and other particles, and this makes it a very rarefied gas with its own index of refraction. In fact, in interstellar space, the density is an average of 1 atom per 10 centimeters, while in the vicinity of the Sun, the density is much higher at about 1 atom per centimeter. Light therefore travels through space as it travels through any other medium, at a speed of c/n relative to that medium. More on light.

The Michelson-Morley experiment is often quoted as proving the constancy of the velocity of light, but it was set up to look for (or more correctly, to prove, an aether drift). The light source and the observer were stationary with respect to each other, and the experiment was done on the surface of the Earth, so how could this experiment prove whether c is constant with respect to the source or the observer or both or neither? The way the Michelson-Morley experiment was set up is akin to trying to find the windspeed by setting up an anemometer in a closed room. The anemometer has to be outdoors, and the further away from any obstacles, the more accurate the reading will be. I find it strange that a direct experiment to prove the constancy of the velocity of light has never been done. It has only been inferred and never proven directly.

There is an "aether", but not one in the classical sense, and that is why the Michelson-Morley experiment could not detect it. That experiment was done on Earth where the speed of light is c/n relative to the earth, whatever its direction or the direction of the earth. Let us move the Michelson-Morley experiment into space, and in fact well away from the Earth. We will put it on an interplanetary probe, and on an extension arm some distance from the probe body. Do the experiment well away from any planet or other large body, aligned with one arm on a line pointing to the Sun, and the other arm will then be at right angles to the Sun. One arm is now at right angles to the aether drift and one arm is aligned with it. The result of the experiment will be an aether drift towards the Sun. What has been measured here is the dark matter (DM), and this re-enforces what we already know, that space is not entirely empty, there are particles moving about through all of space. In the region of stars, this DM moves towards them, whilst in a region devoid of stars, it could assume any direction.

The speed of light then is with respect to this "aether", and it has an index of refraction just the same as any other medium, but this medium is rather sparse and very fluid, and if taken over a large enough distance, the currents will cancel themselves out, leaving the speed of light as c/n relative to the "aether". c/n is of course 300,000,000m/s or what is commonly known as c.

7. THE TWIN PARADOX. This is an update of the travelling versus the stay at home twin.

In the first preliminary step we take Earth and relocate it far into intergalactic space. It will be far enough out such that gravity from the nearest galaxy is a trillion times less than Earth’s surface gravity. The reason for doing this is firstly so that we don’t need to consider the gravity of surrounding stellar bodies, and secondly to remove the motion of the Earth around the Sun and Milky Way from consideration. Next we prevent the Earth from rotating. Likewise we do this to avoid having to consider the SR/GR effects of the rotation speed and the small amount of centrifugal force it provides.

Now to begin the story.

A rocket sits on the Earth’s surface with a large supply of fuel. Inside it is a room with living facilities and enough food and oxygen to support an occupant for many months. It also contains an accurate atomic clock. Beside the launch pad is an identically-fitted room. It contains a similar clock that has been synchronised with the one aboard the rocket. There is also a third clock on the opposite side of the Earth that is synchronised with the other two.

Two identical twins agree to take part in the experiment. Each will spend the next several months either in the rocket or the Earth room, but neither will know which. Prior to launch, they are both given a sedative to put them to sleep. Each twin is then randomly assigned to be moved into either the rocket or the stationary room.

The rocket lifts off. At first, very slowly so as not to apply much acceleration. Then as it moves further from Earth and gravity decreases, the rocket adjusts its acceleration to ‘fill in’ what is missing from Earth’s gravity. This acceleration will be controlled so that the gravity felt at all times will be exactly equal to 1G. That is, the gravity measured by an on-board accelerometer (as the sum of real plus artificial gravity) will measure the same as on Earth. Assume that the rocket engine is silent and acceleration is smooth.

Shortly after launch, when the acceleration is a steady 1G, the twins wake up. Neither of them know which room they are in. The rooms are identical in layout and both experience what appears to be gravity. If they drop something it will accelerate toward the floor at 9.81m/s^{2}, i.e. at 1G.

Now according to the Principle of Equivalence (also called the strong equivalence principle), as proposed by Einstein and frequently described by using falling elevators and rising rockets, the situation inside the two rooms is essentially identical. That is, there is no experiment you could devise that would allow either of the twins to determine which room they are in. We will also assume the rooms are not very tall. This is to prevent an occupant in the Earth-room from measuring slightly less gravity near the ceiling.

/b> According to the combined rules of SR and GR, will one of the clocks be ahead of the other, and if so, what is the reason for selecting that clock instead of the other?

