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Posts Tagged ‘Relativity’

At number 10 in “The Guardian’s” 10 best physicists is English theoretical physicist Paul Dirac.

 

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Dirac’s brief biography

Dirac was born in Bristol in the south-west of England in 1902. He died in 1984. He was brought up in Bristol. His father was Swiss-French, his mother was English. He did his undergraduate degree at Bristol University studying engineering. However, he was unable to find work as an engineer, and so instead undertook a second degree, this time in mathematics, at the same institute. He then went to Cambridge to do his PhD, working on General Relativity and Quantum Mechanics, under the supervision of Ralph Fowler. The title of his PhD thesis was simply “Quantum Mechanics”.

 

The front cover of Paul Dirac's PhD Thesis, submitted in 1927 to Saint John's College, Cambridge.

The front cover of Paul Dirac’s PhD Thesis, submitted in 1927 to Saint John’s College, Cambridge.

 

Dirac’s main achievements

Dirac’s place in this top 10 list is due to two main things, his prediction of the existence of antimatter, and for the equation which describes the motion of a fundamental particle such as an electron when it is travelling near the speed of light. Both of these will be described in more detail in future blogs. Dirac won the Nobel prize for Physics in 1933, he shared it with Erwin Shrödinger “for the discovery of new productive forms of atomic theory”.

Antimatter

The theoretical prediction for which Dirac is most famous to people outside of physics is his idea of antimatter, which of course has become a firm favourite of science fiction. His basic idea was that every fundamental particle has an anti-particle. So, for example, an electron has an anti-particle which would have the same mass and the opposite electric charge. We call this anti-electron a positron. A proton would have an anti-proton and so on. Anti-matter was predicted by Dirac in 1928 and was experimentally verified in 1932 with the discovery of the positron.

The Dirac equation

Dirac is most famous amongst physicists for what is now known as “Dirac’s equation”. This is an equation which describes the relativistic behaviour of an electron, and therefore unified quantum mechanics with special relativity. Relativistic means travelling near the speed of light.

 

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The terms in this equation need a little explaining. Rather than explaining them in this blog, I will do so in a series of future blogs, as I will need to give some background. Not only do I need to explain the terms in this equation, but this equation cannot be understood in isolation, one has to also understand Schrödinger’s equation.

For example, the term \psi(x,t) is the so-called “wave-function” of the particle, and \nabla^{2} is the so-called Laplacian. i \text{ is the imaginary number, that is } \sqrt{-1}. Now you see why I need to give some background!!

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You can read more about Paul Dirac and the other physicists in this “10 best” list in our book 10 Physicists Who Transformed Our Understanding of the Universe. Click here for more details and to read some reviews.

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Ten Physicists Who Transformed Our Understanding of Reality is available now. Follow this link to order

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Quite a few months ago now I derived the so-called Galilean transformations, which allow us to relate one frame of reference to another in the case of Galilean Relativity.

\boxed {\begin{array}{lcl} x^{\prime} & = & x + vt \\ y^{\prime} & = & y \\ z^{\prime} & = & z \\ t^{\prime} & = & t \end{array} }

It had been shown that for experiments involving mechanics, the Galilean transformations seemed to be valid. To put it another way, mechanical experiments were invariant under a Galiean transformation. However, with the development of electromagnetism in the 19th Century, it was thought that maybe results in electrodynamics would not be invariant under the Galilean transformation.

The electrostatic force between two charges

If we have two charges which are stationary, they experience a force between them which is given by Coulomb’s law.

\vec{F}_{C} = \frac{ Q^{2} }{ 4\pi\epsilon_{0}\vec{r}^{2} } where Q is the charge of each charge, r is the distance between their centres, and \epsilon_{0} is the permittivity of free space, which determines the strength of the force between two charges which have a charge of 1 Coulomb and are separated by 1 metre.


Coulomb's law gives us the force between two charges. If the charges are the same sign the force is repulsive, if the charges are opposite in sign the force is attractive.

Coulomb’s law gives us the force between two charges. If the charges are the same sign the force is repulsive, if the charges are opposite in sign the force is attractive.


Moving charges produce a magnetic field

If charges are moving we have an electric current. An electric current produces a magnetic field. The strength of this field is given by Ampère’s law

\oint \vec{B} \cdot d\vec{\l} = \mu_{0}I where d\vec{l} is the length of the wire, \vec{B} is the magnetic field, \mu_{0} is the permeability of free space and I is the current. So, if the two charges are moving, each will be surrounded by its own magnetic field.


