A Roller Coaster Ride through Relativity/Mass inflation

Mass Inflation

If two perfectly elastic bodies of equal mass m travel towards each other with equal and opposite speeds u, they will bounce off each other with identical but reversed velocity. This is a straightforward consequence of the laws of conservation of energy and momentum. But what if one of the bodies is also travelling across the line of the collision at a speed of v? The collision will look something like this:

Colliding balls

In Newtonian dynamics the sideways motion of one body is irrelevant and does not change the behaviour of the two balls in the perpendicular direction. Nor should it in Special Relativity either, even if v is a sizeable fraction of the speed of light. The problem comes when we consider how the body is given its sideways velocity u (assumed to be much smaller than v).

You see, from B's point of view, the mechanism which propels ball A sideways is running slow because of time dilation and u_A appears to be less than u_B by a factor of $\gamma$. Naturally from A's point of view it is u_B which appears to be less than u_A. All other factors being equal, A and B will therefore predict different outcomes. In fact, since the whole situation must be symmetrical with respect to A and B, we know that what actually happens is that the balls bounce off each other with unchanged speeds.

So what is the solution? Clearly, while the speed of the two balls appear to be different, the momentum of the balls must be the same. Now, momentum is mass × speed and if the speed is apparently reduced by a factor of $\gamma$ because of time dilation, the mass must be increased by a factor of $\gamma$ to compensate. This phenomenon is called mass inflation and the appropriate formula is:

$M=\gamma M_{0}={M_{0} \over {\sqrt {1-v^{2}/c^{2}}}}$

where M₀ is the rest mass of the object - ie its mass when measured at rest. Likewise the momentum p of an object moving with a relativistic speed v is equal to:

$p=mv=\gamma M_{0}v$

This gives us another explanation as to why the speed of the 1g rocket never gets greater than the speed of light. From the point of view of an observer on Earth, the faster the rocket goes, the more massive it gets. Since we assume that the thrust of the rocket is constant, the effective acceleration (as seen from a stationary observer outside the craft) must decrease more and more. In fact this is exactly what we observe when we accelerate electrons in a linear accelerator over millions of volts. When they get up to 99% of the speed of light, the extra volts don't make them go much faster but they do impart extra momentum and of course, energy to the electrons.

Ok, so I accept that mass increases and that the momentum of a particle of rest mass M₀ travelling at a speed v is $\gamma M_{0}v$. What about kinetic energy? I suppose that is ${\tfrac {1}{2}}\gamma M_{0}v^{2}$. Is that right?'

Well that is an excellent guess. At low speeds, $\gamma$ = 1 so the formula reduces to ${\tfrac {1}{2}}\gamma M_{0}v^{2}$ which is correct and at high speeds $\gamma$ tends towards infinity so the kinetic energy also becomes infinite which is correct again. Nevertheless, I am afraid it still isn't quite right. The correct answer is:

$KE_{r}=M_{0}c^{2}(\gamma -1)$

where KE_r is the relativistic kinetic energy. (for the proof of this equation see Appendix F)

When the two functions are plotted on top of one another you can see just how little difference there is:

Relativistic kinetic energy

The importance of the distinction lies not in the slight differences in numerical values that are generated but in the whole new perspective that the equation throws on the nature of mass and energy.

Let's rewrite the equation in the following way:

$KE_{r}=M_{0}c^{2}(\gamma -1)$
$KE_{r}=\gamma M_{0}c^{2}-M_{0}c^{2}$
$KE_{r}=Mc^{2}-M_{0}c^{2}$
$Mc^{2}=M_{0}c^{2}+KE_{r}$

Rewriting an equation in a different way doesn't prove anything but it can suggest ideas. Lets give the terms some meaningful names.

Mc² is a kind of energy term formed from the total relativistic mass M. It seems sensible therefore to call this term the total relativistic energy of the moving body E.

Similarly, since M₀ is the rest-mass of the body, M₀c² should rightly be called the rest-mass energy of the body E₀.