The fact that the clocks are moving away from each other means there must be a velocity present, otherwise they would remain a fixed distance apart.
Therefore according to SR, time dilation should be occurring and the faster-moving clock should be running more slowly. But since the relative speed between the Earth and rocket is at all times exactly equal from both viewpoints, there appears to be no way of determining which is ‘faster’. As for GR, since the acceleration-slash-gravity situation of both rooms is exactly equal at all times (other than the brief lift-off, when it was marginally more than 1G), according to the Equivalence Principle it would appear we are also unable to favour one clock over the other.

So we are left with a conundrum: either we find a way of favouring one clock over the other or we agree that no time difference accumulates between them.

Now an objection might be that we have no way of comparing the clocks without one of them stopping and reversing, which would destroy the symmetry of the situation. And so the question of which of them runs faster up until that point is somehow hypothetical or meaningless. But this avoids the issue because the question here is about which clock *according to the theory of SR and GR*, runs slower. Unlike the ‘Copenhagen Interpretation’ of quantum mechanics, relativity does not depend on observers to determine the reality of a situation. So the answer to this question won’t depend on
the clocks ever being compared or not.

Still, this objection can be overcome and will be addressed in the remainder of this essay

After travelling for 10 months, and using a simple classical mechanics calculation, we could determine that the rocket is moving at 87% the speed of light (relative to Earth, which is now relocated outside our galaxy). At this speed we get a Lorentz factor of 2. This might mean that either the rocket or Earth clock is running half the speed of the other. These numbers however are not so important because we mainly care about which clock is ahead of the other, and not by how much (although we are also interested in that!). So let’s just pick 10 months as an arbitrary duration and assume a rough Lorentz factor of 2 at that point. This factor will be sufficient to override minor clock-drift errors, measurement errors, and brief periods where the acceleration of the rocket is not 1G, such as the launch and rotation (as described later). It should also cause noticeable differences in what the twins remember about the duration of their journey, assuming that one is ‘running’ half the speed of the other.

So after travelling for 10 months (according to the local clock) the occupant aboard the rocket will take a sedative. The same will occur at the Earth-room (according to their clock). Both twins will then sleep for a while. The rocket engine will then be stopped, allowing the craft to drift freely in space with no acceleration. It will then be gently rotated 180 degrees to face the opposite direction – now pointing at Earth. The engine will then be started again, applying an acceleration force of exactly 1G. Both twins will then wake up.

When the rocket-twin awakes, he notices no difference. Just as before, he experiences what feels like a gravitational force of 1G toward the floor.
The Earth-twin experiences the same. The rocket is facing the opposite direction and is now decelerating, but by all accounts everything according to the Equivalence Principle is the same. There is still no experiment either twin could perform to determine which is experiencing gravity.

Therefore it would seem that according to GR, both clocks should still be running at the same rate. And since the relative velocity is still identical – that aspect never changes – the clocks’ situation is still symmetrical according to SR.

An objection here might be that there is a difference because the clocks are experiencing ‘gravity’ in opposite directions, therefore the clock on the rocket will now be faster or slower (pick one!) than the one on the Earth. For those who raise this objection, refer instead to that third clock placed on the other side of the Earth. It is still in-synch with the first Earth clock and now experiencing gravity in exactly the same direction as the rocket.

To continue the story, the deceleration process continues for the same time as the original acceleration process (10 months), at which point the rocket comes to rest relative to Earth. However the engine doesn’t stop. Instead it continues to apply exactly the same amount of force. Deceleration becomes acceleration and the occupant notices nothing unusual.

The acceleration continues for the next 10 months (according to the local clock) until the rocket reaches (presumably) the original rotation point. At this point (according to their own clocks), both twins are put to sleep, the rocket is rotated 180 degrees, and then starts to decelerate while pointed away from Earth. Both twins awake and notice nothing unusual in their ‘gravity’ situation.

The rocket continues its deceleration in a perfect reverse of its original departure, steadily coming to a stop relative to Earth, and all the while carefully adjusting its acceleration to give an on-board experience of
1G. Just prior to landing, both twins are put to sleep and then woken up after landing.

The rocket has now landed beside the replica Earth room. Neither twin has yet to emerge, and neither still has any idea which one of them was aboard the rocket.

Not that it matters. This story was never about the twins, it was about the atomic clocks. The twins were just there to make it interesting and to bring it in line with the historical thought experiments such as falling elevators and the ‘Twins Paradox’.**
So to state the obvious question:** allowing for minor clock-drift errors and the brief periods of launch, landing, and rotation, when the clocks are compared side-by-side, which of them will have recorded more time? And why not the other way around?

And while we’re at it, which of the twins will be older?

This discussion of the twins' paradox can also be found on Bernard Burchell's web site :-

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