A wire carrying a current produces a magnetic field as given by Ampère's law.

A wire carrying a current produces a magnetic field as given by Ampère’s law.


The Lorentz force

If the two charges are moving and hence producing magnetic fields around each of them then there will be an additional force between the two charges due to the magnetic field each is producing. This force is called the Lorentz force and is given by the equation

\vec{F}_{L} = Q\vec{v}\times\vec{B}. If r is the distance between the two wires, and they are carrying currents I_{1} and I_{2} respectively, and are separated by a distance r, we can write B=\frac{\mu_{0}I}{2\pi r} which then gives us that the Lorentz force F_{L} = \frac{ I_{1} \Delta L \mu_{0} I_{2} }{2 \pi r } and so the Lorentz force per unit length due to the magnetic field in the other wire that each wire feels is given by \boxed{ \frac{ F_{L} }{\Delta L} = \frac{ \mu_{0} I_{1} I_{2} }{ 2 \pi r} }. Writing the currents in terms of the rate of motion of the charges, we can write this as

F_{L} = \frac{ \mu_{0} Q_{1} Q_{2} }{ 4\pi r^{2} } v^{2}


The Lorentz force is the force on a wire due to the magnetic field produced in the other wire from the current flowing in it.

The Lorentz force is the force on a wire due to the magnetic field produced in the other wire from the current flowing in it.


Putting it all together

Let us suppose the two charges are sitting on a table in a moving train. This would mean that someone on the train moving with the charges would measure a different force between the two charges (just the electrostatic force) compared to someone who was on the ground as the train went past (the electrostatic force plus the Lorentz force).

The force measured on one of the charges by the person on the train, for whom the charges are stationary, which we shall call F will be

F = \frac{ Q_{1}Q_{2} }{ 4 \pi \epsilon_{0}r^{2} }.

The force measured on one of the charges by the person on the ground, for whom the charges are moving with a velocity v, which we shall call F^{\prime} will be

F^{\prime} = \frac{ Q_{1}Q_{2} }{4 \pi \epsilon_{0}r^{2} } + \frac{ \mu_{0} Q_{1} Q_{2} }{ 4\pi r^{2} } v^{2}.

These two forces are clearly different, and so it would seem that the laws of Electrodymanics are not invariant under a Galilean transformation, or to put it another way that one would be able to measure the force between the two charges to see if one were at rest or moving with uniform motion because the forces differ in the two cases.

As I will explain in a future post, Einstein was not happy with this idea. He believed that no experiment, be it mechanical or electrodynamical, should be able to distinguish between a state of rest or of uniform motion. His solution to this problem, On the Electrodynamics of Moving Bodies, was published in 1905, and led to what we now call his Special Theory of Relativity. This theory revolutionised our whole understanding of space and time.

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Relativity has been in the news quite a bit recently with the detection of neutrinos apparently travelling faster than the speed of light. Although most people don’t know the details of Einstein’s theory of Relativity, many are aware of the “cosmic speed limit” predicted in it, and also the most famous equation in physics which came from his theory – E=mc^{2}.

Many people, however, are unaware that the idea of relativity had been around long before Einstein. In fact, we can trace the idea of relativity back to Galileo. Galileo was one of the first scientists to do experiments on the motions of bodies (what we would now call mechanics), and was also one of the first scientists to use “thought experiments” to make scientific arguments.

Galileo

Galileo started thinking about whether mechanical experiments would behave differently if one were in motion or at rest. For example, if a ship is anchored in the port and one were to drop a stone from the top of the mast, we all know that it would strike the deck at the bottom of the mast, i.e. vertically below the place from where it was dropped (as long as we were careful not to give it any sideways motion). This is, of course, a pretty obvious statement.

But, what would happen if the ship were in motion? Let us suppose the ship is sailing at 5 metres per second (5m/s) in some direction on a perfectly smooth lake. If someone were now to drop a stone from the mast, surely it would fall behind the mast because the ship has moved forwards whilst the stone was dropping. If the stone were to take 1 second to drop to the deck, surely the stone would land 5 metres behind the bottom of the mast, rather than at the bottom, because the ship has moved 5 metres forwards in that 1 second.

NO, Galileo argued, this would not be the case. He argued that it would hit the deck at the bottom of the mast, just as in the case when the ship is not moving. If you think about it carefully you can see why.