This permits us to say the following:

Total relativistic energy = rest-mass energy + relativistic kinetic energy

or in symbols:

$E=E_{0}+KE_{r}=\gamma M_{0}c^{2}$

It was a stroke of genius on Einstein's part to see that the two terms E = Mc² and E₀ = M₀c² were not just mathematical junk, they actually had physical meaning. An object at rest really does have an incredible amount of energy locked up inside it. Special Relativity does not prove that a mass at rest has energy since there is no process dealt with by the theory which could possibly release this energy but Einstein speculated on the possibility that one day such a process might be discovered, a speculation which became all too true in his own life time.

So now we have arrived at:

Profound consequence number 13
All massive objects contain energy according to the famous relation $E=Mc^{2}$

The significance of this equation cannot be exaggerated. It ranks alongside the discovery of the law of gravity, the idea that zero is a number, the invention of language and the discovery of fire as a turning point in human history - and as an icon for the achievements of the twentieth century, it cannot be surpassed.

It hardly needs saying that since the speed of light is quite large, the rest-mass energy of a kilogram of matter is very large indeed. In fact it is equal to 90,000,000,000,000,000 J. The average human being converts energy at a rate of about 100 W so this quantity of energy would keep you alive for about 28 million years! More realistically, it is equal to the total energy output of a large modern power station in 2 years of continuous operation.

On the other hand, the Sun (which outputs a prodigious 4 x 10²⁶ W) eats up 4 million tonnes of matter every second!

Mind you, that is only a titbit compared to the most energetic objects in the known universe, the quasars, which probably emit something like 10³⁹ W and eat up the mass of the Moon every second!

Incredible! So everything that has mass has energy does it?

Absolutely right. But that is only really half the story. Einstein's equation works the other way round as well and everything which has energy, also has mass. What I am saying is that a new battery is more massive than an old flat battery, a wound up watch is heavier than a run-down watch, a hot cup of tea weighs more than a cold cup of tea. And things which lose energy get lighter: the total mass of the products of an exothermic chemical reaction is less (after the heat has escaped) than the total mass of the reagents, two magnets stuck together weigh less than the two magnets separately, and most important of all, the mass of an atom is measurably less than the sum of the masses of all its constituent particles.

Even photons, which since they travel at the speed of light have zero rest mass, must have (relativistic) mass by virtue of the energy they contain. Not only that, they have momentum as well, but before we work that out, we have one more important consequence to state which is absolutely true for all objects under all circumstances. It relates the relativistic energy E of a body to its momentum p and it looks like this:

Important consequence number 14
The total relativistic energy E of a body and its relativistic momentum p are related by the equation: $E^{2}-p^{2}c^{2}=M_{0}^{2}c^{4}$

(you will find the simple proof in Appendix G)

As we mentioned earlier, this relation has special relevance for the photon (whose rest-mass is zero and for which the usual equations for energy and momentum are undefined). Putting M₀ = 0 we get

Important consequence number 15
The total energy E of a photon is related to its momentum by the equation: $E=pc$

What do you mean when you say that a photon has momentum? Surely something which doesn't have mass can't have momentum, can it?

Well, yes, it can. If you like you can think of it as the momentum of the mass of the energy which the photon carries. To be sure a photon does not have a lot of momentum. If you shine a 1 W laser beam (or an ordinary torch) at a black surface, every second the surface absorbs E/c units of momentum ie 3.3 x 10⁻⁹ Ns. Since rate of change of momentum equals force, the laser beam exerts a force of 3.3 nN (nano-Newtons) on the surface. That is a trillionth of the weight of a can of baked beans. Not a lot!

On the other hand, the energy density of solar radiation here on Earth is about 3,000 W m⁻² so the force on a solar wind satellite whose sail has dimensions 30 m × 30 m would be 9 mN (9 x 10⁻³ N). This doesn't sound much either but the force of gravity from the Sun on a 1 kg mass at the same distance from the Sun is only 6 mN (6 x 10⁻³ N).