When the person drops the stone from the mast, they are moving forwards with the ship. So the stone is actually given a forwards motion as it is dropped, and it is this forwards motion which leads it to land at the bottom of the mast, not behind it. As the ship moves forwards at 5 m/s, so does the stone. By performing this simple mechanical experiment one would not be able to tell whether the ship were anchored in the port, or moving on a smooth lake.

If the person at the port were able to see the motion of the stone against some sort of background, he would see the stone move in a parabola, which is exactly the motion a falling object which is also given some sideways velocity has. But, at every point of its travel down towards the deck, it will be next to the mast, as this is moving forwards as the stone falls.

Galileo then went on to generalise this specific thought experiment to say that there was no mechanical experiment that one could perform which would be able to tell the difference between being at rest or moving with a constant velocity (that is, with no acceleration). This principle is know as Galilean relativity, and we define a set of equations known as the Galilean transforms which allow us to switch between what we would see in two frames of reference, for example what someone standing on the shore would measure and what someone on a moving ship would measure.

If the ship is moving with a constant velocity v then in time t it will move a distance v t (distance = velocity x time). To make it easier for ourselves we will set up the x,y,z axes so that the ship is moving only along our x-axis. If we refer to the position and time of any event in the person on the shore’s frame of reference as (x,y,z,t) and those in the frame of reference of someone on the ship as (x^{\prime},y^{\prime},z^{\prime},t^{\prime}) then the equations which relate the two (known as the Galilean transforms) are:

\begin{array}{lcl} x^{\prime} & = & x + vt \\  y^{\prime} & = & y \\  z^{\prime} & = & z \\  t^{\prime} & = & t \end{array}

What these equations mean is that the only variable which is different in the two frames of reference is the x-displacement. The y and z-displacements are unaltered (as the ship is only moving in the x-direction), and time is the same for the two frames of reference. Let us look at how the x-displacement is transformed in going from one frame of reference to the other.

Suppose the ship is moving in the positive x-direction at 5 m/s. We want to measure the position of an object which is on the deck of the ship, let’s say the mast, as time goes by. For the person on the ship, it’s position is say, 15m in front of the stern of the ship. This is clearly not going to change with time, the mast does not move relative to the ship! So, we shall call this x.

For the person on the shore, the position of the mast is going to change as the ship sails away from him. So if the ship is sailing away at 5 m/s and the mast is initially 15m away from the person on the shore, then after 1 second it will be 15+(5 \times 1)=15+5=20m away. This is the x-position x^{\prime}, the x-position for the person in the other frame of reference, as given by the Galilean transformation equations above.

You can get this straight from using the equation x^{\prime} = x + v t = 15 + (5)(1) = 20m

The Galilean transforms are mathematically very simple, and conceptually simple too. As I will discuss in a future blog, the idea of performing experiments to determine between a state of rest or uniform motion, which Galileo argued could not be done, haunted scientists for centuries. In the 19th Century, with the development of electrodynamics (the study of the electricity and magnetism of moving bodies), physicists thought they could devise experiments to distinguish one’s state of uniform motion. They were wrong in thinking this, and it led to Einstein overthrowing the whole ideas of absolute time and absolute space in his Special Theory of Relativity.

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Last night the BBC showed a programme on the faster than light neutrino experiment which has made the physics world go into overdrive in the past few weeks. If you haven’t see the programme, it is available on iPlayer here. I don’t see any bits of it on YouTube yet, but keep a look out for it as, I realise, only people within the Disunited Kingdom can watch programmes via the iPlayer.

The programme was presented by the mathematician Marcus du Sautoy, whom I have seen present some excellent programmes on mathematics in the past. As a physicist/astrophysicst, I appreciate the fact that du Sautoy said right up front that he was not a physicist. I am curious why the BBC chose du Sautoy instead of e.g. the darling of the media at the moment Brian Cox, but then again maybe the BBC feel BC is suffering from over exposure.

I put a post on FaceBook alerting people to the programme going out at 9pm last night, and a colleague of mine commented “Gosh. So they can make science tv progs quick when they need to!” (I’ll excuse her poor grammar, this time 🙂 ). Indeed, it is amazing how quickly the BBC have put the programme together. And, considering how quickly it has been put together, I thought it was excellent. Maybe du Sautoy and the film crew were able to send their finished product from 2 years in the future back in time by using faster than light neutrinos to bring the video to October 2011!

I wanted to go into a lot more details about this programme, but I don’t have time today. I will be returning to the topic of relativity in the near future – I am in the process of writing some lectures on the whole historical development of relativity, from Galileo through to Einstein, so will post bits of that story on this blog over the next few weeks.

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