What this means is that, if you could make a reflecting 'sail' large enough (and light enough) you could balance the force of gravity against the force of the solar wind and 'sail' around the solar system for free!

An even more important consequence of the fact that photons have momentum is the fact that the Sun itself is supported by the enormous pressure of all those photons fighting to get out. The instant the Sun runs out of nuclear fuel in its core, the photon pressure will disappear and the Sun's interior will collapse causing a massive explosion called a supernova which will engulf the Earth and obliterate all traces of life on it.

That doesn't sound too good. How long have we got?

Don't worry. It won't happen for another 4 billion years yet!

The second roller coaster

At last, the roller coaster jerks to a halt in the station, and you stagger off the train, your mind reeling with important principles and bizarre consequences.

Well, I survived! you exclaim but oh how my head is spinning!

I am afraid we have not quite finished yet though. Gently, I lead you by the arm round a corner where we stop in front of a huge gateway over which is blazoned in scary letters the single word OBLIVION.

What's this? you ask.

It is another roller coaster.

Hell's teeth! Is it as bad as the other one?

Well, not so bad really - but I have to admit it is a bit of a cheat. There is another roller coaster which is such a rough ride that hardly anyone can stomach it but this one has been specially designed to give you just a flavour of what the real one is like. Are you ready to try it?

I suppose so. But why is it called OBLIVION?

Wait and see! But first, while we wait in the queue, I must tell you about Einstein's second Big Idea. The Fundamental Principle of Special Relativity (Einstein's first Big Idea) dealt with observers in uniform motion. The second Fundamental Principle (on which the General Theory of Relativity is based) deals with observers in accelerated motion. It is this.

The Fundamental Principle of General Relativity
The laws of physics in a uniformly accelerated frame of reference are identical to the laws of physics in a gravitational field.

or, to put it another way:

It is impossible to carry out any experiment inside a closed laboratory which will determine whether the laboratory is being accelerated uniformly or whether it is situated in a uniform gravitational field.

Well that's silly. Surely you can always tell if there is gravity. All you have got to do is drop something. If it falls - there is gravity!

Not at all. Suppose you are in deep space on a journey to a distant star perhaps. You may be moving, you may be stationary, it doesn't matter. All that matters is that there are no stars or planets nearby so there is no gravity. You have just finished reading a chapter of your favourite book – 'The Hitch-hikers Guide To The Galaxy' – and, as is your custom, you simply park it in front of you where, because of the weightless conditions inside the spaceship, it hovers obediently. Suddenly, it falls to the back of the ship, accelerating as it goes. At the same time you feel your own body pressed into the couch on which you have been lying. What are you to think? Have you suddenly arrived at your destination where gravity is normal again? Possible perhaps. But it is a lot more likely that the captain of the ship has fired the rocket engines and that you and the rocket are simply accelerating. The book did not fall because of gravity. In fact it didn't really fall at all. It simply stayed where it was and the rocket accelerated forwards and caught up with it. Likewise, you weren't pressed into the couch by the force of gravity, no, it was simply the thrust of the rocket engines, transferred to you through the couch, which made you accelerate forwards with the rocket.

So the effects of acceleration look, on the face of it, to be just like the effects of gravity. But are they? Is there any way to tell the difference? To answer this question, we must have a closer look at exactly what we mean by mass.

Has it ever struck you as rather odd that two objects of different mass should fall with the same acceleration in a gravitational field? If not, then you must be either very clever or you haven't thought about it at all. After all, the great Greek scientist and philosopher Aristotle thought that it was self evidently obvious that heavy objects would fall faster than light ones and he carried the weight of intellectual opinion with him for over two thousand years. It was Galileo who first saw the illogicality in Aristotle's theory. His argument went something like this.

"According to Aristotle, a heavy stone should fall twice as fast as a light stone. Now, if you tie a heavy stone to a light stone, the heavy stone will pull the light one down faster while the light one will tend to stop the heavy one from travelling so fast. The combination will, therefore, fall at a speed which is intermediate between the speeds of the two stones on their own. But hang on a minute – if you tie a light stone to a heavy one, surely you make a stone which is heavier than either and therefore, according to the theory, should fall faster than either? There is an inconsistency here, therefore the original premise is false."

Fifty years later, Isaac Newton vindicated Galileo by setting out a wonderful theory of mechanics based on two central ideas - the idea of a force and the idea of mass. which are brought together in the central relation

$F=ma$

In principle, you can measure the mass of an object by applying a standard force to it (eg using a 'STANDARD' firework) and measuring its acceleration.

A mass strapped to a firework

The greater the mass, the greater its inertia will be and hence the smaller the acceleration. Mass measured in this way should properly be called inertial mass.

But Newton did not stop at explaining all the laws of dynamics. Like Einstein after him, he also went on to explain the laws of gravity using the same fundamental concepts of force and mass. His second Big Idea was this: between two (small) masses M and m separated by a distance r there exists an attractive force F which is proportional to the product of the masses and inversely proportional to their separation. In symbols:

$F=G{Mm \over r^{2}}$

where G is a constant equal to 6.67 x 10⁻¹1 kg⁻¹m³s⁻². This equation gives us an alternative and quite independent method of measuring mass. All we have to do is measure the force of gravity on the mass (eg by hanging it from a standard spring) in a standard gravitational field (eg the Earth's field.)

A mass on a spring

Mass measured in this way should properly be called gravitational mass.

It stands to reason (though it is neither obvious nor necessarily true) that, if you strap two identical objects together, you will double both the inertial mass and the gravitational mass – and it is this assumption that is at the heart of Galileo's argument. The heavy stone has twice the force of gravity on it - but then it needs twice the force in order to accelerate the same amount!

While it may stand to reason that two identical objects with the same inertial mass will have the same gravitational mass, there is absolutely nothing in Newton's theory which prevents two objects with the same inertial mass, but made of different materials, from having different gravitational masses. It is conceivable, for example, that dense materials like lead or gold would weigh more (or less) than their inertial mass would suggest.

That's amazing! May be there are substances which we don't know about that have inertial mass but don't weigh anything at all!

As far as Newton's theory is concerned, that is certainly possible but all the evidence we have to date strongly suggests that, whatever substance an object is made of, inertial and gravitational mass are exactly the same. Even if there were tiny differences, we would notice it immediately. It would, for example, mean that the giant gaseous planets would orbit the Sun at a different rate from the smaller rocky ones (over and above the normal differences that is); it would mean that an aluminium satellite would have a different orbital period from the titanium shuttle which launches it; it would mean that the Earth would have a different orbit round the Sun than its own oceans which would, consequently, appear to fly off into space of their own accord! It is fairly evident that none of these things happen.

There is, therefore, plenty of evidence to show that the ratio of inertial mass to gravitational mass is the same for all materials, everywhere. But why? According to Newton's theory, there is no reason why, they just are. But according to Einstein, there is a very good reason: it is simply the Fundamental Principle of General Relativity - The laws of Physics in a uniformly accelerated frame of reference are identical to the laws of physics in a uniform gravitational field.

So just as the Fundamental Principle of Special Relativity explains the observed constancy of the speed of light, so the Fundamental Principle of General Relativity explains the observation that all objects fall with the same acceleration in a gravitational field. We can summarise this conclusion as follows:

Reassuringly normal consequence number 16
Inertial mass and gravitational mass are one and the same thing.

But you haven't paid good money to the roller coaster owners just to discover Reassuringly Normal Consequences so, now that we have reached the summit of the first hill and are ready to go, here is the next Bizarre one:

Bizarre consequence number 17
Gravity bends light.

That's ridiculous. Light can't bend. And I can tell you why too! When we say 'light travels in straight lines' we aren't stating an experimental fact; the way light travels defines what we mean by a straight line!

Mmm... yes... you've got a good point there...

But before I have a chance to think up a suitable reply the catches on the roller coaster are suddenly released and we are plunging in free fall down into a gaping black hole.

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